Random-Effects Approaches Comparisons in Meta-Analysis

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Random-Effects Approaches Comparisons in Meta-Analysis Data Simulation with Binomial Outcomes
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1 online resource (155 p.)
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english
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Guo,Jennifer D
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University of Florida
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Degree:
Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Pharmaceutical Sciences, Pharmaceutical Outcomes and Policy
Committee Chair:
Hartzema, Abraham G
Committee Members:
Delaney, Joseph A.
Segal, Richard
Shuster, Jonathan J
Kauf, Teresa
Lu, Xiaomin

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Subjects / Keywords:
analysis -- binomial -- meta -- outcome
Pharmaceutical Outcomes and Policy -- Dissertations, Academic -- UF
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Pharmaceutical Sciences thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract:
The empirical (classical) random-effects models provide biased summary effect sizes and generate too narrow 95% CI due to three factors: (a) treating random weights as non-random, (b) a probable association between effect size and weights at low event incidence, (c) use of an asymptotic normal distribution when a small number of studies are combined for a meta-analysis. Two unweighted random-effects approaches , the UW_OR and the UW_MM, were proposed to estimate the unweighted summary effect size by overcoming the issues with the empirical random-effects approach but efficiency of those two approaches are unknown. Additionally, a sample size weighted approach (W_RR) was proposed to estimate the weighted summary effect size. However, the relative performance of the unweighted summary effect size from the UW_RR versus the weighted summary effect size from the W_RR is unknown when they are forced to estimate the same summary effect size. The study aims to evaluate the statistical properties of the UW_OR and the UW_MM, and the performance of the UW_RR and the W_RR using two major data simulations. This simulation approach is taken because it is impossible at present to contrast the methods using mathematical means in that the calculation of the parameter dimensions are well beyond our capabilities. Our findings suggest that the efficiency of the UW_OR and the UW_MM is close when a large number of studies are combined for a meta-analysis and the low event rates are unlikely. Further, the UW_RR generates a similar summary effect size as the W_RR does when the effect size does not depend upon the study sample size. In addition, we demonstrate that a t distribution is more appropriate compared with a normal distribution for all proposed methods, when a small number of studies are combined for a meta-analysis. It might be a good statistical practice to report results from both the sample size weighted (W_RR) and the unweighted (UW_RR) analysis (one as primary and one as secondary). If the two agree qualitatively, the consistent reports add strength to make any inference. If not, since the UW_RR estimates the unweighted summary effect size and the W_RR estimates the weighted summary effect size, the disagreement is not a contradiction, but suggests further investigation of inherent differences between larger and smaller studies.
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In the series University of Florida Digital Collections.
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Includes vita.
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Statement of Responsibility:
by Jennifer D Guo.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Hartzema, Abraham G.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2012-02-29

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1 RANDOM EFFECTS APPROACHES COMPARISONS IN META ANALYSIS: DATA SIMULATION WITH BINOMIAL OUTCOMES By DONGWEN GUO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQU IREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011

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2 2011 D ongwen Guo

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3 T o my mom and dad

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4 ACKNOWLEDGMENTS First and foremost, I would like to express my sincerest gratitude to my supervisor, Dr Abraham, G. Hartzema, for his guidance, encouragement, support, and faith in me This dissertation would not have been possible without his support in p roviding me with an excellent atmosphere for doing research It is a great honor to thank Dr. Jonathan J. Shuster. This diss ertation would not exist without his innovative research findings. He has made available his support in many ways with extreme patience enabl ing me to develop a clear understanding of the subject. I am very grateful to Dr. Richard Segal, who is always wi lling to act as a sounding board. W ithout his constant direction and support, I could not have achieved my goals I would also like to thank Dr. Teresa Kauf for her generous encouragement when I needed it most I am thankful to Dr. Xiaomin Lu, who often m ade time to g i ve her invaluable advice Special thanks go to Dr. Chris Delaney for making himself available to participate in my final defense and offer his excellent suggestions I am indebted to my teachers and professors for their valuable advi ce and special support, staff members for their timely support graduate colleagues and friends for their friendship M y sincere regards and blessings reach out to all of those who supported me in any respect during the completion of my dissertation. Final ly I owe my deepest gratitude to my mother and father as well as other family members for their unconditional love and unwavering support.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 9 LIST OF ABBREVIATIONS ................................ ................................ ........................... 12 ABSTRACT ................................ ................................ ................................ ................... 13 C HA PTE R 1 INTRODUCTION ................................ ................................ ................................ .... 15 Statement of Problem ................................ ................................ ............................. 15 Specific Aims ................................ ................................ ................................ .......... 16 Significance ................................ ................................ ................................ ............ 17 Definition of Terms ................................ ................................ ................................ .. 18 Organization of Study ................................ ................................ ............................. 19 2 REVIEW OF THE LITERATURE ................................ ................................ ............ 21 Definition of Meta analysis ................................ ................................ ...................... 21 Models of Meta analysis ................................ ................................ ......................... 21 Fixed effect Model ................................ ................................ ............................ 22 Random effects Model ................................ ................................ ..................... 22 Selection of Meta analysis Models ................................ ................................ ......... 23 Method 1 Cochran Q test ................................ ................................ ................. 23 Method 2 Clinical Judgment ................................ ................................ ............. 23 Issues Regarding Empirically Weighted Random effects Approach ....................... 24 Solutions ................................ ................................ ................................ ................. 28 Solution 1: Unweighted Random effects Approach I ................................ ........ 28 Solut ion 2: Unweighted Random effects Approach II ................................ ....... 29 Solution 3: Sample size Weighted Random effects Approach ......................... 32 3 METHODS ................................ ................................ ................................ .............. 44 Odds Ratio Data Simulation ................................ ................................ .................... 44 Definition of a Scenario ................................ ................................ .................... 44 Random Sampling for Meta analyses ................................ .............................. 45 Relative Risk Data Simulation ................................ ................................ ................. 46 Definition of a Scenario ................................ ................................ .................... 47 Random Sampling for Meta anal yses ................................ .............................. 47 Statistical Analyses ................................ ................................ ................................ 47

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6 Aim 1 ................................ ................................ ................................ ................ 47 Aim 2 ................................ ................................ ................................ ................ 48 Aim 3 ................................ ................................ ................................ ................ 49 Aim 4 ................................ ................................ ................................ ................ 49 4 RESULTS ................................ ................................ ................................ ............... 59 Aim 1 ................................ ................................ ................................ ....................... 60 Aim 2 ................................ ................................ ................................ ....................... 62 Aim 3 ................................ ................................ ................................ ....................... 65 Aim 4 ................................ ................................ ................................ ....................... 66 5 DISCUSSION ................................ ................................ ................................ ....... 132 Specific Findings ................................ ................................ ................................ ... 132 Aim 1 ................................ ................................ ................................ .............. 132 Aim 2 ................................ ................................ ................................ .............. 134 Aim 3 ................................ ................................ ................................ .............. 135 Aim 4 ................................ ................................ ................................ .............. 136 Meta Analysis in General ................................ ................................ ...................... 137 Weighting Issues and Summary Effect Size Estimates ................................ .. 138 Low Event Issue and Continuity Adjustment ................................ .................. 140 Asymptotic Properties and Large Sample Theory ................................ .......... 141 Selection of Measurement Metrics for Binomial Outcomes ............................ 141 Elaborating Example on Mortality for Erythropoiesis Stimulating Agents .............. 142 Strengths ................................ ................................ ................................ .............. 143 Limitations ................................ ................................ ................................ ............. 144 Further studies ................................ ................................ ................................ ...... 144 Conclusions ................................ ................................ ................................ .......... 145 A PP EN D IX T TABLE ................................ ................................ ................................ ...................... 148 LIST OF REFERENCES ................................ ................................ ............................. 149 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 155

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7 LIST OF TABLES Table page 2 1 Illustration of issues with empirically weighted random effects approach .......... 39 2 2 Nomenclature for 2 by 2 table of outcome by treatment ................................ .... 41 2 3 Illustration of advantages of three proposed random effects approaches ......... 43 3 1 Parameter list and val ues for OR data simulation ................................ .............. 52 3 2 Examples with combination of parameters for scenarios ................................ ... 52 3 3 Random sampling based upon parameters under a scenario ........................... 52 3 4 Example of studies structure for a ran dom sampling in a meta analysis ........... 52 4 1 Aim 1: Comparisons of UW_OR vs. UW_MM ................................ .................... 70 4 2 Aim 1: Comparisons of UW_OR vs. UW_MM by event rate in treatment arm and number of studies ................................ ................................ ........................ 71 4 3 Aim 2 1: Comparisons of t distribution vs. normal distribution for UW_OR ........ 76 4 4 Aim 2 1: Comparisons of t distribution vs. normal distribution for UW_OR by event rate in treatment arm and number of studies ................................ ............ 77 4 5 Aim 2 2: C omparisons of t distribution vs. normal distribution for UW_MM ....... 83 4 6 Aim 2 2: Comparisons of t distribution vs. normal distribution for UW_MM by event rate in treatment arm and number of studies ................................ ............ 84 4 7 Aim 3: Comparisons of UW_RR vs. W_RR ................................ ....................... 89 4 8 Aim 3: Comparisons of UW_RR vs. W_RR by event rate in tr eatment arm and number of studies ................................ ................................ ........................ 90 4 9 Aim 4 1: Comparisons of t distribution vs. normal distribution for UW_RR ........ 96 4 10 Aim 4 1: Comparisons of t distribution vs. normal distribution for UW_RR by event rate in treatment arm and number of stu dies ................................ ............ 97 4 11 Aim 4 2: Comparisons of t distribution vs. normal distribution for W_RR ........ 102 4 12 Aim 4 2: Comparisons of t distribution vs. normal distribution for W_RR by event rate in treatment arm and number of studies ................................ .......... 103

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8 5 1 Mortality for erythropoiesis stimulating agents in cancer population using different random effects approaches in meta analyses ................................ .... 147

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9 LIST OF FIGURES Figure page 1 1 Comparison of normal distribution vs. t distribution[16]. ................................ .... 20 1 2 Binomial distribution (Y=Probability, X=N) by R software V2.1 .......................... 20 2 1 Evidence Hierarchy ................................ ................................ ........................... 35 2 2 Overview of systematic review ................................ ................................ .......... 36 2 3 Overview of meta analysis ................................ ................................ ................ 37 2 4 Illustration of empirically weighted random effects approach (W_DL) ............... 38 2 5 Illustration of unwei ghted random effects approach (UW_MM) ......................... 40 2 6 Illustration of unweighted random effects approach (UW_OR/UW_RR) ........... 41 2 7 Illustration of weighted random effects approach (W_RR) ................................ 42 3 1 Odds ratio data simulation flow chart. ................................ ................................ 51 3 2 Relative risk data simulation flow chart. ................................ ............................. 53 3 3 Illustration of analysis steps for aim 1 ................................ ................................ 55 3 4 Illustration of analysis steps for aim 2 ................................ ................................ 56 3 5 Illustration of analysis steps for aim 3 ................................ ................................ 57 3 6 Illustration of analysis steps for aim 4 ................................ ................................ 58 4 1 Aim 1: Comparisons of UW_OR vs. UW_MM in terms of difference f or both mean coverage and mean length 95% CI ................................ ........................ 108 4 2 Aim 1: Comparisons of UW_OR vs. UW_MM in terms of difference for both mean coverage and mean length 95% CI by event rate in treatment arm and number of studies ................................ ................................ ............................. 109 4 3 Aim 1: Comparisons of UW_OR vs. UW_MM in terms of both mean coverage and mean length 95% CI ................................ ................................ .. 110 4 4 Aim 1: Comparisons of UW_OR vs. UW_MM in terms of both mean coverage and mean length 95% CI by event rate in treatment arm and number of studies ................................ ................................ ............................. 111

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10 4 5 Aim 2 1: Comparisons of t distribution vs. normal distribution for UW_OR in terms of difference for both mean coverage and mean length 95% CI ............. 112 4 6 Aim 2 1: Comparisons of t distribution vs. normal distribution for UW_OR in terms of difference for both mean coverage and mean length 95% CI by event rate in treatment arm and numb er of studies ................................ .......... 113 4 7 Aim 2 2: Comparisons of t distribution vs. normal distribution for UW_OR in terms of both mean covera ge and mean length 95% CI ................................ ... 114 4 8 Aim 2 2: Comparisons of t distribution vs. normal distribution for UW_OR in terms of both mean coverage and mean length 95% CI by event rate in treatment arm and number of studies ................................ ............................... 115 4 9 Aim 2 2: Comparisons of t distribution vs. normal distribution for UW_MM in terms of difference for both mean coverage and mean length 95% CI ............. 116 4 10 Aim 2 2: Comparisons of t distribution vs. normal distribution for UW_MM in terms of difference for both mean coverage and mean length 95% CI by event rate in treatment arm and number of studies ................................ .......... 117 4 11 Aim 2 2: Comparisons of t distribution vs. normal distribution for UW_MM in terms of both mean coverage and mean length 95% CI ................................ ... 118 4 12 Aim 2 2: Comparisons of t distribution vs. normal distribution for UW_MM in terms of both mean coverage and mean length 95% CI by event rate in trea tment arm and number of studies ................................ ............................... 119 4 13 Aim 3: Comparisons of UW_RR vs. W_RR in terms of difference for both mean coverage and mean length 95% CI ................................ ........................ 120 4 14 Aim 3: Comparisons of UW_RR vs. W_RR in terms of difference for both mean coverage and mean length 95% CI by event rate in treatment arm and number of studies ................................ ................................ ............................. 121 4 15 Aim 3: Comparisons of UW_RR vs. W_RR in terms of both mean coverage and mean length 95% CI ................................ ................................ .................. 122 4 16 Aim 3: Comparisons of UW_RR vs. W_RR in terms of b oth mean coverage and mean length 95% CI by event rate in treatment arm and number of studies ................................ ................................ ................................ .............. 123 4 17 Aim 4 1: Comparisons of t distribution vs. normal distribution for UW_RR in terms of difference for both mean coverage and mean length 95% CI ............. 124 4 18 Aim 4 1: Comparisons of t distribution vs. normal distribution for UW_RR in terms of difference for both mean coverage and mean length 95% CI by event rate in treatment arm and number of studies ................................ .......... 125

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11 4 1 9 Aim 4 1: Comparisons of t distribution vs. normal distribution for UW_RR in terms of both mean coverage and mean length 95% CI ................................ ... 126 4 20 Aim 4 1: Comparisons of t distribution vs. normal distribution for UW_RR in terms of both mean coverage and mean length 95% CI by event rate in trea tment arm and number of studies ................................ ............................... 127 4 21 Aim 4 2: Comparisons of t distribution vs. normal distribution for W_RR in terms o f difference for both mean coverage and mean length 95% CI ............. 128 4 22 Aim 4 2: Comparisons of t distribution v s. normal distribution for W_RR in terms of difference for both mean coverage and mean length 95% CI by event rate in treatment arm and number of studies ................................ .......... 129 4 23 Aim 4 2: Comparisons of t distribution vs. normal distribution for W_RR in terms of both mean coverage and mean length 95% CI ................................ ... 130 4 24 Aim 4 2: Comparisons of t distribution vs. normal distribution for W_RR in terms of both mean coverage and mean length 95% CI by event rate in treatment arm and number of studies ................................ ............................... 131 5 1 Illustration of associations among W_DL, W_RR and UW_RR ....................... 147

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12 LIST OF ABBREVIATION S CI Conf idence i nterval D Diversity between studies M Number of studies combined for a meta analysis study NHigh Highest number of subjects in each arm NLow Lowest number of subjects in each arm OR Odds Ratio PC ( ) Global true event probability in the control ar m PT ( ) Global true event probability in the treatment arm RD Risk d ifference RR Relative r isk STD Standard d eviation UW_MM Unweighte d mean of mean s meta analysis random effects approach in terms of odds ratio UW_OR Unweighted summary proportions meta analysis random effects approach in terms of odds ratio UW_RR Unweighted summary proportions meta analysis random effects approach in terms of relative risk W_DL DerSimonian Laird model or empirical meta analysis random effects model W_RR S ample size we ighted meta analysis random effects approach in terms of relative risk

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13 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosoph y R ANDOM EFFECTS APPROACHES COMPARISONS IN META ANALYSIS: DATA SIMULATION WITH BINOMIAL OUTCOMES By D ongwen Guo August 2011 Chair: Abraham G. Hartzema Major: Pharma ceutical Science s T he empirical (classical) random effects models provide biased summar y effect size s and generate too narrow 95 % CI due to three factors: (a) treating random weights as non random, (b) a probable association between effect size and weights at low event incidence (c) use of an asymptotic normal distribution when a small numb er of studies are combined for a meta analysis T wo unweighted random effects approaches the UW_OR and the UW_MM were proposed to estimate the unweighted summary effect size by overcoming the issues with the empirical random effects approach but effici ency of those two approaches are unk n own. A ddition ally a sample size weighted approach (W_RR) was proposed to estimate the weighted summary effect size. However, the relative performance of the unweighted summary effect size from the UW_RR versus the wei ghted summary effect size from the W_RR is unknown when they are forced to estimate the same summary effect size The study aims to evaluate the statistical properties of the UW_OR and the UW_MM, and the performance of the UW_RR and the W_RR using two ma jor data simulation s This simulation approach is taken b ec ause i t is impossible at present to

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14 contrast the methods using mathematical means in that the calculation of the parameter dimensions are well beyond our capabilities. Our findings suggest that t h e efficiency of the UW_OR and the UW_MM is close when a large number of studies are combined for a meta analysis and the low event rates are unlikely Further, the UW_RR generate s a s imilar summary effect size as the W_ RR does when the effect size does not depend upon the study sample size In addition, w e demonstrate that a t distribution is more appropriate compared with a n ormal distribution for all proposed methods when a small number of studies are combined for a meta analysis. It might be a good sta tistical practice to report results from both the sample size weighted (W_RR) and the unweight ed (UW_RR) analysis (one as primary and one as secondary). If the two agree qualitatively, the consistent reports add strength to make any inference. If not, sin ce the UW_RR estimates the unweighted summary effect size and the W_RR estimates the weighted summary effect size, the disa greement is not a contradiction but suggests further investigation of inherent differences between larger and smaller studies.

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15 CH APTER 1 INTRODUCTION Statement of Problem Meta analys e s are systematic reviews which quantitatively combine results of previous research [1] The F ederal Drug and Food Administration (F DA ) requires review s of clinical trials on safety and eff icacy in marketing applications [2] I n safety assessment, meta analysis studies are viewed as primary evidence in support of regulatory decision making [3, 4] In regard s to meta analy tical methods, the most widely used models are the empirical (classical) fixed effect model and the empirical (classical) random effects model s [5] Under these two models, each study within a meta analysis is weighted by the inverse of that [5] T he empirically weighted random effects approach suffers from three issues that have been identified in recent years [6] These issues lead to biased summary effect size estimates and the corresponding smaller estimates of standard errors [6] In a meta analysis with a binomial outcome, Shuster et al [7 9] prop osed t wo unwe ighted random effects approaches ( UW _MM, UW _OR / UW_ RR ) and one sample size weighted random effects approach (W_RR) to overcome i ssues that the empirical random effects model has Theoretically, the above proposed approaches both provide unbiased summary ef fect size estimates [6 9] H owever, we are un certain which one is more efficient. Our research goal is to compare the efficiency of the two unweighted random effects approaches (UW_ MM vs UW_ OR ) using the odds ratio data simulation, and to compare the efficiency of the unweighted random effect s and the sample size weighted

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16 random effects approach (UW_RR vs W_RR) when they estimate the same summary effect size using the relative risk d ata simulation. Specific A ims T he following specific aims are proposed : Specific Aim #1: Evaluate efficiency for the UW_OR and the UW_MM in a meta analysis of binomial trials when synthesizing a large number of studies. Hypothesis 1 : It is hypothesized that there is no precision difference between the UW_OR and the UW_MM when synthesizing a large number of studies. Specific Aim #2: Evaluate e fficiency of t approximation vs normal approximation in a meta analysis of binomial trials when synthesizing a small number of studies. H ypothesis 2 1 : It is hypothesized that a t approximation is more accurate than a normal approximation in terms of closeness to the 95% coverage for the UW_ OR when synthesizing a small number of studies Hypothesis 2 2 : It is hypothesized that a t approxima tion is more accurate than a normal approximation in terms of closeness to the 95% coverage for the UW_ MM when synthesizing a small number of studies. Specific Aim # 3 : Evaluate efficiency for the UW_RR and the W_RR when they estimate same summary effect s ize in a meta analysis of binomial trials. Hypothesis 3 : It is hypothesized that efficiency of the UW_RR and the W_RR is similar when they estimate the same summary effect size. Specific Aim # 4 : Evaluate efficiency of t approximation vs normal approxim ation in a meta analysis of binomial trials when synthesizing a small number of studies.

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17 Hypothesis 4 1 : It is hypothesized that a t approximation is more accurate than a normal approximation in terms of closeness to the 95% coverage for the UW_ R R when syn thesizing a small number of studies. Hypothesis 4 2 : It is hypothesized that a t approximation is more accurate than a normal approximation in terms of closeness to the 95% coverage for the W_ R R when synthesizing a small number of studies. Significance E vidence from meta analyses represents the highest authority in the development of guidelines and helps reach consensus when conflicting evidence occurs [1] In addition to selecting the appropriate source studies for a meta analysis the appropriate selection of t he meta analysis method is a very importan t step in ensur ing the validity of synthesized evidence e specially when a meta analysis is adopted to summarize the low incidence of adverse drug event s studies At the completion of this series of experiments we expect to provide more solid supporting evidenc e on the efficiency of the UW_OR and the UW_MM, and on the efficiency of the UW_RR and the W_RR The successful completion of this study will help medical researchers select appropriate meta analysis methods in the future Moreover, the evidence fro m an appropriate meta analys i s will help the FDA to evaluate the efficacy and safety of new drugs. Further more formulary managers and healthcare officials will be able to use the evidence from appropriate meta analyse s to develop treatment guidelines or d ecide upon the inclusion of a new drug in a formulary [1] in a more rigorous manner than previously

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18 Definition of Terms E FFICIENCY In statistics, efficiency refers to the degree to which a statistic is stable from one sample to another sample. In other word s a statistic is more efficient when it i s less subject to sampling variation and bias The efficiency o f a statistic can be treated as a synonym for the precision of an estimate. The more efficient a statistic is the more precise the statistic is as an estimator for a parameter [10] P RECISION OF METHODS The precision of methods is a general term for the foll owing statistics of variance, standard error and confidence interval [5] R ANDOM VARIABLE A r andom variable denotes a variable whose values m ay be generated from a random experiment and the values cannot be predicted with certainty before a specific sample is selected [11] L ARGE SAMPLE T HEORY As a branch of statistics, the large sample theory states that it is relatively easy to obtain good approximate results if the sample size is large [12] C ENTRAL LIMIT THEOREM As a probability theory, the theorem states conditi ons under which the means of a sufficiently large number of independent random variables are approximately normally distributed [13] D ELTA METHOD The delta method is a technique which is used to approximate expected values of func tions of random variables when it is not feasible to obtain the expect ed value directly, such as compute the moments of an approximating asymptotic distribution [14] P ROBABILITY DISTRIBUT ION The probability distribution of a random variable provides the frequencies of each value of the random variable in a population [11] N ORMAL DISTRIBUTION The n ormal distribution is characterized as a symmetrical ly bell shaped curve. The parameters of (the mean describing the center) and (the standard deviation describing the spread ) are used to describe the distribution of a population ( Figure 1 1) [11] S TUDENT S T DISTRIBUTION a symmetrically bell shaped curve It appears similar to the normal distribution but it differs because it has an additional parameter of degrees of freedom (df) which c hange the shape of the curve. The degrees of freedom are equal to the sample size minus one. The small er the degrees of freedom the large r the area under the tail of the curve is In contrast, the larger the degrees of freedom the closer it is to the cur ve of standard normal distribution. df= ( Figure 1 1) [15]

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19 B INOMIAL DISTRIBUTION The binomial distribution describes the number of occurrences of a particular event in a series of n trials. It needs to meet four conditions. First, t he trials are identical. Second the outcome of each trial is independent, i.e. the outcome is not associated with the outcome from other trials. Third, the outcome h The last condition is that the probability of success ( ) stays the same for all trials. The binomial distribution is described by the sample size ( n ) and the probability of success ( ) ( Figure 1 2) [11] Organizati on of Study The remainder of the study is organized into four chapters. Chapter two presents a review of the related literature cover ing the concepts proposed in this chapter Chapter three describes the research design and statistical analysis plan of the study. Chapter four delineates the results of the study Cha pter five contains discussion and conclusions of this study

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20 Figure 1 1 Comparison of n ormal distribution vs. t distribution [16] Figure 1 2 Binomi al distribution (Y=Probability, X=N) by R software V2.1

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21 CHAPTER 2 REVIEW OF THE LITERA TURE This chapter provides the background and the rationale for conducting this study. We first define meta analysis Next we review the most popular ly used statistical meta analy tical models for this study Third, we explain the selection of the models Fo u rth, we summarize the criticism with the empirical rando m effects model, as stated by DerSimonian Laird ( W_ DL) [17] Fift h we in troduce the solutions that overcome the issues with the W_ DL model. Finally we propose questions that remain unknown Definition of Meta analysis Meta analyses for combining data from independent studies to obtain a numerical estimate of the overall effect of a particular procedure or variable on a defined outcome. [18] In [1, 19] an accepted term in Medline In 1989 [1] The evidence from syste matic reviews and meta analyse s is often placed on the top of a pyramid in terms of authority ( Fig ure 2 1 [20] ) a lthough there is no single, universally ac cepted hierarchy of evidence. There are two main steps in conduct ing a meta analysis : a critical review of the literature and synthesizing data from each individual study into a summary (pooled or combined) effect size [1] ( Figure 2 2 ) Models of Meta analysis In regards to meta analysis models resea rchers usually have two choices, a fixed effect model and a random effects model [5] ( Figure 2 3) The common underlying assumptions are that each individual study is independent and each one produces an [6] As for a meta analysis with binomial

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22 outcomes, most meta analysis models presume that individual effect size s in the form of odds ratio s or relative risk s are an independent identically normal distribution on a log scale [7, 17, 21, 22] while some work with other prior distribution, i. e. the Bayesian approach [23 25] We elaborate next upon the most popular used models : fixed effect and random effects m odels [5] Fixed effect Model The fixed effect model was proposed by Mantel and Haenszel in 1959 [1, 26] The specific underlying assumption for the fixed effect model is that only one common true value of effect size exists for any study included in a meta ana lysis [5] The variation in the meta analysis comes from sampling error within each individual primary study. The weight is usually the inver estimated variance. Random effect s Model Another meta analysis model is the random effects model developed by William Cochran in 1954 [27] but was not used in the medical field until 1986 by DerSimonian and Laird [1, 17] The specific underlying assumption for the random effects model is that the effect size of each individual study is normal ly distributed as opposed to shari ng a common true value for the fixed effect model. Under this model, t he source of variation in the meta analysis comes from the sampling error within each study and the difference across studies combined for the meta analysis [28] In summary, the commonly use d models for a meta analysis are the fixed effect and the random effects models which have different specific underlying assumptions The fixed effect model is a speci al case of the empirical random effects model. Researchers need to choose an appropriat e model before conducting a meta analysis.

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23 Selection of Meta analysis Models Two ways can help a researcher to select the fixed effect mode l or the random effects model. One is using a Cochran Q test. The other one is based upon clinical judgment. Method 1 Cochran Q test One method of selecti ng a meta analysis model is to use the Cochran Q test and corresponding p value. T he Cochran Q test is a diagnostic tool that is used to determine which model is appropriate, the fixed effect model or the ra ndom effects model If the Q statistic is statistically significant with the p value less than 0. 05 the random effect s model is the choice. However, if the number of studies is small the Q statistic may not have sufficient power to detect the between stu dy variance. Therefore, after the test, we only have two possible conclusions : (a) study effect sizes are statistically diverse (p<0. 05), suggesting choosing the random effects model or (b) it is inconclusive as to whether study effect sizes are diverse i e. we do not know which model is appro pri ate Therefore, the Cochran Q test is not recommended as a diagnostic tool. Method 2 Clinical J udgment Another means of select ing a meta analysis model is based on the distribution of the effect size s of in dividual stud ies and the relevant sources of errors [5] In other words, we should make our decision based on the nature of the studies being combined for a meta analysis instead of using a data driven test such as the Cochran Q test. Fixed effect m odel The fixed effect model is the choice when (a) we believe that all the studies examine the same hypothesis, and (b) the research goal is to cal culate the common effect size, which intend s to be generalized to the same population. Under these conditions, the fixed effect mode l is a better choice for a multicenter clinical trial

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24 This is b ecause when a drug compan y conduct s several studies with th e same way of recruiting patient s using the same researchers, and using the same dosing regimens [28] Random effects m odel In contrast, the random effects model is more easily justified than the fixed effect model when a researcher combines data from a seri es of studies performed by different investigators [28] First, it is inappropriate to assume that all the studies examine the common effect size in that these studies would differ in many ways that impact the results, such as different study design s different study population s different comparison group s different dosing regimens different statistical method s etc. Additionally, the goal of the meta analysis is often to generalize to a range of populations. In summary, although the two models are popularly u sed, the random effects model is a more appropriate approach when a meta analysis is conducted at a study level [28] Among available random effects models, the empirically weighted random effects approach (W_DL) is usually used [6] Issues Regarding Empirical ly Weighted R andom effects A pproach In this section, we describe the empirically weighte d random effects approach ( W_ DL) first, and then identify the issue s with this approach As for the most popular ly used random effects model, the empirical approach is a weighted approach E ach study is weighted by the inverse of estimat ed variance including the within stud y variance and the between study variance T he rationale behind the weights is that it i ntends to emphases the impact from larger sample size studies on the summary effect size because t he larger the sample size, the l ower within study random error.

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25 The conceptual formula s for the summary weighted mea n ( ) and corresponding 95% CI are computed below [28] The outcome metrics could be odds ratio, relative risk or risk difference. If the outcome i s measured in the forms of o dds ratio or relative risk, they need to be transformed to a log scale then apply the following formulas. An illustration diagram in the form of odds ratio is displayed in Figure 2 4. The conceptual formula for the summary weig hted mean ( ) Where refers to the weighted summary mean refers to the refers to th study under a meta analysis refers a weight for the th study refers to the variance of the th study including the within study variance for the t h study plus the between stud ies variance ( ) Where

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26 Where (Tau squared) refers to the between study variance estimate Q refers to the Q statistic. df refers to the degrees of freedom, which is equal to M 1 Where M refers to t he number of studies for a meta analysis 95% CI is computed blow: Where LL refers to the low limit of 95% confidence interval UL refers to the upper limit of 95% confidence interval 1. 96 refe rs to the z score for a normal distribution Where However, the empirical random effects model ( W_ DL) has t hree issues ( Table 2 1 ) First, the W_ DL is widely used with weights [28] As we mentioned earlie r the

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27 w eight refers to the inverse of the estimated variance of each study including both vari ance from within study and between studies Empirically the weighted random effects analysis treats its weights as non random against its random nature which is shown above in the conceptual formula for the weighted summary mean ( ) Therefore the estimate of the summary effect size is not only biased but also does not estimate what it claims to estimate of the unweighted summary eff ect size In addition the confidence intervals derived from empirical weights are narrower than they should be [29] The second and often the cas e is that t he low event rates make the estimate of the effect size for that individual study highly biased including its standard error. The reason is that the estimate s of the effect size and the standard error for an individual study are calculated acc ording to a large sample theory [12] This situation is similar to that of the low expected cell number problem s in two by two contingency tables, which explain s why the is popular Moreover at low event rates, the weights are generally associ ated with the effect size s of studies [6] The third issue occurs when the numb er of studies combined for a meta analysis is small This is because the large sample theory plays a role not only in each individual study but also in the number of studies combined for a meta analysis. The estimator of between study variations is based upon In other word s when a small number of studies are combined for a meta analysis, the large sample approach is unreliable. Therefore, the z score of 1. 96 that is based upon the standard normal distribution appl ying to calculate t he 95% CI is not appropriate. Instead a t distribution with a degree of freedom (i. e. the number of studies minus one ) might be used But a s

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28 the number of studies increases the summary effect size with a t distribution is close r to the summary effect si ze with a normal distribution In conclusion, although the empirical weighted random effects model (W_DL) often suffers from three issues of the empirical weighting, the low event rates and the asymptotic normal distribution assumption it has been widel y used [6] Solutions To overcome the issues using the empirically weighted random effects approach, Shuster et a l. proposed the two unweighted random effects approaches [7, 8] and one sample size weighted approach [6] The advantages for each method are listed in Table 2 3. Now we introduce how a summary effect size is computed in the following order : UW_MM, UW_OR/ UW_ RR and W_RR Solution 1 : Unweighted Random effects Approach I Shuster et al proposed an un weighted random effects approach (UW_MM) in 2010 [8] By using this approach, we compute the odds ratio for each study first, and then tr ansfo rm the odds ratio into its log form In the log scale, we assume that each log odds ratio is normally distributed Therefore, we derive the mean log odds ratio by adding each log odds ratio together and divi ding the summation by the number of studies. L a st, we take the antilog on the mean log odds ratio back to the original odds ratio scale We illustrate the steps in Figure 2 5 The detailed formulas are displayed as follows. The operational unweighted estimate for a summary odds ratio for a sample is: Where

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29 M=number of studies and denote the number of events and sample size for treatment and trial =1 represents control arm, =2 represents treatment ar m 95% CI for in a log scale is Where has an asymptotic t distribution with M 1 degrees of freedom 95% CI for in the original scale is Solution 2 : Unweighted R an dom effects Approach II Shuster et al [7, 9] proposed an other un weighted random effects approach that completely avoid s the low event issue with the weighted random effects approach by taking ratios only after comb in ing the proportions of each study ( Table 2 2, Figure 2 6 )

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30 T he only place that the large sample theory is ever employed is at the number of studies being combined. This method provides an unbiased summary effect size by avoiding taking the log o f the ef fect size for each individual study and the antilog of the summary effect size back to the original scale Moreover this method avoid s the chance of not being able to calculate individual effect size because of zero event in one or both arms. Odds Ratio Estimation: The operational unweighted estimate for a summary odds ratio for a sample is: Where and denote th e number of events and sample size for treatment and trial =1 represents control arm, =2 represents treatment arm M=number of studies 95% CI for in a log scale is Where has an asymptotic t distribution with M 1 degrees of freedom

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31 95% CI for in the original scale is Relative Risk Estimation: The operational unweighted estimate for a summary relative risk for a sample is: Where and denote the number of events and sample size for treatment and trial =1 represents control arm, =2 represents treatment arm M=number of studies 95% CI for in a log scale is has an asymptotic t distribution with M 1 degrees of freedom. Where Where ,

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32 Where k=1,2; l=1,2 95% CI for in the original scale is In all, theoretically the above two unweighted random effects approaches (UW_MM and UW_OR), which are both in odds ratio fashion provide valid summary effect size estimates for a meta analysis. H o wever the efficiency of both methods remain s unknown Solution 3 : Sample size Weighted Random effects Approach This method was proposed by Shuster et al. in 2011 [6] The basic idea is that we conceptually draw a subject at random from the target population of past, present, and future subjects. T he probability of selecting a given study t o which this subject belongs is proportional to the total sample size of the study. The inferential framework differs from the solution 2 of the unweighted approach (UW_R R) in that this sample size weighted approach is at a patient level, not at a study le vel. This approach is illustrated in Figure 2 7. The conceptual formula for the weighted summary effect size is Where represents the fraction of subjects in the universe which belong s t o t rial j. is effect size estimate for th study

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33 is completely different from the defined for solution 1 2 unless there is no assoc iation between the sample size and the effect size The formula for t he weighted summary effect size for a sample of studies is Where denotes the estimated effect size for the th study The operational formula for the weighted summary effect size for a sample is Where is the adjusted number of events i=1,2 1 indicating the control arm 2 indicating the tr eatment arm s denotes the estimate of risk for th arm and th study 95% CI for in a log scale is Where is asymptotically t distribution with (M 1) degrees of freedom

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34 Where S() represents the sample standard deviation and C(, ) re presents the sample covariance. 95% CI for in the original scale is In all, although the two unweighted random effects approaches (U W_OR/UW_RR) have been prove n to be valid we do not know the ir efficiency In addition another solution of the sampl e size weighted random effects approach (W_RR) was proposed but it estimate s the weighted summary effect size which is different from the unweighted summary effect size It is unknown if we force them to estimate the same effect size, which one would be more efficient.

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35 Figure 2 1. Evidence Hierarchy

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36 Figure 2 2 Overview of systematic revi ew

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37 Figure 2 3 Overview of meta analysis

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38 Figure 2 4 Illustration of empirically weighted random effects approach (W_DL)

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39 Table 2 1. Illustrat ion of issues with empirically weighted random effects approach No. of Issues Issues Summary effect size SE 95% CI 1 Empirical weigh ts are random but treated as non random Does not estimate the claimed unweighted effect size B iased summary effect size I naccurate SE Too narrow 2 Estimate of point effect size for an individual study is computed based upon the large sample theory This becomes a problem when low event occurs. B iased effect size for individual study B ias from log transformation on each individual effect size Inaccurate SE S kewed 3 Low number of studies is combined for a meta analysis but with a normal distribution assumption (z=1.96) to calculate 95% CI N/A N/A Too narrow

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40 Figure 2 5 Illustration of unweighted random effects approach ( UW_MM )

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41 Table 2 2. Nomenclature for 2 by 2 table of outcome by treatment Ou tcomes (+) Outcomes ( ) N Treatment a b n 1 Control c d n 2 Figure 2 6 Illustration of unweighted random effects approach ( UW_OR /UW_RR )

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42 Figure 2 7 Illustration of weighted random effects approach ( W_RR )

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43 Table 2 3 Illustration of advantage s of three proposed random effect s approaches No. of Issues Issues w/DL UW_MM UW_OR/ U W_ RR W_RR 1 Empirical weights are random but treated as non random Using no weight Using no weight Using sample size as weight s and treating them as random plus 2 Estimate of point effect size for an individual study is computed based upon the large sample theory. This becomes a problem when low event occurs. N/A Using summation of proportion s for each individual study and doing ratio at summary level plus A void ing log transformation for each point effect size Using summation of weighted proportio ns for each individual study plus A void ing log transformation for each point effect size 3 Low number of studies is combined for a meta analysis but with a normal distribution assumption (z=1. 96) to calculate 95% CI A t distribution is close to a norm al distribution when a large number of studies are combined Same as UW_MM Same as UW_MM

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44 CHAPTER 3 METHODS Given the issues of the empirical random effects model (W_DL) the two unweighted random effects approaches and one weighted random effects approach were proposed [7, 8] Ai m 1 and aim 2 of this study compare the efficiency of the two unweighted random effects approaches (UW_OR vs UW _MM ) using an odds ratio data simulation Aim 3 and aim 4 of this study compare the efficiency of the UW_RR and the W_RR when they estimate the same effect size using a relative risk data simulation. T wo sections are addressed below First, we delineate the data simulation process including the odds ratio data simu lation and the relative risk data simulation Second we elaborate statistical analys es for each aim Odds Ratio Data S imulation In this section, two components are included. First we define scenarios by using predetermined parameters Second w e repe atedly simulate random sampl e s 10,000 times [30 32] under each scenario to conduct corresponding s eries of meta analyses ( Figure 3 1) The concept of a scenario can be thought as a source target populati on in term s of individual studies Definition of a S cenario We use six parameters to define a scenario including the global true probability of an event on the treatment arm ( ), the global true probabil ity of an event on the control arm ( ) the minimum number of subjects in each arm ( NLow ) the m aximum number of subjects in each arm (N High ) the diversity across studies (D) and the number of studies (M) requested for condu cting a meta analysis under the scenario.

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45 A series of scenarios are generated from the combination of these six par ameters of interest ( Figure 3 1) Additionally, each scenario needs to me et the following three criteria: an d ( 2) (1+D) is less than or equal to the value of one and (3) i f NLow is not equal to NHigh The reasons for the first two criteria are two folds as follows. a) These ensure the probability of the event of interest is not higher than the value of one on each arm b) L imit ing the values for the metric of odds ratio to be less than one is for reducing the redundancy of scenarios b ecause the failure rate and success rate are symmetric for the metric of odds ratio. The third criterion is to ensure variation in the sample size s across studies in the predefined range between the lowest number of subjects and the highest number of subjects in an individual study In this study we assume that the number of subjects is equally distributed in both arms. The valu es for each parameter are listed in Table 3 1. The considerations of selecting these values are based upon empirical clinical trial s combined for a meta analysis and the efficiency of data simulations We also provide three examples of scenarios with diffe rent parameters in Table 3 2 Random S ampling for M eta analys es Once a scenario is defined i. e. the pool of studies is established we start doing random sampling of studies from the pool. Each selected study is structured with four random variables the number of event s in the treatment arm (treatment event) the number of event s in the control arm (control event) and the number of subjects (treatment N / Control N) in each arm Under a scenario, in order to generate an individual study, t he tr ue event rate in the treatment arm ( ET ) for an individual study is uniformly distributed from (1 D) to (1+D) T he true event rate in the control arm ( EC ) for the same study is uniformly

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46 distributed from (1 D) to (1+D) The number of subjects ( treatment N /control N ) in each arm is uniformly dis tributed between NLow and NHigh The number of eve nts in the treatment arm is binomially distributed based upon the two factors of the sample size in the treatment arm and the true event rate in the treatment arm (ET) for the study while the number of events in the control arm is binomially distributed based upon the two factors of the sample size in the control arm and the true event rate in the control arm (EC) for the same study The attribute s of the random variables are listed in Table 3 3 According to the definition of a scenario, i n order to c onduct a meta analysis, M studies are needed to be chosen randomly under the scenario (i. e. the source of studies). T able 3 4 provides an example with five randomly selected studies based upon scenario 1 in Table 3 2. Once we have a set of M studies, i. e we finish a round of random sampling then we are ready to apply each proposed meta analysis approach to calculate outcomes of interest, which we describe in the next section. Under a scenario, we randomly select 10,000 times M number of studies to prep are for further analyses. Relative Risk Data S imulation The relative risk data simulation is s imilar to the odds ratio simulation ( Figure 3 2 ) The only difference lies in that we use a different outcome metric of relative risk (RR) instead of odds ra tio (OR). We use the RR simulation due to the precision issue with the calculation of standard error for the sample size weighted random effects approach (W_RR) The reason is that the parameters used to calculate the standard error for the odds ratio ar e 14 (4 means, 4 variances, and 6 covariances) whereas 5 parameters are needed for the W_ RR (2 means, 2 variances, and 1 covariance) [6]

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47 Definition of a Scenario The scenarios are generated from the combination of the six parameters of interest ( Table 3 5 ) A scenario needs to meet the same three criteria as we described in the odds ra tio data simulation Random Sampling for M eta analyses The detailed random sampling process is the same as it is for the odds ratio data simulation ( Figure 3 2). In sum to ensure the quality of two data simulations w e randomly selected a couple of sampling s and conducted corresponding meta analyses for each proposed approach manually to ensure the results from data simulations and from manual calculation s reach consensus T he odds ratio data simulation is used for aim 1 and aim 2 of the study whil e the relative risk data simulation is used aim 3 and aim 4 of the study. Statistical A nalyse s The outcomes of interest are measured in two ways in this simulation study. One is the mean coverage of the global true value of odds ratio or relative risk for the 95% confidence intervals (95% CI ) for each scenario Each scenario has one global true OR value or one global true RR value. The other way is the mean length of the 95% CI in the log scale which is derived from a n upper limit minus a lower limit o f the 95% CI for each scenario Th e closer the mean coverage probability to 95% and the narrower the mean length of the 95% CI indicate a better efficiency of a n approach under a scenario. Aim 1 Evaluate efficiency for the UW_OR and the UW_MM in a meta analysis of binomial trials when synthesizing a large number of studies.

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48 Under each scenario, we carr ied out the following meta analysis using both approaches (UW_OR vs. UW_MM) as shown in Figure 3 3 Step 1 for each random sample of M studies, we c onduct ed a meta analysis to calculate the coverage score of the global true odds ratio (OR) The coverage score is a dummy variable with the value of 1 or 0. If the 95% CI contains the global true OR, then the score is equal to 1, otherwise it is scored as 0. Meanwhile we calculate d the length of the 95% CI for each approach. Step 2, after 10,000 random sampling s i. e. 10,000 times meta analysis, we calculate d the mean coverage of the global true OR and the mean length of the 95% CI for the scenario In all, we follow ed the same two steps described above for each scenario At the end, for each scenario, we summarize d two measurements for each of the two approaches of interest (UW_OR vs. UW_MM). A im 2 Evaluate efficiency of t approximation vs. normal approximation in a meta analysis of binomial trials when synthesizing a small number of studies. As we stated in Chapter II when a small number of studies are combined for a meta analysis it is suggested that a t distribution is used instead of a normal distribution for the summary effect size in a log scale for binomial outcome s However the impact of the different distribution s on the efficiency of unweighted random effects approaches (UW_OR vs. UW_MM) remains unknown when a meta analysis contains a s mall number of studies with binomial outcome s Therefore, we modified the two proposed unweighted random effects approaches [7, 8] using a normal distribution instead of a t distribution Next, w e follow ed the same

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49 two steps outlined in aim 1 to compare the UW_OR with a t distribution (original version) vs. the UW_OR with a normal distribution (modified version) and to compare the UW_MM with a t distribution (original version) vs. the UW_MM with a normal distributi on (modified version) ( see Figure 3 4 ) Aim 3 Evaluate efficiency for the UW_RR and the W_RR when they estimate same summary effect size in a meta analysis of binomial trials. Under each scenario, we carried out the following meta analysis using bot h approaches (UW_RR vs. W_RR) as shown in Figure 3 5 The UW_RR selected in the comparison of the W_RR is based upon the following two reasons. First although the UW_ RR and the W_RR do not estimate the same summary effect size (unweighted vs. weighted), we forced them to estimate the same effect size by independently defin ing ET and EC for individual study and assumed that there is no association between the effect size and the sample size for each individual study Second, they both avoid the log transfo rmation on the individual effect size ( Figure 2 6 2 7 ). The steps are similar to those in aim 1. At the end, for each scenario, we have two summarized measurements for each of the two approaches of interest ( UW_R R vs. W_ R R ). A im 4 Evaluate efficien cy of t approximation vs. normal approximation in a meta analysis of binomial trials when synthesizing a small number of studies. As we stated in Chapter II when a small number of studies are combined in a meta analysis, it is suggested that we assume a t distribution over a normal distribution for the summary effect size in a log scale for binomial outcomes However the impact of

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50 the different distribution s on the efficiency of random effects approaches ( UW_R R vs. W_ RR ) remains unknown when a meta analys is contains a small number of studies with binomial outcome s Therefore, we modif ied the two random effects approaches the UW_RR and the W_RR from a t distribution to a normal distribution. Next, we follow ed the same two steps in aim 1 to compare the U W_ R R with a t distribution (original version) vs. the UW_R R with a normal distribution (modified version) and to compare the W_ RR with a t distribution (original version) vs. the W_RR with a normal distribution (modified version) (see Figure 3 6 ). In summary the data simulations and statistical analyses are performed using the statistical SAS software (9. 2, Cary). W e expect that our research aims of interest can be achieved through this study The results are reported in Chapter IV in the ord er of specific aims

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51 Figure 3 1. Odds ratio d ata simulation flow chart.

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52 Table 3 1. Parameter list and values for OR data s imulation Minimum Value Maximum Value Incremental Value No of Values M 5 5 5 10 6 0. 02 0. 22 0. 05 5 0. 02 0. 22 0. 05 5 NLow in each arm 100 4 00 1 00 4 NHigh in each arm 1 00 5 00 1 00 5 D 0. 1 0. 2 0. 3 3 Table 3 2. Example s with combination of parameters for scenarios M NLow NHigh D Scenario 1 5 0. 2 0. 2 100 200 0. 1 Scenario 2 15 0. 2 0. 2 200 300 0. 1 Scenario n 55 0. 2 0. 2 400 500 0. 3 Table 3 3. Random sampling based upon parameters under a scenario Treatment event Control event Event rate in treatment arm Event rate in control arm Treatment N (=Control N) Attributes Rand om Random Random Random Random Distribution Binomi al Binomi al Uniform Uniform Uniform Table 3 4. Example of studies structure for a random sampling in a meta analysis M Treatment event Control even t Treatment N Control N 1 4 3 101 101 2 0 2 123 123 3 4 4 143 143 4 6 0 156 156 5 2 1 178 178

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53 Figure 3 2. Relative risk d ata simu lation flow chart.

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54 Table 3 5 Parameter list and values for RR data s imulation Minimum Value Maximum Value Incremental Value No of Values M 5 5 5 10 6 0. 02 0. 22 0. 05 5 RR 0. 1 1 0. 2 5 NLow in each arm 100 4 00 1 00 4 NHigh in each arm 1 00 5 00 1 00 5 D 0. 1 0. 2 0. 3 3

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55 Figure 3 3. Illustration of analysis steps for aim 1

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56 Figure 3 4. Illustration of analysis steps for aim 2

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57 Figure 3 5. Illustrat ion of analysis steps for aim 3

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58 Figure 3 6. Illustration of analysis steps for aim 4

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59 CHAPTER 4 RESULTS R esults are reported for each hypothesis under each aim For each hypothesis, the results are presented separately as a primary analysis a subgroup analysis stratified by the global true event rate in the treatment arm ( PT / ) and the number of studies (M) Further more under each primary analysis and subgroup analysis the results are presented in t erms of the two forms of assessment as we explain in the next paragraph Under each scenario, t he efficiency of a method is measured using two metrics: the mean coverage of the global true value by the purported 95% CI in percentage and the mean length of the 95% CI in a log scale. The combination of these two metrics helps to determine which method is more efficient. The closer the mean coverage is to 95% and the lower mean length of the 95% CI indicate a superior efficiency. If one method has a more accurate mean coverage and a wider mean length of the 95% CI or i f one method has a l ess accurate mean coverage and a narr ower mean length of the 95% CI, we consider first whether the mean coverage for the method is close to 95% coverage If the mean coverage for both approaches is all close to 95%, then the narrow er the mean length of the 95% CI the better that approach is Two forms of figures are adopted to compare the efficiency of any two methods of interest One form of figures is us e the difference of the mea n coverage for two comparators on the y axis and the difference of the mean length of the 95% CI for two compar ators on the x axis One red d ot indicates one scenario. R ed dots fall ing into the top left or bottom right quadrant indicate the superior efficiency of a method The other figu re is us e the mean coverage on the y axis and the mean length of the 95% CI o n the

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60 x axis Therefore, the blue and red dots differentiate the comparators of interest for each scenario. In this section, concepts of the mean coverage or the mean length of the 95% CI are used within a scenario with a divisor of 10,000 instances of me ta analysis. Otherwise the concepts of the average of the mean coverage or the average of the mean length 95% CI are adopted for all scenarios with a divisor of the number of scenarios of interest. As we mentioned in Chapter III, the method s section, the odds ratio simulation is performed for aim 1 and aim 2 A total of 6,300 scenarios are generated based upon the combination of six predetermined parameters, 2,700 of which me e t the predefi ned criteria in the odds ratio simulation R esults of aim 1 and ai m 2 are generated based upon these qualified scenarios. At the same time, the relative risk simulation is performed for aim 3 and aim 4. A total of 6,300 scenarios are generated based upon the combination of six predetermined parameters, 4,200 of which me e t the same predefined criteria as in the odds ratio simulation The reason that the more scenarios 4,200, meet the predefined criteria is that there are 24 values for the global true event probability in the control arm ( c) in the relative risk simulation, but 5 values in c in the odds ratio simulation. The r esults of aim 3 and aim 4 are generated based upon these qualified scenarios. Aim 1 Hypothesis 1 : It is hypothesized that there is no precision difference bet ween the UW_OR and the UW_MM when synthesizing a large number of studies.

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61 The difference in the mean coverage is derived f rom the mean coverage of the U W_OR minus that of the UW_MM, which is represented on the y axis ( Figure 4 1, 4 2 ). The difference of the mean length of the 95% CI is derived from the mean length of the 95% CI of the UW_OR minus that of the UW_MM which is repre sented on the x axis ( Figure 4 1 4 2 ). Primary analysis We note that the majo rity of scenarios in the form of dot s, 1,850 ou t of 2700 (68. 52%) f a ll into the top left qua drant It means that t he UW_OR approach has a higher mean coverage and a narrower mean length of 95 % CI indicating that the UW_OR is more efficient than the UW_MM ( Figure 4 1). T he average of the mean covera ge of the UW_ OR for scenarios is 95. 1% (range 94. 4% to 95. 9%) whereas that of the UW_MM is 9 3 1 % (range 62. 0% to 96. 2%) which is away to the expected 95% coverage ( Table 4 1 ). Moreover, t he average of the mean length of the 95% CI of the UW_OR is 0. 369 w hich is shorter than that of the UW_MM, 0. 4 00 Subgroup analysis When the number of studies is as small as five the efficiency of both method s is close regardless of the event rate in the treatment arm ( Figure 4 2, 4 4) A t low event rate s in the treatm ent arm (PT< = 0. 12) as the number of studies ( M ) increase s the UW _OR is more efficient than the UW _MM in terms of the higher mean coverage difference between the UW_OR and the UW_MM ( Figure 4 2) while the mean length of 95% CI for both approaches is cl ose ( Figure 4 2 Table 4 2 ). The reason for the higher mean coverage difference is that the mean coverage of the UW_MM drops as the number of studies increases while that of the UW _OR is robust and close to the expected 95% coverage ( Figure 4 4)

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62 At high event rate s in the treatment arm ( PT > 0. 12 ) the UW _OR is as efficient as the UW _MM ( Figure 4 2) regardless of the number of studies (M) For both methods, the mean length of the 95% CI is short er as the event rate in the treatment arm is increased or the numbe r of studies is increased. The average of mean length of the 95% CI for the UW_OR and the UW_MM under all scenarios is dropped from 1. 232 and 1. 382 respectively at the low event of 0. 02 in the treatment arm and the number of studies of five to 0. 129 and 0. 131 respectively at the high event rate of 0. 22 in the treatment arm and the number of studies of 55 ( Table 4 2 ) Aim 2 Hypothesis 2 1: It is hypothesized that a t approximation is more accurate than a normal approximation in terms of closeness to the 95% coverage for the UW_OR when synthesizing a small number of studies. The difference in the mean coverage is derived from th e mean coverage of the UW_OR with a t distribution minus that of the UW_ OR with a normal distribution which is represente d on the y axis ( Figure 4 5, 4 6). The difference in the mean length of the 95% CI is derived from the mean length of the 95% CI from the UW_OR with a t distribution minus that of the UW_ OR with a normal distribution as represented on the x axis ( Figure 4 5, 4 6). Primary analysis All 2 700 scenarios f a ll into the top right quadrant ( Figure 4 5) T he average of the mean coverage of the UW_OR with a t distribution is 95. 1% (range 94. 4% to 95. 9%) ( Table 4 3) w hereas the average of the mean coverage of th e UW_OR with a normal distribution is 9 3 1% (range 87. 1% to 95. 2%) which is away from the expected 95% coverage ( Table 4 3). The finding s indicat e that the assumption of a t

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63 distribution is more accurate than the assumption of a normal distribution for the UW_OR Subgroup analysis When the number of studies is as low as five the average of the mean coverage of the UW_OR with a t distribution is 95. 5% (range 95. 1% to 95. 9%), whereas the average of the mean coverage of the UW_OR with a norm al distributio n is as low as 88. 6% (range 87. 9% to 89. 3%) ( Figure 4 8, Table 4 4 ) which is far away from the expected 95% coverage A s the number of studies ( M ) increase s the mean coverage for the UW_ OR with a normal distribution is close to that of the UW_OR with a t distribution regardless o f the event rate in the treatment arm ( Figure 4 8 ) w hereas the mean coverage of a t distribution is robust within the range of 94. 4% to 95. 9% regardless of the number of studies or the event rate in the treatment arm ( Table 4 3 Figure 4 8) T he mean length of the 95% CI for both a t distribution and a normal distribution for the UW_OR is shorter when the number of studies increases or the event rate in the treatment arm increases ( Figure 4 8 ). The average of the mean length of the 95% CI for the UW_OR with a t distribution and the UW _OR with a normal distribution is shorter from 1. 232 and 0. 87 0 respectively at the low event of 0. 02 in the treatment arm and the number of studies of five to 0. 129 and 0. 1 26 respectively at the h igh event rate of 0. 22 in the treatment arm and the number of studies of 55 ( Table 4 4 ). Hypothesis 2 2: It is hypothesized that a t approximation is more accurate than a normal approximation in terms of closeness to the 95% coverage for the UW_MM when syn thesizing a small number of studies.

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64 The difference of the mean coverage is derived f rom the mean coverage of the UW_MM with a t distribution minus that of the UW_MM with a normal distribution which is represented on the y axis ( Figure 4 9 4 10 ). The dif ference of the mean length of the 95% CI is derived f rom the mean length of the 95% CI of the UW_MM with a t distribution minus that of the UW_MM with a normal distribution which is represented on the x axis ( Figure 4 9 4 10 ). Primary analysis A ll 2 7 00 scenarios f a ll into the top right quadrant ( Figure 4 9 ). The average of the mean coverage of the UW_MM with a t distribution is 9 3. 1 % (range 62. 0% to 96. 2%), which is closer the expected 95% coverage whereas that of the UW_MM with a normal distributio n is 9 0 8 % (range 60. 4% to 95. 2%), which is lower than the expected 95% coverage ( Table 4 5 ). The finding s indicat e that the assumption of a t distribution is more accurate than the assumption of a normal distribution for the UW_MM. Subgroup analysis When the number of studies is as low as five the average of the mean coverage for the UW_MM with a t distribution is 95. 5% (range 94. 9% to 96. 2%) whereas that of the UW_MM with a normal distribution is 88. 4% (range 87. 4% to 89. 5%) that is far away from th e expected 95% coverage ( Figure 4 1 2, Table 4 6 ) A s the number of studies ( M ) increase s the UW_ MM with a normal distribution is close to the UW_MM with a t distribution regardless of the event rat e in the treatment arm ( Figure 4 1 2 ). At the lower event r ate of 0. 02 in the treatment arm as the number of studies increases from 5 to 55 the average of the mean coverage of the UW_ MM drop s dramatically from 95. 5% to 82. 6% for the one with a t distribution and from 88. 4% to 81. 3% for the one with a n ormal di stribution ( Table 4 6 ) W hereas at the high event

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65 rate of 0. 22 in the treatment arm as the number of studies increases from 5 to 55 the average of the mean coverage of the UW_MM stay stably from 95. 3 % to 95. 1 % for a t distribution and from 88. 1 % to 94. 6 % for a normal distribution ( Table 4 6 ). The mean length of the 95% CI for the UW_MM with a t distribution and the UW_MM with a normal distribution is shorter when the number of studies increases or the event rate in the treatment arm increases ( Figure 4 1 2 ). The average of the mean length of the 95% CI of the UW_MM from 1. 382 for a t distribution and 0. 976 for a normal distribution at the low event of 0. 02 in the treatment arm an d the number of studies of five drop to 0. 131 for a t distribution and 0. 12 8 for a normal distribution at the high event rate of 0. 22 in the treatment arm and number of studies of 55 ( Table 4 6 ). Aim 3 Hypothesis 3: It is hypothesized that efficiency of the UW_RR and the W_RR is similar when they estimate the same summary effec t size. The difference in the mean coverage is derived from the mean coverage of the UW_RR minus that of the W_RR which is represented on the y axis ( Figure 4 13 4 14 ). The difference of the mean length of the 95% CI is derived from the mean length of t he 95% CI from the UW_RR minus that of the W_RR as is represented on the x axis ( Figure 4 13 4 14 ). Primary analysis T he average of the mean coverage for both the UW_RR and the W_RR is closer to the expected 95% coverage which is 95. 1 % (range 94. 4 % t o 95. 9%) vs. 95. 0 % ( range 94. 0% and 95. 9% ) ( Table 4 7 ). However, the W_RR provides a narrower average of mean length of the 95% CI of 0. 291 (range 0. 0 58 to 1. 976) as compared to 0. 299 for the UW_RR (range 0. 058 to 2. 014) ( Table 4 7 ). As we note 3,062 out of 4,200 (72. 90%) f a ll into the top right quadrant while 1,110 out of 4 200

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66 (26. 43%) f a ll into the bottom right quadrant ( Figure 4 13) indicating that the UW_RR and the W _RR have similar efficiency in terms of the mean coverage, but the W_RR is more accu rate as compared with the UW_RR in some scenarios Subgroup analysis In regards to the mean coverage, as the number of studies increases, the mean coverage difference between the UW_RR and the W_RR decreases r egardless of the e vent rate in the treatment arm ( Figure 4 14) indicating that the effic iency of both approaches is closer as the number of studies increases In addition, the mean length of the 95% CI for both the UW_RR and the W_RR is getting narrower as the number of studies increases or the eve nt rate in the treatment arm increases ( Figure 4 16) Aim 4 Hypothesis 4 1: It is hypothesized that a t approximation is more accurate than a normal approximation in terms of closeness to the 95% coverage for the UW_RR when synthesizing a small number of studies. The difference of the mean coverage is derived from the mean coverage of the UW_RR with a t distribution minus that of the U W_RR with a normal distribution a s represented on the y axis ( Figure 4 1 7, 4 18 ). The difference of the mean length of the 95% CI is derived from the mean length of the 95% CI from the UW_RR with a t distribution minus that of the U W_RR with a normal distribution which is represented on the x axis ( Figure 4 1 7, 4 18 ). Primary analysis All 4,200 scenarios f a ll into t he t op right quadrant ( Figure 4 17 ). The average of the mean coverage of the UW_RR with a t distribution is 95. 1% (range 94. 4% to 95. 9%) whereas that of the U W_RR with a normal distribution is 93 1 % (range 87. 0% to 95. 2%), which is lower than the expected 95% coverage ( Table 4 9 ).

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67 The finding s indicat e that the assumption of a t distribution is more accurate than the assumption of a normal distribution for the UW_ R R. Subgroup analysis When the number of studies is as low as five the difference of the mean cov erage between the UW_RR with a t distribution and the UW_R R with a normal distribution i s large ( Figure 4 18). The average of the mean coverage of the UW_RR with a t distribution is 95. 5% (range 95. 1% to 95. 9%), whereas the average of the mean coverage of the UW_RR with a normal distribution is 88. 6% (range 87. 8% to 89. 4%) which is far from the expected 95% coverage ( Table 4 10). A s the number of studies ( M ) increase s the mean coverage for the UW_ RR with a normal distribution is close to that of the UW_R R with a t distribution reg ardless of the event rate in the treatment arm ( Figure 4 20 ). Whereas the mean coverage of the UW_RR with a t distribution is robust within the range of 94. 4% to 95. 9% regardless of the number of s tudies or the event rate in the treatment arm ( Table 4 9 Figure 4 20 ) The mean length of the 95% CI for the UW_RR with both a t distribution and a normal distribution is shorter when the number of studies increases or the event rate in the treatment arm increases ( Figure 4 20 ). The av erage of the mean length of the 95% CI for the UW_ R R with a t distribution and with a normal distribution is shorter from 1. 312 and 0. 926 respectively at the low event of 0. 02 in the treatment arm and t he number of studies of five to 0. 092 and 0. 0 9 0 respec tively at the high event rate of 0. 22 in the treatment arm and the number of studies of 55 ( Table 4 10 ). Hypothesis 4 2: It is hypothesized that a t approximation is more accurate than a normal approximation in terms of closeness to the 95% coverage for th e W_RR when synthesizing a small number of studies.

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68 The difference in the mean coverage is derived f rom the mean coverage of the W_RR with a t distribution minus that of the W_RR with a normal distribution, as represented on the y axis ( Figure 4 2 1, 4 22 ) The difference in the mean length of the 95% CI is derived from the mean length of the 95% CI from the W_RR with a t distribution minus that of the W_RR with a normal distribution as represented on the x axis ( Figure 4 21 4 22 ). Primary analysis All 4 ,200 scenarios f a ll into the top right quadrant ( Figure 4 21 ). The average of the mean coverage of the W_RR with a t distribution is 9 5. 0% (range 94. 0% to 95. 9%) wh ereas that of the W_RR with a normal distribution is 92. 9% (range 86. 0% to 95. 2%) which is lower than the expected 95% coverage ( Table 4 11 ). The finding s indicat e that the assumption of a t distribution is more accurate than the assumption of a normal distribution for the W_RR. Subgroup analysis When the number of studies is as low as five the difference in the mean coverage between the W_RR with a t distribution and the W_RR with a normal distribution is large ( Figure 4 22 ). The average of the mean coverage of the W_RR with a t distribution is 95. 2 % (range 9 4. 4 % to 95. 9%), whereas the avera ge of the mean coverage of the W_RR with a normal distribution is 88. 2 % (range 8 6 8% to 89. 2 %), which is far away from the expected 95% coverage ( Table 4 1 2 ). A s the number of studies ( M ) increase s the W_ RR with a normal distribution is close to the W_RR with a t distribution rega rdless of the event rate in the treatment arm ( Figure 4 24 ). Whereas the mean coverage for the W_RR with a t distribution is robust within the range of 9 4 0 % to 95. 9% regardless of the number of studies or the event rate in the treatment arm ( Table 4 1 1 Figure 4 2 4 )

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69 The mean length of the 95% CI for the W_RR with both a t distribution and a normal distribution is shorter when the number of studies increase s or the event rate in the treatment arm increases ( Figure 4 2 4 ). The ave rage of the mean length of the 95% CI for the W_RR with a t distribution and with a normal distribution decreases from 1. 273 and 0. 899 respectively at the low event of 0. 02 in the treatment arm and the number of studies of five to 0. 09 1 and 0. 0 89 respect ively at the high event rate of 0. 22 in the treatment arm with the number of studies of 55 ( Table 4 1 2 ). In summary through the odds ratio simulation, we f ind that the efficiency of the UW_OR and the UW_MM is close as long as no rare event occ urs suggesting that we cannot reject the null hypothesis for aim 1 In addition, when a small number of studies are combined for a meta analysis both the UW_OR and the UW_MM show that the assumption with a t distribution is more accurate than the assumpt ion with a normal distribution in terms of closeness to the 95% coverage which suggest s that we cannot reject the null hypotheses for aim 2 In regards to the relative risk simulation, we f ind that when the UW_RR and the W_RR estimate the same summary e ffect size both of which are efficient in terms of closeness to the 95% coverage but the W_RR is marginally precise in terms of the mean length of the 95% CI, suggesting that we cannot reject the null hypothesis for aim 3. Moreover, when a small number of studies are combined for a meta analysis, both the UW_ R R and the W_ RR show that the assumption with a t distribution that is more accurate than the assumption with a normal distribution in terms of closeness to the 95% coverage It suggest s that we can not reject the null hypotheses for aim 4 In next Chapter V, we discuss our findings and related issues in more depth.

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70 Table 4 1. Aim 1 : C omparison s of UW_OR vs. UW_MM Outcomes Mean STD Minimum Maximum Mean coverage for UW_OR 0. 951 0. 002 0. 944 0. 959 Me an coverage for UW_MM 0. 931 0. 047 0. 62 0. 962 Mean 95% CI l ength for UW_OR 0. 369 0. 279 0. 091 2. 053 Mean 95% CI l ength for UW_MM 0. 400 0. 312 0. 092 2. 289

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71 Table 4 2. Aim 1 : C omparisons of UW_OR vs. UW_MM by e vent rate in treatment arm and number of studies Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 02 5 Mean coverage for UW_OR 150 0. 955 0. 002 0. 951 0. 959 Mean coverage for UW_MM 150 0. 955 0. 002 0. 949 0. 962 Mean 95% CI length for UW_OR 150 1. 232 0. 2 51 0. 856 2. 053 Mean 95% CI length for UW_MM 150 1. 382 0. 278 0. 955 2. 289 15 Mean coverage for UW_OR 150 0. 952 0. 002 0. 947 0. 957 Mean coverage for UW_MM 150 0. 939 0. 013 0. 902 0. 957 Mean 95% CI length for UW_OR 150 0. 567 0. 115 0. 394 0. 937 Mean 95% CI length for UW_MM 150 0. 656 0. 129 0. 457 1. 071 25 Mean coverage for UW_OR 150 0. 950 0. 002 0. 946 0. 955 Mean coverage for UW_MM 150 0. 911 0. 031 0. 826 0. 957 Mean 95% CI length for UW_OR 150 0. 425 0. 086 0. 295 0. 701 Mean 95% CI le ngth for UW_MM 150 0. 495 0. 096 0. 347 0. 803 35 Mean coverage for UW_OR 150 0. 950 0. 002 0. 946 0. 956 Mean coverage for UW_MM 150 0. 885 0. 049 0. 760 0. 958 Mean 95% CI length for UW_OR 150 0. 354 0. 072 0. 246 0. 583 Mean 95% CI length for UW_MM 15 0 0. 413 0. 079 0. 290 0. 669 45 Mean coverage for UW_OR 150 0. 950 0. 002 0. 945 0. 955 Mean coverage for UW_MM 150 0. 856 0. 067 0. 689 0. 955 Mean 95% CI length for UW_OR 150 0. 310 0. 063 0. 215 0. 510 Mean 95% CI length for UW_MM 150 0. 363 0. 069 0. 2 55 0. 586 55 Mean coverage for UW_OR 150 0. 951 0. 002 0. 944 0. 957 Mean coverage for UW_MM 150 0. 826 0. 087 0. 620 0. 957 Mean 95% CI length for UW_OR 150 0. 279 0. 056 0. 194 0. 459 Mean 95% CI length for UW_MM 150 0. 327 0. 062 0. 230 0. 528

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72 Table 4 2. C o ntinued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 07 5 Mean coverage for UW_OR 120 0. 953 0. 003 0. 946 0. 958 Mean coverage for UW_MM 120 0. 954 0. 003 0. 945 0. 959 Mean 95% CI length for UW_OR 120 0. 757 0. 130 0. 521 1. 129 Mean 95% CI length for UW_MM 120 0. 784 0. 144 0. 531 1. 208 15 Mean coverage for UW_OR 120 0. 952 0. 002 0. 947 0. 956 Mean coverage for UW_MM 120 0. 950 0. 002 0. 946 0. 956 Mean 95% CI length for UW_OR 120 0. 352 0. 061 0 243 0. 525 Mean 95% CI length for UW_MM 120 0. 368 0. 068 0. 249 0. 570 25 Mean coverage for UW_OR 120 0. 951 0. 001 0. 948 0. 954 Mean coverage for UW_MM 120 0. 947 0. 004 0. 936 0. 952 Mean 95% CI length for UW_OR 120 0. 264 0. 045 0. 183 0. 394 M ean 95% CI length for UW_MM 120 0. 277 0. 051 0. 188 0. 428 35 Mean coverage for UW_OR 120 0. 950 0. 001 0. 948 0. 955 Mean coverage for UW_MM 120 0. 944 0. 007 0. 924 0. 953 Mean 95% CI length for UW_OR 120 0. 221 0. 038 0. 153 0. 328 Mean 95% CI length for UW_MM 120 0. 231 0. 043 0. 157 0. 357 45 Mean coverage for UW_OR 120 0. 950 0. 003 0. 944 0. 955 Mean coverage for UW_MM 120 0. 942 0. 009 0. 911 0. 955 Mean 95% CI length for UW_OR 120 0. 193 0. 033 0. 134 0. 287 Mean 95% CI length for UW_MM 120 0. 203 0. 038 0. 137 0. 313 55 Mean coverage for UW_OR 120 0. 951 0. 002 0. 945 0. 953 Mean coverage for UW_MM 120 0. 939 0. 012 0. 902 0. 954 Mean 95% CI length for UW_OR 120 0. 174 0. 030 0. 120 0. 259 Mean 95% CI length for UW_MM 120 0. 183 0. 034 0. 124 0 282

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73 Table 4 2. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 12 5 Mean coverage for UW_OR 90 0. 953 0. 003 0. 945 0. 957 Mean coverage for UW_MM 90 0. 953 0. 002 0. 945 0. 957 Mean 95% CI lengt h for UW_OR 90 0. 642 0. 103 0. 446 0. 910 Mean 95% CI length for UW_MM 90 0. 654 0. 109 0. 451 0. 942 15 Mean coverage for UW_OR 90 0. 952 0. 002 0. 948 0. 956 Mean coverage for UW_MM 90 0. 951 0. 002 0. 948 0. 956 Mean 95% CI length for UW_OR 90 0. 299 0. 048 0. 209 0. 424 Mean 95% CI length for UW_MM 90 0. 306 0. 051 0. 211 0. 442 25 Mean coverage for UW_OR 90 0. 951 0. 001 0. 948 0. 953 Mean coverage for UW_MM 90 0. 950 0. 001 0. 947 0. 953 Mean 95% CI length for UW_OR 90 0. 225 0. 036 0. 157 0. 318 Mean 95% CI length for UW_MM 90 0. 230 0. 039 0. 159 0. 331 35 Mean coverage for UW_OR 90 0. 950 0. 001 0. 948 0. 954 Mean coverage for UW_MM 90 0. 949 0. 002 0. 943 0. 953 Mean 95% CI length for UW_OR 90 0. 187 0. 030 0. 131 0. 265 Mean 95% CI length for UW_MM 90 0. 192 0. 032 0. 133 0. 277 45 Mean coverage for UW_OR 90 0. 950 0. 003 0. 945 0. 955 Mean coverage for UW_MM 90 0. 948 0. 004 0. 939 0. 955 Mean 95% CI length for UW_OR 90 0. 164 0. 026 0. 115 0. 232 Mean 95% CI length for UW_MM 90 0. 168 0. 028 0. 116 0. 242 55 Mean coverage for UW_OR 90 0. 951 0. 002 0. 946 0. 955 Mean coverage for UW_MM 90 0. 949 0. 003 0. 939 0. 955 Mean 95% CI length for UW_OR 90 0. 148 0. 024 0. 103 0. 209 Mean 95% CI length for UW_MM 90 0. 152 0. 025 0. 105 0. 218

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74 T able 4 2. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 17 5 Mean coverage for UW_OR 60 0. 953 0. 002 0. 947 0. 957 Mean coverage for UW_MM 60 0. 953 0. 002 0. 947 0. 957 Mean 95% CI length for UW_ OR 60 0. 588 0. 093 0. 413 0. 811 Mean 95% CI length for UW_MM 60 0. 596 0. 096 0. 416 0. 829 15 Mean coverage for UW_OR 60 0. 951 0. 002 0. 948 0. 956 Mean coverage for UW_MM 60 0. 951 0. 002 0. 948 0. 955 Mean 95% CI length for UW_OR 60 0. 274 0. 044 0. 1 93 0. 378 Mean 95% CI length for UW_MM 60 0. 279 0. 045 0. 195 0. 388 25 Mean coverage for UW_OR 60 0. 950 0. 001 0. 947 0. 953 Mean coverage for UW_MM 60 0. 950 0. 001 0. 946 0. 953 Mean 95% CI length for UW_OR 60 0. 206 0. 033 0. 145 0. 284 Mean 95 % CI length for UW_MM 60 0. 209 0. 034 0. 146 0. 291 35 Mean coverage for UW_OR 60 0. 950 0. 001 0. 948 0. 954 Mean coverage for UW_MM 60 0. 950 0. 001 0. 947 0. 953 Mean 95% CI length for UW_OR 60 0. 172 0. 027 0. 121 0. 236 Mean 95% CI length for UW_MM 60 0. 175 0. 028 0. 122 0. 243 45 Mean coverage for UW_OR 60 0. 950 0. 002 0. 946 0. 954 Mean coverage for UW_MM 60 0. 950 0. 003 0. 944 0. 954 Mean 95% CI length for UW_OR 60 0. 151 0. 024 0. 106 0. 207 Mean 95% CI length for UW_MM 60 0. 153 0. 025 0. 107 0. 213 55 Mean coverage for UW_OR 60 0. 951 0. 002 0. 946 0. 954 Mean coverage for UW_MM 60 0. 951 0. 002 0. 946 0. 955 Mean 95% CI length for UW_OR 60 0. 136 0. 021 0. 096 0. 187 Mean 95% CI length for UW_MM 60 0. 138 0. 022 0. 096 0. 192

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75 Table 4 2 Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 22 5 Mean coverage for UW_OR 30 0. 953 0. 002 0. 948 0. 956 Mean coverage for UW_MM 30 0. 953 0. 002 0. 948 0. 956 Mean 95% CI length for UW_OR 30 0. 558 0. 089 0. 395 0. 759 Mean 95% CI length for UW_MM 30 0. 564 0. 092 0. 397 0. 772 15 Mean coverage for UW_OR 30 0. 951 0. 002 0. 948 0. 954 Mean coverage for UW_MM 30 0. 951 0. 002 0. 948 0. 954 Mean 95% CI length for UW_OR 30 0. 261 0. 042 0. 185 0. 354 Mean 95% CI length for UW_MM 30 0. 264 0. 043 0. 186 0. 361 25 Mean coverage for UW_OR 30 0. 951 0. 002 0. 946 0. 953 Mean coverage for UW_MM 30 0. 950 0. 002 0. 946 0. 953 Mean 95% CI length for UW_OR 30 0. 196 0. 031 0. 139 0. 266 Mean 95% CI len gth for UW_MM 30 0. 198 0. 032 0. 140 0. 271 35 Mean coverage for UW_OR 30 0. 950 0. 001 0. 948 0. 953 Mean coverage for UW_MM 30 0. 950 0. 001 0. 948 0. 954 Mean 95% CI length for UW_OR 30 0. 163 0. 026 0. 116 0. 222 Mean 95% CI length for UW_MM 30 0. 16 5 0. 027 0. 117 0. 226 45 Mean coverage for UW_OR 30 0. 950 0. 002 0. 946 0. 954 Mean coverage for UW_MM 30 0. 950 0. 002 0. 947 0. 954 Mean 95% CI length for UW_OR 30 0. 143 0. 023 0. 102 0. 194 Mean 95% CI length for UW_MM 30 0. 145 0. 024 0. 102 0. 198 55 Mean coverage for UW_OR 30 0. 951 0. 002 0. 946 0. 955 Mean coverage for UW_MM 30 0. 951 0. 002 0. 948 0. 955 Mean 95% CI length for UW_OR 30 0. 129 0. 021 0. 091 0. 175 Mean 95% CI length for UW_MM 30 0. 131 0. 021 0. 092 0. 179

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76 Table 4 3. Aim 2 1: C omparisons of t distribution vs. normal distribution for UW_OR Outcomes Mean STD Minimum Maximum Mean coverage for normal distribution 0. 931 0. 022 0. 871 0. 952 Mean coverage for t distribution 0. 951 0. 002 0. 944 0. 959 Mean 95% CI length for normal dis tribution 0. 315 0. 191 0. 089 1. 449 Mean 95% CI length for t distribution 0. 369 0. 279 0. 091 2. 053

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77 Table 4 4. Aim 2 1: C omparisons of t distribution vs. normal distribution for UW_OR by event rate in treatment arm and number of studies Event rate in treat ment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 02 5 Mean coverage for normal distribution 150 0. 886 0. 003 0. 879 0. 893 Mean coverage for t distribution 150 0. 955 0. 002 0. 951 0. 959 Mean 95% CI length for normal distributi on 150 0. 870 0. 177 0. 604 1. 449 Mean 95% CI length for t distribution 150 1. 232 0. 251 0. 856 2. 053 15 Mean coverage for normal distribution 150 0. 932 0. 002 0. 925 0. 937 Mean coverage for t distribution 150 0. 952 0. 002 0. 947 0. 957 Mean 95% CI length for normal distribution 150 0. 518 0. 105 0. 360 0. 856 Mean 95% CI length for t distribution 150 0. 567 0. 115 0. 394 0. 937 25 Mean coverage for normal distribution 150 0. 939 0. 002 0. 933 0. 943 Mean coverage for t distribution 150 0. 950 0. 002 0. 946 0. 955 Mean 95% CI length for normal distribution 150 0. 403 0. 082 0. 280 0. 665 Mean 95% CI length for t distribution 150 0. 425 0. 086 0. 295 0. 701 35 Mean coverage for normal distribution 150 0. 941 0. 002 0. 938 0. 949 Mean coverage for t distribution 150 0. 950 0. 002 0. 946 0. 956 Mean 95% CI length for normal distribution 150 0. 341 0. 069 0. 237 0. 562 Mean 95% CI length for t distribution 150 0. 354 0. 072 0. 246 0. 583 45 Mean coverage for normal distribution 150 0. 943 0. 002 0. 939 0 949 Mean coverage for t distribution 150 0. 950 0. 002 0. 945 0. 955 Mean 95% CI length for normal distribution 150 0. 301 0. 061 0. 209 0. 496 Mean 95% CI length for t distribution 150 0. 310 0. 063 0. 215 0. 510 55 Mean coverage for normal distrib ution 150 0. 945 0. 002 0. 939 0. 952 Mean coverage for t distribution 150 0. 951 0. 002 0. 944 0. 957 Mean 95% CI length for normal distribution 150 0. 273 0. 055 0. 189 0. 449 Mean 95% CI length for t distribution 150 0. 279 0. 056 0. 194 0. 459

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78 Table 4 4. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 07 5 Mean coverage for normal distribution 120 0. 882 0. 004 0. 873 0. 891 Mean coverage for t distribution 120 0. 953 0. 003 0. 946 0. 958 Mean 9 5% CI length for normal distribution 120 0. 534 0. 092 0. 368 0. 797 Mean 95% CI length for t distribution 120 0. 757 0. 130 0. 521 1. 129 15 Mean coverage for normal distribution 120 0. 931 0. 002 0. 927 0. 936 Mean coverage for t distribution 120 0. 952 0. 002 0. 947 0. 956 Mean 95% CI length for normal distribution 120 0. 322 0. 056 0. 222 0. 480 Mean 95% CI length for t distribution 120 0. 352 0. 061 0. 243 0. 525 25 Mean coverage for normal distribution 120 0. 939 0. 001 0. 936 0. 942 Mean coverage for t distribution 120 0. 951 0. 001 0. 948 0. 954 Mean 95% CI length for normal distribution 120 0. 251 0. 043 0. 174 0. 374 Mean 95% CI length for t distribution 120 0. 264 0. 045 0. 183 0. 394 35 Mean coverage for normal distribution 120 0. 942 0. 002 0. 938 0. 947 Mean coverage for t distribution 120 0. 950 0. 001 0. 948 0. 955 Mean 95% CI length for normal distribution 120 0. 213 0. 036 0. 147 0. 316 Mean 95% CI length for t distribution 120 0. 221 0. 038 0. 153 0. 328 45 Mean coverage for normal di stribution 120 0. 944 0. 003 0. 938 0. 949 Mean coverage for t distribution 120 0. 950 0. 003 0. 944 0. 955 Mean 95% CI length for normal distribution 120 0. 188 0. 032 0. 130 0. 279 Mean 95% CI length for t distribution 120 0. 193 0. 033 0. 134 0. 287 5 5 Mean coverage for normal distribution 120 0. 946 0. 002 0. 940 0. 949 Mean coverage for t distribution 120 0. 951 0. 002 0. 945 0. 953 Mean 95% CI length for normal distribution 120 0. 170 0. 029 0. 118 0. 253 Mean 95% CI length for t distribution 120 0. 174 0. 030 0. 120 0. 259

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79 Table 4 4. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 12 5 Mean coverage for normal distribution 90 0. 881 0. 003 0. 872 0. 890 Mean coverage for t distribution 90 0. 9 53 0. 003 0. 945 0. 957 Mean 95% CI length for normal distribution 90 0. 453 0. 073 0. 315 0. 642 Mean 95% CI length for t distribution 90 0. 642 0. 103 0. 446 0. 910 15 Mean coverage for normal distribution 90 0. 931 0. 002 0. 927 0. 935 Mean coverage for t distribution 90 0. 952 0. 002 0. 948 0. 956 Mean 95% CI length for normal distribution 90 0. 273 0. 044 0. 191 0. 387 Mean 95% CI length for t distribution 90 0. 299 0. 048 0. 209 0. 424 25 Mean coverage for normal distribution 90 0. 939 0. 001 0. 936 0. 942 Mean coverage for t distribution 90 0. 951 0. 001 0. 948 0. 953 Mean 95% CI length for normal distribution 90 0. 213 0. 034 0. 149 0. 302 Mean 95% CI length for t distribution 90 0. 225 0. 036 0. 157 0. 318 35 Mean coverage for normal distribut ion 90 0. 942 0. 002 0. 939 0. 945 Mean coverage for t distribution 90 0. 950 0. 001 0. 948 0. 954 Mean 95% CI length for normal distribution 90 0. 181 0. 029 0. 126 0. 256 Mean 95% CI length for t distribution 90 0. 187 0. 030 0. 131 0. 265 45 Mean cove rage for normal distribution 90 0. 944 0. 003 0. 938 0. 949 Mean coverage for t distribution 90 0. 950 0. 003 0. 945 0. 955 Mean 95% CI length for normal distribution 90 0. 160 0. 026 0. 111 0. 226 Mean 95% CI length for t distribution 90 0. 164 0. 026 0. 115 0. 232 55 Mean coverage for normal distribution 90 0. 946 0. 002 0. 941 0. 949 Mean coverage for t distribution 90 0. 951 0. 002 0. 946 0. 955 Mean 95% CI length for normal distribution 90 0. 145 0. 023 0. 101 0. 204 Mean 95% CI length for t distr ibution 90 0. 148 0. 024 0. 103 0. 209

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80 Table 4 4. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 12 5 Mean coverage for normal distribution 90 0. 881 0. 003 0. 872 0. 890 Mean coverage for t distribu tion 90 0. 953 0. 003 0. 945 0. 957 Mean 95% CI length for normal distribution 90 0. 453 0. 073 0. 315 0. 642 Mean 95% CI length for t distribution 90 0. 642 0. 103 0. 446 0. 910 15 Mean coverage for normal distribution 90 0. 931 0. 002 0. 927 0. 935 Mea n coverage for t distribution 90 0. 952 0. 002 0. 948 0. 956 Mean 95% CI length for normal distribution 90 0. 273 0. 044 0. 191 0. 387 Mean 95% CI length for t distribution 90 0. 299 0. 048 0. 209 0. 424 25 Mean coverage for normal distribution 90 0. 939 0 001 0. 936 0. 942 Mean coverage for t distribution 90 0. 951 0. 001 0. 948 0. 953 Mean 95% CI length for normal distribution 90 0. 213 0. 034 0. 149 0. 302 Mean 95% CI length for t distribution 90 0. 225 0. 036 0. 157 0. 318 35 Mean coverage for norma l distribution 90 0. 942 0. 002 0. 939 0. 945 Mean coverage for t distribution 90 0. 950 0. 001 0. 948 0. 954 Mean 95% CI length for normal distribution 90 0. 181 0. 029 0. 126 0. 256 Mean 95% CI length for t distribution 90 0. 187 0. 030 0. 131 0. 265 4 5 Mean coverage for normal distribution 90 0. 944 0. 003 0. 938 0. 949 Mean coverage for t distribution 90 0. 950 0. 003 0. 945 0. 955 Mean 95% CI length for normal distribution 90 0. 160 0. 026 0. 111 0. 226 Mean 95% CI length for t distribution 90 0. 1 64 0. 026 0. 115 0. 232 55 Mean coverage for normal distribution 90 0. 946 0. 002 0. 941 0. 949 Mean coverage for t distribution 90 0. 951 0. 002 0. 946 0. 955 Mean 95% CI length for normal distribution 90 0. 145 0. 023 0. 101 0. 204 Mean 95% CI length for t distribution 90 0. 148 0. 024 0. 103 0. 209

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81 Table 4 4. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 17 5 Mean coverage for normal distribution 60 0. 881 0. 004 0. 871 0. 888 Mean coverage for t distribution 60 0. 953 0. 002 0. 947 0. 957 Mean 95% CI length for normal distribution 60 0. 415 0. 066 0. 292 0. 572 Mean 95% CI length for t distribution 60 0. 588 0. 093 0. 413 0. 811 15 Mean coverage for normal distribution 60 0. 931 0. 002 0. 927 0. 93 4 Mean coverage for t distribution 60 0. 951 0. 002 0. 948 0. 956 Mean 95% CI length for normal distribution 60 0. 251 0. 040 0. 176 0. 346 Mean 95% CI length for t distribution 60 0. 274 0. 044 0. 193 0. 378 25 Mean coverage for normal distribution 60 0. 939 0. 001 0. 935 0. 942 Mean coverage for t distribution 60 0. 950 0. 001 0. 947 0. 953 Mean 95% CI length for normal distribution 60 0. 196 0. 031 0. 138 0. 269 Mean 95% CI length for t distribution 60 0. 206 0. 033 0. 145 0. 284 35 Mean coverage for normal distribution 60 0. 942 0. 001 0. 939 0. 945 Mean coverage for t distribution 60 0. 950 0. 001 0. 948 0. 954 Mean 95% CI length for normal distribution 60 0. 166 0. 026 0. 117 0. 228 Mean 95% CI length for t distribution 60 0. 172 0. 027 0. 121 0. 236 45 Mean coverage for normal distribution 60 0. 944 0. 003 0. 938 0. 949 Mean coverage for t distribution 60 0. 950 0. 002 0. 946 0. 954 Mean 95% CI length for normal distribution 60 0. 146 0. 023 0. 103 0. 202 Mean 95% CI length for t distribut ion 60 0. 151 0. 024 0. 106 0. 207 55 Mean coverage for normal distribution 60 0. 946 0. 002 0. 941 0. 949 Mean coverage for t distribution 60 0. 951 0. 002 0. 946 0. 954 Mean 95% CI length for normal distribution 60 0. 133 0. 021 0. 093 0. 182 Mean 95% CI length for t distribution 60 0. 136 0. 021 0. 096 0. 187

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82 Table 4 4. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 22 5 Mean coverage for normal distribution 30 0. 881 0. 003 0. 873 0. 887 Mean co verage for t distribution 30 0. 953 0. 002 0. 948 0. 956 Mean 95% CI length for normal distribution 30 0. 394 0. 063 0. 279 0. 536 Mean 95% CI length for t distribution 30 0. 558 0. 089 0. 395 0. 759 15 Mean coverage for normal distribution 30 0. 931 0. 002 0. 927 0. 934 Mean coverage for t distribution 30 0. 951 0. 002 0. 948 0. 954 Mean 95% CI length for normal distribution 30 0. 238 0. 038 0. 169 0. 324 Mean 95% CI length for t distribution 30 0. 261 0. 042 0. 185 0. 354 25 Mean coverage for normal di stribution 30 0. 939 0. 002 0. 935 0. 942 Mean coverage for t distribution 30 0. 951 0. 002 0. 946 0. 953 Mean 95% CI length for normal distribution 30 0. 186 0. 030 0. 132 0. 252 Mean 95% CI length for t distribution 30 0. 196 0. 031 0. 139 0. 266 35 Me an coverage for normal distribution 30 0. 942 0. 001 0. 939 0. 944 Mean coverage for t distribution 30 0. 950 0. 001 0. 948 0. 953 Mean 95% CI length for normal distribution 30 0. 157 0. 025 0. 112 0. 214 Mean 95% CI length for t distribution 30 0. 163 0 026 0. 116 0. 222 45 Mean coverage for normal distribution 30 0. 944 0. 002 0. 939 0. 948 Mean coverage for t distribution 30 0. 950 0. 002 0. 946 0. 954 Mean 95% CI length for normal distribution 30 0. 139 0. 022 0. 099 0. 189 Mean 95% CI length for t distribution 30 0. 143 0. 023 0. 102 0. 194 55 Mean coverage for normal distribution 30 0. 946 0. 002 0. 941 0. 950 Mean coverage for t distribution 30 0. 951 0. 002 0. 946 0. 955 Mean 95% CI length for normal distribution 30 0. 126 0. 020 0. 089 0. 171 Mean 95% CI length for t distribution 30 0. 129 0. 021 0. 091 0. 175

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83 Table 4 5. Aim 2 2: C omparisons of t distribution vs. normal distribution for UW_MM Outcomes Mean STD Minimum Maximum Mean coverage for normal distribution 0. 908 0. 049 0. 604 0. 952 Mean coverage for t distribution 0. 931 0. 047 0. 620 0. 962 Mean 95% CI length for normal distribution 0. 343 0. 217 0. 090 1. 616 Mean 95% CI length for t distribution 0. 400 0. 312 0. 092 2. 289

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84 Table 4 6. Aim 2 2: C omparisons of t distribution vs. normal distribu tion for UW_MM by event rate in treatment arm and number of studies Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 02 5 Mean coverage for normal distribution 150 0. 884 0. 004 0. 874 0. 895 Mean coverage for t distribution 150 0. 955 0. 002 0. 949 0. 962 Mean 95% CI length for normal distribution 150 0. 976 0. 196 0. 674 1. 616 Mean 95% CI length for t distribution 150 1. 382 0. 278 0. 955 2. 289 15 Mean coverage for normal distribution 150 0. 912 0. 017 0. 862 0 938 Mean coverage for t distribution 150 0. 939 0. 013 0. 902 0. 957 Mean 95% CI length for normal distribution 150 0. 599 0. 118 0. 417 0. 979 Mean 95% CI length for t distribution 150 0. 656 0. 129 0. 457 1. 071 25 Mean coverage for normal distrib ution 150 0. 892 0. 036 0. 795 0. 943 Mean coverage for t distribution 150 0. 911 0. 031 0. 826 0. 957 Mean 95% CI length for normal distribution 150 0. 470 0. 091 0. 329 0. 762 Mean 95% CI length for t distribution 150 0. 495 0. 096 0. 347 0. 803 35 Mea n coverage for normal distribution 150 0. 869 0. 053 0. 735 0. 951 Mean coverage for t distribution 150 0. 885 0. 049 0. 760 0. 958 Mean 95% CI length for normal distribution 150 0. 399 0. 077 0. 280 0. 646 Mean 95% CI length for t distribution 150 0. 41 3 0. 079 0. 290 0. 669 45 Mean coverage for normal distribution 150 0. 842 0. 072 0. 669 0. 949 Mean coverage for t distribution 150 0. 856 0. 067 0. 689 0. 955 Mean 95% CI length for normal distribution 150 0. 353 0. 067 0. 248 0. 570 Mean 95% CI lengt h for t distribution 150 0. 363 0. 069 0. 255 0. 586 55 Mean coverage for normal distribution 150 0. 813 0. 090 0. 604 0. 952 Mean coverage for t distribution 150 0. 826 0. 087 0. 620 0. 957 Mean 95% CI length for normal distribution 150 0. 320 0. 061 0. 225 0. 517 Mean 95% CI length for t distribution 150 0. 327 0. 062 0. 230 0. 528

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85 Table 4 6. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 07 5 Mean coverage for normal distribution 120 0. 882 0. 004 0. 8 73 0. 891 Mean coverage for t distribution 120 0. 954 0. 003 0. 945 0. 959 Mean 95% CI length for normal distribution 120 0. 554 0. 101 0. 375 0. 853 Mean 95% CI length for t distribution 120 0. 784 0. 144 0. 531 1. 208 15 Mean coverage for normal dis tribution 120 0. 929 0. 002 0. 922 0. 934 Mean coverage for t distribution 120 0. 950 0. 002 0. 946 0. 956 Mean 95% CI length for normal distribution 120 0. 337 0. 063 0. 228 0. 521 Mean 95% CI length for t distribution 120 0. 368 0. 068 0. 249 0. 570 25 Mean coverage for normal distribution 120 0. 934 0. 004 0. 921 0. 941 Mean coverage for t distribution 120 0. 947 0. 004 0. 936 0. 952 Mean 95% CI length for normal distribution 120 0. 263 0. 049 0. 178 0. 406 Mean 95% CI length for t distribution 120 0. 277 0. 051 0. 188 0. 428 35 Mean coverage for normal distribution 120 0. 935 0. 007 0. 911 0. 945 Mean coverage for t distribution 120 0. 944 0. 007 0. 924 0. 953 Mean 95% CI length for normal distribution 120 0. 223 0. 041 0. 151 0. 345 Mean 95% CI l ength for t distribution 120 0. 231 0. 043 0. 157 0. 357 45 Mean coverage for normal distribution 120 0. 935 0. 010 0. 902 0. 949 Mean coverage for t distribution 120 0. 942 0. 009 0. 911 0. 955 Mean 95% CI length for normal distribution 120 0. 197 0. 037 0 134 0. 305 Mean 95% CI length for t distribution 120 0. 203 0. 038 0. 137 0. 313 55 Mean coverage for normal distribution 120 0. 933 0. 012 0. 894 0. 949 Mean coverage for t distribution 120 0. 939 0. 012 0. 902 0. 954 Mean 95% CI length for normal d istribution 120 0. 179 0. 033 0. 121 0. 276 Mean 95% CI length for t distribution 120 0. 183 0. 034 0. 124 0. 282

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86 Table 4 6. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 12 5 Mean coverage for normal distribution 90 0. 881 0. 004 0. 874 0. 891 Mean coverage for t distribution 90 0. 953 0. 002 0. 945 0. 957 Mean 95% CI length for normal distribution 90 0. 462 0. 077 0. 318 0. 665 Mean 95% CI length for t distribution 90 0. 654 0. 109 0. 451 0. 942 15 Mean coverage for normal distribution 90 0. 930 0. 002 0. 926 0. 935 Mean coverage for t distribution 90 0. 951 0. 002 0. 948 0. 956 Mean 95% CI length for normal distribution 90 0. 280 0. 047 0. 193 0. 403 Mean 95% CI length for t distribution 90 0. 30 6 0. 051 0. 211 0. 442 25 Mean coverage for normal distribution 90 0. 938 0. 001 0. 934 0. 941 Mean coverage for t distribution 90 0. 950 0. 001 0. 947 0. 953 Mean 95% CI length for normal distribution 90 0. 219 0. 037 0. 151 0. 315 Mean 95% CI length f or t distribution 90 0. 230 0. 039 0. 159 0. 331 35 Mean coverage for normal distribution 90 0. 941 0. 002 0. 933 0. 946 Mean coverage for t distribution 90 0. 949 0. 002 0. 943 0. 953 Mean 95% CI length for normal distribution 90 0. 185 0. 031 0. 128 0. 267 Mean 95% CI length for t distribution 90 0. 192 0. 032 0. 133 0. 277 45 Mean coverage for normal distribution 90 0. 942 0. 004 0. 932 0. 950 Mean coverage for t distribution 90 0. 948 0. 004 0. 939 0. 955 Mean 95% CI length for normal distribution 90 0. 164 0. 027 0. 113 0. 236 Mean 95% CI length for t distribution 90 0. 168 0. 028 0. 116 0. 242 55 Mean coverage for normal distribution 90 0. 944 0. 004 0. 934 0. 950 Mean coverage for t distribution 90 0. 949 0. 003 0. 939 0. 955 Mean 95% CI length f or normal distribution 90 0. 148 0. 025 0. 102 0. 214 Mean 95% CI length for t distribution 90 0. 152 0. 025 0. 105 0. 218

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87 Table 4 6. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 17 5 Mean coverage f or normal distribution 60 0. 881 0. 003 0. 874 0. 888 Mean coverage for t distribution 60 0. 953 0. 002 0. 947 0. 957 Mean 95% CI length for normal distribution 60 0. 420 0. 068 0. 294 0. 585 Mean 95% CI length for t distribution 60 0. 596 0. 096 0. 416 0. 829 15 Mean coverage for normal distribution 60 0. 930 0. 001 0. 927 0. 935 Mean coverage for t distribution 60 0. 951 0. 002 0. 948 0. 955 Mean 95% CI length for normal distribution 60 0. 255 0. 041 0. 178 0. 355 Mean 95% CI length for t distributio n 60 0. 279 0. 045 0. 195 0. 388 25 Mean coverage for normal distribution 60 0. 939 0. 001 0. 935 0. 942 Mean coverage for t distribution 60 0. 950 0. 001 0. 946 0. 953 Mean 95% CI length for normal distribution 60 0. 199 0. 032 0. 139 0. 277 Mean 95% CI length for t distribution 60 0. 209 0. 034 0. 146 0. 291 35 Mean coverage for normal distribution 60 0. 942 0. 001 0. 939 0. 945 Mean coverage for t distribution 60 0. 950 0. 001 0. 947 0. 953 Mean 95% CI length for normal distribution 60 0. 169 0. 027 0. 1 18 0. 234 Mean 95% CI length for t distribution 60 0. 175 0. 028 0. 122 0. 243 45 Mean coverage for normal distribution 60 0. 944 0. 003 0. 938 0. 949 Mean coverage for t distribution 60 0. 950 0. 003 0. 944 0. 954 Mean 95% CI length for normal distri bution 60 0. 149 0. 024 0. 104 0. 207 Mean 95% CI length for t distribution 60 0. 153 0. 025 0. 107 0. 213 55 Mean coverage for normal distribution 60 0. 946 0. 002 0. 941 0. 950 Mean coverage for t distribution 60 0. 951 0. 002 0. 946 0. 955 Mean 95% CI length for normal distribution 60 0. 135 0. 022 0. 094 0. 188 Mean 95% CI length for t distribution 60 0. 138 0. 022 0. 096 0. 192

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88 Table 4 6. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 22 5 Mean coverage for normal distribution 30 0. 881 0. 003 0. 874 0. 886 Mean coverage for t distribution 30 0. 953 0. 002 0. 948 0. 956 Mean 95% CI length for normal distribution 30 0. 398 0. 065 0. 281 0. 545 Mean 95% CI length for t distribution 30 0. 564 0. 09 2 0. 397 0. 772 15 Mean coverage for normal distribution 30 0. 931 0. 002 0. 928 0. 934 Mean coverage for t distribution 30 0. 951 0. 002 0. 948 0. 954 Mean 95% CI length for normal distribution 30 0. 241 0. 039 0. 170 0. 330 Mean 95% CI length for t d istribution 30 0. 264 0. 043 0. 186 0. 361 25 Mean coverage for normal distribution 30 0. 938 0. 002 0. 933 0. 942 Mean coverage for t distribution 30 0. 950 0. 002 0. 946 0. 953 Mean 95% CI length for normal distribution 30 0. 188 0. 031 0. 133 0. 257 M ean 95% CI length for t distribution 30 0. 198 0. 032 0. 140 0. 271 35 Mean coverage for normal distribution 30 0. 942 0. 001 0. 940 0. 945 Mean coverage for t distribution 30 0. 950 0. 001 0. 948 0. 954 Mean 95% CI length for normal distribution 30 0. 160 0. 026 0. 113 0. 218 Mean 95% CI length for t distribution 30 0. 165 0. 027 0. 117 0. 226 45 Mean coverage for normal distribution 30 0. 944 0. 003 0. 940 0. 948 Mean coverage for t distribution 30 0. 950 0. 002 0. 947 0. 954 Mean 95% CI length for nor mal distribution 30 0. 141 0. 023 0. 099 0. 193 Mean 95% CI length for t distribution 30 0. 145 0. 024 0. 102 0. 198 55 Mean coverage for normal distribution 30 0. 946 0. 002 0. 942 0. 950 Mean coverage for t distribution 30 0. 951 0. 002 0. 948 0. 955 M ean 95% CI length for normal distribution 30 0. 128 0. 021 0. 090 0. 175 Mean 95% CI length for t distribution 30 0. 131 0. 021 0. 092 0. 179

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89 Table 4 7 Aim 3 : C omparisons of UW_RR vs. W_RR Outcomes Mean STD Minimum Maximum Mean coverage for UW_RR 0. 951 0. 002 0. 944 0. 959 Mean coverage for W_RR 0. 950 0. 002 0. 940 0. 959 Mean 95% CI length for UW_RR 0. 299 0. 261 0. 058 2. 014 Mean 95% CI length for W_RR 0. 291 0. 253 0. 058 1. 976

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90 Table 4 8. Aim 3 : C omparisons of UW_ R R vs. W_ RR by event rate in treatment arm an d number of studies Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 02 5 Mean coverage for UW_RR 150 0. 955 0. 002 0. 951 0. 959 Mean coverage for W_RR 150 0. 952 0. 003 0. 944 0. 959 Mean 95% CI length for UW _RR 150 1. 312 0. 252 0. 862 2. 014 Mean 95% CI length for W_RR 150 1. 273 0. 238 0. 861 1. 976 15 Mean coverage for UW_RR 150 0. 953 0. 002 0. 947 0. 957 Mean coverage for W_RR 150 0. 951 0. 002 0. 945 0. 958 Mean 95% CI length for UW_RR 150 0. 603 0. 116 0. 397 0. 918 Mean 95% CI length for W_RR 150 0. 585 0. 109 0. 396 0. 901 25 Mean coverage for UW_RR 150 0. 951 0. 002 0. 946 0. 955 Mean coverage for W_RR 150 0. 950 0. 002 0. 944 0. 955 Mean 95% CI length for UW_RR 150 0. 452 0. 087 0. 297 0. 687 M ean 95% CI length for W_RR 150 0. 438 0. 081 0. 297 0. 674 35 Mean coverage for UW_RR 150 0. 950 0. 002 0. 946 0. 956 Mean coverage for W_RR 150 0. 950 0. 002 0. 945 0. 957 Mean 95% CI length for UW_RR 150 0. 376 0. 072 0. 248 0. 571 Mean 95% CI length f or W_RR 150 0. 365 0. 068 0. 247 0. 560 45 Mean coverage for UW_RR 150 0. 950 0. 003 0. 945 0. 955 Mean coverage for W_RR 150 0. 950 0. 002 0. 945 0. 956 Mean 95% CI length for UW_RR 150 0. 330 0. 063 0. 217 0. 500 Mean 95% CI length for W_RR 150 0. 320 0 059 0. 217 0. 491 55 Mean coverage for UW_RR 150 0. 951 0. 002 0. 945 0. 957 Mean coverage for W_RR 150 0. 951 0. 002 0. 945 0. 956 Mean 95% CI length for UW_RR 150 0. 297 0. 057 0. 195 0. 450 Mean 95% CI length for W_RR 150 0. 288 0. 053 0. 195 0. 442

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91 Table 4 8. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 02 5 Mean coverage for UW_RR 150 0. 955 0. 002 0. 951 0. 959 Mean coverage for W_RR 150 0. 952 0. 003 0. 944 0. 959 Mean 95% CI length for U W_RR 150 1. 312 0. 252 0. 862 2. 014 Mean 95% CI length for W_RR 150 1. 273 0. 238 0. 861 1. 976 15 Mean coverage for UW_RR 150 0. 953 0. 002 0. 947 0. 957 Mean coverage for W_RR 150 0. 951 0. 002 0. 945 0. 958 Mean 95% CI length for UW_RR 150 0. 603 0. 11 6 0. 397 0. 918 Mean 95% CI length for W_RR 150 0. 585 0. 109 0. 396 0. 901 25 Mean coverage for UW_RR 150 0. 951 0. 002 0. 946 0. 955 Mean coverage for W_RR 150 0. 950 0. 002 0. 944 0. 955 Mean 95% CI length for UW_RR 150 0. 452 0. 087 0. 297 0. 687 Mean 95% CI length for W_RR 150 0. 438 0. 081 0. 297 0. 674 35 Mean coverage for UW_RR 150 0. 950 0. 002 0. 946 0. 956 Mean coverage for W_RR 150 0. 950 0. 002 0. 945 0. 957 Mean 95% CI length for UW_RR 150 0. 376 0. 072 0. 248 0. 571 Mean 95% CI length for W_RR 150 0. 365 0. 068 0. 247 0. 560 45 Mean coverage for UW_RR 150 0. 950 0. 003 0. 945 0. 955 Mean coverage for W_RR 150 0. 950 0. 002 0. 945 0. 956 Mean 95% CI length for UW_RR 150 0. 330 0. 063 0. 217 0. 500 Mean 95% CI length for W_RR 150 0. 320 0. 059 0. 217 0. 491 55 Mean coverage for UW_RR 150 0. 951 0. 002 0. 945 0. 957 Mean coverage for W_RR 150 0. 951 0. 002 0. 945 0. 956 Mean 95% CI length for UW_RR 150 0. 297 0. 057 0. 195 0. 450 Mean 95% CI length for W_RR 150 0. 288 0. 053 0. 195 0. 442

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92 Table 4 8. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 12 5 Mean coverage for UW_RR 150 0. 952 0. 003 0. 945 0. 958 Mean coverage for W_RR 150 0. 949 0. 003 0. 942 0. 954 Mean 95% CI length for UW_RR 150 0. 526 0. 096 0. 326 0. 802 Mean 95% CI length for W_RR 150 0. 512 0. 091 0. 325 0. 789 15 Mean coverage for UW_RR 150 0. 951 0. 002 0. 947 0. 956 Mean coverage for W_RR 150 0. 950 0. 002 0. 945 0. 955 Mean 95% CI length for UW_RR 150 0. 245 0. 0 45 0. 152 0. 373 Mean 95% CI length for W_RR 150 0. 239 0. 043 0. 152 0. 367 25 Mean coverage for UW_RR 150 0. 950 0. 001 0. 946 0. 954 Mean coverage for W_RR 150 0. 949 0. 002 0. 944 0. 953 Mean 95% CI length for UW_RR 150 0. 184 0. 033 0. 115 0. 280 Mean 95% CI length for W_RR 150 0. 180 0. 032 0. 114 0. 276 35 Mean coverage for UW_RR 150 0. 950 0. 001 0. 947 0. 954 Mean coverage for W_RR 150 0. 949 0. 002 0. 943 0. 954 Mean 95% CI length for UW_RR 150 0. 154 0. 028 0. 096 0. 233 Mean 95% CI length for W_RR 150 0. 150 0. 026 0. 096 0. 230 45 Mean coverage for UW_RR 150 0. 950 0. 002 0. 945 0. 954 Mean coverage for W_RR 150 0. 950 0. 003 0. 944 0. 955 Mean 95% CI length for UW_RR 150 0. 135 0. 024 0. 084 0. 204 Mean 95% CI length for W_RR 150 0. 131 0. 023 0. 084 0. 201 55 Mean coverage for UW_RR 150 0. 951 0. 002 0. 945 0. 955 Mean coverage for W_RR 150 0. 951 0. 002 0. 945 0. 955 Mean 95% CI length for UW_RR 150 0. 121 0. 022 0. 075 0. 184 Mean 95% CI length for W_RR 150 0. 118 0. 021 0. 075 0. 181

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93 Table 4 8. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 12 5 Mean coverage for UW_RR 150 0. 952 0. 003 0. 945 0. 958 Mean coverage for W_RR 150 0. 949 0. 003 0. 942 0. 954 Mean 95% CI length fo r UW_RR 150 0. 526 0. 096 0. 326 0. 802 Mean 95% CI length for W_RR 150 0. 512 0. 091 0. 325 0. 789 15 Mean coverage for UW_RR 150 0. 951 0. 002 0. 947 0. 956 Mean coverage for W_RR 150 0. 950 0. 002 0. 945 0. 955 Mean 95% CI length for UW_RR 150 0. 245 0 045 0. 152 0. 373 Mean 95% CI length for W_RR 150 0. 239 0. 043 0. 152 0. 367 25 Mean coverage for UW_RR 150 0. 950 0. 001 0. 946 0. 954 Mean coverage for W_RR 150 0. 949 0. 002 0. 944 0. 953 Mean 95% CI length for UW_RR 150 0. 184 0. 033 0. 115 0. 280 Mean 95% CI length for W_RR 150 0. 180 0. 032 0. 114 0. 276 35 Mean coverage for UW_RR 150 0. 950 0. 001 0. 947 0. 954 Mean coverage for W_RR 150 0. 949 0. 002 0. 943 0. 954 Mean 95% CI length for UW_RR 150 0. 154 0. 028 0. 096 0. 233 Mean 95% CI leng th for W_RR 150 0. 150 0. 026 0. 096 0. 230 45 Mean coverage for UW_RR 150 0. 950 0. 002 0. 945 0. 954 Mean coverage for W_RR 150 0. 950 0. 003 0. 944 0. 955 Mean 95% CI length for UW_RR 150 0. 135 0. 024 0. 084 0. 204 Mean 95% CI length for W_RR 150 0. 1 31 0. 023 0. 084 0. 201 55 Mean coverage for UW_RR 150 0. 951 0. 002 0. 945 0. 955 Mean coverage for W_RR 150 0. 951 0. 002 0. 945 0. 955 Mean 95% CI length for UW_RR 150 0. 121 0. 022 0. 075 0. 184 Mean 95% CI length for W_RR 150 0. 118 0. 021 0. 075 0. 18 1

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94 Table 4 8. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 17 5 Mean coverage for UW_RR 130 0. 952 0. 003 0. 945 0. 957 Mean coverage for W_RR 130 0. 949 0. 003 0. 942 0. 954 Mean 95% CI length for UW_RR 130 0. 447 0. 083 0. 265 0. 673 Mean 95% CI length for W_RR 130 0. 436 0. 079 0. 265 0. 663 15 Mean coverage for UW_RR 130 0. 951 0. 002 0. 947 0. 956 Mean coverage for W_RR 130 0. 950 0. 002 0. 945 0. 956 Mean 95% CI length for UW_RR 130 0. 209 0. 039 0. 124 0. 314 Mean 95% CI length for W_RR 130 0. 204 0. 037 0. 124 0. 310 25 Mean coverage for UW_RR 130 0. 950 0. 001 0. 946 0. 954 Mean coverage for W_RR 130 0. 949 0. 002 0. 944 0. 953 Mean 95% CI length for UW_RR 130 0. 157 0. 029 0. 093 0. 235 Mean 95% CI length for W_RR 130 0. 153 0. 028 0. 093 0. 232 35 Mean coverage for UW_RR 130 0. 950 0. 002 0. 947 0. 954 Mean coverage for W_RR 130 0. 949 0. 002 0. 944 0. 954 Mean 95% CI length for UW_RR 130 0. 131 0. 024 0. 078 0. 196 Mean 95% CI le ngth for W_RR 130 0. 128 0. 023 0. 078 0. 194 45 Mean coverage for UW_RR 130 0. 950 0. 002 0. 946 0. 955 Mean coverage for W_RR 130 0. 950 0. 002 0. 945 0. 954 Mean 95% CI length for UW_RR 130 0. 114 0. 021 0. 068 0. 172 Mean 95% CI length for W_RR 130 0 112 0. 020 0. 068 0. 170 55 Mean coverage for UW_RR 130 0. 951 0. 002 0. 946 0. 954 Mean coverage for W_RR 130 0. 951 0. 002 0. 946 0. 954 Mean 95% CI length for UW_RR 130 0. 103 0. 019 0. 061 0. 155 Mean 95% CI length for W_RR 130 0. 101 0. 018 0. 061 0. 153

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95 Table 4 8. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 22 5 Mean coverage for UW_RR 120 0. 952 0. 003 0. 944 0. 957 Mean coverage for W_RR 120 0. 949 0. 003 0. 940 0. 955 Mean 95% CI length for UW_RR 120 0. 400 0. 070 0. 249 0. 593 Mean 95% CI length for W_RR 120 0. 391 0. 068 0. 248 0. 584 15 Mean coverage for UW_RR 120 0. 951 0. 002 0. 948 0. 955 Mean coverage for W_RR 120 0. 950 0. 002 0. 944 0. 954 Mean 95% CI length for UW_RR 120 0. 18 7 0. 033 0. 116 0. 276 Mean 95% CI length for W_RR 120 0. 183 0. 032 0. 116 0. 273 25 Mean coverage for UW_RR 120 0. 950 0. 002 0. 945 0. 955 Mean coverage for W_RR 120 0. 949 0. 002 0. 944 0. 952 Mean 95% CI length for UW_RR 120 0. 140 0. 024 0. 088 0. 207 Mean 95% CI length for W_RR 120 0. 138 0. 024 0. 087 0. 205 35 Mean coverage for UW_RR 120 0. 950 0. 002 0. 946 0. 954 Mean coverage for W_RR 120 0. 949 0. 002 0. 944 0. 955 Mean 95% CI length for UW_RR 120 0. 117 0. 020 0. 073 0. 173 Mean 95% CI l ength for W_RR 120 0. 115 0. 020 0. 073 0. 171 45 Mean coverage for UW_RR 120 0. 950 0. 002 0. 945 0. 954 Mean coverage for W_RR 120 0. 949 0. 002 0. 945 0. 954 Mean 95% CI length for UW_RR 120 0. 103 0. 018 0. 064 0. 151 Mean 95% CI length for W_RR 120 0. 101 0. 017 0. 064 0. 150 55 Mean coverage for UW_RR 120 0. 951 0. 002 0. 946 0. 955 Mean coverage for W_RR 120 0. 951 0. 002 0. 945 0. 956 Mean 95% CI length for UW_RR 120 0. 092 0. 016 0. 058 0. 136 Mean 95% CI length for W_RR 120 0. 091 0. 016 0. 058 0 135

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96 Table 4 9. Ai m 4 1: C omparisons of t distribution vs. normal distribution for UW_RR Outcomes Mean STD Minimum Maximum Mean coverage for normal distribution 0. 931 0. 022 0. 870 0. 952 Mean coverage for t distribution 0. 951 0. 002 0. 944 0. 959 Mean 95% CI length for normal distribution 0. 255 0. 186 0. 056 1. 422 Mean 95% CI length for t distribution 0. 299 0. 261 0. 058 2. 014

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97 Table 4 10. Aim 4 1: C omparisons of t distribution vs. normal distribution for UW_RR by event rate in treatment arm and number of stu dies Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 02 5 Mean coverage for normal distribution 150 0. 886 0. 003 0. 878 0. 894 Mean coverage for t distribution 150 0. 955 0. 002 0. 951 0. 959 Mean 95% CI len gth for normal distribution 150 0. 926 0. 178 0. 609 1. 422 Mean 95% CI length for t distribution 150 1. 312 0. 252 0. 862 2. 014 15 Mean coverage for normal distribution 150 0. 932 0. 002 0. 926 0. 937 Mean coverage for t distribution 150 0. 953 0. 002 0. 9 47 0. 957 Mean 95% CI length for normal distribution 150 0. 551 0. 106 0. 362 0. 839 Mean 95% CI length for t distribution 150 0. 603 0. 116 0. 397 0. 918 25 Mean coverage for normal distribution 150 0. 939 0. 002 0. 933 0. 944 Mean coverage for t dis tribution 150 0. 951 0. 002 0. 946 0. 955 Mean 95% CI length for normal distribution 150 0. 429 0. 082 0. 282 0. 652 Mean 95% CI length for t distribution 150 0. 452 0. 087 0. 297 0. 687 35 Mean coverage for normal distribution 150 0. 942 0. 002 0. 937 0. 949 Mean coverage for t distribution 150 0. 950 0. 002 0. 946 0. 956 Mean 95% CI length for normal distribution 150 0. 363 0. 069 0. 239 0. 551 Mean 95% CI length for t distribution 150 0. 376 0. 072 0. 248 0. 571 45 Mean coverage for normal distributio n 150 0. 944 0. 003 0. 939 0. 949 Mean coverage for t distribution 150 0. 950 0. 003 0. 945 0. 955 Mean 95% CI length for normal distribution 150 0. 321 0. 061 0. 211 0. 486 Mean 95% CI length for t distribution 150 0. 330 0. 063 0. 217 0. 500 55 Mean co verage for normal distribution 150 0. 946 0. 002 0. 939 0. 952 Mean coverage for t distribution 150 0. 951 0. 002 0. 945 0. 957 Mean 95% CI length for normal distribution 150 0. 290 0. 056 0. 191 0. 440 Mean 95% CI length for t distribution 150 0. 297 0. 057 0. 195 0. 450

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98 Table 4 10. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 07 5 Mean coverage for normal distribution 150 0. 882 0. 004 0. 872 0. 892 Mean coverage for t distribution 150 0. 953 0. 0 03 0. 945 0. 958 Mean 95% CI length for normal distribution 150 0. 488 0. 090 0. 311 0. 742 Mean 95% CI length for t distribution 150 0. 691 0. 127 0. 441 1. 051 15 Mean coverage for normal distribution 150 0. 931 0. 002 0. 927 0. 935 Mean coverage for t distribution 150 0. 952 0. 002 0. 947 0. 956 Mean 95% CI length for normal distribution 150 0. 294 0. 054 0. 188 0. 446 Mean 95% CI length for t distribution 150 0. 322 0. 059 0. 206 0. 488 25 Mean coverage for normal distribution 150 0. 939 0. 002 0. 935 0. 942 Mean coverage for t distribution 150 0. 951 0. 002 0. 947 0. 955 Mean 95% CI length for normal distribution 150 0. 229 0. 042 0. 147 0. 348 Mean 95% CI length for t distribution 150 0. 241 0. 044 0. 155 0. 366 35 Mean coverage for normal distr ibution 150 0. 942 0. 002 0. 939 0. 946 Mean coverage for t distribution 150 0. 950 0. 002 0. 947 0. 955 Mean 95% CI length for normal distribution 150 0. 194 0. 036 0. 125 0. 294 Mean 95% CI length for t distribution 150 0. 201 0. 037 0. 129 0. 305 45 M ean coverage for normal distribution 150 0. 944 0. 003 0. 939 0. 949 Mean coverage for t distribution 150 0. 950 0. 003 0. 945 0. 955 Mean 95% CI length for normal distribution 150 0. 172 0. 032 0. 110 0. 260 Mean 95% CI length for t distribution 150 0. 176 0. 032 0. 113 0. 267 55 Mean coverage for normal distribution 150 0. 946 0. 002 0. 940 0. 949 Mean coverage for t distribution 150 0. 951 0. 002 0. 946 0. 954 Mean 95% CI length for normal distribution 150 0. 155 0. 029 0. 099 0. 235 Mean 95% CI len gth for t distribution 150 0. 159 0. 029 0. 102 0. 241

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99 Table 4 10. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 12 5 Mean coverage for normal distribution 150 0. 881 0. 003 0. 872 0. 890 Mean coverag e for t distribution 150 0. 952 0. 003 0. 945 0. 958 Mean 95% CI length for normal distribution 150 0. 372 0. 067 0. 230 0. 566 Mean 95% CI length for t distribution 150 0. 526 0. 096 0. 326 0. 802 15 Mean coverage for normal distribution 150 0. 931 0. 002 0. 926 0. 935 Mean coverage for t distribution 150 0. 951 0. 002 0. 947 0. 956 Mean 95% CI length for normal distribution 150 0. 224 0. 041 0. 139 0. 341 Mean 95% CI length for t distribution 150 0. 245 0. 045 0. 152 0. 373 25 Mean coverage for normal distribution 150 0. 939 0. 002 0. 934 0. 943 Mean coverage for t distribution 150 0. 950 0. 001 0. 946 0. 954 Mean 95% CI length for normal distribution 150 0. 175 0. 032 0. 109 0. 266 Mean 95% CI length for t distribution 150 0. 184 0. 033 0. 115 0. 280 35 Mean coverage for normal distribution 150 0. 942 0. 002 0. 937 0. 946 Mean coverage for t distribution 150 0. 950 0. 001 0. 947 0. 954 Mean 95% CI length for normal distribution 150 0. 148 0. 027 0. 092 0. 225 Mean 95% CI length for t distribution 1 50 0. 154 0. 028 0. 096 0. 233 45 Mean coverage for normal distribution 150 0. 944 0. 003 0. 938 0. 948 Mean coverage for t distribution 150 0. 950 0. 002 0. 945 0. 954 Mean 95% CI length for normal distribution 150 0. 131 0. 024 0. 081 0. 199 Mean 95% C I length for t distribution 150 0. 135 0. 024 0. 084 0. 204 55 Mean coverage for normal distribution 150 0. 946 0. 002 0. 939 0. 949 Mean coverage for t distribution 150 0. 951 0. 002 0. 945 0. 955 Mean 95% CI length for normal distribution 150 0. 119 0. 02 1 0. 074 0. 180 Mean 95% CI length for t distribution 150 0. 121 0. 022 0. 075 0. 184

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100 Table 4 10. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 17 5 Mean coverage for normal distribution 130 0. 881 0 003 0. 870 0. 889 Mean coverage for t distribution 130 0. 952 0. 003 0. 945 0. 957 Mean 95% CI length for normal distribution 130 0. 316 0. 058 0. 187 0. 475 Mean 95% CI length for t distribution 130 0. 447 0. 083 0. 265 0. 673 15 Mean coverage for no rmal distribution 130 0. 931 0. 002 0. 925 0. 934 Mean coverage for t distribution 130 0. 951 0. 002 0. 947 0. 956 Mean 95% CI length for normal distribution 130 0. 191 0. 035 0. 113 0. 287 Mean 95% CI length for t distribution 130 0. 209 0. 039 0. 124 0. 3 14 25 Mean coverage for normal distribution 130 0. 938 0. 002 0. 935 0. 942 Mean coverage for t distribution 130 0. 950 0. 001 0. 946 0. 954 Mean 95% CI length for normal distribution 130 0. 149 0. 027 0. 089 0. 224 Mean 95% CI length for t distribut ion 130 0. 157 0. 029 0. 093 0. 235 35 Mean coverage for normal distribution 130 0. 942 0. 002 0. 938 0. 946 Mean coverage for t distribution 130 0. 950 0. 002 0. 947 0. 954 Mean 95% CI length for normal distribution 130 0. 126 0. 023 0. 075 0. 189 Mean 95% CI length for t distribution 130 0. 131 0. 024 0. 078 0. 196 45 Mean coverage for normal distribution 130 0. 944 0. 003 0. 939 0. 949 Mean coverage for t distribution 130 0. 950 0. 002 0. 946 0. 955 Mean 95% CI length for normal distribution 130 0. 111 0. 021 0. 066 0. 167 Mean 95% CI length for t distribution 130 0. 114 0. 021 0. 068 0. 172 55 Mean coverage for normal distribution 130 0. 946 0. 002 0. 941 0. 949 Mean coverage for t distribution 130 0. 951 0. 002 0. 946 0. 954 Mean 95% CI length for normal distribution 130 0. 101 0. 019 0. 060 0. 151 Mean 95% CI length for t distribution 130 0. 103 0. 019 0. 061 0. 155

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101 Table 4 10. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 22 5 Mean coverage f or normal distribution 120 0. 880 0. 003 0. 873 0. 888 Mean coverage for t distribution 120 0. 952 0. 003 0. 944 0. 957 Mean 95% CI length for normal distribution 120 0. 283 0. 049 0. 176 0. 418 Mean 95% CI length for t distribution 120 0. 400 0. 070 0. 24 9 0. 593 15 Mean coverage for normal distribution 120 0. 931 0. 002 0. 927 0. 935 Mean coverage for t distribution 120 0. 951 0. 002 0. 948 0. 955 Mean 95% CI length for normal distribution 120 0. 171 0. 030 0. 106 0. 253 Mean 95% CI length for t dist ribution 120 0. 187 0. 033 0. 116 0. 276 25 Mean coverage for normal distribution 120 0. 938 0. 002 0. 934 0. 943 Mean coverage for t distribution 120 0. 950 0. 002 0. 945 0. 955 Mean 95% CI length for normal distribution 120 0. 133 0. 023 0. 083 0. 197 Mean 95% CI length for t distribution 120 0. 140 0. 024 0. 088 0. 207 35 Mean coverage for normal distribution 120 0. 942 0. 002 0. 939 0. 946 Mean coverage for t distribution 120 0. 950 0. 002 0. 946 0. 954 Mean 95% CI length for normal distribution 120 0. 113 0. 020 0. 071 0. 167 Mean 95% CI length for t distribution 120 0. 117 0. 020 0. 073 0. 173 45 Mean coverage for normal distribution 120 0. 944 0. 002 0. 939 0. 948 Mean coverage for t distribution 120 0. 950 0. 002 0. 945 0. 954 Mean 95% CI length for normal distribution 120 0. 100 0. 017 0. 062 0. 147 Mean 95% CI length for t distribution 120 0. 103 0. 018 0. 064 0. 151 55 Mean coverage for normal distribution 120 0. 946 0. 002 0. 941 0. 950 Mean coverage for t distribution 120 0. 951 0. 002 0. 946 0. 955 Mean 95% CI length for normal distribution 120 0. 090 0. 016 0. 056 0. 133 Mean 95% CI length for t distribution 120 0. 092 0. 016 0. 058 0. 136

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102 Table 4 11. Aim 4 2: C omparisons of t distribution vs. normal distribution for W_RR Outcomes Mean ST D Minimum Maximum Mean coverage for normal distribution 0. 929 0. 024 0. 860 0. 952 Mean coverage for t distribution 0. 950 0. 002 0. 940 0. 959 Mean 95% CI length for normal distribution 0. 249 0. 180 0. 056 1. 395 Mean 95% CI length for t distribution 0. 291 0. 25 3 0. 058 1. 976

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103 Table 4 12. Aim 4 2: C omparisons of t distribution vs. normal distribution for W_RR by event rate in treatment arm and number of studies Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 02 5 Mean coverage for normal distribution 150 0. 882 0. 005 0. 868 0. 892 Mean coverage for t distribution 150 0. 952 0. 003 0. 944 0. 959 Mean 95% CI length for normal distribution 150 0. 899 0. 168 0. 608 1. 395 Mean 95% CI length for t distribution 150 1. 273 0. 238 0. 861 1. 976 15 Mean coverage for normal distribution 150 0. 931 0. 003 0. 923 0. 937 Mean coverage for t distribution 150 0. 951 0. 002 0. 945 0. 958 Mean 95% CI length for normal distribution 150 0. 535 0. 100 0. 362 0. 824 Mean 95% CI length for t distrib ution 150 0. 585 0. 109 0. 396 0. 901 25 Mean coverage for normal distribution 150 0. 938 0. 003 0. 932 0. 943 Mean coverage for t distribution 150 0. 950 0. 002 0. 944 0. 955 Mean 95% CI length for normal distribution 150 0. 416 0. 077 0. 282 0. 640 Mean 95% C I length for t distribution 150 0. 438 0. 081 0. 297 0. 674 35 Mean coverage for normal distribution 150 0. 942 0. 002 0. 937 0. 949 Mean coverage for t distribution 150 0. 950 0. 002 0. 945 0. 957 Mean 95% CI length for normal distribution 150 0. 352 0. 065 0. 2 38 0. 541 Mean 95% CI length for t distribution 150 0. 365 0. 068 0. 247 0. 560 45 Mean coverage for normal distribution 150 0. 943 0. 002 0. 939 0. 949 Mean coverage for t distribution 150 0. 950 0. 002 0. 945 0. 956 Mean 95% CI length for normal distributi on 150 0. 311 0. 058 0. 211 0. 477 Mean 95% CI length for t distribution 150 0. 320 0. 059 0. 217 0. 491 55 Mean coverage for normal distribution 150 0. 946 0. 002 0. 940 0. 952 Mean coverage for t distribution 150 0. 951 0. 002 0. 945 0. 956 Mean 95% CI length for normal distribution 150 0. 281 0. 052 0. 191 0. 432 Mean 95% CI length for t distribution 150 0. 288 0. 053 0. 195 0. 442

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104 Table 4 12. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 07 5 Mean covera ge for normal distribution 150 0. 878 0. 005 0. 866 0. 887 Mean coverage for t distribution 150 0. 950 0. 003 0. 944 0. 955 Mean 95% CI length for normal distribution 150 0. 474 0. 085 0. 311 0. 729 Mean 95% CI length for t distribution 150 0. 672 0. 120 0. 440 1. 033 15 Mean coverage for normal distribution 150 0. 930 0. 003 0. 922 0. 935 Mean coverage for t distribution 150 0. 950 0. 002 0. 945 0. 956 Mean 95% CI length for normal distribution 150 0. 286 0. 051 0. 188 0. 439 Mean 95% CI length for t distribution 150 0. 313 0. 056 0. 205 0. 480 25 Mean coverage for normal distribution 150 0. 938 0. 002 0. 931 0. 943 Mean coverage for t distribution 150 0. 950 0. 002 0. 944 0. 955 Mean 95% CI length for normal distribution 150 0. 223 0. 040 0. 147 0. 342 Mean 95% CI leng th for t distribution 150 0. 235 0. 042 0. 154 0. 360 35 Mean coverage for normal distribution 150 0. 941 0. 002 0. 935 0. 946 Mean coverage for t distribution 150 0. 949 0. 002 0. 945 0. 954 Mean 95% CI length for normal distribution 150 0. 189 0. 034 0. 124 0. 2 89 Mean 95% CI length for t distribution 150 0. 196 0. 035 0. 129 0. 300 45 Mean coverage for normal distribution 150 0. 944 0. 003 0. 938 0. 950 Mean coverage for t distribution 150 0. 950 0. 003 0. 945 0. 955 Mean 95% CI length for normal distribution 150 0. 167 0. 030 0. 110 0. 255 Mean 95% CI length for t distribution 150 0. 172 0. 031 0. 113 0. 263 55 Mean coverage for normal distribution 150 0. 946 0. 002 0. 941 0. 949 Mean coverage for t distribution 150 0. 951 0. 002 0. 946 0. 954 Mean 95% CI length for n ormal distribution 150 0. 151 0. 027 0. 099 0. 231 Mean 95% CI length for t distribution 150 0. 155 0. 028 0. 101 0. 237

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105 Table 4 12. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 12 5 Mean coverage for normal distribution 150 0. 876 0. 005 0. 861 0. 886 Mean coverage for t distribution 150 0. 949 0. 003 0. 942 0. 954 Mean 95% CI length for normal distribution 150 0. 362 0. 064 0. 230 0. 557 Mean 95% CI length for t distribution 150 0. 512 0. 091 0. 325 0. 789 15 Mean coverage for normal distribution 150 0. 929 0. 003 0. 923 0. 935 Mean coverage for t distribution 150 0. 950 0. 002 0. 945 0. 955 Mean 95% CI length for normal distribution 150 0. 219 0. 039 0. 139 0. 336 Mean 95% CI length for t distribution 150 0. 239 0. 043 0. 152 0. 367 25 Mean coverage for normal distribution 150 0. 937 0. 002 0. 932 0. 942 Mean coverage for t distribution 150 0. 949 0. 002 0. 944 0. 953 Mean 95% CI length for normal distribution 150 0. 171 0. 030 0. 109 0. 262 Mean 95% CI length for t distribution 150 0. 180 0. 032 0. 114 0. 276 35 Mean coverage for normal distribution 150 0. 941 0. 002 0. 935 0. 946 Mean coverage for t distribution 150 0. 949 0. 002 0. 943 0. 954 Mean 95% CI length for normal distribution 150 0. 145 0. 026 0. 092 0. 222 Mean 95% CI length for t distribution 150 0. 150 0. 026 0. 096 0. 230 45 Mean coverage for normal distribution 150 0. 943 0. 002 0. 938 0. 949 Mean coverage for t distribution 150 0. 950 0. 003 0. 944 0. 955 Mean 95% CI length for normal distribution 150 0. 128 0. 023 0. 081 0. 196 Mean 95% CI length for t distribution 150 0. 131 0. 023 0. 084 0. 201 55 Mean coverage for normal distribution 150 0. 946 0. 002 0. 939 0. 950 Mean coverage for t distribution 150 0. 951 0. 002 0. 945 0. 955 Mean 95% CI length for normal distribution 150 0. 116 0. 020 0. 074 0. 177 Mean 95% CI length for t distribution 150 0. 118 0. 021 0. 075 0. 181

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106 Table 4 12. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 17 5 Mean coverage for norma l distribution 130 0. 876 0. 005 0. 861 0. 886 Mean coverage for t distribution 130 0. 949 0. 003 0. 942 0. 954 Mean 95% CI length for normal distribution 130 0. 308 0. 056 0. 187 0. 468 Mean 95% CI length for t distribution 130 0. 436 0. 079 0. 265 0. 663 15 M ean coverage for normal distribution 130 0. 929 0. 003 0. 923 0. 933 Mean coverage for t distribution 130 0. 950 0. 002 0. 945 0. 956 Mean 95% CI length for normal distribution 130 0. 186 0. 034 0. 113 0. 283 Mean 95% CI length for t distribution 130 0. 204 0. 037 0. 124 0. 310 25 Mean coverage for normal distribution 130 0. 937 0. 002 0. 932 0. 941 Mean coverage for t distribution 130 0. 949 0. 002 0. 944 0. 953 Mean 95% CI length for normal distribution 130 0. 145 0. 027 0. 088 0. 221 Mean 95% CI length for t dis tribution 130 0. 153 0. 028 0. 093 0. 232 35 Mean coverage for normal distribution 130 0. 941 0. 002 0. 935 0. 946 Mean coverage for t distribution 130 0. 949 0. 002 0. 944 0. 954 Mean 95% CI length for normal distribution 130 0. 123 0. 022 0. 075 0. 187 Mean 9 5% CI length for t distribution 130 0. 128 0. 023 0. 078 0. 194 45 Mean coverage for normal distribution 130 0. 943 0. 002 0. 938 0. 949 Mean coverage for t distribution 130 0. 950 0. 002 0. 945 0. 954 Mean 95% CI length for normal distribution 130 0. 109 0. 020 0. 066 0. 165 Mean 95% CI length for t distribution 130 0. 112 0. 020 0. 068 0. 170 55 Mean coverage for normal distribution 130 0. 946 0. 002 0. 941 0. 949 Mean coverage for t distribution 130 0. 951 0. 002 0. 946 0. 954 Mean 95% CI length for normal distri bution 130 0. 099 0. 018 0. 060 0. 149 Mean 95% CI length for t distribution 130 0. 101 0. 018 0. 061 0. 153

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107 Table 4 12. Continued Event rate in treatment No. of studies Outcomes No. of scenarios Mean STD Minimum Maximum 0. 22 5 Mean coverage for normal dist ribution 120 0. 875 0. 005 0. 860 0. 884 Mean coverage for t distribution 120 0. 949 0. 003 0. 940 0. 955 Mean 95% CI length for normal distribution 120 0. 276 0. 048 0. 175 0. 412 Mean 95% CI length for t distribution 120 0. 391 0. 068 0. 248 0. 584 15 Mean co verage for normal distribution 120 0. 929 0. 003 0. 923 0. 934 Mean coverage for t distribution 120 0. 950 0. 002 0. 944 0. 954 Mean 95% CI length for normal distribution 120 0. 167 0. 029 0. 106 0. 249 Mean 95% CI length for t distribution 120 0. 183 0. 032 0. 116 0. 273 25 Mean coverage for normal distribution 120 0. 937 0. 002 0. 930 0. 943 Mean coverage for t distribution 120 0. 949 0. 002 0. 944 0. 952 Mean 95% CI length for normal distribution 120 0. 131 0. 023 0. 083 0. 195 Mean 95% CI length for t distribut ion 120 0. 138 0. 024 0. 087 0. 205 35 Mean coverage for normal distribution 120 0. 941 0. 002 0. 935 0. 946 Mean coverage for t distribution 120 0. 949 0. 002 0. 944 0. 955 Mean 95% CI length for normal distribution 120 0. 111 0. 019 0. 070 0. 165 Mean 95% CI length for t distribution 120 0. 115 0. 020 0. 073 0. 171 45 Mean coverage for normal distribution 120 0. 943 0. 002 0. 938 0. 947 Mean coverage for t distribution 120 0. 949 0. 002 0. 945 0. 954 Mean 95% CI length for normal distribution 120 0. 098 0. 017 0. 062 0. 146 Mean 95% CI length for t distribution 120 0. 101 0. 017 0. 064 0. 150 55 Mean coverage for normal distribution 120 0. 946 0. 002 0. 940 0. 951 Mean coverage for t distribution 120 0. 951 0. 002 0. 945 0. 956 Mean 95% CI length for normal distribution 120 0. 089 0. 015 0. 056 0. 132 Mean 95% CI length for t distribution 120 0. 091 0. 016 0. 058 0. 135

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108 Figure 4 1. Aim 1 : C omparison s of UW_OR vs. UW_MM in terms of difference for both mean coverage and mean length 95% CI

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109 Figure 4 2. Aim 1 : C ompariso n s of UW_OR vs. UW_MM in terms of difference for both mean coverage and mean length 95% CI by e vent rate in treatment arm and number of studies

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110 Figure 4 3. Aim 1 : C omparison s of UW_OR vs. UW_MM in terms of both mean coverage and mean length 95% CI

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111 Figure 4 4. Aim 1 : C omparison s of UW_OR vs. UW_MM in terms of both mean coverage and mean length 95% CI by e vent rate in treatment arm and number of studies

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112 Figure 4 5. Aim 2 1 : C omparison s of t distribution vs. normal distribution for U W_ OR in terms o f difference for both mean coverage and mean length 95% CI

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113 Figure 4 6. Aim 2 1 : C omparison s of t distribution vs. normal distribution for U W_ OR in terms of difference for both mean coverage and mean length 95% CI by e vent rate in treatment arm and numb er of studies

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114 Figure 4 7. Aim 2 2 : C omparison s of t distribution vs. normal distribution for U W_ OR in terms of both mean coverage and mean length 95% CI

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115 Figure 4 8. Aim 2 2 : C omparison s of t distribution vs. normal distribution for U W_ OR in terms o f both mean coverage and mean length 95% CI by e vent rate in treatment arm and number of studies

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116 Figure 4 9. Aim 2 2 : C omparison s of t distribution vs. normal distribution for U W_ MM in terms of difference for both mean coverage and mean length 95% CI

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117 Figure 4 1 0. Aim 2 2 : C omparison s of t distribution vs. normal distribution for U W_ MM in terms of difference for both mean coverage and mean length 95% CI by e vent rate in treatment arm and number of studies

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118 Figure 4 11. Aim 2 2 : C omparison s of t distribution vs. normal distribution for U W_ MM in terms of both mean coverage and mean length 95% CI

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119 Figure 4 12. Aim 2 2 : C omparison s of t distribution vs. normal distribution for UW_MM in terms of both mean coverage and mean length 95% CI by e vent ra te in treatment arm and number of studies

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120 Figure 4 13. Aim 3 : C omparison s of UW_RR vs. W_RR in terms of difference for both mean coverage and mean length 95% CI

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121 Figure 4 14. Aim 3 : C omparison s of UW_RR vs. W_RR in terms of difference for both mean coverage and mean length 95% CI by e vent rate in treatment arm and number of studies

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122 Figure 4 15. Aim 3 : C omparison s of UW_RR vs. W_RR in terms of both mean coverage and mean length 95% CI

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123 Figure 4 16. Aim 3 : C omparison s of UW_RR vs. W_RR in terms o f both mean coverage and mean length 95% CI by e vent rate in treatment arm and number of studies

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124 Figure 4 17. Aim 4 1 : C omparison s of t distribution vs. normal distribution for U W_RR in terms of difference for both mean coverage and mean length 95% CI

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125 Figure 4 18. Aim 4 1 : C omparison s of t distribution vs. normal distribution for U W_RR in terms of difference for both mean coverage and mean length 95% CI by e vent rate in treatment arm and number of studies

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126 Figure 4 19. Aim 4 1 : C omparison s of t d istribution vs. normal distribution for U W_RR in terms of both mean coverage and mean length 95% CI

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127 Figure 4 20. Aim 4 1 : C omparison s of t distribution vs. normal distribution for U W_RR in terms of both mean coverage and mean length 95% CI by e vent rat e in treatment arm and number of studies

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128 Figure 4 21. Aim 4 2 : C omparison s of t distribution vs. normal distribution for W_RR in terms of difference for both mean coverage and mean length 95% CI

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129 Figure 4 22. Aim 4 2 : C omparison s of t distribution v s. normal distribution for W_RR in terms of difference for both mean coverage and mean length 95% CI by e vent rate in treatment arm and number of studies

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130 Figure 4 23. Aim 4 2 : C omparison s of t distribution vs. normal distribution for W_RR in terms of b oth mean coverage and mean length 95% CI

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131 Figure 4 24. Aim 4 2 : C omparison s of t distribution vs. normal distribution for W_RR in terms of both mean coverage and mean length 95% CI by e vent rate in treatment arm and number of studies

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132 CHAPTER 5 DISCUSSI ON To investigate post marketing safety signals researchers usually adopt three approaches, meta analys e s, large clinical safety trials and observational studies. Due to the practical limitations of the clinical trials and observational studies [33] the evidence from a meta analysis on existing clinical trials often is viewed as the h ighest authority, especially in mak ing consensu s on a controversial issue T he FDA first issued regulatory guidelines to industry on conducting a meta analysis on clinical trials w hich w ere especially for cardiovascular risks of antidiabetics in 2008 [34] In March, 201 1, t he FDA co sponsored the Drug Information Association (DIA) conference on the DIA/FDA best practices for regulatory information synthesis of randomized controlled trials for product safety evaluation [35] Considering the highest authority given to evidence arrived from a meta analysis, selecting the appropriate meta analy tic al method is imperative and important. This chapter is organized as follows. First, we discuss the specific findings Second, we elabor ate in depth regarding the stat istical meta analysis methods. Third, we provide an empirical example using the UW_RR and the W_RR as compared with the empirical weighted method (W_ DL ) Fourth, we highlight the strengths of this study. Fifth we address the limitations of this study. Sixth we identify the need for further studies. At the end, we draw our conclusions based upon this study Specific Findings Aim 1 Finding s for hypothesis 1 When a small number of studies, such as five, are combined for a meta analysis, the results of both approaches of the UW_OR and the

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133 UW_MM are similar regardless of event rate. H owever, at low event rate s in the treatment arm (PT), for example, PT=0. 02 as the number of studies (M) increase s the mean coverage of the UW_MM drop s wh ereas at high event rate s in the treatment arm (PT), such as PT=0. 22, the UW_MM does not drop apparently. The reason is that the individual study estimates of the odds ratio are biased at low event rate s This bias remains the same reg ardless of the number of studies combined for a meta analysis study A s the number of studies ( M ) goes up the standard errors get smaller. Therefore, the confidence intervals are tightened so that the 95% CIs are more likely to exclude the global true OR value At M=5 the 95 % CIs are wider, and are more likely to cover the global true value of OR than at M= 55 The UW_OR provides a robust estimate of the summary effect size in that the UW_OR estimates the summary effect size using first the summation o f each proportion for each arm in each study i. e. risk for each arm in each study then do ing ratio for the relative risk or the odds ratio on the summary estimates of proportions This provides evidence of the robustness of the UW_OR when applied to a l arge numbe r of studies contributing to a meta analysis. Since each study is weighted equally, no single trial dominates this unweighted analysis. This also lessens, but does not eliminate, concerns about publication bias involving a small fraction of misse d studies. I n responses to two letters to Statistics in Medicine by Carpenter et al. [36] and Rucker et al. [29] Shuster e t al. [37] demonstrated, at least empirically, that the UW_OR is robust to both (a) exclusion of a small number of trials [37] and (b) to early termination of a large trial [29] As for a meta analysis, in order to have more accurate estimate s of summary effect size s the large sample theory comes into play in two ways One is a large sample

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134 size f or each individual study The other is a large number of studies combined for conducting a meta analysis The theory explains that the efficiency of both the UW_OR and the U W _MM is getting closer as the number of studies increases or event rate s increase. Implication for finding s 1 There is no precision difference between the UW_OR and the UW_MM when rare event s are unlikely regardless of the number of studies Aim 2 Finding s for hypothesis 2 1 The UW_OR is developed with the assumption that the s ummary effect size is a t distribution instead of the widely used normal distribution. Therefore, the 95% CI is calculated using the t cutoff point with the number of studies (M) 1 degrees of freedom. When M is small, the t score is large (appendix A) Th erefore, the 95% CI is wider, lead ing to the more accurate coverage of 95% On the other hand, the UW_OR with a normal distribution has a narrower 95% CI which leads the underestimation of the coverage. However, a s the number of studies increases the t score decreases towards the normal z score (appendix A) This is the reason that the UW_OR with a t distribution is close to the UW_OR with a normal distribution as the number of studies combined for a meta analysis increases. Implication for finding s 2 1 T he UW_OR with a t approximation is more accurate which provides the mean coverage closer to the expected 95% coverage than the UW_OR with a normal approximation when a small number of studies are combined for a meta analysis regardless whether or not t here are low event rates Finding s for hypothesis 2 2 The UW_MM is developed with the assumption that the summary effect size is with a t distribution instead of the widely used normal distribution. Therefore, the 95% CI is calculated using the t cutoff p oint with the number

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135 of studies ( M ) 1 degrees of freedom. When M is small, the t score is large. Therefore, the 95% CI is wider lead ing to the more accurate coverage of 95%. On the other hand, the UW_ MM with a normal distribution has a narrower 95% CI w hich leads to the underestimation of the coverage. However, a s the number of studies increases, the t score decreases towards the normal z score (appendix A). This is the reason that the UW_MM with a t distribution is close to the UW_MM with a normal dis tribution as the number of studies increases. As we discussed in aim 1, the 95% coverage for the UW_MM with a t distributi on drop s dramatically when the event rate is low in that t he estimates of the odds ratio s from individual stud ies are biased at low event rate s This bias remains the same regardless of the number of studies combined in a meta analysis study Moreover, as M increases, the t score and standard errors become smaller. Therefore, the 95% CIs are tightened so that they are more likely to e xclude the true value of OR Implication for finding s 2 2 T he UW_ MM with a t distribution is more a ccurate provid ing the mean coverage closer to the expected 95% coverage than the one with a normal distribution when a small number of studies are combin ed for a meta analysis and rare rates are unlikely Aim 3 Finding s for Hypothesis 3 The study show s that the method with the sample size weighting on the proportion (W_RR) has similar summary effect size as compared with the UW_RR but the W_RR shows shorter mean length of the 95% CI for certain number of scenarios We know that these two approaches do not estimate the same summary effect s which the UW_RR estimates the unweighted summary effect while the W_RR estimates the weighted summary effect. In this study, we forced two

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136 methods to estimate the same summary effect size by assuming that there is no association between the effect size and the sample size. One important observation centers on the rate of convergence of the asymptotic distributio ns for the weighted (W_RR) vs. the unweighted method (UW_RR) While we cannot mathematically prove superior convergence, the central limit theorem works more rapidly when the risk of outliers is relatively low. The UW_RR uses the bivariate summary proport ions, which are relatively low, and unlikely to be prone to large outliers. On the other hand, the W_RR uses the adjusted number of events in its ratio estimate of relative risk. As such, these will have much higher variability, since large scale trials te nd to have many more adverse events than small er ones. Implication for Finding 3 As a special case, t he UW_RR and the W_RR estimate the same summary effect size when there is no association between the effect size and the sample size But the W_RR is mo re preci se in terms of the shorter 95% CI. As the number of studies combined increases, the precision of both approaches is increased simultaneously. Aim 4 Finding s for hypothesis 4 1 The UW_RR, like the UW_OR, is developed with the assumption tha t the summary effect size is a t distribution instead of the widely used normal distribution. Therefore, the 95% CI is calculated using the t cutoff point with the number of studies (M) 1 degrees of freedom. When M is small, the t score is large ( A ppendix A). Therefore, the 95% CI is wider lead ing to the more accurate coverage of 95%. On the other hand, the UW_RR with a normal distribution has a narrower 95% CI which lead s to underestimate the coverage. However, a s the number of studies increase s the t score decreases toward s the normal z score (appendix A). This is the

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137 reason that the UW_RR with a t distribution is close to the UW_RR with a normal distribution as the number of studies combined for a meta analysis increases. Implication for finding s 4 1 T he UW_ R R with a t approximation is more accurate, provid ing the mean coverage closer to the expected 95% coverage than the UW_RR with a normal approximation when a small number of studies are combined for a meta analysis regardless whether or not there are low event rates Finding s for Hypothesis 4 2 The W_RR is developed with the assumption that the summary effect size is a t distribution instead of the widely used normal distribution Therefore, the 95% CI is calculated using the t cutoff point with the number of studies (M) 1 degrees of freedom. When M is small, the t score is large. Therefore, the 95% CI is wider leading to the more accurate coverage of 95%. On the other hand, the W_RR with a normal dis tribution has a narrower 95% CI which lead s t o the underestimation of the coverage. However, a s the number of studies increases, the t score decreases towards the normal z score (appendix A). This is the reason that the W_RR with a t distribution is close to the W_RR with a normal distribution as the number of studies combined for a meta analysis increases. Implication for Finding s 4 2 T he W_ R R with a t approximation is more accurate, and provides the mean coverage closer to the 95% coverage than the one with a normal approximation when a small numb er of studies are combined for a meta analysis regardless whether or not there are low event rates Meta Analysis in General As mentioned in C hapter II ( Review of the Literature ) the most widely used meta analysis method is the random effects model.

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138 U nder the umbrella of the random effects model, two categories of weighted and unweighted approaches are frequently used both of which are non Bayesian In regards to the weighted random effects approaches, there are the empirical DerSimonian Laird approa [17] and the new approach developed by Shuster et al [6] with sample size weighting on the proportion s T here are also two unweighted random effects approaches, one with the summarized proportion s in the form of odds ratio or relative risk [7, 9 ] and one with the mean of mean s in the form of odds ratio [8] In this subsection, we focus on issues related to the random effects model in depth. We start with the weighting issue and the estimated summary effect size. Next we discuss the low event rates and continuity adjustment wh en zero event cells occur Thirdly we address asymptotic properties when a small number of studies are combined for a meta analys is. F inally we discuss the three typical metrics for binomi al outcomes, odds ratio, relative risk and risk difference Weighting Issues and Summary Effect Size Estimates T he weights for the empirical random effects model (W_ DL ) are volatile random variables because the weights are derived from both the wit hin study variance and the between study variance [5] The W_ DL needs to be avoided in low event rate applications for following two reasons. First, it is unclear as to what the W_DL is estimating, given that the weights are random variables but treated as non random Second, in low event rate applications, the effect size s and weights probably are correlated [8, 38] For example a) I n drug development, early smaller studies may be pure (drug only vs placebo), while later larger studies may use the drug vs placebo in an adjuvant setting Therefore larger studies expect to have a smaller difference in

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139 efficacy. b) For side effects however, an adjuvant therapy interaction with the experimental drug may trigger a larger differential, yielding the opposite correlation. c) Better designed studies may lower the sampling error, thereby increasing the weight, while the greater skills of these investigators may lead to larger advantages for the experimental therapy over the controls. This is especially problematic in surgery device trials. d) Unknown to a meta analyst, some studies may have been terminated earl y for efficacy, yielding smaller weights than those that run to completion. These arguments suggest that the correlations of weights and effect sizes can be expected, and it is statistically risky to assume they do not exist especially at low event case A positive correlation may produce a positive bias and a negative correlation may produce a negative bias. The W_RR is one solution to overcome the issues from the empirical approach (W_DL) which the weights are derived from the sample size a random va riable. The weights are applied to the proportion before the summary effect size is calculated and they are treated as random variables In contrast, some sample size based methods such as Emerson et al [39, 40] use weights proportional to the product of the sample size in each arm divided by the sum of the sample sizes of both arm s This implies that m ore balanced randomizations carry more weight. However, i t is hard to interpret its results. The advantage of th e weighted approach (W_RR) is that it tend s to have narrower confidence limits at the cost of slower convergence to normality as the number of studies being pooled becomes large. As alternative solution s the methods of UW_OR/ UW_ RR and UW_ MM estimate the unweighted summary effect size. These methods can be viewed unweighted or

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140 weighted equally, i. e. each study carries 1/M weight. The only method that has a similar spirit was developed by Follmann et al [41] This method weight s studies equally, but they apply their test to the mean of the individual study effect size, rather than a ratio estimate approach. This approach is complemen tary to the UW_OR/ UW_ RR approach in that its application is to a meta analysis of a small number of larger studies [7] However, when rare event occurs the method would be problematic for relative risk estimates when any study has zero event in either arm, which yields an undefined individual effect size estimate. Low Event Issue and Continuity Adjustment Despite warnings in DerSimonian Laird, Bradburn et al. [42] and Shuster et al. [7] the W_ DL remains widely used for meta analys e s with low event rate binomial trials [43 56] When zero event s occur on one or both arms, a continuity adj ustment is needed to compute the effect size for the individual study. There are several ways to adjust for continuity [57] As for the W_ DL, the continuity corr ection [57] is used by add ing 0. 5 to each of the four cells of the 2 by 2 outcome table that has zero event cells and studies with no event i n both arms are excl uded This way of continuity adjustment is the default option for the two most popular software packages (Comprehensive Meta analysis 2. 0 and RevMan 5. 0) [5] In o ne of the proposed solution s, the UW_MM use s the sample size adjusted continuity method when zero event happen s on one or both arms [57] The refore, no studies are excluded In this simulation study, since the number of subjects in both arms is the same the sample size adjusted continuity is the same as adding 0. 5 in the studies with zero in one or both arms.

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141 By contrast, other proposed sol utions of the unweighted method (UW_OR/ UW_ RR) and the weighted method (W_RR) d o not need continuity adjustment. The reason is that the summary effect size is not computed until the proportion (risk) in each arm from each study i s added together. Asy mptotic Properties and Large Sample Theory As we mentioned before, for a meta analysis, the large sample theory should apply at two levels of the individual study and the number of individual studies being combined for a meta analysis However, the m ost r andom effects meta analysis studies do not have a large number of studies to combine but presume s a normal distribution on the summary effect size on a log scale Methods along these lines can be found in DerSimonian and Laird [17] Smith et al. [23] Hartung and Knapp [58] Brockwell and Gordon [21] ] Warn et al. [24] and Burr and Doss [25] Considering that we do not often have a large number of studies availa ble for a meta analysis we shall use t approximations, rather than normal approximations. The t approximations are equivalent to normal cutoffs in the limit, as the number of studies being combined tends to infinity. The advantage of using a t distribut ion is that the mean coverage of the 95% CI is closer to the 95% coverage, while a method with a normal distribution often tend s to underestimate ( narrow confidence limits ). Selection of Measurement Metrics for Binomi al Outcomes As for a meta analy sis with binomial outcome s a researcher often selects the metrics risk ratio or odds ratio over risk difference. The reason is that the risk ratio and odds ratio are relative measures, and therefore tend to be relatively insensitive to differences in bas eline events [5] By contrast, the risk difference is an absolute measure and it is very sensitive to the baseline risk. However the risk di fference is

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142 someti me s more clinically meaningful [5] In any event, the risk difference provides an unbiased metric for a meta analysis due to no issue s with low event and no assumption for the summary effect size. T he risk difference can be predicted afterwards for any given baseline risk [5] Elaborating Example on Mortality for Erythropoiesis Stimulating Agents In this subsection, we illustrate how the empirically weighted model (W_DL), the sample size weighted model (W_RR), and the unweighted (UW_RR) work on the study of the mortality risk of erythropoiesis stimulating agents in cancer population. Bohlius et al. [59] conducted a patient level meta analysis with respect to potentially increased mortality risk of erythropoiesis stimulatin g agents in cancer population. This study combined 53 randomized cancer trials and found a highly significant increase in mortality with these agents as compared to controls. The FDA, in part on the basis of these data, issued a recommendation that these agents should be used under a risk management program known as a risk evaluation and mitigation strategy (REMS) [60] Table 5 1 provides the summary effect size through four meta analysis approaches: the published hazard ratio (instantaneous relati ve risk) analysis, the empirical weighted DL approach (W_DL) the weighted analysis of relative risk (W_RR), and the unweighted analysis of relative risk (UW_RR). We note the close agreement between the published results [59] and the other two weighted (W_DL, W_RR) results that use study level data. We discuss the findings as follows. First, although the analysis of Bohlius et al. [59] was based upon actual patient level data, we have to utilize study level data due to the lack of access to the patient level data However, a s long as the follow up time is

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143 equally rigorous on both treatment arms and event rates are low, in such situations, the ratio of person years at risk with in a study (Treatment 2: Treatment 1) is closely approximated by the ratio of the sample sizes, which explains that the result s from the actual patient le vel data are close to the result from the empirically weighted approach (W_DL). Second, the reason th at the result s from the W_RR are different from the results from the UW_RR is because they do not estimate the same summary effect size (weighted vs. unweighted) (Figure 5 1). Third, the situations that the results from the W_DL are close to the results fr om the UW_RR are either when (a) the diversity between studies goes to infinity, (b) sample size goes to infinity, or (c) there is no relationship between effect size s and weights. In this case, the results from the W_DL are different from the results fr om the UW_RR, indicating that the W_DL does not estimate the unweighted summary effect size as claimed. In summary, we learn two points from this example as follows. First, the W_DL does not estimate the unweighted summary effect size as claimed. Instead the W_DL may estimate the weighted summary effect size in this case (though not in general), i. e. the results from the W_DL are close to the results from the W_RR. Second, the results from the W_RR (1. 14, 95% CI: 1. 04 1. 24) are different from the results f rom the UW_RR (1. 05, 95% CI: 0. 95 1. 17). The contradicting results are fine since they estimate different summary effect size (unweighted for the UW_RR vs. weighted for the W_RR). The consensus of the W_RR and the UW_RR provides more confidence to make the inference, but no consensus was obtained here. Strengths Given the existing theoretical issues with the empirical random effects meta analysis, Shuster et al. proposed several ways to overcome their obstacles the

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144 UW_MM, the UW_OR /UW_RR and the W_R R. This is the first study to evaluat e the efficiency of these methods by using data simulation s The findings of this study are built upon thousands of simulated scenarios provid ing solid evidence to guide a researcher to select an appropriate meta anal ysis statistical method in the future Limitations T his study also has some limitations. First t he results are generated based upon data simulation s with somewhat limited scenarios. More scenarios with data simulations will be more helpful to make a general statement Second, in all of these simulations, we keep the number of subjects the same for both treatment and control arm s in each individual study. Therefore the impact of variation of the number of subjects in the treatment arm and in the contr ol arm is not reflected in existing scenarios. Finally since scenarios are generated based upon predetermined parameters, each scenario is not randomly determined Therefore, statistical test s are not helpful in determin ing the difference between compa red methods of interest. We use graphics and tables to describe the efficiency difference s across methods of interest Further studies Since w e draw conclusions based upon limited scenarios we can extend this study by adding more scenarios For exampl e, we could add variation in the number of subjects between the treatment arm and the control arm. As for the relative risk data simulation, although the UW_RR estimates the unweighted summary effect size whereas the W_RR estimates the weighted summary ef fect size o ur study forced the summary effect size from both the UW_RR and the W_RR to be mathematically equal Further more, data simulation is needed to compare

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145 the UW_RR vs the W_RR w hen the association between the effect size and the sample size exi sts A lso a s a further study, the empirical validation on both the sample size weighted approach (W_RR) and the unweighted on proportion (UW_RR) may apply to some known association s of exposures and outcomes. Conclusions Aim 1: The UW _OR is more effici en t and robust regardless of the number of studies combined and with or without low event s compared with the UW_MM. The UW_MM can be used when low events are unlikely Aim 2 1: The UW_OR with a t distribution provides a more accurate coverage regardless of the number of studies combined and with or without low event s compared with the UW_OR with a normal distribution. The UW_OR with a t distribution and the UW_OR with a normal distribution are very close when synthesizing a large number of studies. Ai m 2 2: The UW_MM with a t distribution has a more accurate coverage than the UW_MM with a normal distribution when a small number of studies are combined and no rare events occur. The UW_MM with a t distribution and the UW_MM with a normal distribution ar e very close when synthesizing a large number of studies. Aim 3: Both the UW RR and the W_RR are efficient when they are forced to estimate the same summary effect size. The UW_RR provides more accurate and more stable coverage than the W_RR, whereas the W_RR also provides an accurate coverage and a relatively narrow confidence interval in some scenarios Aim 4 1 : The UW_ R R with a t distribution provides a more accurate coverage regardless of the number of studies combined and with or without low event s compared

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146 with the UW_ R R with a normal distribution. The UW_ R R with a t distribution and the UW_ R R with a normal distribution are very close when synthesizing a large number of studies. Aim 4 2: The W_RR with a t distribution provides a more accurate coverage regardless of the number of studies combined and with or without low events, compared with the W_RR wi th a normal distribution. The W_RR with a t distribution and the W_RR with a normal distribution are very close when synthesizing a large number of studies. In summary, when in doubt, it might be a good statistical practice to report both the sample size weighted (W_RR) and the unweighted analyses (UW_RR) (one as primary and one as secondary). If the two approaches agree qualitativel y, this will add strength to make inference. But since they estimate completely different summary effect size (weighted vs. unweighted) it is not a contradiction when the qualitative conclusions are different.

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147 Table 5 1. Mortality for erythropoiesis s t imulating agents in cancer population using different random effects approaches in meta analyses Approaches Summary effect size 95% CL P value (2 sided) HR as Published [59] 1. 17 1. 06 1. 30 0. 001 W_DL (RR) 1. 15 1. 0 5 1. 26 0. 003 W_RR 1. 14 1. 04 1. 24 0. 005 UW_RR 1. 05 0. 95 1. 17 0. 33 0 Figure 5 1. Illustration of associations among W_DL, W_RR and UW_RR

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148 APPENDIX T TABLE

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154 60. FDA. FDA Drug Safety Communication: Erythropoiesis Stimulating Agent s (ESAs): Procrit, Epogen and Aranesp. In FDA Drug Safety Communication: Erythropoiesis Stimulating Agents (ESAs): Procrit, Epogen and Aranesp. 2010. http://www.fda.gov/Drugs/DrugSafety/PostmarketDrugSafetyInformationforPatientsandP roviders/ucm200297.htm

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155 BIOGRAPHICAL SKETCH The author was born and grew up in Beijing, China. She hold s a m degree in pharmaceutical outcome and policy at the University of North Carolina at Chapel Hill with a prior pharmacology background In 2008, s he j oin ed the D epartment of Pharmaceutical Outcomes and P olicy for a doctorate and she m ajor s in pharmacoepidemology Her research interest is pharma coepidemiology study design s and analys e s including meta analysis p harmacovigilance and risk management.