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## Material Information- Title:
- Accuracy of Estimates, Empirical Type I Error Rates, and Statistical Power Rates for Testing Mediation in Latent Growth Modeling in the Presence of Nonnormal Data
- Creator:
- Koo, Namwook
- Place of Publication:
- [Gainesville, Fla.]
Florida - Publisher:
- University of Florida
- Publication Date:
- 2012
- Language:
- english
- Physical Description:
- 1 online resource (113 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Research and Evaluation Methodology
Human Development and Organizational Studies in Education - Committee Chair:
- Algina, James J
- Committee Members:
- Leite, Walter
Miller, M David Brownell, Mary T - Graduation Date:
- 8/11/2012
## Subjects- Subjects / Keywords:
- Bootstrap resampling ( jstor )
Estimation bias ( jstor ) Estimation methods ( jstor ) False positive errors ( jstor ) Kurtosis ( jstor ) Mathematical variables ( jstor ) Sample size ( jstor ) Sampling bias ( jstor ) Skewed distribution ( jstor ) Standard error ( jstor ) Human Development and Organizational Studies in Education -- Dissertations, Academic -- UF growth -- latent -- longitudinal -- mediation -- sem - Genre:
- bibliography ( marcgt )
theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) born-digital ( sobekcm ) Electronic Thesis or Dissertation Research and Evaluation Methodology thesis, Ph.D.
## Notes- Abstract:
- The mediated effect is generally nonnormally distributed. Therefore, nonnormal data could cause more serious bias in estimating and testing the mediated effect. This study aimed to investigate the impact of nonnormality on estimating and testing the mediated effect in longitudinal mediation analysis using a parallel process latent growth curve model. In this Monte Carlo simulation study, the design factors were the degree of nonnormality, effect size of the mediated effect, sample size, and the R^2 value of the observed variables The dependent variables were the relative bias of the mediated effect and of the standard error estimates, empirical Type I error and power rates. In this study, accurate estimates of the mediated effect and standard error and the adequate statistical power were found when the effect size of the mediated effect, R^2, and sample size were larger. Also, it was found that nonnormality had little effect on the accuracy of the estimates of the mediated effect and standard error, empirical Type I error, and power rates except in a few conditions. Furthermore, this study found that relatively small sample sizes (e.g., 100 and 200) frequently caused outliers of the mediated effect and standard error estimates, and also standard error estimates were more frequently and inconsistently biased when both paths (a and b) were zero. In terms of empirical Type I error and power rates, the bias-corrected bootstrap performed best and then, the asymmetric distribution and Sobelâ€™s methods followed. Also, it was found that Sobelâ€™s method produced very conservative Type I error rates when the estimated mediated effect and standard error had a relationship, but when the relationship was weak or did not exist the Type I error was closer to the nominal 0.05 value. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2012.
- Local:
- Adviser: Algina, James J.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-08-31
- Statement of Responsibility:
- by Namwook Koo.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Copyright Koo, Namwook. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 8/31/2014
- Resource Identifier:
- 857767193 ( OCLC )
- Classification:
- LD1780 2012 ( lcc )
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PAGE 1 1 A CCURACY OF ESTIMATES EMPIRICAL TYPE I ERROR RATES AND STATISTICAL POWER RATES FOR TESTING MEDIATION IN LATENT GROWTH MODELING IN THE PRESENCE OF NONNORMAL DATA By NAMWOOK KOO A DISSERTATION PRESENTED TO THE GRADUATE SCH OOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2012 PAGE 2 2 2012 N amwook Koo PAGE 3 3 To my parents PAGE 4 4 ACKNOWLEDGMENTS I would like to thank my dissertation commi ttee members I would like to express my gratitude to my advisor and committee chair, Dr. Algina for his in valuable guidance during my doctoral study. My dissertation started from the independent study with him because he introduced me to the statistical m odel and program which are the basis of my dissertation I also appreciate his priceless courses which improved my understanding and knowledge in the areas of the structural equation and multilevel modeling. I would like to thank Dr. Leite T hrough his cou rses and independent studies, I learned how to conduct a Monte Carlo Simulation study and wrote a research paper, which helped me complete my dissertation Also, I am extremely grateful to Dr. Miller Director of CAPES and a prominent scholar in program ev aluation and psychometrics, for his financial support during my doctoral study As his research assistant, I worked happily with nice people and had a chance to apply what I learned in my coursework. I am thankful to Dr. Brownell for agreeing to be a part of my committee member and giving me support and valuable suggestions Finally, I would like to thank Dr. Kim, my for his support during my study in the United States W hen I took his courses, I de cided to pursue a doctoral degree in this field of study PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ .......... 9 ABSTRACT ................................ ................................ ................................ ................... 10 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 12 2 LITERATURE REVIEW ................................ ................................ .......................... 18 Causal Steps Approach ................................ ................................ .......................... 19 Methods for Testing the Mediated Effect ................................ ................................ 20 Causal Steps Approach ................................ ................................ .................... 21 Product of Coefficients Methods ................................ ................................ ....... 22 order solut ion ................................ .............................. 22 Asymmetric distribution method ................................ ................................ 23 Bootstrap Methods ................................ ................................ ........................... 26 Perce ntile confidence intervals ................................ ................................ .. 28 Bias corrected percentile confidence intervals ................................ ........... 30 Accelerated bias corrected percentile intervals ................................ .......... 31 Latent Growth Modeling ................................ ................................ .......................... 32 Univariate Latent Growth Model ................................ ................................ ....... 33 Parall el Process Latent Growth Model (Bivariate Latent Growth Model) .......... 33 Parallel Process Latent Growth Model for Mediation Analysis ......................... 35 S ignificance of This Study ................................ ................................ ....................... 37 Research Questions ................................ ................................ ............................... 38 3 METHOD ................................ ................................ ................................ ................ 40 Popu lation Values ................................ ................................ ................................ ... 40 Data Generation ................................ ................................ ................................ ..... 41 Data Analysis and the Dependent Variables of this Study ................................ ...... 42 Design Factors ................................ ................................ ................................ ........ 43 Sample Sizes and Mediated Effect Sizes ................................ ......................... 44 Degree of Nonnormality ................................ ................................ .................... 45 Population Skewness and Kurtosis Values ................................ ...................... 47 Simulation Program Check ................................ ................................ ..................... 48 4 RESULTS ................................ ................................ ................................ ............... 54 PAGE 6 6 Accuracy of Estimates of the Mediated Effect and Standard Error ......................... 54 Results for the Small Mediated Effect Size ................................ ....................... 54 Results for the Medium Mediated Effect Size ................................ ................... 55 Results for the Large Mediated Effect Size ................................ ...................... 56 Results When the Mediated Effect Size Is Zero ................................ ............... 56 Empirical Type I Error Rates ................................ ................................ ................... 59 Empirical Power Rates ................................ ................................ ............................ 63 Small Effect Size of the Mediated Effect ................................ ........................... 63 Medium Effect Size of the Mediated Effect ................................ ....................... 63 Large Effect Size of the Mediated Effect ................................ .......................... 64 5 DISCUSSION AND CONCLUSION ................................ ................................ ........ 95 Accuracy of Estimates for the Mediated Effects and Standard Errors .................... 95 Empirical Type I Error Rates ................................ ................................ ................... 97 Empirical Power Rates ................................ ................................ .......................... 101 Conclusions ................................ ................................ ................................ .......... 103 Limitations and Suggestions for Future Research ................................ ................ 104 LIST OF REFERENCES ................................ ................................ ............................. 108 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 113 PAGE 7 7 LIST OF TABLES Table page 3 1 Constant parameter and regression coefficient values over conditions .............. 50 3 2 The generation of the degree of nonnormality ................................ .................... 51 3 3 Cumulants of Normal, Bernoulli, and Chi square distribution s ........................... 51 3 4 Population skewness, and kurtosis values of latent and observed variables ...... 52 3 5 Relative bias of skewness and kurtosis w ith the mean estimates of skewness and kurtosis with 1,000,000 sample size and 100 replications ........................... 53 4 1 Median and mean relative biases of the mediated effect estimates ( ) and standard error estimates ( ) for a small mediated effect size ........................... 6 5 4 2 Median and mean relative biases of the mediated effect estimates ( ) and standard error estimates ( ) for a medium mediated effec t size ...................... 66 4 3 Median and mean relative biases of the mediated effect estimates ( )and standard error estimates ( ) for a large mediated effect size ........................... 67 4 4 Median and mean mediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect across when ................................ ................................ ........................ 68 4 5 Median and mean mediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect when ................................ ................................ ................................ ... 69 4 6 Median and mean mediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect when ................................ ................................ ................................ ... 70 4 7 Median and mean mediated effect es timates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect when ................................ ................................ ................................ .... 71 4 8 Median and mean mediated effect estimates ( ) and media n and mean relative biases for the standard error estimates ( ) for no mediated effect when ................................ ................................ ................................ ..... 72 4 9 Median and mean mediated effect estimates ( ) and median and mean relative biase s for the standard error estimates ( ) for no mediated effect when ................................ ................................ ................................ ..... 73 PAGE 8 8 4 10 Median and mean mediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect when ................................ ................................ ................................ 74 4 11 order solution, the asymmetric distribution, and bias corrected bootstrap methods: and .............. 75 4 12 order solution, the asymmetric distribution, and bias corrected bootstrap methods: and ............. 76 4 13 order solution, the distribution, and bias corrected bootstrap methods: and ................................ 77 4 14 order solution, the asymmetric distribution, and bias corrected bootstrap methods: and ........... 78 4 15 Estimated Type I error rates for Sobe order solution, the asymmetric distribution, and bias corrected bootstrap methods: and ........... 79 4 16 order solution, the asymmet ric distribution, and bias corrected bootstrap methods: and ........... 80 4 17 order solution, the asymmetric distribution, and bias corrected bootstrap methods: and ................ 81 4 18 order solution, the asymmetric distribution, and bias corrected bootstrap methods for a small effect size ......... 82 4 19 order solution, the asymmetric distribution, and bias corrected bootstrap methods for a medium effect size ..... 83 4 20 order solution, the asymmetric distribution, and bias corrected bootstrap methods for a large effect size .......... 84 PAGE 9 9 LIST OF FIGURES Figure page 2 1 Path Diagram for mediation analysis ................................ ................................ .. 39 2 2 A parallel process latent growth model in longitudinal mediation analysis .......... 39 4 1 Empirical Type I error rates when and ................................ ........ 85 4 2 Empirical Type I error rates when and ................................ ........ 86 4 3 Empirical Type I error rates when and ................................ ........ 87 4 4 Empirical Type I error rates when and ................................ ........ 88 4 5 Empirical Type I error rates when and ................................ ........ 89 4 6 Empirical Type I error rates when and ................................ ........ 90 4 7 Empirical Type I error rates when and ................................ ............ 91 4 8 Empirical power rates for a small effect size of the mediated effect ................... 92 4 9 Empirical power rates for a medium effect size of the mediated effect ............... 93 4 10 Empirical power rates for a large effect size of the mediated effect .................... 94 5 1 Relationship between the estimated mediated effect error when and under normality ................................ ....... 106 5 2 error when and under normality .............................. 107 PAGE 10 10 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 Philosophy ACCURACY O F ESTIMATES, EMPIRICAL TYPE I ERROR RATES AND STATISTICAL POWER RATES FOR TESTING MEDIATION IN LATENT GROWTH MODELING IN THE PRESENCE OF NONNORMAL DATA By Namwook Koo August 2012 Chair: James Algina Major: Research and Evaluation Methodology The media ted effect is generally nonnormally distributed. Therefore, nonnormal data could cause more serious bias in estimating and testing the mediated effect. This study aimed to investigate the impact of nonnormality on estimating and testing the mediated effect in longitudinal mediation analysis using a parallel process latent growth curve model. In this Monte Carlo simulation study, the design factors were the degree of nonnormality, effect size of the mediated effect, sample size, and the value of the observed variables T he dependent variables were the relative bias of the mediated effect and of the standard error estimates, empirical Type I error and power rates. In this study, accurate estimates of the mediated effect and standard e rror and the adequate statistical power were found when the effect size of the mediated effect, and sample size were larger. Also, it was found that nonnormality had little effect on the accuracy of the estimates of the mediated effect and standard error empirical Type I error, and power rates except in a few conditions. Furthermore, this study found that relatively small sample sizes (e.g., 100 and 200) frequently caused outliers of the PAGE 11 11 mediated effect and standard error estimates, and also standa rd error estimates were more frequently and inconsistently biased when both paths ( and ) were zero. In terms of empirical Type I error and power rates, t he bias corrected bootstrap performed best and then, the asymmetric distribution and Sobel s methods followed. Also, it was found that method produced very conservative Typ e I error rates when the estimated mediated effect and s tandard error had a relationship but when the relationship was weak or did not exist the Type I error was closer to the nominal 0.05 value PAGE 12 12 CHAPTER1 INTRODUCTION Research about methodological issues in mediation analysis have focused on one of three points: defining the mediated effect (Baron & Kenny, 1986; Collins et al., 1998; Judd & Kenny, 1981; Zhao et al., 2009), developing new mediation models to analyze data (Cheong, MacKinnon, & Khoo, 2003; K rull & MacKinnon, 1999; 2001; Li, 2011; Preacher, Zyphur, & Zhang, 2010; Preacher, 2011; von Soest & Hagtvet, 2011), and developing and evaluating new statistical tests of the mediated effect (Bollen & Stine, 1990, MacKinnon et al., 2002; MacKinnon et al., 2004; MacKinnon et al., 2007; Pituch, Stapleton, & Kang, 2006; Pituch & Stapleton, 2008; Sobel, 1982). The causal steps approach suggested by Judd and Kenny (1981) and Baron and Kenny (1986) provides conceptual guidelines (Cole & Maxwell, 2003; MacKinnon et al., 2002) for mediation analysis. Basically, the causal steps approach indicates the causal relationship, that is, an independent variable ( ) causes a mediator ( ), which, in turn, causes a dependent variable ( ) (Baron & Kenny, 1986). The focus of the causal steps approach is on how affects rather than just testing the significant relationship between and (Judd & Kenny, 1981). A shortcoming of the causal steps strategy is that it does not provide a statistical test of the mediated effec t of on through (i.e., the product of two regression coefficients of (path from X to m) and (path from to ) usually denoted as or ) but it offers a series of tests for individual paths of and (MacKinnon et al., 2002; Pituch & Stapleton, 2008). When Judd and Kenny (1981) and Baron and Kenny (1986) proposed mediation analysis with regression models, they pointed out the potential advantages of structural equation modeling, such as simultaneously test ing more than one eq uation, handling PAGE 13 13 measurement errors and conducting overall model fit test (e.g., chi squared test). Recently, new mediation models have been developed within structural equation and multilevel modeling frameworks such as mediation models combined with par allel process latent growth models (Cheong et al., 2003; Cheong, 2011; von Soest & Hagtvet, 2011), multilevel models (Bauer, Preacher, & Gil, 2006; Krull & MacKinnon, 1999; 2001, Pituch et al., 2006; Pituch & Stapleton, 2008), and multilevel structural equ ation models (Preacher et al., 2010; Preacher, 2011). Mediation models such as a single level mediation model and a multilevel mediation model (Krull & MacKinnon, 1999; 2001) and a multilevel mediation model and a multilevel structural equation mediation m odel (Li, 2011) have been compared under varied conditions through Monte Carlo simulations. Because the mediated effect (i.e., usually, the product of two regression coefficients) does not always follow a normal distribution, conventional normal theory te sts (i.e., and test) are not correct for testing the mediated effect. Thus, various new statistical methods such as resampling methods using different types of bootstrapping or j ackknife and methods based on the (asymmetric) distribution of products have been d eveloped and compared (MacKinnon et al., 2002; MacKinnon et al., 2004; MacKinnon et al., 2007; Pituch et al., 2006; Pituch & Stapleton, 2008; Sobel, 1982). Pituch et al. (2006) and Pituch and Stapleton (2008) examined several methods for testing the mediat ed effect with multilevel models, and fou nd that a resampling method (e.g ., bias corrected bootstrap ) and the asymmetric distribution method (i.e., the test based on the asymmetric distribution of the product ) consistently performed best under normality an d nonnormality. PAGE 14 14 Recently, longitudinal mediation studies have been conducted in many areas based on the structural equation modeling framework. Longitudinal studies using latent growth modeling have been popular because latent growth modeling can test indi vidual differences over time and also flexibly capture the varied growth trends (Leite, 2007; Leite & Stapleton, 2011; Meredith and Tisak, 1990; Ferrer, Hamagami, & McArdle, 2004; Flora, 2008). Furthermore, a parallel process latent growth model can be use d for a longitudinal mediation analysis when the hypothesis of the study is to assess whether the mediator accounts for the relationship between the independent variable (e.g., interventions) and the dependent variable over time. Longitudinal mediation stu dies using a parallel process growth model have been conducted in developmental and preventive studies after Cheong et al. (2003) introduced a parallel process growth model which can investigate how affects the growth rate (i.e., slope factor) of through the growth rate (i.e., slope factor) of For example, Jagers et al. (2007) conducted a longitudinal study of Aban Aya Youth Project (AAYP) and found the AAYP interventions ( ) improved the g rowth of the empathy ( ) which, in turn, reduced the growth of the youth violence ( ). Also, von Soest and Wichstrm (2009) studied how emotional problems ( ) mediated the relationship between gender ( ) and the development of dieting ( ), and Littlef ield, Sher, and Wood (2010) found that changes in personality ( ) affected changes in alcohol problems ( ) through changes in motives ( ). The focus of these studies was on how affects the growth rate (i.e., slope factor) of through the growth rat e (i.e., slope factor) of using a parallel latent growth model. Moreover, we can test different hypotheses of mediation processes using a parallel process latent growth model, for example, ) affects an initial status (i.e., intercept PAGE 15 15 factor) of which, in turn, affects growth rates of or ) affects an initial status of which affects an initial status of and so on. The purpose of this study wa s to evaluate the effects of nonnormality on estimating and testing the mediated effect us ing a parallel process latent growth model for l ongitudinal mediation analysis. Through Monte Carlo simulations under varied conditions, we can find when the nonnormality would cause more or less bias in estimating and testing the mediated effect. Althoug h there has not been much research about the effects of nonnormality in longitudinal mediation analysis, previous mediation studies investigated the effects of nonnormality with different statistical methods (e.g., tests for the mediated effects and estima tion methods). For example, in structural equation modeling, Finch, West, and MacKinnon (1997) compared three estimation methods (e.g., the maximum likelihood, robust maximum likelihood, and asymptotically distribution free methods) under the degree of non normality (e.g., for all indicators, normal (skewness = 0 kurtosis = 0), moderately nononrmal (skewness = 2, kurtosis = 7), and substantially nonnormal (skewness = 3, kurtosis = 21)) and found that nonnormality affected the standard errors of direct and in direct (i.e., the mediated effect) parameter estimates, and also a robust maximum likelihood estimation was performed best. Pituch and Stapleton (2008) compared varied statistical tests (e.g., the asymmetric distribution and bootstrap methods) for the medi ated effect with multilevel models under the degree of nonnormality (i.e., normal, moderately nonnormal, and substantially normal) and found that the bias corrected parametric percentile bootstrap method and the empirical M test (asymmetric distribution me thod) performed best. Thus, it is interesting to assess the PAGE 16 16 impact of nonnormality in longitudinal mediation studies using a parallel process latent growth model. In the proposed Monte Carlo simulation study, a parallel process latent growth model for medi ation analysis was investigated, and based on this model, normal and nonnormal data were generated and linearly combined. Relative bias of the parameter estimates and of the standard error estimates were used to evaluate the impact of nonnormality on estim ating and testing the direct and indirect (mediated) effects. Also, the statistical power and Type I error rates were computed for three methods for testing the significance of the mediated effect: order solution (Sobel, 1 982), the asymmetric distribution method ( i.e., asymmetric distribution of the product test; MacKinnon et al., 2002; MacKinnon et al., 2004; MacKinnon et al., 2007; Tofighi & MacKinnon, 2011) and the bias corrected bootstrap method (Carpenter & Bithell, 20 00; Manly, 2007). The test adjusted by order solution (Sobel, 1982) is one of the most commonly used methods to compute standard errors of the mediated effect and is available in structural equation modeling software such as EQS, LISREL, and M plus (Cheong et al ., 2003; Cheung & Lau, 2007; MacKinnon et al 2002; MacKinnon et al 2004). The asymmetric distribution method has performed well with normal and nonnormal data (MacKinnon et al., 2002; MacKinnon et al., 2004; Pituch et al., 2006; Pituch & Stapleton, 200 8). The bias corrected bootstrap method is available in M plus Moreover, recent longitudinal meditational studies using a parallel process latent growth model have used the asymmetric distribution method to test for the mediated effect (Cheong et al., 2003 ; Jager et al., 2007; Littlefield et al., 2010). In this simulation study, the design factors were the degree of nonnormality (i.e., normal ity moderate and PAGE 17 17 substantial nonnormality), mediated effect sizes (i.e zero, small, medium, and large mediated eff ect sizes), values of measured variables (i.e., 0.5 and 0.8), and sample sizes (i.e., 100, 200, 500, and 1000). The dependent variables were the relative bias of parameter estimates, relative bias of standard error estimates, and Type I error and statistical pow er rates. PAGE 18 18 CHAPTER 2 LITERATURE REVIEW Mediation analysis concerns a causal relation in which an independent variable causes a mediator, which in turn causes a dependent variable. Mediation analysis has been conducted in different disciplines under diffe in epidemiology (MacKinnon et al., 2002). There are many examples of mediation studies. In child psychology, sensitivity to critic ism is thought to mediate the relationship between theory of mind (i.e., ability to understand how mental states govern behaviors) and academic achievement (Lecce, Caputi, & Hughes, 2011). In business, job satisfaction is thought to be a mediator of the re lationship between leadership behavior and task performance (Liang et al., 2011). In public health, it was investigated if pain mediate s the relationship between depression and quality of life, and also if depression mediate s the relationship between pain and quality of life (Wong et al., 2010). One statistical issue in mediation analysis is that the mediated effect is not usually normally distributed and various methods have been developed to correctly test the mediated effect. There have been studies abou t mediation analysis (MacKinnon et al., 2002; MacKinnon et al., 2004; Pituch et al., 2006; Pituch and Stapleton, 2008) which compared different methods for testing the mediated effect, for example, conventional methods such as and tests and new methods such as the asymmetric distribution method and resampling methods. In this section, first, the concept of mediation analysis is presented based on the causal steps approach proposed by Judd and Kenny (1981) and Baron and Ke nny PAGE 19 19 (1986) and different methods for testing the mediated effect are discussed. Then, a longitudinal mediation model using a parallel process growth model is presented. Causal Steps Approach Judd and Kenny (1981) and Baron and Kenny (1986) presented the ca usal steps approach to mediation analysis although there is slight difference between two papers. The causal steps approach is focused on the conditions for mediation rather than estimating and testing the mediated effect (MacKinnon et al., 2002). Judd an d Kenny explained their causal steps approach using a concept of a causal chain which links the treatment at one end with the outcome variable at the other end, and multiple mediators can be inserted in this causal chain. The point of Judd and Kenny is how the treatment produces the outcome through mediators. Baron and Kenny allowed the independent variable to be either a measured variable or a categorical variable indicating a treatment and described the causal steps approach based on three variables (i.e. mediator, independent and dependent variables) instead of incorporating multiple mediators The steps introduced by Baron and Kenny are based on the following models (2 1) (2 2) (2 3 ) Equations 2 1 to 2 3 are depicted in Figure 2 1. The causal steps of Judd and Kenny (1981) are: Step 1) the independent variable ( ) affects the dependent variable ( ) (i.e., Path is significantly different from zero); Step 2) the independent varia ble ( ) affects the mediators ( s) (i.e., Path is significant), and the mediators ( s) affect the dependent variable ( ) (i.e., Path is PAGE 20 20 significant); and Step 3) when mediators ( s) are controlled, there is no effect of the independent variable ( ) on ( ) (i.e., is zero). The steps in Baron and Kenny (1986) are: Step 1) the independent variable significantly accounts for the mediator (path ); Step 2) the mediator significantly accounts for the dependent variable (path ); and Step 3) in Figure 2 1, when the paths and are controlled, the path decreases but is not equal to zero (partial mediation) or decreases to zero (complete mediation). According to MacKinnon et al. (2002), the main difference between two approaches is that Judd and Kenny emphasized the complete mediation (i.e., ) but Baron and Kenny argued that partial mediation is acceptable. MacKinnon et al. (2002) pointed out some limitations of the causal steps approach. First, it does not p rovide a direct estimate of the mediated effect of on (i.e., in Figure 2 1). Second, it is difficult to model and explain multiple mediators. Third, the first step in Baron and Kenny (1981) and Judd and Kenny (1986) (i.e., significant in Fig ure 2 1) is not correct when the mediated effect and direct effect with the opposite signs cancel out. Zhao et al. (2009) provided a decision tree for establishing and understanding mediation and non mediation, where they started from test ing the mediated effect instead of testing the path coefficient in Figure 2 1. Methods for Testing the Mediated Effect The mediated effect can be estimated in two different ways: 1) the difference in the estimates of independent variable ( ) coefficie nts ( ) and 2) the product of two parameter estimates Two estimates are equivalent when the dependent variable is continuous and ordinary least squares regression is used to estimate and in PAGE 21 21 Equations 2 2 and 2 3 above (Ma cKinnon, Warsi, & Dwyer, 1995; MacKinnon et al., 2004). Currently the product of two parameter estimates is most commonly used with order solution (MacKinnon et al., 2004). Varied methods have been developed for testing the mediated e ffect. MacKinnon et al. (2002) classified fourteen methods into three categories (i.e., first, the causal steps, second, difference in coefficients, and third, product of coefficients). Although the causal steps method is commonly used (MacKinnon et al., 2 002), this approach does not provide the estimate of the mediated effect such as and The causal steps approach is more focused on the conditions for mediation than on the statistical tests of the mediated effect (MacKinnon et al., 2 002). However, the causal steps approach can order solution (e.g., von Soest and Wichstrm, 2009) and the asymmetric distribution method (e.g., Littlefield, Sher, and Wood, 2010). Previous studies found that the bias corrected bootstrap and the asymmetric distribution method performed best in terms of Type I error and statistical power rates among the various methods (e.g., a series of tests in causal steps approach, test order solution, resampling meth ods such as bootstrap methods, j ackknife, and Monte Carlo, and the distribution of the product methods (MacKinnon et al., 2002; MacKinnon et al., 2004; Pituch et al., 2006; Pituch & Stapleton, 2008). Causal Steps Approach The c ausal steps approach used a series of t tests for testing the mediated effect. In the above Equations 2 1, 2 2, and 2 3 and Figure 2 1, Judd and Kenny (1981) and PAGE 22 22 Baron and Kenny ( t tests (i.e., ) and Judd and Kenny included one more test, for testing a complete mediation effect (MacKinnon, 2002). Product of Coefficients Methods MacKinnon et al. (2002) summarized and compared seve n methods of the product of coefficients. For example, three methods are based on adjustments to the test order exact solut unbiased solution), and other three methods are based on the distribution (i.e., distribution of the product of two variables, distribution of the two variable divided by their covariance, and asymmetric distribution of the produc ts of two variables), and the last method is the product of correlation coefficients (Bobko & Rieck, 1980). Among the order solution and the asymmetric distribution method) are common ly used. Statistical software packages such as EQS and LISREL (and M plus ) estimate the standard error of the order solution (MacKinnon et al., 2004). Also, SAS, R, and SPSS programs that implement the asymmetric distribu tion method have been prepared (MacKinnon et al., 2007). order solution Sobel (1982) pointed out that previous research ignored the distribution of the mediated effect and derived the asymptotic normal distribution of the mediated effe ct because the mediated effect is estimated from a nonlinear function of the coefficients and the small sample distribution of the effect does not follow the normal distribution. Based on the multivariate delta method which is a method to estimate the var iance of PAGE 23 23 functions of random variables that follow a multivariate normal distribution (MacKinnon et al., 1995), Sobel (1982; 1986) derived the estimator of the variance of the mediated effect shown in Equation 2 4 (2 4) where and are independent because Equations 2 2 and 2 3 comprise a recursive model without cross equation constraints and therefore the covariance of and (i.e., ) is zero. The square root of in Equation 2 4 is the standard error of the mediated effect. Equation 2 5 below is the confidence interval of the mediated effect ( 2 5) where is the value on the standard normal distribution that cuts off ( ) below it. Asymmetric distribution method A potential problem with using E quation 2 4 is that the distribution of is not necessarily normal and therefore is not the correct critical value for a confidence. The asymmetric distribut ion method attempts to address this problem by using the actual distribution of the Craig (1936) derived the distribution function and moment generating function of the product of two normally distributed random variables. These results are appli ed to the distribution of under the assumption that and are normally distributed which is true in small samples when and are normally distributed and more generally in large samples under commonly used estimation procedures su ch as ordinary least PAGE 24 24 squares and maximum likelihood. Craig provided results for when the two random variables are correlated and the special case in which they are not correlated. The latter results are relevant because, as noted earlier, Equations 2 2 and 2 3 comprise a recursive model without cross equation constraints and therefore the sampling correlation of and is zero. Let and be two normally distributed random variables with means and and variances and and correlation coefficient Define The mean and variance of the r andom variable are and (Meeker et al., 1981). W hen the above equations f or the mean a nd variance can be rewritten as (2 6) (2 7) The measure of the skewness is (2 8) and the measure of the kurtosis is (2 9) Based in Equation 2 8, the distribution of the product of two normal random variables is asymmetric either or which implies that there is no mediated effect. Thus, if and are in f act normally distributed and independent and PAGE 25 25 there is a mediated effect the sampling distribution of the mediated effect will be asymmetric. In the asymmetric distribution method, the estimates of and are t reated as population values (MacKinnon et al., 2004). Let and denote the estimated percentile points of the distribution of Standardized values of and are (2 10) These values of and are then used as critical values in the interval that is (2 11) (2 12) In Equations 2 11 and 2 12, is Aroi 47) second order exact solution (2 13) order solution by the additional term, order order solution is based on the first and second or der derivatives and the use of the second order derivatives lead to the additional term in Equation 2 11 which is omitted in Equation 2 4 (MacKinnon, 2008). Although Equation 2 11 is based on a more elaborate derivation than Equa tion 2 4, the additional term is typically trivial (Baron & Kenny, 1986; MacKinnon et al, 2002; MacKinnon et al, 2007; Preacher & Hayes, 2004) and the two methods usually result in very similar 1982) PAGE 26 26 first order solution (i.e., Equation 2 4) is the most commonly used standard error (MacKinnon et al, 2002), but the asymmetric distribution second order solution (i.e., Equation 2 11) (MacKinnon et al, 2004; Tofighi & Ma cKinnon, 2011). MacKinnon et al. (2 002; 2004) conducted extensive Monte Carlo simulations to compare various methods for testing the mediated effect and found that methods based on the distribution of the product had more accurate Type I error and highe r statistical power rates than normal theory based methods (e.g., t test and z (1982) method). Computer programs such as PRODCLIN (MacKinnon et al., 2007) and RMediation (Tofighi & MacKinnon, 2011) use the asymmetric distribution of the p roducts of two variables and they have been implemented in popular statistical software packages (e.g., SAS, SPSS, and R). Bootstrap Methods Previous methods (i.e., causal steps approach and the product of the coefficients methods) are focused on testing the mediated effect rather than estimating the mediated effect with a specific mediation model (e.g., OLS regression mediation models, multilevel mediation models, structural equation mediation models); however, bootstrap methods involve both estimating an d testing the mediated effect because bootstrap methods estimate and test the mediated effect based on the estimates from numerous bootstrap data sets (e.g., 1,000). The bootstrap is computer intensive method because it involves numerous repetitive computa tions. The basic idea of bootstrap (Bollen & Stime, 1990; Shrout & Bolger, 2002) is that when we have a data set with a sample size equal to we can take independent draws with replacement from the original data set. Then, we have bootstrap sampl es PAGE 27 27 (usually, ) and compute the mean and variance of a parameter estimate and then, construct confidence intervals with the estimates from bootstrap samples. Bollen and Stine (1990) introduced bootstrap methods in mediation analysis with regre ssion models, and compared parameter estimates, standard errors, and confidence intervals between the classical (e.g., test and and bootstrap methods. Shrout and Bolger (2002) recommended bootstrap methods in mediation analy sis when we have small to moderate sample sizes ( MacKinnon et al. (2004) conducted extensive simulation study with regression models to compare different methods for estimating and testing the mediated effect including different resampling methods such as bootstrap (e.g., percentile bootstrap, bias corrected bootstrap, bootstrap t, and bootstrap Q), jackknife, and Monte Carlo. Pituch et al. (2006) and Pituch and Stapleton (2008) compared more bootstrap methods (e.g., parametric percentile bootstrap bias corrected parametric percentile bootstrap, nonparametric percentile bootstrap, bias corrected nonparametric percentile bootstrap, stratified nonparametric percentile bootstrap, bias corrected stratified nonparametric percentile bootstrap) in their s imulations in multilevel mediation analysis. Previous simulation studies (MacKinnon et al., 2004; Pituch et al., 2006; Pituch & Stapleton, 2008) found that the bias corrected bootstrap method performed best among the bootstrap methods. However, in Monte Carlo Simulation studies, bootstrap methods are sometimes time consuming because of the extensive computations when statistical models and/or simulation conditions are complex (MacKinnon et al., 2004; Pituch et al., 2006; Pituch & Stapleton, 2008). Moreove r, software availability is important to use bootstrap methods. PAGE 28 28 SAS programming language can be used to implement various bootstrap methods and thus, the previous simulation studies in mediation analysis (e.g., MacKinnon et al., 2004; Pituch et al., 2006; Pituch & Stapleton, 2008) used SAS program to assess different bootstrap methods (e.g., percentile bootstrap, bias corrected bootstrap, bootstrap bootstrap and so on) with single level and multilevel mediation models. However, in multilevel mediation modeling, according to Pituch et al. (2006) and Pituch and Stapleton (2008), the iterated bootstrap methods (e.g., the iterated parametric perce ntile bootstrap and the iterated bias corrected bootstrap) and the parametric percentile residual bootstrap method are only available in the multilevel program MLwiN (Rasbash et al., 2002; Rasbash et al., 2005). Also, structural equation modeling programs implement bootstrap methods for testing the mediated effect, for example, AMOS (Arbuckle & Wothke, 1999) includes the bias corrected bootstrap method, and M plus (Muthn & Muthn, 2010) implements the percentile bootstrap and bias corrected bootstrap method s. In this dissertation M plus was used to implement the bias corrected confidence interval, which was designed as an improvement on the percentile confidence interval. Therefore the percentile and bias corrected methods are presented in the following. In a ddition the accelerated bias corrected percentile confidence interval is briefly discussed. Percentile confidence intervals In percentile method, bootstrap samples of size where is the total sample size are drawn at random with replacem ent from the available sample. Typically is 1000 or larger. Then the parameters are estimated from each bootstrap sample and the PAGE 29 29 resulting estimates are used to obtain percentile confidence intervals. In the case of the indirect effect, estimates of would be obtained. Let be an estimate of the percentile of the bootstrap distribution and let be an estimate of the percentile. Then the bootstrap percentile confidence interval is (2 14) According to Manly (2007) the method is b ased on the assumption that there is a monotonic increasing function such that is normally distributed with mean and standard deviation 1. Based on the assumption, we can construct Equation 2 15 (2 15) which can be rearranged as (2 1 6 ) The percentile method is a monotonically increasing functi on of Therefore, if we order the sample estimates from smallest to largest, we do not need to know and can find the values of lower and upper limits and ) in a sampling distribution of which can be written as (2 1 7 ) Now Equation 2 1 7 can be rewritten with bootstrap upper and lower limits and respective ly bootstrap percentile confidence limits can be written as Equation 2 14. The advantage of this method is simplicity because we do not need to estimate PAGE 30 30 however, the di sadvantage is that may not exist, which causes a substantial coverage error (Carpenter & Bithell, 2000). Bias corrected percentile confidence intervals This method assumes that there is a monotonic increasing function exists such that is normally distributed with mean and standard deviation 1 (Manly, 2007). The quantity corrects the bias, and thus When is symmetric we can construct the following Equation 2 18 (Manly, 2007) (2 18 ) W here follows the standardized normal distri bution and can be rearranged as (2 19 ) Let be the probability less than in a bootstrap distribution of and then is the following Also, the lower confidence limit is Let be the probability less than this value in the bootstrap distribution of and then is the following: PAGE 31 31 The quantity is estimated as the score value of the proportion of the bootstrap samples below from the original sample (MacKinnon et al., 2004 ). For example, if 600 out of 1,000 bootstrap estimates of the indirect effect are below the original estimate then is 0.25 which is the value of the proportion of 0.6 and if 500 out of 1,000 bootstrap estimates of the in direct effect are below the estimate that is, the median of the bootstrap distribution is equal to the original sample estimate then is 0 Let be an estimate of the percentile of the bootstrap distri bution and let be an estimate of the percentile of the bootstrap distribution. Then the bootstrap percentile confidence interval is ( 2 20) This method adjusts for an asymmetric sampling distribution, which is an improvement to the percentile method; however, there is still substantial coverage error because does not commonly exi st (Carpenter & Bithell, 2000). This method is also variable in M plus which is the most popular structural equation modeling software. Accelerated bias corrected percentile intervals This method assumes that is normally distributed with mean and standard deviation where and are constants (Manly, 2007). The constant is known as the acceleration constant and is usually denoted by The symbol is use d in this dissertation to avoid confusion with the path coefficient Thus, is normally distributed with mean and standard deviation This is a less restrictive assumption than the assumpt ions made in the PAGE 32 32 percentile and bias correct percentile method. The coverage error of this method is smaller than the percentile and bias corrected percentile methods; however, the coverage error increases as (Carpenter & Bithell, 2000). The complicat ion in using this method is estimating the constant It can be estimated by using the jackknife, but this would require an additional estimations of the model, where is the sample size, for each replication of the simulation. Therefore, the accel erated bias corrected percentile method was not investigated in this dissertation. Latent Growth Modeling Latent growth models in structural equation modeling have been popularly used in longitudinal data analysis (Ferrer, Hamagami, & McArdle, 2004; Flora 2008; Leite, 2007; Leite & Stapleton, 2011). According to Bollen and Curran (2006), current latent growth models were first proposed by Meredith and Tisak (1984; 1990), which were based on the previous growth models and exploratory factor analytic models (e.g., Rao, 1958; Tucker, 1958; 1966). Meredith and Tisak (1990) pointed out that standard repeated measures ANOVA or MANOVA models are the special cases of their growth model (i.e., latent growth model) and also their approach can simultaneously model an d test both individual differences and different types of growth trends, which is the main advantage over traditional growth models. Researchers have presented different types of latent growth modeling such as multivariate latent growth models (e.g., McArd le, 1988; Leite, 2007), multilevel latent growth models (e.g., Muthn, 1997), growth mixture models (e.g., Muthn & Shedden, 1999; Bauer & Curran, 2003), latent growth models for mediation analysis (Cheong et al., 2003; Cheong, 2011; von Soest & Hagtvet, 2 011) and so on. PAGE 33 33 Univariate Latent Growth Model The univariate unconditional latent growth model ( Bollen & Curran, 2004; 2005) is (2 2 1 ) where is the observed variable for the th individual at time and are the random intercept and random slope for an individual respectively. The matrix represents factor loadings for the random slope at time and can allow linear and nonlinear growth trends by constraining or estimating the values of and also for a specific case such as a linear gr owth trend, the values of indicates the elapsed time from the initial time point (reference point) to (Bollen & Curran, 2004; Shi, 2009). The variable is an error for the th individual at time Bollen and Curran (2004; 2005) presen ted the assumptions for the latent growth model: for all and for all and for each and for all and and for The individual differences in the intercept and slope are modeled as follows (2 22 ) (2 23 ) where and are the mean of the intercept and slope factors, respectively. The variables and are errors of the inte rcept and slope factors, and are uncorrelated with but and can be correlated. Parallel Process Latent Growth Model (Bivariate Latent Growth Model) The univariate latent growth model is focused on the growth of one outcome var iable. However, the parallel process latent growth model (bivariate latent growth PAGE 34 34 model, Bollen & Curran, 2004; 2005) is a model for the growth and interrelation of two outcome variables. The models for the two outcome variables in parallel process latent growth model (B ollen & Curran, 2004; 2005) are (2 24 ) ( 2 25 ) where each latent growth equation has the same assumptions as those for Equation 2 2 1 and errors in Equations 2 24 and 2 25 are uncorrelated. The random intercepts and slopes for are (2 26 ) (2 27 ) and those for are (2 28 ) (2 29 ) where the errors of random intercepts and slopes (i.e., ) in Equations 2 26 2 27 2 28 and 2 29 can be correlated. We can model the structural relationship among the random intercepts and slopes of two variables of and For example, if is measured earlier than the initial status (i.e., ) and growth rate (i.e., ) of can affect the growth ra te (i.e., ) of a relationship that can be represented by Equation 2 3 0 : (2 30 ) PAGE 35 35 where a positive indicates that persons receiving higher test scores at the initial measurement time point for will grow faster on and a positive indicates that the person who is growing faster on will also grow faster on Parallel Process Latent Growth Model for Mediation Analysis Mediation analysis is usually based on the three variables (i.e., the mediator, independent and dependent variables) to assess if the independent variable causes the mediator, which in turn causes the dependent variable. Accordingly, the parallel process latent growth model for mediation analysis consists of the independent variable, mediator, and dependent variable. With the parallel process latent growth model, we can test different types of the mediation process and so on). Recent longitudinal mediation studies using a parallel process latent growth model (e.g., Cheong et al., 2003; Jagers et al., 2007; Lit tlefield et al., 2010 ) have investigated whether the impact of an independent variable on the growth rate of the dependent variable is mediated through the growth rate of the mediator ( from Equations 2 3 1 2 3 2 2 3 3 and 2 34 below) Equations 2 30 and 2 31 represent the measurement models for the mediator and dependent variable. If we include the independent variable to explain the structural relationship among random intercepts and slopes, the parallel process latent growth model can be repr esented by following equations (2 3 1 ) (2 3 2 ) (2 3 3 ) PAGE 36 36 (2 34 ) These equations are depicted in Figure 2 2. Figure 2 2 (see Cheong et al., 2003; Cheong, 2011) shows how a longitudinal mediation process (i.e., ) occurs. In Figure 2 2 and Equations 2 3 1 to 2 3 4, path coefficients and indicate the impact of on the initial status (i.e., random intercepts) of the mediator and dependent variable respectively. In experimental design, can be a dichotomous variable indicating treatment and control conditions, and and are zero when individuals are randomly assigned to each lev el of and the initial measurement occasion precedes the onset of the treatment. Path coefficient (or ) represent the effects of the initial status of (or ) on the growth rates of (or ). The path coefficient indicates the impact of on the growth rate of controlling for the random intercepts of in the Equation 2 32 The path coefficient represents the impact of the growth rate of on the growth rate of controlling for the effects of the and the random interce pts of in Equation 2 34 and the product of two path coefficients (i.e., ) is the mediated effect. Path coefficient is the direct effect of on controlling random intercept and slopes of (see Cheong, 2011; von Soest, T. & Hagtvet, K. A., 2011). Monte Carlo simulation study t he design factors were the effect size of mediated effect (i.e., small, medium, and large effect sizes), value (i.e., 0.5 and 0.8), number of measurement occasions (i.e., 3 and 5 occasions ), and sample size (i.e., 100, 200, 500, 1000, 2000 and 5000) Therefore, Cheong considered 72 conditions The dependent variables were relative biases for the mediated effect and standard error estimates and the empirical power rates. Cheong estimated sta ndard PAGE 37 37 errors using order solution (i.e., Equation 2 4) and calculated empirical power rates using three methods: Sobel s first order solution (i.e., Equations 2 4 and 2 5 ) the asymmetric distribution confidence interval (i.e., Equation s 2 4 2 11 and 2 12 ) and the joint significant test (i.e., two tests: and ). The following are Cheong s findings: The accuracy of the mediated effect and standard error estimates increased as the sample size, and number of measurement occasions increased. However, the increase in effect size of the mediated effect did not improve the accuracy of the estimates in many conditions. Empirical power rates increased as all the design factors increased. The asymmet ric distribution confidence interval and the joint significant test produced equivalent empirical power rates t hat were higher than those from Sobel s first order solution. Significance of This Study In longitudinal mediation analysis, a parallel process l atent growth model can be useful to test the longitudinal mediated effect. Previous longitudinal research has used a parallel process latent growth model to investigate whether the growth rate of the mediator mediated the relationship between the independe nt variable (e.g., intervention) and the growth rate of the dependent variable (e.g., Cheong et al., 2003; Jagers et al., 2007; Littlefield et al. 2010; von Soest & Wichstrm, 2009). Perhaps these studies assumed normal data, which is rarely met in practi ce. Furthermore, the distribution or test of the mediated effect has been one of the main issues in mediation analysis; however, there is not much information about the effects of nonnormality. Thus, it would be informative to investigate the effects of no nnormality using a parallel process latent growth model. PAGE 38 38 Research Questions This study aims to assess the effects of nonnormality on estimating and testing the mediated effect using a parallel process latent growth model. The research questions of this stu dy are: 1. What are the effects of nonnormality (depende nt variable or mediator) on the parameter estimates and standard error estimates of (mediated effect)? 2. What are the effects of nonnormality on the statistical power and Type I error rates for the test of ? 3. Among corrected bootstrap, which procedures provide adequate Type I error rates for the test of ? 4. Among nd the bias corrected bootstrap, which p rocedures provide the best powe r for the test of ? PAGE 39 39 Figure 2 1. Path Diagram for mediation analysis Figure 2 2. A parallel process latent growth model in longitudinal mediation analysis PAGE 40 40 CHAPTER 3 METHOD Population Values Cheong (2011) investigated the accuracy of the parameter and standard error estimates and the statistical power of the mediated effect using a parallel process latent growth model. s study extended by assessing the effects of nonnormal data. Thus, the population values in this study, such as effect size of the mediated effect, regression coefficient, the number of time points, sample size, and of measured variables, were primarily selected from Cheong (2011). However, Cheong (2011) did not specify some population values (e.g., means and variances of the latent and observed variables) and thus those values were selected from Thoemmes, MacKi nnon, and Reise (2010) where the Monte Carlo simulation study using a parallel process latent growth model was clearly demonstrated by a M plus program for the simulation study. This study includes an independent variable and mediator and dependent variable s which are measured across five time points and each have two latent variables, the intercept and the slope. The independent variable is a dichotomous variable, representing an experimental and control treatment. The mediator and dependent variables are q uantitative variables. Equations 2 24 and 2 25 represent measurement models and Equations 2 3 1 to 2 34 represent structural models for the parallel process latent growth model. In Equations 2 3 1 to 2 34 the mean and variance of the independent variable ( ) is 0.5 and 0.25 respectively. In Equations 2 3 1 and 2 3 3 and the mean and variance of and are zero and one respectively. In Equation s 2 3 2 and 2 34 and the mean and variance of and PAGE 41 41 are 0 and 0.1. The four variables and are uncorrelated (i.e., the correlations between residuals for intercept and slope factors for the mediator and dependent variable are all zero). Cheong (2011) pointed out that in Equations 2 3 1 and 2 3 2 and in Figure 2 2, indicates the random assignment in experiment design. In Equations 2 24 and 2 25 the mean of and are all zero and each variance of and are selected to yield two different values of (i.e., 0.5 and 0.8). According to Tisak and Tisak (1996), longitudinal reliability at e ach measurement occasion is (3 1) where is the variance of the intercept fact or, is the variance of the slope factor, is the covariance between the intercept and slope factors, is the variance of error at time point Thus, for the mediator and dependent variable, we can find the error variance at each time point with the population values of the intercept and slope variances and covariance so that at each time point is either 0.5 or 0.8 (i.e., 1) if and then ; 2) if and then ). The parameters that do no t vary over the conditions in the study are recorded in Table 3 1. Data Generation In this study, data were generated by R version 2.12 (R Development Core Team, 2010). Three types of random variables (e.g., normal, Bernoulli, and chi squared PAGE 42 42 random variab les) were generated and then linearly combined. Equations 2 24 2 25 2 3 1 2 3 2 2 3 3 and 2 34 and Figure 2 2 present a parallel process latent growth model for mediation, where the independent variable ( ) is Bernoulli random variable which represents the treatment and control groups (i.e., 0 and 1 values), and the errors (e.g., ) in the latent and observed variables are norm al or chi squared random distributions with df = 1 or 2 which represent the degree of nonnormality (i.e., normality, moderate and substantial nonnormality). As shown in Equations 2 24 2 25 2 3 1 2 3 2 2 3 3 and 2 34 the independent variable (Bernoulli r andom variable) and errors (normal and chi squared random distributions) are linearly combined with means (e.g., and ) and regression coefficients (e.g., and ). Data Analysis and the Dependent Variables of this Study Simulated data were analyzed by using the M plus software. Th e parameter and standard error estimates of were evaluated by relative bias (Hoogland & Boomsma, 1998) (3 2) (3 3) In Equation 3 2, is the population parameter value, is the parameter estimate and is the mean of across a certain number of replications in simulation study. In Equation 2 2 1 is the average of standard error estimates (Shi, 2009) across replications and is the empirical standard error which is calculated by the standard deviation of across replications. If and t he PAGE 43 43 parameter and standard error estimates are acceptable respectively (Hoogland &Boomsma, 1998). The Type I error and statistical power order s olution, the asymmetric distribution method and bias corrected bootstrap (MacKinn on et al., 2002; MacKinnon et al., 2007; Tofighi & MacKinnon, 2011) were evaluate (1982) first order s olution was computed by R software based on the results (i.e., and ) from M plus analyses, the confidence intervals of the bias corrected bootstrap were obtained from M plus and the confidence intervals of the asymmetric distribution method were estimated through the R package of RMediation (Tofighi & MacKinnon, 2011) order solution is the most popular method, and the asymmetric distribution method has been used in recent longitudinal mediation analysis using a parallel process latent growth model (Cheong et al., 2003; Jager et al., 2007; Littl efield et al., 2010), and also the bias corrected bootstrap ha s been identified as the best method in previous studies (e.g., Mackinnon et al., 2004; Pituch et al, 2006; Pituch & Stapleton, 2008). Therefore, the depende nt variables of this study were relative bias of parameter and standard error estimates, empirical Type I error, and statistical power rates. Relative biases were computed based on the analyses of 5,000 replications of each condition and Type I error and p ower rates were estimated based on 1,000 replications of each condition. Design Factors The design factors of this study were based on Cheong (2011). The extended normally distributed variables. The total number of PAGE 44 44 conditions in the study wa s 240, i.e., sample sizes (4 conditions) the combination of and values (10 conditions) the degree of nonnormality of (3 conditions) values (2 conditions). The number of occasions of measurement occasions was held con stant at five. Sample Sizes and Mediated Effect Sizes Four different sample sizes of 100, 200, 500, and 1000 were selected based on Cheong (2011) who claimed that these sample sizes are commonly used in social science and prevention studies. In this study the direct effect (i.e., in Equation 2 34 ) is 0.25 and the mediated effect expressed as a percentage of the total effect is computed by For the statistical power and Type I error rates of the mediated effect, nine combinations of and were assessed: 1. for a small mediated effect size ( ) (Cheong, 2011), 2. for a medium mediated effect size ( ) (Cheong, 2011), 3. for a large mediated effect size ( ) (Ch eong, 2011), 4. for no mediated effect ( ), 5. for no mediated effect ( ), 6. for no mediated effect ( ), 7. for no mediated effect ( ), 8. for no mediated effect ( ), 9. for no mediated effect ( ) 10. for no mediated effect ( ). In the above combinations, the negative direct effect ( is considered because when (mediated effect) is positive (e.g., or ), the distribut ion of is positively skewed, and when (mediated effect) is negative (e.g., or ), the distribution of is negatively skewed (Craig, 1936; Pituch et al, 2006). Thus, it is interesting to assess the combined effect of the direction of the skewness of nonnormal data (e.g., positive or negative skewness of data) with PAGE 45 45 the direction of the skewness of the (e.g., the positive or negative mediated effect). When both and are zero, Type I error rates of the mediated e ffect tend to be lower than the nominal Type I error rates, and thus combinations when one of them is not zero such as (0.18, 0) and (0.31, 0) should be investigated (Pituch & Stapleton, 2008). Degree of Nonnormality Let and denote the third and fourth central moments. According to Muthn and Asparouhov (2002), in structural equation modeling, nonnormal data with skewness values of about 2 and kurtosis values between 3 and 4 are commonly found. Thus, the proposed study includes normal distributions and two degrees of nonnormality: moderate and substantial nonnormality (e.g., Finch et al., 1997; Lei & Lomax, 2005; Pituch & Stapleton, 2008). In the struct ural equation modeling framework, based on Monte Carlo approach, Finch et al. (1997) studied the impact of nonnormality on the direct and indirect (mediated) effects with three latent variables, each with three indicators, and thus the distributions of the nine indicators were normal (skewness = 0 kurtosis = 0), moderately nononrmal (skewness = 2, kurtosis = 7) and severely nonnormal (skewness = 3, kurtosis = 21). Lei and Lomax (2005) investigated the impact of nonnormality on the parameter estimates and da ta model fit indexes, where they generated nonnormal data with varied skewness and kurtosis values, that is, the distribution of six observed variables are normal (i.e., all the skewness and kurtosis are zero), slightly nonnormal (i.e., the range of skewne ss is [0, 0.44] and the range of kurtosis is [0, 1.2]), and severely nonnormal (i.e., the range of skewness is [0, 1.76] and PAGE 46 46 the range of kurtosis is [0, 3.9]). In a multilevel modeling framework, Pituch and Stapleton (2008) compared various statistical te sts (e.g., the asymmetric distribution method and bootstrap methods) for the multilevel mediated effect under various degrees of nonnormality: The authors generated errors in the level one and two equations with normal and chi squared distributions with 1 and 2 degree of freedoms. The chi squared distribution with 1 degree of freedom modeled severe nonnormality (i.e., the skewness and kurtosis are 2.828427 and 12 respectively) and the chi squared distribution with 2 degrees of freedom modeled moderate nonno rmality (i.e., the skewness and kurtosis are 2 and 6 respectively). In the measurement model in Lei and Lomax (2005), each latent variable was measured by three observed variables, and in the structural model two latent variables were a function (linear c ombination) of other latent variables. Also, in Pituch and Stapleton (2008), some variables in the level one equation were linear combination of other variables in level one and level two equations. Therefore, in these simulation studies, the skewness and kurtosis of the combined variables could be changed by the coefficients or variances of the other variables. Thus, it would be informative to provide the population skewness and kurtosis values of all the variables to see the degree of nonnormality in the data. Skewness and kurtosis of all latent and observed variables are provided in the following section. In this study, based on Pituch and Stapleton (2008), nonnormal distribution were generated using chi squared distributions with 1 and 2 degree of freedo ms; Table 3 2 summarizes how the distributions were generated with normal and chi squared distributions. PAGE 47 47 Population Skewness and Kurtosis Values Population skewness and kurtosis values can be defined as a function of cumulants which are characteristics of distributions like moments. Cumulants are denoted by where is the order of the cumulant. Skewness and kurtosis are defined by (3 4) (3 5) The properties of cumulants are: If and if and thus, changing the means of the variables (e.g., the independent variable and errors in the latent and observed variables in this study) does not affect the population skewness and kurtosis values. The cumulant for is which is related to the variance transformation of the variables (e.g., errors for the latent variables and mediator and dependent variable in this study). Thus, when we compute population skewness and kurtosis values of a variable which is a linea r combination of the other variables, we need to consider how each variable is transformed to have a specific variance. If and are independent random variables, Based on the above three properties of cumulants and Equations 3 4 and 3 5, we can compute the population skewness and kurtosis values of all the variables in this study except for the slope factor of the dependent variable and dependent variables at time 2, 3, 4, 5 which are influenced by the slope fact or of the dependent variable. In Equation 2 40, the slope factor of the dependent variable ( is a function of the PAGE 48 48 intercept factor of the mediator ( ), independent variable ( and the slope factor of the mediator ( ) which also is the function of the independent variable ( Thus, the independence among var iables (predictors) is not me t in Equation 2 34 which is related to the multicollinearity issue in mediation analysis. Table 3 3 shows the second, third, and fourth order cumulants of normal, Bernoulli, and chi square random variables. Table 3 4 shows po pulation skewness and kurtosis values for latent and observed variables which were computed from the cumulants in Table 3 3, the properties of cumulants, and Equations 3 4 and 3 5. As noted previously, the cumulants of the slope factor of the dependent var iable and the dependent variables at time 2, 3, 4, 5 cannot be computed because the intercept factor of the mediator ( ) and independent variable ( are not independent. However, estimates of skewness and kurtosis can be calculated based on a large sample size Under normality (i.e., all the variables except for ), there was almost no impact of which follows the Bernoulli distribution, and the population skewness and kurtosis values of all variables except for the independent variable are zero and the cumulants, skewness and kurtosis values under normality are not presented in Table 3 4. Simulation Program Check The sample kurtosis and skewness values can be estimated b y using the following equations (3 6) (3 7) PAGE 49 49 where is the sample size. With a very large sample size such as we can estimate the population skewness and kurtosis (Muthn & Asparouhov, 2002). In this study, using the Equations 3 6 and 3 7, aver age skewness and kurtosis values were estimated with and 100 replications using R program. Then, the relative biases in Equation 3 2 were computed with the population values in Table 3 2. As expected, the relative biases were almost zero. Ther efore, the data generation by R program was correct. PAGE 50 50 Table 3 1. Constant parameter and regression coefficient values over conditions Parameter Mediator Dependent variable Intercept mean Intercept error mean Intercept error variance Slope mean Slope error mean Slope error variance Error variance at time 1 Error variance at time 2 Error variance at time 3 Error variance at time 4 Error variance at time 5 Error variance at time 1 Error variance at time 2 Error variance at time 3 Error variance at time 4 Error variance at time 5 Regression coefficients and the mean and variance of the independent variable In E quation 2 3 1 In E quation 2 3 2 In E quation 2 3 3 In E quation 2 34 Independent variable and PAGE 51 51 Table 3 2. The generation of the degree of nonnormality Normality Moderately Nonnormality Substantially Nonnormality (i.e., in Equations 2 3 1 to 2 34 is generated to follow a Bernoulli distribution with p = 0.5). her variables are generated to follow normal distributions: 1) Errors in measurement equations (i.e., in Equation 2 24 and in Equation 2 25 ). 2) Errors in structural equations ( in Equation 2 3 1 in Equation 2 3 2 in Equation 2 3 3 in Equation 2 34 ). dependent variable are the linear combination of the above variables. (i.e., in Equations 2 3 1 to 2 34 is generated to follow a Berno ulli distribution with p = 0.5). generated to follow chi squared distribution with df = 2, and then transformed to have the mean of zero and specific variance at each time point: 1) Errors in measurement equations (i.e., in Equation 2 24 and in Equation 2 25 ). 2) Errors in structural equations ( in Equation 2 3 1 in Equation 2 3 2 in Equation 2 3 3 in Equation 2 34 ). dependent variable are t he linear combination of the above variables. (i.e., in Equations 2 3 1 to 2 34 is generated to follow a Bernoulli distribution with p = 0.5). generated to follow chi squared distribution with df=1, an d then transformed to have the mean of zero and specific variance at each time point: 1) Errors in measurement equations (i.e., in Equation 2 24 and in Equation 2 25 ). 2) Errors in structural equations ( in Equation 2 3 1 in Equation 2 3 2 in Equation 2 3 3 in Equation 2 34 ). dependent variable are the linear combination of the above variables. Table 3 3. Cumulants of Normal, Bernoulli, and Chi square distribut ions Normal distribution Bernoulli distribution Chi squared distribution where, in this study. where, df = 1 (substantially nonnormal) and df = 2 (moderately nonnormal) in this study. PAGE 52 52 Table 3 4. Population skewness, and kurtosis values of latent and observed variables S ubstantial nonnormality (Chi Squared with 1 degree of freedom) Moderate nonnormality (Chi Squared with 2 degree of freedom) Mediated effect size Small Medium Large No Small Medium Large No Intercepts of mediator and DV Skewness 2.828 2.828 2.828 2.828 2 2 2 2 Kurtosis 12 12 12 12 6 6 6 6 Slo pe of mediator Skewness 2.27 4 1.88 1 1.23 3 2.529 1.60 8 1.3 30 0.87 2 1.788 Kurtosis 8.680 6.683 3.55 3 10.02 5 4.335 3.309 1.631 5.008 Slope of DV 2) Skewness 1.92 6 1.56 9 1.151 1.810 1.36 3 1.10 9 0.81 5 1.28 2 Kurtosis 6.880 4.999 2.99 3 6.027 3.427 2.45 8 1.3 97 3.01 7 Mediator at time 1 Skewness 2.27 7 2.27 7 2.27 7 2.27 7 1.61 1.61 1.61 1.61 Kurtosis 8.16 8.16 8.16 8.16 4.08 4.08 4.08 4.08 Mediator at time 2 Skewness 2.024 1.990 1.901 2.042 1.431 1.407 1.344 1.444 Kurtosis 6.713 6.56 2 6.170 6.792 3.35 7 3.28 1 3.083 3.396 Mediator at time 3 Skewness 1.68 8 1.603 1.404 1.73 4 1.193 1.13 4 0.99 3 1.226 Kurtosis 4.639 4.326 3.595 4.809 2.319 2.16 1 1.780 2.40 5 Mediator at time 4 Skewness 1.54 5 1.42 3 1.159 1.613 1.092 1.00 6 0.8 20 1.141 Kurtosis 3.806 3.398 2.518 4 .036 1.902 1.69 3 1.2 20 2.018 Mediator at time 5 Skewness 1.534 1.382 1.073 1.622 1.08 5 0.977 0.7 60 1.147 Kurtosis 3.836 3.320 2.268 4.136 1.91 7 1.6 50 1.076 2.068 DV at time 1 Skewness 2.27 7 2.27 7 2.27 7 2.27 7 1.61 1.61 1.61 1.61 Kurtosis 8.16 8.16 8. 16 8.16 4.08 4.08 4.08 4.08 DV at time 2 Skewness 2.007 1.98 1 1.91 3 2.01 1 1.420 1.399 1.35 4 1.42 2 Kurtosis 6.66 3 6.574 6.27 4 6.70 7 3.335 3.27 6 3.146 3.353 DV at time 3 Skewness 1.637 1.570 1.433 1.64 2 1.158 1.109 1.01 4 1.16 1 Kurtosis 4.505 4.29 3 3.780 4.561 2.252 2.13 4 1.879 2.280 DV at time 4 Skewness 1.45 6 1.35 5 1.18 1 1.450 1.030 0.957 0.835 1.025 Kurtosis 3.529 3.197 2.59 8 3.50 8 1.76 5 1.58 5 1.27 6 1.754 DV at time 5 Skewness 1.407 1.27 3 1.074 1.384 0.99 6 0. 900 0.7 60 0.97 9 Kurtosis 3.38 1 2.880 2.192 3.225 1.68 6 1.428 1.0 60 1.612 Note: 1) DV is the dependent variable. 2) for the s mall mediated effect size ; for the medium mediated effect size; for large mediated effect; for no mediated effect. 3 ) indicates the estimate of skewnesss and kurtosis from 1,000,000 sam ple size and 100 Replications because independence assumption is not met (i.e., ). PAGE 53 53 Table 3 5. Relative bias of skewness and kurtosis with t he mean estimates of skewness and kurtosis with 1,000,000 sample s ize and 100 replications First, second, third, and fourth cumulants, skewness, and kurtosis Substantial nonnormality: Chi Squared Distribution with 1 Degree of Freedom Moderate nonnormality: Chi Squared Distribution with 2 Degree of Freedom Small Medium Large No Small Medium Large No Intercept Mediator Skewness 0.00 1 0.000 0.000 0 0.00 1 0.000 0.000 0.000 Kurtosis 0.00 2 0.000 0.00 1 0.000 0.003 0.000 0.00 3 0.001 Intercept DV 1) Skewness 0.000 0.00 1 0.00 1 0.000 0.000 0.00 1 0.000 0. 000 Kurtosis 0.00 1 0.00 1 0.002 0.00 2 0.00 1 0.00 3 0.00 1 0.00 1 Slope Mediator Skewness 0.000 0.00 1 0.00 1 0.000 0.000 0.001 0.00 1 0.000 Kurtosis 0.00 1 0.001 0.001 0.000 0.000 0.00 4 0.003 0.00 1 Mediator at Time 1 Skewness 0.00 1 0.000 0.000 0 0.001 0 0.00 1 0.000 Kurtosis 0.00 3 0.00 1 0.000 0.000 0.00 4 0.000 0.003 0.00 2 Mediator at Time 2 Skewness 0.000 0.000 0.000 0.000 0.00 1 0.000 0.00 1 0.00 3 Kurtosis 0.001 0.000 0.00 1 0.001 0.00 3 0.000 0.003 0.0 0 2 Mediator at Time 3 Skewness 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Kurtosis 0.00 1 0.00 1 0.001 0.00 2 0.00 2 0.001 0.002 0.00 1 Mediator at Time 4 Skewness 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 Kurtosis 0.00 1 0.000 0.000 0.001 0.001 0.001 0.00 1 0.00 2 Mediator at Time 5 Skewness 0.00 1 0.00 1 0.000 0.000 0.000 0.00 1 0.00 1 0.000 Kurtosis 0.001 0.001 0.00 1 0.00 2 0.000 0.00 3 0.00 3 0.00 1 DV at Time 1 Skewness 0.000 0.00 1 0.000 0.000 0 .000 0.001 0.000 0.000 Kurtosis 0.000 0.001 0.00 2 0.00 2 0.00 1 0.00 4 0.00 1 0.000 Note: 1) DV is the dependent variable. 2) for the small mediated effect size; for the medium mediated effect size; for large mediated effect; for no mediated effect. 3 ) Bold indicates greater relative bias (> 0.05) PAGE 54 54 CHAPTER 4 RESULTS This chapter consists of three sections based on the results of two rounds of simulations. The first section provides mean and median relative biases to evaluate the accuracy of the estimated mediated effects and standard errors. Relative biases were computed based on the analyses of 5,000 replications of each condition. It was found that in some conditions with small sam ple sizes of 100 and 200, mean relative biases were misleading because of outlying estimates, and thus median relative biases as well as mean relative biases are provided for all sample sizes. The second and third sections present empirical Type I error an d power rates, based on 1,000 replications of each condition. Three methods were used to compute empirical Type I error and power order solution, the asymmetric distribution, and the bias corrected bootstrap Accuracy of Estimates of t he Mediated Effect and Standard Error Tables 4 1 to 4 10 show the median and mean relative biases of the mediated effect estimates and standard error estimates based on 5,000 replications of order solution with robust standard error est imates ( and ) which were provided by using the MLR (Maximum Likelihood with Robust standard errors) estimator option in M plus program. Results for the Small Mediated Effect Size Table 4 1 shows relative biases when the mediated effec t size was small. Estimates in most conditions were negatively biased. The median relative biases PAGE 55 55 indicated the mediated effect estimates were seriously biased (median relative bias ) when was 0.5 and the sample size was 500 or smaller and when was 0.8 and the sample size was 200 or smaller. The relative biases for the standard error were larger than 0.1 when was 0.5 and the sample sizes was 200 or smaller, and under substantial nonnormality conditions when was 0.5 and the s ample size was 500 and when was 0.8 and the sample size was 200 or smaller. Table 4 1 shows that bias of estimates tended to decrease as the sample size and increased. Nonnormality did not cause serious bias except in two conditions: when was 0.8 and sample sizes were 100 and 200. The median and mean relative biases for standard error estimates were very different (i.e., negative median and positive mean values) when was 0.5 and sample sizes were 100 and 200 because of outliers. Results for the Medium Mediated Effect Size Table 4 2 shows median and mean relative biases when the mediated effect size was medium. The mediated effect estimates were seriously biased (median relative bias > 0.05) when was 0.50 or 0.80 and the sam ple size was 100. Also, the standard error estimates were more seriously biased when was .50 than when it was .80, whereas the mediated effect estimates were not affected by different values. Nonnormality did not result in more bias in estima ting the mediated effect and standard error in most conditions, but the relative bias of standard error estimates increased as departure from normality increased in two conditions in which was 0.5 and the sample size was 500 and when was 0.8 and the sample size was 100. Most of the PAGE 56 56 median relative biases for the standard error were negative but some mean relative biases were positive; these differences were due to positive outliers. Results for the Large Mediated Effect Size Table 4 3 shows t he relative biases of the mediated effect and standard error estimates for the large mediated effect size. Median relative biases indicated that estimated mediated effects were not biased and there were almost no effects of or departures from normality on the bias of the mediated effect estimates. However, the bias and relative bi a s of the standard error estimates was similar to those for the small and medium mediated effects. Nonnormality did not affect the relative bi as of the mediated effect. But for the standard error estimates, substantial nonnormality resulted in the large median relative bias ( ) when was 0.5 and the sample size was 500 and when was 0.8 and the sample size was 200. When the effect size was large, standard error estimates were negatively biased in most conditions. Comparison of median and mean relative bias for the standard error estimates indicates outliers which were detected when was 0.5 and the sample size was 200 or smal ler. Results When the Mediated Effect Size Is Zero Tables 4 4 to 4 10 show the median and mean mediated effect estimates. In Tables 4 4 to 4 10, the median and mean of the mediated effect estimates were presented instead of the relative bias because the population mediated effect is zero. Tables 4 4 to 4 10 also show the median and mean relative biases of the standard error estimates when there was no mediated effect. Tables 4 4 to 4 6 show the results of different values of path (= 0.18, 0.31, and 0.52) when path was zero, and Tables 4 7 PAGE 57 57 to 4 9 show the results of different values of path (= 0.16, 0.35, and 0.49) when path was zero, and Table 4 10 shows the results when both paths and were zero. Results when Table 4 4 displays the results when and Across all conditions the median and mean mediated effect estimates were almost zero, which indicates that the mediated effects were correctly estimated under all conditions. However, standard er ror estimates were seriously biased (median relative bias > 0.1) when was low (= 0.5) and the sample size was relatively small (= 100 and 200). Nonnormality did not cause more bias across conditions, but the median relative biases of standard error estimates increased as departure from normality increased in two conditions in which was 0.8 and the samples size was 100 or 200. Also, outliers were found when was 0.5 and the sample size was 100 or 200. Table 4 5 displays relative biases w hen and Most standard error estimates were seriously biased when was 0.5 and the sample size was 500 or smaller; when was 0.8, biased standard error estimates were found with the sample sizes of 100 and 200, whereas median a nd mean mediated effect estimates were almost zero. Nonnormality did not cause serious bias in estimating the mediated effects and standard errors. Outliers were detected when was 0.5 and the sample sizes were 100 and 200. The results in Table 4 5 a re similar to those in Table 4 4. Table 4 6 displays the results when and The results in Table 4 6 are similar to the previous results in Tables 4 4 and 4 5, which are, in turn, similar to each other. Therefore, we can conclude that ther e is no significant impact of the different values of (i.e., 0.18, 0.31, and 0.52) in estimating the mediated effect and PAGE 58 58 standard error when there is no mediated effect (i.e., ). Moreover, departures from normality did not cause more bias. Result s when Tables 4 7 to 4 9 show the results under different values of path (= 0.16, 0.35, and 0.49) when path was zero. In Tables 4 7 to 4 9, the median and mean mediated effects were almost zero across all conditions; however, standard error es timates were biased (median relative bias > 0.1) with relatively small sample sizes of 100 and 200 when was 0.5. Because the results in Tables 4 7 to 4 9 are similar, we can conclude that there is little effect of different values of in estimatin g the mediated effect and standard error. The exception to this generalization was for the estimated standard errors when was .50 and the sample size was 100 or 200. In these conditions the median bias tended to decrease as increased. Furthermor e, there was no impact of nonnormality on estimating the mediated effect and standard error when path was zero. Results when Table 4 10 shows the results when both paths and were zero. Median and mean mediated effect estimates were zero in all conditions, which indicates that all the bias es for the mediated effect estimates were acceptable However, standard error estimates were biased more frequently and inconsistently than the previous results when only one path was zero. Also, median a nd mean relative biases of standard error estimates were very different ( ) indicating outlying standard error estimates in many conditions. Therefore, when both paths and were zero, standard error estimates were not stably estimated although the mediated effects were correctly estimated. PAGE 59 59 Empirical Type I Error Rates Tables 4 11 to 4 13 and Figures 4 1 to 4 3 show estimated Type I error rates when path was zero with different values of path (i.e., 0.18, 0.31, and 0.52 respectively). Tables 4 14 to 4 16 and Figures 4 4 to 4 6 show estimated Type I error rates when path was zero with different values of path (i.e., 0.16, 0.35, and 0.49 respectively). Tabl e 4 17 and Figure 4 7 are results when both paths and were zero. of robustness which is the [.025, .075] interval. If an estimated T ype I error rate is in this cri terion interval then the estimated Type I error rate is considered acceptable. Results w hen In Table 4 11, in which path was small ( first order solution produces estimated Type I error rates that are below .05 in all conditions an d outside the [.025, .075] interval when was .50 and the sample size was 500 or smaller and when was .80 and the sample size was 200 or smaller. In the remaining conditions estimated Type I error rates were below .05 but in the criterion inte rval. For the asymmetric distribution method, estimated Type I error rates were outside the criterion interval ( ) when was .50 and the sample size was 200 or smaller and when was .80 and the sample size was 100, but in the remaining con ditions estimated Type I error rates were in the criterion interval. For the bias corrected bootstrap, Type I error rates tended to be above .05 and in the criterion interval. The exceptions to the generalization that the estimated Type I error rates were larger than .05 occurred when the sample size was 100 and when was .80 and the sample size was 1,000. The exceptions to generalization about the criterion interval PAGE 60 60 occurred when was .50 and the sample size was 500 and when was .80 and the sample size was 200. Table 4 12 shows the estimated Type I error rates when path was medium ( order solution tended to produce estimated Type I error rates that were below or equal to 0.05 in all conditions and in the criterion interval. The exceptions to generalization about the criterion interval occurred when was 0.5 and the sample size was 200 or smaller and when was 0.8 and the sample size was 100. Similarly to rates that were below 0.05 in most conditions and in the criterion interval, but estimated Type I error rates were outside the criterion interval (i.e., ) when was 0.5 and the sample size was 200 or smaller. For the bias corrected bootstrap method, estimated Type I error rates were in the criterion interval except when was 0.8, the sample size was 100, and the data were nonnormal. Results when path was large ( ) are presented in Table 4 order solution and the asymmetric distribution methods produced estimated Type I error rates that were in the criterion interval except when was 0.5 and the sample size was 200 or smaller. The bias corrected bootstrap produced acceptable estimated Type I error rates except when was 0.5 or 0.8 and the sample size was 100 and was 0.5 and the sample size was 500. Then, the estimated Type I error rates tended to be above .075 when the data were non normal. When path was zero, the bias first order solution and the asymmetric distribution method tended to produce PAGE 61 61 conservative Type I error rates ( ). Trends of the estimated Type I error rates in Table 4 11 to 4 13 can also be observed in Figure 4 1 to 4 3 respectively. Results when In Tabl e 4 14, in which path was small ( first order solution produces estimated Type I error rates that were below 0.05 and outside the [.025, .075] interval in all the conditions except when was .80, the sample size was 1,000, and the d ata were substantially nonormal. For the asymmetric distribution method, estimated Type I error rates tended to be below 0.05 and outside the criterion interval, but estimated Type I error rates in the criterion interval occurred when was .80 and th e sample size was 500 or larger. The bias corrected bootstrap produced estimated Type I error rates that were in the criterion interval when was 0.5 and the sample size was 1,000 and when was 0.8 and the sample size was 200 or larger. In othe r conditions the estimated Type I error rates were too small. Table 4 15 shows Type I error rates when path was medium ( first order solution produced estimated Type I error rates that were below 0.05 in all conditions, and estimated Type I error rates are in the criterion interval when was 0.5 and the sample size was 1,000 and when was 0.8 and sample size was 200 or larger. The asymmetric distribution method produced estimated Type I error rates that were in the criterion in terval except when was 0.5 and the sample size was 200 or smaller. For the bias corrected bootstrap, estimated Type I error rates were above 0.05 in most conditions and in the criterion interval when was 0.5 and the sample size was 500 or lar ger and when was 0.8, but estimated Type I errors were below 0.05 when was 0.5 and sample size was 200 or smaller. However, when was 0.5 and PAGE 62 62 sample size was 200, the estimated Type I error rate was in the criterion interval except the data were normal. In Table 4 16, in which path was large ( order solution produces estimated Type error rates that were in the criterion interval when was 0.5 and the sample size was 1,000 and when was 0.8 and the samp le size was 500 or larger, whereas estimated Type I error rates were below 0.025 when was 0.5 and the sample size was 500 or smaller and when was 0.8 and the sample size was 200 or smaller. For the asymmetric distribution method, estimated Ty pe I error rates were in the criterion interval in most conditions, but they were below 0.05 and outside the criterion interval when was 0.5 and the sample size was 200 or smaller. For the bias corrected bootstrap, all the estimated Type I error rat es were in the criterion interval. Trends of estimated Type I error rates in Tables 4 14 to 4 16 can also be observed in Figures 4 4 to 4 6 respectively. When path was zero, the overall estimated Type I error rates were slightly smaller than those when path was zero. Based on the criterion interval, the bias methods tended to produce estimated Type I error rates that were below 0.05 in many conditions. Results w hen Table 4 20 and Figure 4 10 display estimated Type I error rates when both paths and were zero. Across all conditions, empirical Type I error rates were almost zero and thus they were outside the criterion interval. These are similar to the results of previous mediation studies (MacKinnon et al., 2004; Pituch & PAGE 63 63 Stapleton, 2008; Pituch et al., 2006). Trends in Table 4 20 can be observed in Figure 4 10. Empirical Power Rates Tables 4 18 to 4 20 show the empirical power rates when the effect sizes of th e mediated effect were small, medium, and large respectively. Also, Figures 4 8 to 4 10 show power curves as a function of sample size, distribution, and Small Effect Size of the Mediated Effect In Table 4 18, across all conditions, power increases as the sample size and increase. However, nonormality did not affect empirical power rates except in the condition in which was 0.8 and the sample size was 200. Then, the bias corrected bootstrap produced larger empirical power rates under nonnormality. Empirical power rates indicated the bias order solution and the asymmetric distribut ion method. On the other hand, when was 0.8 and the sample size was 1,000, the three methods provided very high power rates ( ). The bias corrected bootstrap and the asymmetric distribution method produced empirical power rates that are almost 0.8 when was 0.8 and the sample size was 500, whereas the three methods produced empirical power rates that were below 0.8 when was 0.5. The trends observed in Table 4 18 can also be observed in Figure 4 8. Medium Effect Size of the Mediated Effect Table 4 19 shows empirical power rates when the mediated effect size was medium. Across all conditions, empirical power rates increased as the sample size and increased. Empirical power rates were near 1.00 when was 0.5 and the sample PAGE 64 64 size was 1,000 and when was 0.8 the sample size was 500 or more. Nonnormality did not affect empirical power rates for any condition. To have 0.8 power with a medium effect size, the sample size should be 200 when is 0.8 and 500 when is 0.5. Figure 4 9 displays power curves observed in Table 4 19. Large Effect Size of the Mediated Effect Table 4 20 shows empirical power rates when the mediated effect size was large. When was 0.8, the power was greater than 0.8 across all conditions. However, when was 0.5, powe r rates were greater than 0.8 in the conditions in which the sample size was 500 or above. The bias corrected bootstrap resulted in much greater power than other methods when was 0.5 and the sample size was 200 or smaller. For the bias corrected boo tstrap, the sample size of 200 was needed to have almost 0.80 power with the low value of 0.5. Departures from normality produced similar power rates, and therefore based on the results in Tables 4 19 and 4 20, we can conclude that nonnormality did not affect empirical power rates. PAGE 65 65 Table 4 1. Median and mean relative biases of the mediated effect estimates ( ) and standard error estimates ( ) for a small mediated effect size Path Path Sample Size Degree of Nonnor mality of Median (Mean) Relative Bias for Median (Mean) Relative Bias for 0.18 0.16 0.5 100 Normal 0.352 ( 0.327) 0.529 (0.522) Moderate 0.399 ( 0.385) 0.527 (0.706) Subst antial 0.363 ( 0.270) 0.547 (0.965) 200 Normal 0.235 ( 0.185) 0.337 (0.040) Moderate 0.234 ( 0.191) 0.468 (0.087) Substantial 0.193 ( 0.167) 0.392 (0.121) 500 Normal 0.091 ( 0.044) 0.065 ( 0.009) Moderate 0.112 ( 0.064) 0.095 ( 0.024) Substantial 0.086 ( 0.053) 0.109 ( 0.033) 1000 Normal 0.044 ( 0.013) 0.025 (0.000) Moderate 0.043 ( 0.016) 0.029 ( 0.007) Substantial 0.042 ( 0.019) 0.030 (0.001) 0.8 100 Normal 0.157 ( 0.023) 0.080 ( 0.046) Moderate 0.185 ( 0.043) 0.095 ( 0.037) Substantial 0.211 ( 0.058) 0.208 ( 0.099) 200 Normal 0.076 ( 0.024) 0.018 ( 0.005) Moderate 0.089 ( 0.019) 0.076 ( 0.043) Substantial 0.097 ( 0.021) 0.120 ( 0. 068) 500 Normal 0.034 ( 0.007) 0.002 (0.004) Moderate 0.033 (0.002) 0.032 ( 0.019) Substantial 0.046 ( 0.009) 0.056 ( 0.031) 1000 Normal 0.017 ( 0.007) 0.005 (0.008) Moderate 0.018 ( 0.003) 0.017 ( 0.009) Substantial 0 .018 ( 0.003) 0.031 ( 0.013) Note: 1) A mediated effect size is small 2) Standard errors were (1982) first o rder Solution. 4) Relative biases with are not acceptable (for and for ). PAGE 66 66 Table 4 2. Median and mean relative biases of the mediated effect estimates ( ) and standard error estimates ( ) for a medium mediated effect size Path Path Sample Size Degree of Nonnormality of Median (Mean) Relativ e Bias for Median (Mean) Relative Bias for 0.31 0.35 0.5 100 Normal 0.109 (0.012) 0.524 (0.688) Moderate 0.124 (0.006) 0.526 (0.572) Substantial 0.140 ( 0.052) 0.538 (0 .751) 200 Normal 0.004 (0.004) 0.363 (0.107) Moderate 0.004 (0.003) 0.394 (0.059) Substantial 0.005 (0.002) 0.428 (0.330) 500 Normal 0.018 (0.004) 0.077 ( 0.017) Moderate 0.032 ( 0.001) 0.106 ( 0.026) Subs tantial 0.019 (0.003) 0.108 ( 0.026) 1000 Normal 0.013 ( 0.001) 0.023 ( 0.002) Moderate 0.011 (0.002) 0.051 ( 0.025) Substantial 0.011 (0.001) 0.046 ( 0.010) 0.8 100 Normal 0.053 ( 0.006) 0.069 ( 0.039) Mod erate 0.051 (0.006) 0.158 ( 0.081) Substantial 0.059 (0.001) 0.166 ( 0.074) 200 Normal 0.030 (0.000) 0.033 ( 0.021) Moderate 0.009 (0.009) 0.049 ( 0.023) Substantial 0.027 (0.000) 0.094 ( 0.049) 500 Normal 0. 008 (0.000) 0.017 ( 0.014) Moderate 0.010 (0.000) 0.040 ( 0.029) Substantial 0.015 (0.000) 0.049 ( 0.024) 1000 Normal 0.008 ( 0.004) 0.008 ( 0.005) Moderate 0.006 ( 0.001) 0.002 (0.013) Substantial 0.003 (0.004) 0.033 ( 0.020) Note: 1) A mediated effect size is medium 2) Standard errors we re order s olution based on robust standard er rors. 4) Relative biases with are not acceptable (for and for ). PAGE 67 67 Table 4 3. Median and mean relative biases of the mediated effect estimates ( )and standard error estimates ( ) for a large mediated effect size Path Path Sample Size Degree of Nonnormality of Me dian (Mean) Relative Bias for Median (Mean) Relative Bias for 0.52 0.49 0.5 100 Normal 0.018 (0.110) 0.527 (0.521) Moderate 0.038 (0.091) 0.544 (0.665) Substantial 0.025 (0.0 73) 0.547 (0.751) 200 Normal 0.011 (0.099) 0.431 (0.088) Moderate 0.002 (0.067) 0.432 (0.157) Substantial 0.008 (0.069) 0.450 (0.124) 500 Normal 0.000 (0.022) 0.081 ( 0.011) Moderate 0.007 (0.018) 0.086 ( 0.01 0) Substantial 0.000 (0.028) 0.220 ( 0.102) 1000 Normal 0.007 (0.014) 0.035 ( 0.005) Moderate 0.002 (0.009) 0.041 ( 0.008) Substantial 0.001 (0.011) 0.066 ( 0.026) 0.8 100 Normal 0.016 (0.011) 0.103 ( 0.065) Moderate 0.023 (0.012) 0.154 ( 0.083) Substantial 0.024 (0.016) 0.211 ( 0.091) 200 Normal 0.000 (0.009) 0.051 ( 0.032) Moderate 0.001 (0.013) 0.077 ( 0.046) Substantial 0.017 (0.000) 0.110 ( 0.056) 500 Normal 0.004 (0.003) 0.027 ( 0.020) Moderate 0.002 (0.004) 0.039 ( 0.024) Substantial 0.009 (0.000) 0.039 ( 0.016) 1000 Normal 0.003 (0.001) 0.020 ( 0.018) Moderate 0.001 (0.003) 0.028 ( 0.018) Substantial 0.001 (0.00 2) 0.035 ( 0.020) Note: 1) A mediated effect size is large. 2) Standard errors we (1982) first order s olution based on robust standard errors. 4) Relative biases with are not acceptable (for and for ). PAGE 68 68 Table 4 4. Median and mean mediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect across when Path Path Sample Size Degree of Nonnormality o f Median (Mean) for Median (Mean) Relative Bias for 0.18 0 0.5 100 Normal 0.002 ( 0.024) 0.586 (0.479) Moderate 0.001 ( 0.018) 0.607 (0.455) Substantial 0.001 ( 0.020) 0. 603 (0.602) 200 Normal 0.003 ( 0.012) 0.440 ( 0.003) Moderate 0.002 ( 0.013) 0.484 (0.027) Substantial 0.002 ( 0.012) 0.532 (0.159) 500 Normal 0.001 ( 0.003) 0.099 ( 0.020) Moderate 0.001 ( 0.003) 0.101 ( 0.018) Substantial 0.001 ( 0.003) 0.142 ( 0.039) 1000 Normal 0.000 ( 0.001) 0.037 ( 0.007) Moderate 0.001 ( 0.002) 0.027 (0.012) Substantial 0.000 ( 0.001) 0.069 ( 0.023) 0.8 100 Normal 0.001 ( 0.002) 0.091 ( 0.036) Moderate 0.001 ( 0.002) 0.191 ( 0.098) Substantial 0.001 ( 0.002) 0.261 ( 0.143) 200 Normal 0.000 ( 0.001) 0.043 ( 0.024) Moderate 0.001 ( 0.001) 0.063 ( 0.025) Substantial 0.001 ( 0.001) 0.112 ( 0.059) 500 Normal 0.000 ( 0.001) 0.027 ( 0.020) Moderate 0.000 ( 0.001) 0.044 ( 0.024) Substantial 0.000 (0.000) 0.051 ( 0.024) 1000 Normal 0.000 (0.000) 0.017 ( 0.013) Moderate 0.000 (0.000) 0.014 ( 0.005) Substantial 0.000 (0.000) 0.022 ( 0.009) Note: 1) A mediated effect size is zero 2) Standard errors we re order s olution based on robust standard errors. 4) Relative biases with are not acceptable (for and for ). PAGE 69 69 Table 4 5. Median and mean mediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no med iated effect when Path Path Sample Size Degree of Nonnormali ty of Median (Mean) for Median (Mean) Relative Bias for 0.31 0 0.5 100 Normal 0.004 ( 0.032) 0.563 (0.480) Moderate 0.006 ( 0.037) 0.564 (0.590) Substantial 0.004 ( 0.037) 0.573 (0.748) 200 Normal 0.004 ( 0.018) 0.457 (0.009) Moderate 0.005 ( 0.022) 0.495 (0.077) Substantial 0.003 ( 0.020) 0.504 (0.062) 500 Normal 0.002 ( 0.005) 0.097 ( 0.009) Moderate 0.002 ( 0.005) 0.106 ( 0.017 ) Substantial 0.002 ( 0.006) 0.167 ( 0.054) 1000 Normal 0.001 ( 0.002) 0.030 (0.005) Moderate 0.001 ( 0.002) 0.032 (0.005) Substantial 0.001 ( 0.002) 0.082 ( 0.038) 0.8 100 Normal 0.003 ( 0.005) 0.113 ( 0.069) Moderate 0.003 ( 0.004) 0.165 ( 0.089) Substantial 0.002 ( 0.003) 0.259 ( 0.128) 200 Normal 0.001 ( 0.002) 0.045 ( 0.027) Moderate 0.001 ( 0.002) 0.097 ( 0.060) Substantial 0.001 ( 0.001) 0.113 ( 0.060) 500 Nor mal 0.000 ( 0.001) 0.022 ( 0.014) Moderate 0.001 ( 0.001) 0.040 ( 0.025) Substantial 0.001 ( 0.001) 0.053 ( 0.025 1000 Normal 0.001 ( 0.001) 0.020 ( 0.015) Moderate 0.000 (0.000) 0.028 ( 0.019) Substantial 0.001 ( 0 .000) 0.011 (0.002) Note: 1) A mediated effect size is zero 2) Standard errors we re order s olution based on robust standa rd errors. 4) Relative biases with are not acceptable (for and f or ). PAGE 70 70 Table 4 6. Median and mean mediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect when Path Path Sample Size Degree of Nonno rmality of Median (Mean) for Median (Mean) Relative Bias for 0.52 0 0.5 100 Normal 0.013 ( 0.063) 0.560 (0.517) Moderate 0.012 ( 0.059) 0.560 (0.585) Substantial 0.006 ( 0.051) 0.586 (0.683) 200 Normal 0.008 ( 0.031) 0.458 (0.019) Moderate 0.011 ( 0.038) 0.511 (0.027) Substantial 0.008 ( 0.033) 0.506 (0.119) 500 Normal 0.005 ( 0.010) 0.102 ( 0.024) Moderate 0.005 ( 0.011) 0.156 ( 0.057) Substantial 0.001 ( 0.007) 0.153 ( 0.034) 1000 Normal 0.001 ( 0.004) 0.037 ( 0.004) Moderate 0.001 ( 0.004) 0.059 ( 0.018) Substantial 0.003 ( 0.005) 0.067 ( 0.023) 0.8 100 Normal 0.004 ( 0.008) 0.141 ( 0.085) Moderate 0.004 ( 0.005) 0.190 ( 0.110) Substantial 0.005 ( 0.008) 0.245 ( 0.139) 200 Normal 0.003 ( 0.004) 0.053 ( 0.035) Moderate 0.004 ( 0.003) 0.074 ( 0.041) Substantial 0.003 ( 0.003) 0.115 ( 0.062) 500 Normal 0.000 (0.000) 0.010 ( 0.004) Moderate 0.001 (0.000) 0.035 ( 0.018) Substantial 0.001 ( 0.001) 0.062 ( 0.036) 1000 Normal 0.001 ( 0.001) 0.013 ( 0.010) Moderate 0.000 (0.000) 0.011 ( 0.004) Substantial 0. 001 ( 0.001) 0.054 ( 0.037) Note: 1) A mediated effect size is zero 2) Standard errors we re order s olution based on robust standard errors. 4) Relative biases with are not acceptable (for and for ). PAGE 71 71 Table 4 7. Median and mean mediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect when Path Path Sample Size Degree o f Nonnormality of Median (Mean) for Median (Mean) Relative Bias for 0 0.16 0.5 100 Normal 0.000 (0.001) 0.507 (0.434) Moderate 0.000 ( 0.001) 0.503 (0.328) Substantial 0.000 ( 0.001) 0.527 (0.615) 200 Normal 0.000 (0.000) 0.235 (0.221) Moderate 0.000 (0.000) 0.203 (0.249) Substantial 0.000 (0.000) 0.290 (0.247) 500 Normal 0.000 (0.000) 0.053 (0.106) Moderate 0.000 (0.000) 0.034 (0.090) Substantial 0.000 (0.000) 0.008 (0.051) 1000 Normal 0.000 (0.000) 0.009 (0.025) Moderate 0.000 (0.000) 0.010 (0.030) Substantial 0.000 (0.000) 0.005 (0.028) 0.8 100 Normal 0.000 (0.000) 0.009 (0.065) Moderate 0.000 (0.00 0) 0.036 (0.041) Substantial 0.000 (0.000) 0.091 (0.032) 200 Normal 0.000 (0.000) 0.006 (0.023) Moderate 0.000 (0.000) 0.015 (0.015) Substantial 0.000 (0.000) 0.030 (0.014) 500 Normal 0.000 (0.000) 0.003 (0.003) Mode rate 0.000 (0.000) 0.001 (0.008) Substantial 0.000 (0.000) 0.005 (0.015) 1000 Normal 0.000 (0.000) 0.006 ( 0.007) Moderate 0.000 (0.000) 0.007 (0.009) Substantial 0.000 (0.000) 0.008 (0.007) Note: 1) A mediated effect size is zero 2) Standard errors we re order s olution based on robust standa rd errors. 4) Relative biases with are not acceptable (for and for ). PAGE 72 72 Table 4 8. Median and mean m ediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect when Path Path Sample Size Degree of Nonnormality of Median (Mean) for Median (Mean) Relative Bias for 0 0.35 0.5 100 Normal 0.000 ( 0.003) 0.391 (0.471) Moderate 0.000 ( 0.002) 0.407 (0.545) Substantial 0.000 (0.000) 0.428 (0.522) 200 Normal 0.000 (0.001) 0.169 (0.049) Moderate 0.000 (0.000) 0.171 (0.429) Substantial 0.000 (0.000) 0.161 (0.164) 500 Normal 0.000 (0.000) 0.014 ( 0.003) Moderate 0.000 (0.000) 0.016 (0.036) Substantial 0.000 (0.000) 0.037 ( 0.016) 1000 Normal 0.000 (0.000) 0.022 ( 0.019) Moderate 0.000 (0.000) 0.019 ( 0.017) Substantial 0.000 (0.000) 0.009 (0.013) 0.8 100 Normal 0.000 (0.000) 0.041 ( 0.034) Moderate 0.000 (0.000) 0.042 ( 0.017) Substantial 0.000 (0.000) 0 .058 ( 0.023) 200 Normal 0.000 (0.000) 0.012 ( 0.010) Moderate 0.000 (0.000) 0.011 ( 0.012) Substantial 0.000 (0.000) 0.019 ( 0.010) 500 Normal 0.000 (0.000) 0.017 (0.016) Moderate 0.000 (0.000) 0.015 ( 0.014) Subst antial 0.000 (0.000) 0.006 ( 0.003) 1000 Normal 0.000 (0.000) 0.022 ( 0.023) Moderate 0.000 (0.000) 0.008 ( 0.012) Substantial 0.000 (0.000) 0.015 ( 0.015) Note: 1) A mediated effect size is zero 2) Standard error s we re order s olution based on robust standard errors. 4) Relative biases with are not acceptable (for and for ). PAGE 73 73 Table 4 9. Median and mean mediated effect estimates ( ) and me dian and mean relative biases for the standard error estimates ( ) for no mediated effect when Path Path Sample Size Degree of Nonnormality of Median (Mean) for Median (Mean) Relative Bias for 0 0.49 0.5 100 Normal 0.000 ( 0.001) 0.332 (0.440) Moderate 0.000 (0.002) 0.351 (0.442) Substantial 0.000 (0.000) 0.357 (0.506) 200 Normal 0.001 (0.000) 0.115 (0.072) Moderate 0.000 (0.000) 0.163 (0.137) Substantial 0.000 ( 0.001) 0.182 (0.194) 500 Normal 0.001 (0.000) 0.010 (0.006) Moderate 0.000 ( 0.001) 0.026 ( 0.005) Substantial 0.000 (0.000) 0.015 (0.008) 1000 Normal 0.000 (0.000) 0.017 ( 0.013) Moderate 0.000 (0.000) 0.012 ( 0.008) Substantial 0.000 (0.000) 0.002 (0.004) 0.8 100 Normal 0.001 ( 0.001) 0.029 ( 0.021) Moderate 0.000 (0.000) 0.039 ( 0.029) Substantial 0.000 (0.001) 0.046 ( 0.028) 200 Normal 0 .000 (0.000) 0.024 ( 0.024) Moderate 0.000 (0.000) 0.034 ( 0.028) Substantial 0.000 (0.001) 0.040 ( 0.034) 500 Normal 0.000 (0.000) 0.002 (0.004) Moderate 0.000 (0.000) 0.017 ( 0.017) Substantial 0.000 (0.000) 0.009 ( 0.01 0) 1000 Normal 0.000 (0.000) 0.008 ( 0.008) Moderate 0.000 (0.000) 0.014 (0.016) Substantial 0.000 (0.000) 0.003 (0.005) Note: 1) A mediated effect size is zero 2) Standard errors we re t order s olution based on robust standa rd errors. 4) Relative biases with are not acceptable (for and for ). PAGE 74 74 Table 4 10. Median and mean mediated effect estimates ( ) and median and mean relative biases for the standard error estimates ( ) for no mediated effect when Path Path Sample Size Degree of Nonnormality of Median (Mean) for Median (Mean) Relative Bias for 0 0 0.5 10 0 Normal 0.000 ( 0.001) 0.473 (0.371) Moderate 0.000 ( 0.001) 0.506 (0.726) Substantial 0.000 ( 0.001) 0.509 (0.516) 200 Normal 0.000 (0.000) 0.371 (0.195) Moderate 0.000 (0.001) 0.455 (0.180) Substantial 0.000 (0.0 00) 0.528 (0.236) 500 Normal 0.000 (0.000) 0.012 (0.133) Moderate 0.000 (0.000) 0.014 (0.146) Substantial 0.000 (0.000) 0.023 (0.133) 1000 Normal 0.000 (0.000) 0.119 (0.225) Moderate 0.000 (0.000) 0.136 (0.246) Subs tantial 0.000 (0.000) 0.071 (0.196) 0.8 100 Normal 0.000 (0.000) 0.062 (0.169) Moderate 0.000 (0.000) 0.049 (0.174) Substantial 0.000 (0.000) 0.090 (0.068) 200 Normal 0.000 (0.000) 0.116 (0.207) Moderate 0.000 (0.000) 0.1 17 (0.204) Substantial 0.000 (0.000) 0.093 (0.192) 500 Normal 0.000 (0.000) 0.163 (0.237) Moderate 0.000 (0.000) 0.133 (0.214) Substantial 0.000 (0.000) 0.096 (0.197) 1000 Normal 0.000 (0.000) 0.165 (0.245) Moderate 0. 000 (0.000) 0.163 (0.260) Substantial 0.000 (0.000) 0.164 (0.241) Note: 1) A mediated effect size is zero 2) Standard errors we re estimated order s olution based on robust standa rd errors. 4) Relative biases with are not acceptable (for and for ). PAGE 75 75 Table 4 order solution, the asymmetric distribution, and bias corrected bootstrap methods: and Path Path Sample Size Degree of Nonnormality of First Order Solution Asymmetric Distribution Bias Corrected Bootstrap 0.18 0 0.5 100 Normal 0 0.001 0.031 Moderate 0.001 0.003 0.033 Substantial 0 0.001 0.041 200 Normal 0.003 0.014 0.053 Moderate 0.002 0.011 0.059 Substantial 0.001 0.01 0.062 500 Normal 0.023 0.04 0.077 Moderate 0.017 0.036 0.068 Substantial 0.011 0.038 0.078 1000 Normal 0.035 0.046 0.067 Moderate 0.029 0.046 0.055 Substantial 0.024 0.037 0.061 0.8 100 Normal 0.004 0.02 0.047 Moderate 0 0.016 0.049 Substantial 0.001 0.018 0.064 200 Normal 0.014 0.044 0.071 Moderate 0.013 0.04 0.080 Substantial 0.013 0.029 0.068 500 Normal 0.038 0.051 0.06 Moderate 0.036 0.053 0.071 Substantial 0.036 0.047 0.068 1000 Normal 0.036 0.044 0.043 Moderate 0.039 0.047 0.054 Substantial 0.045 0.057 0.068 Average Type I error rates across conditions 0.018 0.031 0.060 Not e: Estimated Type I error rates with are unacceptable. PAGE 76 76 Table 4 order solution the asymmetric distribution, and bias corrected bootstrap methods: and Path Path Sample Size Degree of Nonnormality of First Order Solution Asymmetric Distribution Bias Corrected Bootstrap 0.31 0 0.5 100 Normal 0 0.007 0.048 Moderate 0.002 0.005 0.066 Substantial 0.001 0.003 0.065 200 Normal 0.008 0.022 0.061 Moderate 0.005 0.023 0.075 Substantial 0.009 0.017 0.062 500 Normal 0.03 0.042 0.064 Moderate 0.03 0.036 0.055 Substantial 0.024 0.035 0.056 1000 Normal 0.027 0.035 0.041 Moderate 0.041 0.042 0.06 Substantia l 0.039 0.045 0.062 0.8 100 Normal 0.013 0.043 0.062 Moderate 0.016 0.035 0.085 Substantial 0.01 0.025 0.076 200 Normal 0.036 0.047 0.06 Moderate 0.03 0.047 0.071 Substantial 0.031 0.04 0.075 500 Normal 0.05 0.053 0.059 Moderate 0.041 0.048 0.051 Substantial 0.042 0.046 0.058 1000 Normal 0.044 0.046 0.046 Moderate 0.048 0.05 0.049 Substantial 0.04 0.043 0.051 Average Type I error rates across conditions 0.026 0.035 0.060 Note: Estimated Type I error rates with are unacceptable. PAGE 77 77 Table 4 order solution, the distribution, and bias corrected bootstrap methods: and Path Path Sample Size Degree of Nonnormality of First Order Solution Asymmetric Distribution Bias Corrected Bootstrap 0.52 0 0.5 100 Normal 0.002 0.007 0.049 Moderate 0.002 0.003 0.063 Substantial 0.002 0.004 0.076 200 Normal 0.01 0.015 0.041 Moderate 0.015 0.021 0.071 Substantial 0.014 0.015 0.071 500 Normal 0.022 0.029 0.04 Moderate 0.04 0.042 0.05 Substantial 0.046 0.05 0.083 1000 Normal 0.049 0.052 0.055 Moderate 0.042 0.044 0.048 Substantial 0.037 0.038 0.051 0. 8 100 Normal 0.032 0.042 0.063 Moderate 0.028 0.037 0.08 Substantial 0.024 0.027 0.082 200 Normal 0.046 0.052 0.059 Moderate 0.036 0.038 0.057 Substantial 0.039 0.046 0.066 500 Normal 0.036 0.041 0.041 Moderate 0.046 0.0 48 0.057 Substantial 0.047 0.048 0.055 1000 Normal 0.062 0.066 0.062 Moderate 0.048 0.051 0.057 Substantial 0.041 0.041 0.047 Average Type I error rates across conditions 0.032 0.036 0.059 Note: Estimated Type I error rates with are unacceptable. PAGE 78 78 Table 4 14. Estimated Type I error rates for Sob order solution, the asymmetric distribution, and bias corrected bootstrap methods: and Path Path Sample Size Degree of Nonnormality of First Order Solution Asymmetric Distribution Bias Corrected Bootstrap 0 0.16 0.5 100 Normal 0 0 0.004 Moderate 0 0.001 0.009 Substantial 0 0 0.014 200 Normal 0.001 0.004 0.013 Moderate 0 0.003 0.008 Substan tial 0 0.005 0.015 500 Normal 0.002 0.008 0.019 Moderate 0 0.003 0.012 Substantial 0 0.005 0.017 1000 Normal 0.006 0.019 0.032 Moderate 0.003 0.021 0.032 Substantial 0.001 0.021 0.03 0.8 100 Normal 0.001 0. 002 0.011 Moderate 0 0.01 0.026 Substantial 0 0.001 0.016 200 Normal 0.005 0.012 0.024 Moderate 0.002 0.016 0.038 Substantial 0.003 0.011 0.031 500 Normal 0.01 0.034 0.044 Moderate 0.013 0.041 0.052 Sub stantial 0.008 0.029 0.046 1000 Normal 0.019 0.044 0.048 Moderate 0.018 0.051 0.066 Substantial 0.027 0.056 0.059 Average Type I error rates across conditions 0.005 0.017 0.028 Note: Estimated Type I error rates with are unacceptable. PAGE 79 79 Table 4 15. order solution, the asymmetric distribution, and bias corrected bootstrap methods: and Path Path Sample Size Degree of Nonnormality of First Ord er Solution Asymmetric Distribution Bias Corrected Bootstrap 0 0.35 0.5 100 Normal 0 0.002 0.018 Moderate 0 0.002 0.019 Substantial 0.001 0.003 0.027 200 Normal 0 0.004 0.016 Moderate 0.001 0.011 0.045 Substantial 0.0 02 0.009 0.038 500 Normal 0.016 0.034 0.059 Moderate 0.013 0.039 0.059 Substantial 0.004 0.028 0.048 1000 Normal 0.026 0.05 0.061 Moderate 0.02 0.045 0.055 Substantial 0.031 0.049 0.067 0.8 100 Normal 0.004 0.023 0.05 3 Moderate 0.008 0.026 0.053 Substantial 0.003 0.024 0.056 200 Normal 0.016 0.038 0.055 Moderate 0.027 0.052 0.073 Substantial 0.025 0.049 0.071 500 Normal 0.038 0.055 0.061 Moderate 0.035 0.051 0.06 Substantial 0. 043 0.053 0.063 1000 Normal 0.048 0.053 0.056 Moderate 0.047 0.054 0.052 Substantial 0.043 0.051 0.048 Average Type I error rates across conditions 0.019 0.034 0.051 Note: Estimated Type I error rates with are unacceptable. PAGE 80 80 T able 4 order solution, the asymmetric distribution, and bias corrected bootstrap methods: and Path Path Sample Size Degree of Nonnormality of First Order Solution A symmetric Distribution Bias Corrected Bootstrap 0 0.49 0.5 100 Normal 0 0 0.025 Moderate 0 0.003 0.039 Substantial 0 0.003 0.045 200 Normal 0.003 0.022 0.051 Moderate 0.005 0.021 0.055 Substantial 0.003 0.011 0.043 500 Normal 0.032 0.069 0.059 Moderate 0.016 0.046 0.062 Substantial 0.021 0.04 0.058 1000 Normal 0.038 0.05 0.058 Moderate 0.027 0.038 0.04 Substantial 0.038 0.047 0.054 0.8 100 Normal 0.013 0.035 0.057 Moderate 0.017 0 .038 0.064 Substantial 0.014 0.039 0.065 200 Normal 0.029 0.046 0.062 Moderate 0.042 0.061 0.075 Substantial 0.02 0.04 0.061 500 Normal 0.05 0.057 0.058 Moderate 0.047 0.058 0.072 Substantial 0.04 0.053 0.054 1000 Nor mal 0.06 0.061 0.062 Moderate 0.043 0.048 0.043 Substantial 0.041 0.047 0.046 Average Type I error rates across conditions 0.025 0.039 0.055 Note: Estimated Type I error rates with are unacceptable. PAGE 81 81 Table 4 17. Estimated Type I er order solution, the asymmetric distribution, and bias corrected bootstrap methods: and Path Path Sample Size Degree of Nonnormality of First Order Solution Asymmetric Distribution Bias Corr ected Bootstrap 0 0 0.5 100 Normal 0 0 0.007 Moderate 0 0 0.01 Substantial 0 0 0.007 200 Normal 0 0 0.005 Moderate 0 0 0.004 Substantial 0 0 0.008 500 Normal 0 0 0.001 Moderate 0 0 0.004 Substa ntial 0 0.001 0.003 1000 Normal 0 0.002 0.003 Moderate 0 0.002 0.003 Substantial 0 0.001 0.004 0.8 100 Normal 0 0.001 0.005 Moderate 0 0.001 0.009 Substantial 0 0 0.01 200 Normal 0 0.001 0.004 Mod erate 0 0 0.003 Substantial 0 0 0.004 500 Normal 0 0.002 0.004 Moderate 0 0.001 0.003 Substantial 0 0 0.002 1000 Normal 0 0 0.001 Moderate 0 0.001 0.001 Substantial 0 0.004 0.003 Average Type I err or rates across conditions 0 0.000 0.005 Note: Estimated Type I error rates with are unacceptable. PAGE 82 82 Table 4 order solution, the asymmetric distribution, and bias corrected bootstrap methods for a small ef fect size Path Path Sample Size Degree of Nonnormality of First Order Solution Asymmetric Distribution Bias Corrected Bootstrap 0.18 0.16 0.5 100 Normal 0.001 0.002 0.049 Moderate 0 0.003 0.066 Substantial 0.001 0.005 0.07 200 Normal 0.011 0.048 0.133 Moderate 0.013 0.039 0.141 Substantial 0.01 0.047 0.149 500 Normal 0.17 0.279 0.341 Moderate 0.201 0.295 0.381 Substantial 0.162 0.271 0.354 1000 Normal 0.474 0.548 0.577 Moderate 0.48 2 0.553 0.583 Substantial 0.492 0.563 0.597 0.8 100 Normal 0.039 0.106 0.189 Moderate 0.026 0.095 0.189 Substantial 0.015 0.072 0.186 200 Normal 0.187 0.394 0.327 Moderate 0.2 0.343 0.427 Substantial 0.189 0.327 0.434 50 0 Normal 0.714 0.778 0.79 Moderate 0.716 0.772 0.792 Substantial 0.721 0.778 0.801 1000 Normal 0.955 0.961 0.96 Moderate 0.961 0.963 0.962 Substantial 0.965 0.97 0.962 Average power across conditions 0.321 0.384 0.436 PAGE 83 83 Tab le 4 order so lution, the asymmetric distribution, and bias corrected bootstrap methods for a medium effect size Path Path Sample Size Degree of Nonnormality of First Order Solution Asymm etric Distribution Bias Corrected Bootstrap 0.31 0.35 0.5 100 Normal 0.009 0.029 0.251 Moderate 0.007 0.033 0.267 Substantial 0.022 0.045 0.301 200 Normal 0.211 0.315 0.539 Moderate 0.187 0.288 0.549 Substantial 0.181 0.284 0.534 500 Normal 0.858 0.882 0.901 Moderate 0.843 0.863 0.881 Substantial 0.85 0.875 0.886 1000 Normal 0.992 0.994 0.994 Moderate 0.994 0.994 0.994 Substantial 0.994 0.994 0.994 0.8 100 Normal 0.535 0.672 0.745 Moderate 0.488 0.624 0.703 Substantial 0.467 0.603 0.726 200 Normal 0.937 0.954 0.957 Moderate 0.932 0.949 0.948 Substantial 0.918 0.933 0.934 500 Normal 1 1 1 Moderate 1 1 1 Substantial 0.999 0.999 0.999 1000 Normal 1 1 1 Modera te 1 1 1 Substantial 1 1 1 Average power across conditions 0.684 0.722 0.796 PAGE 84 84 Table 4 order solution, the asymmetric distribution, and bias corrected bootstrap methods for a large effect size Path Path Sample Size Degree of Nonnormality of First Order Solution Asymmetric Distribution Bias Corrected Bootstrap 0.52 0.49 0.5 100 Normal 0.081 0.119 0.494 Moderate 0.082 0.109 0.5 Substantial 0.078 0.108 0.472 200 Normal 0.526 0.557 0.777 Moderate 0.441 0.469 0.744 Substantial 0.463 0.493 0.755 500 Normal 0.986 0.987 0.99 Moderate 0.985 0.985 0.985 Substantial 0.987 0.987 0.99 1000 Normal 1 1 1 Moderate 1 1 1 Substantial 1 1 1 0.8 100 Normal 0.894 0.911 0.936 Moderate 0.848 0.866 0.901 Substantial 0.818 0.838 0.884 200 Normal 1 1 1 Moderate 0.993 0.994 0.994 Substantial 0.989 0.991 0.991 500 Normal 1 1 1 Moderate 1 1 1 Substantial 1 1 1 1000 Normal 1 1 1 Moderate 1 1 1 Substantial 1 1 1 Average power across conditions 0.799 0.809 0.892 PAGE 85 85 Figure 4 1. Empirical Type I error rates when and when B) Asymmetric distribution met hod when C) The bias corrected bootstrap when E) Asymmetric distribution method when F) The bias corrected bootstrap when PAGE 86 86 Figure 4 2. Empirical Type I error rates when and when B) Asymmetric distribution method when C) The bias corrected bootstrap when E) Asymmetric distribution method when F) The bias correct ed bootstrap when PAGE 87 87 Figure 4 3. Empirical Type I error rates when and when B) Asymmetric distribution method when C) The bias corrected bootstrap when E) Asymmetric distribution method when F) The bias corrected bootstrap when PAGE 88 88 Figure 4 4. Empirical Type I error rates when and when B) Asymmetric distribution method when C) The bias corrected bootstrap when E) Asymmetric distribution method when F) The bias corrected bootstrap when PAGE 89 89 Figure 4 5. Empirical Type I error rates when and when B) Asymmetric distribution method when C) The bias corrected bootstrap when E) Asymmetric distribution method when F) The bias corrected bootstrap whe n PAGE 90 90 Figure 4 6. Empirical Type I error rates when and when B) Asymmetric distribution method when C) The bias corrected bootstrap when E) Asy mmetric distribution method when F) The bias corrected bootstrap when PAGE 91 91 Figure 4 7. Empirical Type I error rates when and B) Asymmetric distribution method when C) The bias c orrected bootstrap when E) Asymmetric distribution method when F) The bias corrected bootstrap when PAGE 92 92 Figure 4 8. Empirical power rates for a small effect size of the mediated effect A ) B) Asymmetric distribution method when C) The bias corrected bootstrap when when E) Asymmetric distribution method when F) The bias corrected bootstrap when PAGE 93 93 Figure 4 9. Empirical power rates for a medium effect size of the mediated effect A) B) Asymmetric distribution method when C) The bias corrected bootstrap when when E) Asymmetric distribution method when F) The bias corrected bootstrap when PAGE 94 94 Figure 4 10. Empirical power rates for a large effect size of the mediated effect A) B) Asymmetric distributi on method when C) The bias corrected bootstrap when when E) Asymmetric distribution method when F) The bias corrected bootstrap when PAGE 95 95 CHAPTER 5 DISCUSSION AND CONCL USION This Mont e Carlo simulation study investigated the effect of nonnormality in a longitudinal mediation framework using a parallel process latent growth model. The research was undertaken because previous research has not investigated the effect of nonnormality in lo ngitudinal mediation analysis. Interestingly, it was found that among different design factors (i.e., effect size of the mediated effect, sample size, and degree of nonnormality), the degree of nonnormality (i.e., normality, moderate and substantia l nonnormalit y ) did not cause more bias in estimating and testing the mediated effect, whereas, the accuracy of estimates of the mediated effect and statistical power increased as other design factors of the effect size of the mediated effect, sample size, and increased. Accuracy of Estimates for the Mediated Effects and Standard Errors The accuracy of estimates were evaluated with various combinations of paths and First, when there was the mediated effect, (i.e., both population paths and were not zero), the accuracy of the estimated mediated effect improved as the mediated effect size, sample size, and value increased. These results were similar to those in onnormality did not seriously affect the accuracy of the mediated effect estimates. Previous studies in mediation analysis (e.g., Cheong, 2011; MacKinnon et al., 2004; Pituch & Stapleton, 2008) did not assess the accuracy of the mediated effect when the p opulation mediated effect was zero. Interestingly, this study found that all the relative biases for the mediated effect estimates were acceptable when the population mediated effect was zero. PAGE 96 96 irst order solution (i.e., Equation 2 4). When he mediated effect was non zero the accuracy of the standard error estimates improved as the sample size and value increased. However, the mediated effect size did not much affect the standard error es timates, and nonnormality also resulted in more seriously biased standard error estimates in a few conditions. In CFA (confirmatory factor analysis) mediation study, Finch et al. (1997) reported that nonnormality tended to cause more bias than normality in estimating standard errors regardless of the different estimation methods (i.e., the maximum likelihood, the robust maximum likelihood, and the asymptotically distribution free methods), but nonnormality did not affect the mediated effect estimates. Furth ermore, Finch et al. found that the effects of sample size (150, 250, 500, and 1,000) were negligible, in other words, large sample size did not reduce the relative biases of standard error estimates. However, in this study, relatively large sample sizes ( 500 and 1,000) tended to reduce the relative bias of the standard error estimates when there was the population mediated effect. When the population mediated effect was zero, standard error estimates were affected by which path was zero. That is, when both paths and were zero, standard error estimates were more frequently biased than when one of paths was zero. Also, when path was zero, standard error estimates were more frequently biased than when path was zero. In this study, outliers of the mediated effect and standard error estimates were found under the conditions with relatively small sample sizes of 100 and 200, which can be misleading because outliers affect the mean relative bias, and thus both median and PAGE 97 97 mean relative biases were provi ded. Interestingly, when both paths (i.e., ) were zero, outliers of the standard error estimates were more frequently found than when both paths were not zero or when one of paths was zero. Empirical Type I Error Rates o rder solution, the asymmetric distribution method, and the bias corrected bootstrap) produced different estimated Type I error rates across conditions. Based on the estimated Type I error rates, it is difficult to find the effects of the design factors bec ause the estimated Type I error rates were not consistent across order solution and the asymmetric distribution method, the sample size and affected the estimated Type I error rates when only one path ( or ) was zero. Also, the bias corrected bootstrap worked best when only one path was zero. order solution and the asymmetric distribution method produced estimated Type I error rates that increased as the sample size and increased, however, the estimated Type I error rates were not greater than the upper limit of the criterion interval (i.e. 0.075) in all conditions. Based on OLS regression models, MacKinnon et al. (2004) found that estimated Type I error rates by order solution and the asymmetric distribution method tended to increase as the sample size increased (i.e., 50, 100, 200, 500, and 1,000). When both paths were zero, all three methods resulted in very low Type I error rates across all c onditions. This result is consistent with prior studies (Mackinnon et al., 2004; Pituch et al., 2006; Pituch & Stapleton, 2008). PAGE 98 98 Previous studies (MacKinnon et al., 2004; MacKinnon et al., 2007; Pituch & Stapleton, 2008; Tofighi & MacKinnon, 2011) invest order solution, the asymmetric distribution confidence interval, and bias corrected bootstrap methods. MacKinnon et al. (2004), MacKinnon et al. (2007), and Tofighi and MacKinnon studied a model with a continuous independent variable, a continuous mediator, and a continuous dependent variable (see Figure 2 1). In their models path a was always zero and path b was either zero or larger than zero. MacKinnon et al. (2004) concluded that the bias corrected bootstrap worked best in terms of Type I error rate. MacKinnon et al. (2007), and Tofighi and MacKinnon concluded that the asymmetric confidence interval worked best. However, careful review of the results in MacKinnon et al. (2007) and Tofighi and MacKinnon indicated their conclusion was correct only when path b was larger than zero and was medium or large in size. Pituch and Stapleton studied a multilevel model with a dichotomous independent variable, a continuous mediator and a continuous dependent variable. In their models relevant to T ype I error rates, path a, path b, or both were equal to zero. When path b was not equal to zero, it was larger than zero. Pituch and Stapleton included two variations of the asymmetric confidence interval, one that used robust standard errors and one that did not. In the present study, only a version with robust standard errors was used. Pituch and Stapleton also studied two variations of the bias corrected bootstrap: a parametric and a non parametric version. In the present study, only the non parametric bias corrected bootst r ap was used. Pituch and Stapleton concluded that the best control of the Type I error rate was achieved by using the asymmetric confidence interval without robust standard errors and the parametric bias corrected bootstrap. However, among robust methods the best procedure was the PAGE 99 99 non parametric bias order solution, the asymmetric distribution confidence in t erval, and a bias corrected bootstrap confidence in t erval for a para llel process latent growth model and included more extensive conditions (i.e., both paths were zero; three different values of path a were not zero when path b was zero; three different with values of path b were not zero when path a was zero) than previou s studies (i.e., both paths a and b were zero; three different values of path b were not zero when path a was zero ) ( MacKinnon et al., 2004; MacKinnon et al., 2007; Tofighi & MacKinnon, 2011). This study found that the asymmetric distribution and the bias corrected bootstrap methods worked better than order solution produced overall Type I error rates that were low. The bias corrected bootstrap produced estimated Type I error rates that tended to be in the criterion interval (i.e., [0.025, 0.075]) more frequently than the asymmetric distribution method. Therefore, we can conclude that the bias corrected bootstrap worked best in producing accurate Type I error rates. The asymmetric confidence interval method may not have worked as well as in previous studies (MacKinnon et al., 2007; Pituch & Stapleton, 2008; Tofighi & MacKinnon, 2011) due to differences in the models studied, to differences in the type of asymmetric confidence interval method and bias correcte d bootstrap, or to the conditions studied. order solution produced very conservative estimated Type I error rates in some conditions. Figures 5 1 to 5 2 show the relationship between the estimated mediated eff tandard error estimates. Figure 5 1 presents results for conditions in which both paths and were zero. As the absolute value of the estimated mediated effect increased, the standard PAGE 100 100 error of the estimate mediated effect also increase d. As a result the ratio of the Type I error rate tended to be near zero. Figure 5 2 presents results for conditions in which path was zero, path was 0.35, and was 0.8. The four panels varied by sample size. When the sample size was 200 or smaller, the plot of the Sobel standard error against the estimated mediate effect was similar to the plots in F igure 5 1 and hence t he T ype I error rate was near zero Whe n the sample size was 500 or larger, the estimated mediate effect were independently distributed. It is well known that when a test statistic is a ratio of an estimate to a sta ndard error, the test based on the ratio will work poorly when the estimate and the standard error are related. The patterns depicted in Figures 5 1 and 5 2 can be e x plained using (1936) results Based on results, the kurto sis of the mediated effect is This has a maximum value of 6 which occurs when both when both and are zero that is, when As depicted in Figure 5 1, in this case the distribution of the mediated effect is long tailed regardless of the sample size Moreover when the variance tends to be large, As either or increases the kurtosis of the mediated effect declines and the likelihood of extreme values of declines. Increases in or can occur due to a large sample size and will result in a smaller likelihood of extreme values of the mediated effect and accounts for graphs like those in the bottom panels of PAGE 101 101 Figure 5 2. Increases in or als o occur when or respectively increase either or increase, which is consistent with results in Tables 4 11 to 4 17. Although the asymmetric distribution method provided higher Type I error rates nominal value of 0.05 and tended to be outside the criterion interval when was .50 and sample sizes were 200 or smaller ( See Tables 4 11 to 4 17 or Figures 4 1 to 4 7). According to MacKinnon et al. (2007), the asymmetric distribution is based on the tables of the product of two random variables (Meeker et al., 1981), but these tables assume population values of and Therefore, this study suggest s that using sample estimates in place of the population values may have resulted in low estimated Type I error rates when the sample size was small. In this study, the bias corrected bootstrap w orked best and produced acceptable estimated Type I errors in most conditions except when both paths were zero. An important assumption of the bias corrected bootstrap method is that a transformation of the sampling distribution of the mediate effect is no rmal. When this assumption is violated, the bias corrected bootstrap may result in estimated Type I error rates outside the criterion interval. Empirical Power Rates Cheong (2011) pointed out researchers who use a parallel process latent growth model can i mprove statistical power of the mediated effect by increasing sample size, the mediated effect size, the number of measurement occasions, and the value. This study also found that empirical power rates increased as the effect size of the mediated PAGE 102 102 ef fect, sample size, and the value increased. Furthermore, this study considered nonnormality as one of design factors. However, we cannot conclude that nonnorm a lity affects the empirical power rates. Based on the results of empirical power rates of this study, we can find the sample size to have 0.8 power under the different conditions. When the effect size of the mediated effect is small and the asymmetric distribution method and the bias corrected bootstrap are used a sample size of 500 is requir ed to have power near 0.8 when is 0.8; a sample size above the largest sample size used in this study, that is 1000, would be required if were 0.5. When the effect size was medium, the sample sizes of 200 and 500 were sufficient to have 0.8 p ower with the values of 0.8 and 0.5 respectively. With a large effect size, power was greater than 0.8 in all the conditions when was 0.8, whereas the sample size needs to be 200 or above to have 0.8 power when is 0.5. Therefore, the r esults suggest that it is important to have reliable measures of the observed variables and large mediated effect size because statistical power was high enough ( ) with a relatively small sample size of 100 when and the effect size of the mediat ed effect were large. order solution, the asymmetric distribution, and the bias correct bootstrap) were compared and they resulted in different empirical power rates. The bias correcte d bootstrap had power at least as large as the power for the other methods just at it tended have more adequate empirical Type I error rates. Power differences between the bias corrected bootstrap order solution or as ymmetric distribution methods) increased as the sample size and decreased. Using a multilevel mediation model, PAGE 103 103 Pituch and Stapleton (2008) found that the bias corrected bootstrap produced the highest power, followed by the the asymmetric distributio n method, which was confirmed in this study. As previous studies (Bollen & Stine, 1990; MacKinnon et al., 2004; Pituch & Stapleton, 2008; Shrout & Bolger, 2002) have suggested using the bootstrap methods in testing the mediated effect, the results of the s tudy also recommend the bias corrected bootstrap. Conclusions The purpose of this study was to assess the effects of nonnormality on estimating and testing the mediated effect using a parallel process latent growth model through Monte Carlo simulations. Th e design factors of the study were the degree of nonnormality (i.e., normality, moderate and substantial nonnormality), effect size of the mediated effect (i.e., zero, small, medium, and large), sample size (i.e., 100, 200, 300, and 400), and the va lue (i.e., 0.5 and 0.8). The dependent variables were the relative bias of the mediated effect and standard error estimates, empirical Type I and power rates. It was found that nonnormality had little effect on the accuracy of the estimates of the mediate d effect, empirical Type I error, and power rates although the bias of standard error estimates increased in a few conditions of nonnormality. Furthermore, outliers of the mediated effect and standard error estimates were found with relatively small sample sizes (100 and 200). Interestingly, when both paths ( and ) were zero, the standard error estimates were more frequently and inconsistently biased than when only one path was zero with more outlying estimates, which could cause an incorrect Type I err or rate. PAGE 104 104 According to previous studies (MacKinnon et al., 2007; Tofighi & MacKinnon, 2011; Pituch & Stapleton, 2008), the asymmetric distribution method worked better than the bias corrected bootstrap method in terms of empirical Type I error rates. Howeve r, this study found that the bias corrected bootstrap method worked best based on the order solution produced very conservative Type I error rates when the estimates between the med iated effect and standard error have a relationship, but the Type I error increased to almost 0.05 when relationship was not present. To have statistical power of 0.8, it is important to have relatively large values of the mediated effect and value as well as a large sample size, which was found in previous studies (e.g., Cheong, 2011; Pituch & Stapleton, 2008). The bias corrected bootstrap method also worked best in statistical power. Limitations and Suggestions for Future Research The statistical model and simulation conditions of this study was based on the previous studies (Cheong, 2011; Pituch & Stapleton, 2008; Thoemmes et al., 2010), and this study included more conditions (e.g., degree of nonnormality and different types of effect sizes of th e mediated effect). However, this study is still limited by simulation conditions. For example, more distributions need to be investigated in the future such as normal or nonnormal distributions in the presence of outliers or with long tail. A parallel pr ocess latent growth model in this study is relatively simple and this model can be more complex by considering nonlinear growth trends, the different number of measurement occasions and the values and so on. Furthermore, Selig PAGE 105 105 and Preacher (2009) su mmarized different longitudinal mediation models in a structural lagged panel parallel process latent gro wth curve model, and thus we can study other longitudinal mediation models. The results of the previous studies (e.g., Cheong, 2011; MacKinnon et al., 2004; MacKinnon et al., 2006; Pituch & Stapleton, 2008; Tofighi & MacKinnon, 2011) suggested that the as ymmetric distribution and the bias corrected bootstrap methods order solution which is the most popular method (MacKinnon et al, 2002), the asymmetric distribution, and the bias corrected bootstrap methods were compared in this study. The results showed that the bias corrected bootstrap method performed best in terms of controlling empirical Type I error and power rates. However, the bias corrected bootstrap method produced very low Type I error rate when both paths and were zero and therefore, we still need to find or develop a method which can produced more accurate Type I error rates estimates related to the med iated effect estimates when both paths and are zero and when one of path is zero and the sample size is relatively small (100 and 200). PAGE 106 10 6 Figure 5 1. 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