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PAGE 1 1 EFFECTS OF OUTLIER ITEM PARAMETERS ON IRT CHARACTERISTIC CURVE LINKING METHODS UNDER THE COMMONITEM NONEQUIVALENT GROUPS DESIGN By FRANCISCO ANDRES JIMENEZ A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS IN EDUCATION UNIVERSITY OF FLORIDA 2011 PAGE 2 2 2011 Francisco Andres Jimenez PAGE 3 3 To my father, from whom I learned that persistency is one of the greatest qualities that a human being can possess. To my mother, who has always encouraged me to pursue my dreams and goals in life. To my sisters, who since the day they were born, repres ent a continuous source of joy and admiration to me. PAGE 4 4 ACKNOWLEDGMENTS I wish to thank all who have helped me make progress in academia throughout these years at the University of Florida I would specially like to show my gratitude to my advisor, Dr. David Miller, for his guidance, support, and patience during this process. I would also like to thank my committee member Dr. Walter Leite for introducing me to the world of simulation work using the R software. I am also grateful to Dr. James Algina for sharing his experience and knowledge on the application of statistical methods in the social sciences and education I owe my deepest gratitude to my fa mily for their endless encouragement and support during all these years that I have been away from home. This task would have been impossible without their unconditional love and understanding. Last, my thanks to Yasemin Kaya, Fernando Pagliai, and David Home for the unforgettable times and their invaluable friendship. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 7 LIST OF FIGURES .......................................................................................................... 8 ABSTRACT ..................................................................................................................... 9 CHAPTER 1 INTRODUCTION .................................................................................................... 10 2 LITERATURE REVIEW .......................................................................................... 14 2.1 Item Res ponse Theory Parameter Estimation (IRT) ...................................... 14 2.2 IRT Linking Methods Under the CommonItem Nonequivalent Groups Design ............................................................................................................ 16 2.2.1 Commonitem Nonequivalent Groups Design ..................................... 16 2.2.2 Scale Transformation Process ............................................................. 17 2.2.3 Haebara Method .................................................................................. 20 2.2.4 Stocking Lord Method ......................................................................... 21 2.3 Studies Comparing Different IRT Linking Methods ........................................ 22 2.4 Effects of Estimation Errors and Outlier Common Items on IRT Linking and Equating .................................................................................................. 27 3 METHODS .............................................................................................................. 32 3.1 Equating Design ............................................................................................. 32 3.2 Factors Manipulated ....................................................................................... 34 3.2.1 Group Ability Differences ..................................................................... 34 3.2.2 Types of Outliers ................................................................................. 35 3.2.3 Number of Outliers .............................................................................. 36 3.2.4 IRT Characteristic Curve Linking Methods .......................................... 37 3.3 Factors Held Constant .................................................................................... 37 3.3.1 Sample Size ........................................................................................ 37 3.3.2 IRT Model ............................................................................................ 37 3.4 Data Generation ............................................................................................. 37 3.5 Evaluation Criteria .......................................................................................... 41 3.6 Data Analysis ................................................................................................. 43 4 RESULTS ............................................................................................................... 44 5 DISCUSSION ......................................................................................................... 49 PAGE 6 6 5.2 Conclusion ..................................................................................................... 52 5.1 Limitations and Suggestions for Future Research .......................................... 53 REFERENCES .............................................................................................................. 55 BIOGRAPHICAL SKETCH ............................................................................................ 58 PAGE 7 7 LIST OF TABLES Table page 3 1 Item parameters for unique items 1 to 39 on Forms X and Y ( j = 39) ................. 39 3 2 Item parameters for common items 40 to 61 on Forms X and Y ( j = 22) ............ 40 4 1 Mean sq uare errors under the no outliers condition ........................................... 44 4 2 Mean square errors under the second outlier condition ...................................... 45 4 3 Mean square errors under the third outlier condition .......................................... 46 4 4 Mean square errors under the fourth outlier condition ........................................ 46 4 5 Mean square errors under the fifth outlier condition ........................................... 47 4 6 Mean square errors under the sixth outlier condition .......................................... 48 PAGE 8 8 LIST OF F IGURES Figure page 2 1 3PL model ICCs for two items with different lower asymptote values. ............... 16 PAGE 9 9 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Arts in Education EFFECTS OF OUTLIER ITEM PARAMETERS ON IRT CHARACTERISTIC CURVE LINKING METHODS UNDER THE COMMONITEM NONEQUIVALENT GROUPS DESIGN By Francisco Andres Jimenez August 2011 Chair: M. David Miller Major: Research, Evaluation, and Measurement Methodology The comparability of test scores on alternate forms of a test has become a matter of considerable importance for test developers and users. This study used a Monte Carlo simulation to investigate th e effects of common items with outlier b and a parameter estimates on t he Stocking Lord (Stocking & Lord, 1983) and Haebara ( Haebara, 1980) IRT characteristic curve linking methods under the commonitem nonequivalent groups design. The simulation conditions examined in this study included different levels of ability distribution, types of outliers, and number of outliers. Results indicat ed that neither method seemed to be robust to the presence of common items with outlier b parameter or items with an interaction of outlier b and a parameters although the Haebara method tended to perform better than the Stocking Lord method when there w ere outlier common items with extreme a parameter. PAGE 10 10 CHAPTER 1 INTRODUCTION In educational testing, there are many situations in which different groups of examinees are measured with different forms of a test that are supposed to measure the same latent c onstruct (Crocker & Algina, 1986). Since it is very difficult to develop multiple forms of a test in such a way that they are entirely comparable in content, difficulty, or reliability, there is always a possibility that some examinees may be unfairl y advantaged because of differences in the difficulty or reliability of the forms rather than real differences in their achievement levels (Dorans & Holland, 2000; Kolen & Brennan, 2004). Therefore, in these situations the comparability of test scores on alternate forms of a test becomes a matter of considerable importance for test developers and users. Test equating refers to a statistical and psychometric process for adjusting test scores on different forms of a test so that the scores on these forms c an be used interchangeably (Dorans & Holland, 2000; Kolen, 2004; Kolen & Brennan, 2004). The main purpose of test equating is to make scores on alternate forms of a test comparable by eliminating the effect of difficulty differences between such test form s (Cook & Eignor, 1991; Dorans & Holland, 2000; Hambleton & Swaminathan, 1985; Kolen & Brennan, 2004). Thus, if the equating process is successful, then in principle the administration of either form of the test should not make any substantial difference t o the individuals performance or comparability (Lord, 1980). Equating can be accomplished using either classical test theory (CTT) or item response theory (IRT) methods. Since one of the major contributions of IRT to the measurement field has been its ability to place tests and groups of examinees on a common scale of measurement, IRT based equating procedures are widely used PAGE 11 11 ( Baker & Al Karni, 1991; Hambleton & Swaminathan, 1985; Kolen & Brennan, 2004). According to Cook and Eignor (1991), IRT equating methods offer two main theoretical advantages over CTT methods: a) IRT equating be the most appropriate method when nonrandom groups of examinees with different ability levels are administered tests of differing difficulties, and b) because IRT provides i nvariant item parameter estimations, any resulting transformations should be the same regardless of the sample of individuals used to estimate them. Another statistical IRT method similar to equating is linking (Kolen & Brennan, 2004). However, linking r efers to the process of estimating a linear or nonlinear relationship between scores from separate tests that are not necessarily built to the same content or statistical s pecifications (Kim & Kolen, 2006; Kolen, 2004). In practical terms, linking assumes independent test administrations of multiple test forms to two or more samples of examinees (Kim & Kolen, 2006). Although the statistical procedures used for equating and linking are quite similar, their purposes are different. While equating is used to adjust for difficulty differences between forms of a test to be equated, linking refers to finding a correspondence between parameter estimates on a same scale (Dorans & Holland, 2000; Kolen & Brennan, 2004; Lee & Ban, 2010). Equating and linking of scores of examinees on various tests can be conducted only under certain testing circumstances. For instance, two different tests administered to two different groups of examinees cannot be equated (Hambleton & Swaminathan, 1985). Therefore, there is a vari ety of designs that can be used for collecting data for performing linking and equating (Hambleton & Swaminathan, 1985; Kolen & Brennan, 2004). Of these designs, the commonitem nonequivalent groups design is utilized PAGE 12 12 when different forms of a test cannot be administered at once because of test security issues or other practical apprehensions (Kolen & Brennan, 2004). In the commonitem nonequivalent groups design, two forms of a test that share a common set of items are administered to different groups of examinees, which allows to adjust for differences on the difficulty of both test forms while pres erving any ability differences on the separate groups (Kolen & Brennan, 2004). According to Kolen & Brennan (2004) the set of common item s should be a representative sample of the total test forms in terms of content and statistical characteristics and t heir difficulty and discrimination parameters ( a s and b s) should also be similar once they are transformed into the same proficiency scale. However, some outlier item parameters can appear in the calibration of the common items due to reasons such as esti mation errors, previous experience with the common items, context effects, or differential curriculum emphasis (Stocking & Lord, 1983; Hu, Rogers, & Vukmirovic, 2008; Kaskowitz & de Ayala, 2001; Kolen & Brennan, 2004; Michaelides & Haertel 2004). While previous research has examined the effects of items with outlier difficulty parameter estimates ( b parameter) on IRT based equating and linking methods (Bejar & Wingersky, 1981; Cohen & Kim, 1998; Hanson & Feinstein, 1997; Hu et al. 2008; Linn Levine, Has tings, & Wardrop, 1981; Michaelides 2003; 2010; Michaelides & Haertel 2004; Sukin & Keller 2008; Stocking & Lord, 1983), there are no studies addressing the effects of common items with outlier discrimination parameter ( a parameter), or with both outlier difficulty and discrimination parameters, on two popular IRT characteristic curve transformation linking methods: the Stocking Lord method (SL; Stocking & Lord, 1983) and the Haebara (HA; Haebara, 1980) method. PAGE 13 13 Therefore, the purpose of this study is to examine the effects of having commonitem with outlier b and a parameter estimates on the Stocking Lord and Haebara methods under the commonitem nonequivalent groups design using simulated data. The main advant age of using these two IRT characteristic curve methods to determine the effect of outlier common items is that both represent IRT linking methods that consider all of the item parameter estimates simultaneously and generate more stable results than other IRT linking methods (Kim & Cohen, 1998; Hanson & Bguin, 2002; Kolen & Brennan, 2004; Kim & Kolen, 2006; Lee & Ban, 2010). In addition, t he following research questions are addressed: 1. Do the Stocking Lord and Haebara methods differ with respect to bia s in the linking process when there are common items with outlier b parameter? 2. Do the Stocking Lord and Haebara methods differ with respect to bia s in the linking process when there are common items with outlier a parameter? 3. Do the Stocking Lord and Haebara methods differ with respect to bias in the linking process when there are common it ems with an interaction of outlier b and a parameters? 4. Is the performance of the Stocking Lord and Haebara linking methods affected by ability distribution or number of outlier s? PAGE 14 14 CHAPTER 2 LITERATURE REVIEW The present literature review chapter begins with an overview of the IRT models, specifically of the threeparameter logistic model (3PL). Second, a description of the different IRT based linking methods under the common item nonequivalent group designs is given with a focus on the Stocking Lord and Haebara methods Then, a series of studies comparing different IRT linking methods is presented. Finally, a discussion on the research studies examining the effects of estimation errors and outlier common items on IRT linking and equating is given. 2.1 Item Response Theory Parameter Estimation (IRT) Item response theory (IRT) methods are widely used in many largescale testing applications, including test linking and equating. One of the main advantages of the use of IRT equating methods over classical methods of equating (Angoff, 1982; Kolen & Brennan, 2004) relies on the possibility to obtain invariant item parameter estimations from different test forms that can be placed on a common scale of measurement ( Baker & Al Karin, 1991; Cook & Eignor, 1991; Kolen & Brennan, 2004; Lee & Ban, 2010). Most IRT models establish that such common ability scale can be represented by a single latent trait or unidimensional c onstruct, referred to as theta, ( de Ayala, 2009; Kolen & Brennan, 2004). Accordin g to IRT, the ability parameter of an examinee, as well as the item parameters, is invariant across subsets of items or across different groups of examinees. This implies th at, when the IRT model fits the dataset being analyzed, the same item characteristic curve (ICC) will be obtained for the test items regardless of the groups ability distributions or the subsets of items used to estimate the item parameters ( de Ayala, 2009; Hambleton, Swaminathan, & Rogers, 1991). PAGE 15 15 Among the various IRT unidimensional models, the threeparameter logistic model (3PL) is spec ifically used in cases when the chance of success on an item needs to be addressed, as in large scale assessments incor porating multiple choice item tests ( de Ayala, 2009; L ord, 1980). In this model, the probability that an examinee with ability equal to correctly responds to item j is defined as: ( 2 1) where the item parameters aj, bj, and cj are associated with item j ( Hambleton & Swaminathan, 1985; Kolen & Brennan, 2004). The item parameter bj refers to as the difficulty or location parameter for item j This parameter corresponds to the ability level where the probability of correctly responding to an item is equal to 0.5. On the other hand, the item parameter aj represents the discrimination parameter. This parameter is proportional to the slope of the ICC at its inflexion point, bj (de Ayala, 2009; Kolen & Brennan, 2004). Finally, t he item parameter cj is the lower asymptote or pseudoguessing level parameter for item j This parameter represents the probability that an examinee with very low ability level (i.e., = correctly responds to item j An example of the 3PL model is g iven in Figure 21. There are two items that have the same location ( b1 and b2 equal to 0.00 ) and discrimination ( a1 and a2 equal to 1.5 0 ) parameters, although they have different pseudoguessing parameters. For item 1, c1 = 0.1 0 and for item 2, c2 = 0.05. B oth ICCs ha ve nonzero lower asymptotes and are asymptotic with respect to their corresponding aj value (de Ayala, 2009). As said before, th e lower asymptote corresponds to t he lower possible value for the ICC and represents the smallest probability for a correct response (de Ayala, 2009). PAGE 16 16 a1 = 1.50 b1 = 0.00 c1 = 0.10 a2 = 1.50 b2 = 0.00 c2 = 0.05 Figure 21 3PL model ICCs for two items with different lower asymptote values As can be seen in Figure 2 1, the ICC for item 1 is located higher than the ICC for item 2. T his is due to the fact that as cj increases, the probability of correct response also increases, all other things being equal (de Ayala, 2009). T hus, items with larger cjs become easier for examinees than those items with s maller cjs. 2.2 IRT Linking Methods Under the CommonItem Nonequivalent Groups Design 2.2.1 Commonitem Nonequivalent Groups D esign There are several types of equating designs. Some approaches involve the administration of two test forms to a single group of examinees (single group with counterbalancing random groups design, or common item random groups design) and PAGE 17 17 others use two samples of examinees with each sample being administered one of the test forms, as the commonitem nonequivalent groups design (de Ayala, 2009; Kolen & Brennan, 2004). The common item nonequivalent groups design is often used when more than one test form cannot be administered at once due to test security issues or other practical apprehensions (Kolen & Brennan, 2004). In this design, different groups of examinees take two test forms, Form X and Form Y which have a set of items in common. When the score on the chosen set of common items does not contribute to the examinees overall score on the total test, the set of common items is said to be external (Kolen & Brennan, 2004). On the other hand, when the performance of the examinees on the set of common items is taken into account in the examinees overall score on the total test, the set of common items is said to be internal (Kolen & Brennan, 2004). According to Kolen and Brennan (2004) and de Ayala (2009), the set of common items should be proportionally representative of the total test forms in both content and statistical properties. In other words, the set of common items sho uld represent a smaller version of the original test. 2.2.2 Scale Transformation P rocess Under the commonitem nonequivalent groups design, both groups of examinees are not considered to be equivalent and thus the item parameters obtained from the two diff erent test forms ( X and Y ) need to be placed on the same IRT scale in order to conduct the IRT linking process (Kolen & Brennan, 2004). This scale transformation process is possible because the test forms have been created t o have a set of items in common, which is generally known as the anchor test (de Ayala, 2009; Kolen & Brennan, 2004). PAGE 18 18 According to de Ayala (2009) and Kolen and Brennan (2004), the anchor test should measure the same construct, the same content specifications, and the same contextual effects as the noncommon items on the test. T here are two subtypes of common item nonequi valent groups designs: 1) the internal anchor design, which takes into account the score on the set of common items as part of the observed score of the examinees, and 2) the external anchor design, which does not consider the score on the common items as part of the examinees observed score (de Ayala, 2009). If an IRT model fits the responses on the test forms to be linked, then any linear trans formation of the ability scale, based on the set of common items, will also fit the datasets without modifying the probability of correct response given that the item parameters have also been transformed (Kolen & Brennan, 2004). Therefore, one str ategy to perform IRT linking is by using a linear transformation to convert the IRT parameters estimates in to the same scale (de Ayala, 2009, Kolen & Brennan, 2004). The linear transformation from one metric to the other in terms of the IRT parameters can be represented by the following set of equations : (2 1) (2 2) (2 3) (2 4) where aYj, bYj, and cYj represent the item parameters for item j on Form Y and aXj, bXj, and cXj represent the item parameters for item j on Form X Equation 23 shows that PAGE 19 19 the lower asymptote values for both forms remain the same since they are expressed on the probability metric In addition to how the item parameters on both forms are related, Equation 24 shows the relationship between the theta values for examinees on both test forms (Kolen & Brennan, 2004). On the other hand, A and B are referred as the linking coefficients or metric transformation coefficients (de Ayala, 2009). These metric transformation coefficients in terms of groups of items can be expressed as follows: (2 5 ) (2 6 ) and (2 7 ) Several methods for computing the linking coefficients have been proposed. The most direct methods consist in substituting the means and standard deviations of the item parameter estimates of the comm on items for the parameters in E quations 21 and 2 2 (Kolen & Brennan, 2004). One of these methods is the Mean/S igma procedure and was developed by Marco (as cited in Kolen & Brennan, 2004). Such method uses the means and standard deviations of the b parameter estimates from the common items in Equati ons 2 5 and 27 In addition, Loyd and Hoover ( 1980) proposed t he Mean/M ean transformation procedure, which consists in using the mean of the a parameter estimates fr om the common items in Equation 26 to estimate the A constant. Following the mean of the b parameter estimates of the common items is used in place o f the PAGE 20 20 parameters in Equation 27 to calculate the B cons tant (Kolen & Brennan, 2004). According to Kolen and Brennan ( 2004), the Mean/S igma method is preferable over the Mean/M ean method because the estimates of the b parameters are more stable than the estimates of the a parameters A potential limitation of both the Mean/S igma and Mean/M ean methods relates to the use of only means and standard deviations of the item location parameter estimates in the transformation equations When combinations of a b and c parameter esti mates generate almost identical ICCs over the range of ability at which most examinees score, both methods can be excessively influenced by marginal differences in the a or b parameter estimates (Kolen & Brennan, 2004). In contrast to the methods presented above, a second type of scale transformation procedure considers all of the item parameters simultaneously in order to determine the A and B constants (de Ayala, 2009). T hese methods are commonly known as the IRT characteristic curve transformation linking methods and their purpose is to relate as closely as possible the initial total characteristic function (TCF) with that of the new metric (de Ayala, 2009). The most relevant characteristic curve methods were developed by Haebara ( HA; 1980) and Stocking and Lord ( SL; 1983) in the early 80s to overcome the flaws of the Mean/Sigma and Mean/M ean methods. 2.2.3 Haebara M ethod The HA procedure consists in a weighted least squares method involving the use of a loss function and an optimization process (Haebara, 1980). The function used by Haebara (1980) to denote the difference between the ICCs is the sum of the squared difference between the ICCs for each of the test items for examinees located at a PAGE 21 21 particular ability level (Kolen & Brennan, 2004). Then, for a given ability level i, the sum, over items, of the squared difference can be expressed as: (2 8 ) (Kolen & Brennan, 2004). Equation 28 shows that the summation is over the set of common items ( j:V ). As seen, the difference between each ICC on the two scales is squared and summed. This difference is then added up over examinees. The Haebara function aims to find values for A and B such th at they minimize the following condition over examinees : (2 9 ) (Kolen & Brennan, 2004). 2.2.4 StockingLord M ethod Stocking and Lord (1983) developed a robust procedure that gives small weights to those item parameter estimates whose perpendic ular distance from the weighted estimate of the transformation is large. Because of its iterative nature, the weighting procedure is repeated until changes on the perpendicular distances become marginally small. The SL method can be described by the follow ing equation: (2 10) (Kolen & Brennan, 2004). In contrast to the HA method, the SL method summates across items for each set of parameter estimates before it is squared. In this method, the term SLdiff ( i) PAGE 22 22 represents the squared difference between the test characteristic curves for a given theta level of an individual, whereas the term Hdiff ( i) in the HA method is the sum of the squared difference between the ICCs for a given ability level Similar to the HA method the SL method aims to find values for A and B such that they minimize the following condition over examinees: (2.11) (Kolen & Brennan, 2004). 2.3 Studies Comparing D ifferent IRT Linking Methods Several empirical studies have compar ed both characteristic curve methods against one another or against other methods of estimation. Baker and Al Karni (1991) were one of the first authors to compare two IRT linking methods: the SL and Mean/Mean methods. The authors simulated data for four 60i tem tests under the 3PL model and examined three basic types of equating designs: an IRT parameter recovery study, horizontal equating, and vertical equating. In the recovery section of the study, three different normal distributions of ability were used: low ( M = 0.5, SD = 0.25), medium ( M = 0, SD = 1) and high ( M = 0.5, SD = 2.25). Additionally, two ranges for the discrimination parameters were used: low (0.6 to 0.94) and high (1.53 to 1.87), as well as two normal distributions for the difficulty paramet ers: low ( 0.5 to 0.35) and high (0.5 to 0.25). In the horizontal equating section of the study, results from the recovery study were analyzed in order to select two datasets (one with the largest overall root mean square values and another with the lowest values) within each of the three ability levels. In the vertical equating design section of the study, three different normal distributions of ability were specified: low ( M = 0.5 SD = 1), medium ( M = 0, SD = 1), and high ( M = PAGE 23 23 0.5, SD = 1); and for each of the 60 item test, three different distributions of a b and c parameters were defined for a set of 15 common items and for the remaining 45 unique items. Baker and Al Karnis (1991) simulation results indicated that, in the recovery study, the values for the A and B transformation coefficients were very similar under the SL and Mean/Mean methods across conditions and overall the root mean square error values for both methods were small except for the low ability, lowdiscrimination and high difficulty dataset. When comparing the SL method to the Mean/Mean method in terms of performance in this particular dataset, the SL coefficients produced TCFs closer to the underlying TCFs than the Mean/Mean method did (Baker & Al Karni, 1991). In the vertical equating design, the values of the loss function and root mean square errors were quite small for both IRT linking methods, although the SL method showed smaller values than the Mean/Mean method did (Baker and Al Karni, 1991). Kim and Cohen (1998) examined the performance of three different methods for estimating a common metric using simulated data based on the 2PL model: 1) the SL method, 2) concurrent calibration using marginal maximum a posteriori estimation, and 3) concurrent calibration using marginal maximum likelihood estimation. They simulated data for 50 items and 500 examinees and manipulated factors such as the mean of the ability distribution ( M = 0 and 1), number of common items ( j = 5, 10, 25, and 50), and IRT estimation program (BILOG and MULTILOG; Kim & Cohen, 1998). Results indicated that, for the discrimination parameters, the SL method generally produced smaller root mean square differences between the equating coefficients and expected va lues than the other two methods did (Kim & Cohen, 1998). In terms of the item PAGE 24 24 difficulty parameters, the SL method likewise produced better root mean square differences than both concurrent calibration methods, especi ally under the 5and 10common item conditions (Kim & Cohen, 1998). Finally, Kim and Cohen (1998) also reported that all three linking methods performed equally well with larger numbers of common items (i.e., more than 5 common items). In another similar study, Hanson and Bguin (2002) investigated the performance of concurrent versus separate item parameter estimation under the commonitem equating design using simulated data based on the 3PL model. For a 60item dichotomous test, they simulated condi tions under different sample sizes ( n = 1,000 and 3,000), number of commonitems ( j = 10 and 20), equivalent versus nonequivalent groups, and IRT estimation program (BILOG MG and MULTILOG). When comparing the SL and HA methods to the Mean/Mean and Mean/Si gma methods, they concluded that both IRT characteristic curve methods outperformed the Mean/Mean and Mean/Sigma methods in terms of the total error in the estimated true score equating function at a given score (Hanson & Bguin, 2002). On the other hand, the SL and HA methods performed equally well and neither method had consistently lower errors across conditions than the other (Hanson & Bguin, 2002). However, when comparing both IRT characteristic curve methods to the concurrent calibration method the latter resulted in lower errors across conditions except in conditions under a sample size of 3,000, equivalent groups, and 10 common items. In a different set of studies Bguin, Hanson, and Glas (2000) Bguin and Hanson (2001) and Kim and Kolen (2006) evaluated the performance of unidimensional IRT linking methods on multidimensional data. In order to compare the SL method against PAGE 25 25 the concurrent calibration method, Bguin et al. (2000) simulated multidimensional compensatory data for 1) equivalent and nonequivalent groups 2) different levels of covariance between two dimensions and 3) unidimensional and multidimensional parameter estimation. Results showed that under the equivalent groups conditions, the SL method resulted in generally larger mean sq uare errors than the concurrent estimation method, whereas in the nonequivalent groups conditions, both SL and concurrent methods performed poorly Therefore, Bguin et al. (2000) concluded that multidimensionality could affect the performance of unidimens ional IRT linking methods. Similarly, Bguin and Hanson (2001) carried out a simulation study to co mpare the performance of the SL and concurrent calibration methods for unidimensional IRT models when applied to multidimensional noncompensatory data They simulated data based on real item parameters taken from three pairs of test forms of a standardized measur e of language comprehension and examined the following conditions: 1) equivalent and nonequivalent groups, 2) three levels of covariance between the test forms, and 3) multidimensional and unidimensional estimation. Using the same evaluation criteria than Bguin et al. (2000) did, Bguin and Hanson (2001) also concluded that the concurrent estimation method led to smaller mean square errors than the SL estimation method under the equivalent groups condition, although for Bguin et al. (2000) the effect of the type of multidimensionality seemed to vary more as a function of the size of the covariance between dimensions. Kim and Kolen (2006) conducted a simulation study to examine the degree to which four IRT linking methods (Mean/Mean, Mean/Sigma, HA, and SL) and the concurrent calibration method were robust to format effects in mixed format tests under PAGE 26 26 the commonitem nonequivalent groups design. The au thors developed a series of simulations using two types of mixedformat tests as part of the linking design: wide range and narrow range, each of which had a set of 12 multiplechoice (MC) items and five constructedresponse (CR) items in common, had different target information functions, and was based on the item parameters of the science assessment from the 1996 National Assessment of Educational Progress report (as cited in Kim & Kolen, 2006). While the widerange test aimed to replicate common standardized achievement and aptitude tests, the narrow range test aimed to characterize screening tests used other testing situations (Kim & Kolen, 2006). Kim and Kolen (2006) simulated data under three factors: 1) three levels of format effects as correlations between 1 and 2 (0.5, 0.8, and 1), 2) two types of mixedformat tests (wide range and narrow range), and 3) three levels of nonequivalence in linking as values for bivariate distributions ( 1 = 0, 1 = 1, 2 = 0, 1 = 1; 1 = 0.5, 1 = 1, 2 = 0.5, 1 = 1; and 1 = 1, 1 = 1, 2 = 1, 1 = 1). For the narrow range, mixedformat test, r esults indicated that under a combination of nonequivalence and format effects, the Mean/Mean and Mean/Sigma methods had consistency larger mean square errors across point scores than the SL and HA methods, yet there were no substantial differences between these last two. On the other hand, the concurrent calibration method showed consistently lower mean square errors across all score points than the four IRT linking methods (Kim & Kolen, 2006). Nevertheless, the difference between the concurrent calibration method and the SL and HA methods was only slight. For the wide range, mixedformat test, the pattern of results was very similar to what was observed in the narrow range test. However, the widerange showed the m aximum PAGE 27 27 values for the root mean square errors in the middle of the observed score range, whereas the narrow range showed the maximum values in the lower part. In a recent simulation study, Lee and Ban (2010) com pared the performance of four different IRT linking methods in the random groups equating design: 1) concurrent calibration method, 2) SL method, 3) HA method, and 4) proficiency transformation. In order to simulate data, the authors used real item paramet ers from two 75item ACT English form s, each of which was taken for approximately 3,000 examinees and was analyzed using a 3PL model. The authors manipulated four different factor s in their study: 1) combinations of sampling design s ( normal distributions w ith proficiency mean s of 0, 0, and 0; 0.5, and 0.5; and 0, 1, and 1), sample size ( n = 500 and 3,000), and number of items ( j = 25 and 75 ). Lee and Ban (2010) stated that on average, the SL and HA methods outperformed the concurrent calibration and proficiency transformation methods under all conditions, except when the samples were obtained from the same population. On the other hand, mean square errors were substantiall y lower for n = 3,000 and linking errors were prompt to increase as the difference in the proficiency level between the sampling designs also did (Lee & Ban, 2010). With respect to the SL and HA methods compared to one another, the HA method generally tended to perform better than the SL method, producing lower linking errors in average across conditions (Lee & Ban, 2010). 2.4 Effects of Estimation Error s and Outlier Common Items on IRT Linking and Equating Relatively few empirical studies have addressed t he implications of estimation errors and outlier common items on IRT linking and equating methods. Kaskowitz and PAGE 28 28 de Ayala (2001) investigated the effect of error s in item parameter estimates on both the estimation of linking coefficients with the SL method and the subsequent estimation of the examinees true score. Using a horizontal internal anchor equation design, the authors simulated data by generating item parameter estimates for initial and target metrics from known sampling distributions (Kaskowitz & de Ayala, 2001). Four different factors were manipulated: number of common items (5, 15, and 25), IRT model (2PL and 3PL), relationship of the standard error to the parameter estimate (nonrelated error and related error), and error level based on the sample size (low, moderate, and high). Results showed that the SL method was overall robust in relation to the amount of error in the item parameter estimates (Kaskowitz & de Ayala, 2001) In the nonrelated error condition, the bias and root mean square error s for A and B improved as the level of error decreased, and the poorest estimation of A took place in the conditions under the 5common item s factor. In the related error condition, the bias and root mean square errors were higher than for the nonrelated error condition, yet a similar pattern of improvement was seen as the level of error decreased. Interestingly, with the error level being constant, A and B always showed greater levels of estimation error and bias for the 3PL than for the 2PL model (Kaskowi tz & de Ayala, 2001). As part of his dissertation work, Michaelides (2003) conducted a threephase study that examined the effects of misbehaving common items in IRT based equating processes. Of interest i s the first applied stage of his study where the au thor examined the consequences of either retaining or eliminating items identified as outliers by using the deltaplot method proposed by Angoff in 1972 (as cited in Michaelides, 2003). The author analyzed two test administrations (Year 1 and 2) from four statewide PAGE 29 29 assessments (8th grade Mathematics, 11thgrade Science, 6th grade Social Studies, and 6thgrade Science) by fitting the 1PL and 3PL model for dichotomous items, and the graded response model for polytomous items. Michaelides (2003) compared the S L, HA, Mean/Sigma, and Mean/Mean IRT based linking methods. Results indicated that when outliers (three items in this case) were included in the linking process, the Year 2 mean was higher and the 1PLmodel analyses gave positive gains in the standard error of equating for Year 2 that were as twice as large than for Year 1. However, when the 3PL was fitted, there was a slight decline with the SL method, yet a larger drop with the HA and Mean/Mean and a similar drop in magnitude gain with the Mean/Mean metho d. In another study examining the effects of outlier common items on IRT linking Hu et al. (2008) conducted a simulation study to compare four IRT based linking methods when items with outlier b parameters were either ignored or considered under the common item nonequivalent groups design. The authors simulated data under the 2PL (short answer SA, and openended response, OR, items) and 3PL (multiplechoice MC, items) models by using real item parameters from the Massachusetts Comprehensi ve Assessment System mathematics tests (as cited in Hu et al., 2008) T he following conditions were manipulated in their linking design: 1) group ability distribution ( M = 0, SD = 1 and M = 1, SD = 1), 2) number/score points and types of outliers (no outli ers, three MC items wit h three score points from one content area, three MC items randomly distributed across content areas, three MC items with extreme b parameters, five MC and one OR items with nine score points from one content area, five MC and one OR items with nine score points that were randomly distributed across content areas), and 3) IRT based equating designs: concurrent calibration, separate PAGE 30 30 calibration with the SL and HA methods, separate calibration with the Mean/Sigma method, and calibration with fixed common item parameters (FCIP). Study results showed that when no outlier common items were present and groups were equivalent, all IRT based equating methods performed well (Hu et al., 2008) For items with three score points all IRT based linking methods showed similar mean square error results no matter if outliers were present in one content area or across the five content areas. However, for equivalent groups, the concurrent calibration with outliers included and the fixed common item parameter calibration had the smallest mean square errors, followed by the characteristic curve methods, and finally, by the Mean/Sigma method. For nonequivalent groups, the characteristic curve, Mean/Sigma and FCIP methods had moderate mean square errors, whereas the concurrent calibration method had large mean square errors (Hu et al., 2008). Similar to the results obtained with three score points, there were no differences between the IRT based linking methods under the nine score points conditions when outliers were present in one content area or across the five content areas. For equivalent groups, m ean square errors tended to increase as the number/score points of outliers also increased (Hu et al., 2008). For nonequivalent groups when ou tliers were included in the design, the mean square errors for both characteristic curve and Mean/Sigma methods also tended to increase as the number/score points of outliers increased, although for the concurrent calibration method tended to remain the sa me and for the FCIP method they remained small (Hu et al., 2008). Sukin and Keller (2008) also conducted a simulation study to explore the effects of removing or maintaining items with misbehaving b parameters under the commonitem PAGE 31 31 nonequivalent groups design. Using real item parameters from an operational statewide testing program, two test forms were simulated, each of which had a twenty item internal anchor and was analyzed based on the 3PL model. Sukin and Keller (2008) manipulated conditions under two normal ability distributions ( M = 0, SD = 1 and M = 0.2, SD = 1 ) four different IRT based linking methods (Mean/Sigma, Mean/Mean, SL, and HA), three cut scores ( 0.75, 0, and 0.75 on the metric), and two type s of outlier items ( a shift of 0.3 and 0.8 in the b parameter scale) Sukin and Keller (2008) simulation results showed that the overall classification accuracy was not affected by the presence of outlier common items, although the proportion of examinees over and under classified was. When the outlier item was kept in the estimation, about 1% fewer examinees were being identified as under classified in comparison to when the outlier item was excluded from the IRT based linking methods. A similar pattern of results and no real observed differences were obtained across both types of ability distribution, IRT linking methods, and cut scores (Sukin & Keller, 2008). PAGE 32 32 CHAPTER 3 METHODS In the present study, a Monte Carlo simulation was conducted to compare the performances of the Stocking Lord (SL; Stocking & Lord, 1983) and Haebara (HA; Haebara, 1980) IRT linking methods in the presence of outlier common items. Data were simulated for a commonitem nonequivalent groups matrix design. In addition, several factors were manipulated t o reflect different levels of group ability differences, types of outlier s, and number of outliers Following there is a detailed description of the equating design, factors manipulated and factors held constant in this study. Finally, a description of the data generation process evaluation criteria, and data analysis process is given. 3.1 Equating Design A common item nonequivalent groups design using an external anchor test was employed in the current study. Since the results of this study are in part intended to be generalized to large scale achievement tests using IRT based linking methods, the common item nonequivalent groups design was chosen to closest replicate the horizontal equating process utilized in the 2005 and 2006 administration s of the Fl orida Comprehensive Assessment Test (FC AT) mathematics test for 4thGrade (F lorida Department of Education (FDOE), 2006, 2007) In order to maintain the comparability of the FCAT scale year after year, FCAT developers indeed use the SL procedure to establi sh a statistical correspondence bet ween the performance of current year students on the external anchor items and the performance of students on the same set of items in the previous years of testing ( F DOE 2006, 2007). PAGE 33 33 T his st udy employed a linking desig n in which i n Year 1 ( i.e., 2005) a test F orm X was administer ed to a sample of 4th Grade students This test form included a set of unique (or core) items (i.e., UX) and a set of common items (i.e., CZ). In Year 2 ( i.e., 2006), another test Form Y was administered to a nother sample of 4th Grade students as well. The unique items administered on Form Y (i.e., UY) were different from those items administered on Form X in Year 1. Nonetheless, the same set of common items was used on Form Y (i.e., CZ). In this sense, examinees scores on UX and UY can be linked thorough such set of common items, CZ. The FCAT mathematics test composition and format vary depending upon the grade. According to the Sun shine State Standards (SSS; Florida Department of Education, 1996), the FCAT mathematics test s are meant to assess five different content areas: 1) number sense, concepts, and operations, 2) measurement, 3) geometry and spatial sense, 4) algebraic thinking, and 5) data analysis and probability (FDOE 2006, 2007). Likewise, t here are three t ypes of FCAT mathematics item formats : multiple choice (MC), griddedresponse (GR), and performance task (PT; i.e., short and extendedresponse items). The item format used in the FCAT 4th Grade mathematics test only contains dichotomously scored, multiplechoice (MC) items I n 2005 and 2006, each FCAT 4th Grade mathemati cs form had 39 unique items plus 6 to 8 common or fieldte st items that were distributed across 30 different forms. For technical reasons, t he unique and common items in Forms 27 to 30 were finally selected f o r an early return IRT calibration sample of 4thgrade students (F DOE 2006, 2007) PAGE 34 34 Therefore, in the present study the IRT characteristic curve linking process was conducted by using real it em parameters from those 39 unique and 22 common it em s (61 items in total) used in such early return IRT calibration sample (FDOE 2006, 2007) As suggested by Kolen and Brennan (2004), a common item set should be at least 20% of the length of a total tes t containing 40 or more items. In this sense, t he 22 common items thoroughly represented, in statistical, format, and content terms, the same characteristics of the unique items. Given this equating design, the unique items in the test Forms Y and X were li nked to be expressed on the same scale. 3.2 Factors Manipulated 3.2.1 Group Ability D ifferences Similar to Hu et al. (2008) and Lee and Ban (2010) samples of item responses for Form Y were generated by sampling the latent trait ( ) from a normal independent distribution with mean equal to 0 and standard deviation equal to 1 ( NID (0, 1)). In addition, t wo sets of item responses were created for Form X by sampling from a n NID (0, 1) distribution and an NID (1, 1) distribution. Only for comparison purposes, the sa mples with NID (0, 1) for both F orms Y and X examined the specific situation in which t he two groups were equivalent. The samples with NID (0, 1) for Form Y and NID (1, 1) for Form X examined the case in which the groups are nonequivalent. According to Kolen and Brennan (2004), the larger the difference between groups of examinees the more difficult it becomes for the statistical methods to identify differences due to the groups or test forms. Hence, the result of any linking method becomes critical when the groups differ, especially in terms of the representativeness of the content and statistical properties. In addition, the findings of the current study can be compared with similar studies that have also included this condition (Hanson & Bguin, 2002; Hu et al., 2008). PAGE 35 35 3.2.2 Types of O utliers In the current study, an outlier common item was conceptually defined and operationalized primarily based on what Hu et al. (2008) graphically described as an outlier. In terms of its b parameter, Hu et al. (2008) defined a common item as an outlier if in the scatterplot of difficulties the distance between the intersection point drawn from each items xand yaxis position and its predicted position on the straight line was equal to or more than two score points. Thereafter, in the current study all the b parameters for the outliers in Year 2 (i.e., Form Y ) were two score points lower than in Year 1 (i.e., Form X ). This meant that only outliers located on the left side of the straight line were taken into consideration. Hu et al. (2008) gave a theoretical and practical explanation for considering this operati onal definition of an outlier. In educational practice, the presence of outliers is often due to the misbehavior of some common it ems, a change in the instructional emphasis of a certain content area, and item parameter estimation errors (Kaskowitz & de Ayala, 2001; Kolen & Brennan, 2004; Michaelides, 2003). Similarly, the most likely result of the exposure of some common items, and subsequently of the exposure to the instructional emphasis such items measure, is that those items will become easier when they are administered in the second year (Hu et al., 2008). Although Hu et al. (2008) did not examine common items with outlier a p arameter estimates, the current study used a somewhat similar operational definition to identify outlier items based on their a parameter. Theoretically, the discrimination parameter can vary from a parameter may indicate that its performance is severely inconsistent with the IRT model or that it is behaving in a counterintuitive fashion (de Ayala, 2009). Therefore, in this study, an a parameters PAGE 36 36 that w as greater than 1.000 in Year (i.e., Form X ) became an outlier by dropping one point score in Year 2 (i.e., Form Y ). On the other hand, an a parameter that was between 0.500 and 0.999 on Form X became an outlier by dropping 0.5 point scores on Form Y Lastly, an a parameter that was below 0.500 on Form X be came an outlier by dropping to a fixed value of 0.100. Similar to the case of items with outlier b parameter, only outliers with a parameter located on the left side of the straight line were taken into consideration for this study. 3.2.3 N umber of O utlie rs Seven combinations of number of common items with outlier b and a parameters were examined: 1) There were no outliers in the common items, 2) the outliers were two common items with extreme b parameter, 3) the outliers were two common items with extreme a p arameter, 4) the outliers were two common items with extreme b and a parameters (inter action of outlier parameters), 5) the outliers were four common items with extreme b parameter, 6) the outliers were four common items with extreme a parameter, 7) the outliers were four common items with extreme b and a parameters (interaction of outlier parameters). The first condition was used as the baseline against which the other conditi ons were compared. Hu et al. (2008) suggested that the baseline condition should well control the number/score points of outliers and representativeness of the common items. Conditions 2, 3, 5 and 6 simulated the effect of the increase of outlier common items with either extreme b or a parameters due to the impact of differential curriculum emphasis on the common items (Hu et al., 2008). Conditions 4 and 7 reflected the effect of an increase in the presence of outlier common items with an interaction of extreme b and a parameter s. PAGE 37 37 3.2.4 IRT Characteristic C urve Linking M ethods Two commonly used IRT characteristic curve linking methods, the Stocking Lord and Haebara methods, were used to link the tw o test forms with and without outliers. 3.3 Factors Held Constant 3.3.1 Sa mple S ize The sample size for t his study was held constant at 3,000 for each test form According to several research studies on equating methods (Hanson & Bguin, 2002; Hu et al., 2008; Kolen & Brennan, 2004; Lee & Ban, 2010 ), this sample size seems to be adeq uate to obtain stable parameter estimates. T herefore, t he total sample size considering both test forms was 6,000. 3.3.2 IRT M odel The threeparameter logistic model ( 3PL; Lord, 1980) was used to simulate the data for test F orms X and Y A threeparameter logistic model was chosen to account for MC items response data from low ability examinees for which guessing is always an important factor to consider in test performance ( de Ayala; 2009; Hambleton & Swaminathan, 1985). Additionally, the 3PL model was i ndeed used by the FCAT specialists to calibrate and analyze MC items from the 2005 and 2006 FCAT 4th Grade Mathematics tests (FDOE 2006, 2007). 3.4 Data Generation Each of the simulated datasets for Forms X and Y included responses to 61 items in total. Of these, 39 were unique dichotomous items and 22 were common dichotomous items. In order to simulate the item responses, item parameters aj, bj, and cj were taken from the 2005 and 2006 FCAT 4th Grade Mathematics test calibrations (FDOE 2006, 2007) Item parameters corresponding to the 2005 FCAT 4 th Grade PAGE 38 38 Mathematics tests were used to simulate responses on Form X whereas item parameters corresponding to the 2006 FCAT 4 th Grade Mathematics test were utilized to s imulate responses on Form Y Table 31 contains the item parameters used for the unique items on each form in the present simulation study. Table 32 shows the item parameters used for the common items on both forms. On Form X the aj parameters for the unique items used in the simulation ranged from 0.513 to 1.429, the bj parameters ranged from 2.004 to 1.011, and the cj parameters ranged from 0.034 to 0.562. On the other hand, on Form Y the aj parameters for the unique items ranged from 0.530 to 1.491, the bj parameters ranged from 2.640 to 0.802, and the cj parameters ranged from 0.063 to 0.404. With respect to the set of common items on both Forms X and Y the aj parameters ranged from 0.549 to 1.556, the bj parameters ranged from 1.911 to 0.427, and the cj parameters ranged from 0.036 to 0.386. In the first condition, all a and b parameters for the common items remained the same from Form X (Year 1) to Form Y (Year 2) because there were no outliers present in the common items. In Condition 2, the b parameter for item 40 dropped from 0.132 on Form X to 1.868 on Form Y, and the b parameter for item 49 dropped from 1.027 on Form X to 3.027 on Form Y In Condition 3, the a parameter for item 42 dropped from 1.156 on Form X to 0. 156 on Form Y and the a parameter for item 52 dropped from 1.556 on Form X to 0.556 on Form Y In Condition 4, the b parameter for common item 44 dropped from 0.434 to 2.434 and the a parameter did from 1.197 to 0.197 from Form X to Y, respectively. Sim ilarly, the b parameter for common item 54 dropped PAGE 39 39 from 0.001 to 2.001 and the a parameter did from 0.816 to 0.316 from Form X to Y respectively. Table 31. Item parameters for unique items 1 to 39 on Forms X and Y ( j = 39) Item number Form X Form Y a j b j c j a j b j c j 1 1.173 0.634 0.183 0.530 2.022 0.214 2 0.822 0.396 0.309 0.762 2.640 0.149 3 0.513 0.986 0.094 0.751 0.048 0.287 4 0.524 0.202 0.054 0.692 0.490 0.291 5 1.071 0.437 0.122 0.559 1.134 0.143 6 0.977 2.004 0.110 0.608 1.631 0.083 7 0.635 0.357 0.300 0.568 0.772 0.063 8 0.989 1.299 0.208 0.758 0.466 0.068 9 0.779 0.378 0.314 0.905 0.373 0.251 10 0.948 0.296 0.080 0.609 0.210 0.092 11 0.902 1.723 0.113 1.297 0.396 0.145 12 0.804 0.274 0.130 0.733 0.239 0.090 13 0.717 0.769 0.228 1.353 0.065 0.275 14 1.073 0.041 0.124 1.200 0.662 0.115 15 1.134 0.078 0.208 0.580 0.638 0.208 16 0.872 0.116 0.060 0.933 0.276 0.275 17 1.071 0.074 0.095 0.714 1.605 0.280 18 0.912 0.580 0.266 0.532 1.625 0.404 19 0.820 1.350 0.048 0.825 0.194 0.159 20 0.848 0.072 0.060 1.491 0.731 0.229 21 1.221 0.303 0.149 0.715 1.660 0.210 22 1.178 0.082 0.115 1.207 0.643 0.172 23 1.367 1.011 0.348 0.740 1.549 0.152 24 1.273 0.499 0.094 0.620 0.189 0.098 25 1.429 0.525 0.325 0.976 0.519 0.264 26 0.710 0.447 0.141 0.705 1.073 0.209 27 0.559 0.144 0.034 1.029 0.771 0.088 28 0.906 1.173 0.562 0.939 0.802 0.245 29 1.177 0.435 0.084 1.058 0.348 0.161 30 0.936 0.664 0.155 0.831 0.567 0.204 31 0.636 1.036 0.035 0.864 0.130 0.214 32 0.801 1.351 0.162 0.806 1.163 0.190 33 1.066 1.436 0.155 0.808 0.761 0.166 34 0.946 1.078 0.354 0.792 0.081 0.215 35 0.619 1.527 0.093 0.671 0.045 0.296 36 0.892 0.039 0.193 0.805 0.342 0.065 37 1.002 0.208 0.335 0.728 0.212 0.174 38 0.844 0.682 0.185 1.352 0.252 0.273 39 1.233 0.543 0.253 0.984 0.774 0.173 PAGE 40 40 Table 32. Item parameters for common items 40 to 61 on Forms X and Y ( j = 22) Item number Form X and Form Y a j b j c j 40 1.226 0.132 0.076 41 0.561 0.209 0.036 42 1.156 0.579 0.055 43 0.791 0.352 0.268 44 1.197 0.434 0.206 45 0.549 1.228 0.085 46 1.293 0.045 0.132 47 0.706 0.705 0.198 48 1.006 1.911 0.148 49 0.861 1.027 0.289 50 0.603 0.615 0.095 51 0.879 0.846 0.092 52 1.556 0.008 0.232 53 0.511 1.799 0.125 54 0.816 0.001 0.231 55 0.747 0.503 0.193 56 0.685 0.082 0.386 57 1.036 1.148 0.197 58 0.635 1.691 0.158 59 0.654 0.427 0.200 60 0.595 1.533 0.151 61 1.302 0.319 0.106 In Condition 5, the b parameters for item 41 dropped from 2.209 to 2.209 from Form X to Form Y for item 51 did from 0.846 to 2.846, for item 55 did from 0.503 to 2.503, and for item 61 did from 0.319 to 1.681. In Condition 6, the a parameters for i tem 45 dropped from 0.549 to 0. 049 from Form X to Y for item 48 did from 1.006 to 0.006, for item 56 did from 0.685 to 0.185 and for item 60 did from 0.595 to 0.095 In Condition 7, the b parameter for item 43 dropped from 0.352 to 1.648 and the a parameter dropped from 0.791 to 0.291 from Form X to Form Y For item 46, the b parameter dropped from 0.045 to 1.995 and the a parameter dropped from 1.293 to 0.293. For item 53, the b parameter dropped from 1.799 to 3.799 and the a param eter PAGE 41 41 dropped from 0.511 to 0.011. Finally, for item 59 the b parameter dropped from 0.427 to 1.573 and the a parameter did from 0.654 to 0.154. Overall th is study involved a common item nonequivalent groups simulation design composed of seven conditions based on dif ferent outlier common item scenarios ( either the b parameter, a parameter, or an interaction of a and b parameters), two levels of ability distribution ( NID (0, 1) for Form Y and NID (0, 1) and (1, 1) for Form X ), and two IRT characteristic curve linking methods (SL and HA). For each of the conditions, 1000 datasets were generated. Once the datasets were created, the response samples for the two test forms were analyzed with the 3PL model and then linked by using one of the characteristic curve me thods. Following the linking process, test responses to Form X and Form Y were equated by using IRT truescore equating (Kolen & Brennan, 2004). The entire data simulation process was done with the statistical software R, version 2.122 ( R Development Core Team, 2011). 3.5 Evaluation Criteria In order to evaluate the performance of the SL and HA methods in the presence of common items, the unweighted mean square error for the b parameters ( MSEb) and the unweighted mean square error for the a parameters ( MSEa) were used. Equations 3 1 and 32 show each formula respectively: (3 1) and PAGE 42 42 (3 2) (Hu et al., 2008) where b*jr is the b parameter for item j o n the equated test (Form Y ) for replication r bj is the true value for the b parameter for item j on the equated test Form Y a*jr is the a parameter for item j on the equated test (Form Y ) for replication r and aj is the true value for the a parameter for item j on the equated test Form Y In both formulas, such difference is summed across score points (39 points) and across replications (1000 replications) and then divided by the number of replications times the number of score points. Hu et al. (2008) concluded that in order to claim that the size of the unweighted mean square error for a specific condition was considered small, moderate, or large, it was necessary to develop absolute rules for interpreting the sizes of the estimated er ror for each parameter. Such criterion was sustained on the magnitude of the square roots of the unweighted mean square errors (referred to as bias), which correspond to the difference between the observed b and a parameters and their corresponding true b and a parameters. In the case of the b parameters bias values of 0.2500 (one fourth of the standard deviation of the distribution of the b parameters) and 0.5000 (one half of the standard deviation of the distribution of the b parameters) were set as the cutoff scores (Hu et al., 2008) In the specific metric of mean square errors, such values correspond to 0.0625 and 0.2500. Therefore, in this study, the same criteria for interpreting the values of MSEb were used: a) MSEb MSEb 0.25: moderate, and c) MSEb > 0.25: large (Hu et al., 2008) PAGE 43 43 Similar criteria were developed to interpret the size of the MSEa. According to de Ayala (2009), reasonably good values of the a parameter range from approximately 0.8 to 2.5. Under a normal distribution of values for the a parameters, the expected median for this range would then be 1.6500. Therefore, for the a p arameters, bias values of 0.4125 (one fourth of a possible median for the a parameters ) and 0.8250 (one half of the standard deviation of the true a parameters) were established as the cutoff scores. In the metric of mean square errors, such values correspond to 0. 1702 and 0. 6806. Consequently, the criteria for interpreting the values of MSEa were: a) MSEa 0.17: small, b) 0.17 < MSEa 0.68 : moderate, and c) MSEa > 0.68: large. 3.6 Data Analysis The R software, version 2.122 ( R Development Core Team, 2011) was used to fit the 3PL model, conduct IRT linking and IRT truescore equating, as well to estimate the mean square errors for the b and a parameters. The 3PL model was fit ted using the LTM package (Rizopoulos, 2006) through the function tpm The tpm function fit ted the 3PL model on the simulated datasets based on marginal maximum likelihood by using the Gauss Hermite rule for the approximation of the required integrals (Rizopoulos, 2006) and returned the item parameters as part of the function outputs Next, the plink function from the PLINK package (Weeks, 2010) was utilized to conduct link ing with the SL and HA methods to place item parameters on both Forms X and Y onto a common scale. Following, IRT truescore equating was carried out with the equate function from the same statistical package in order to relate number correct scores across Forms X and Y (Weeks, 2010). Finally mean square errors b and a parameters were calculated in R for each replication by applying the formulas described in the E valuation Criteria section ( Section 3.5) PAGE 44 44 CHAPTER 4 RESULTS Convergence rates of the 3PL mo del calibrations and of the characteristic curve linking methods were near 99.8% in all conditions with the exception of the last outlier common item condition (four common items with an interaction of extreme b and a parameters) for which the IRT characteristic curve linking results converged for very few replications, yet generating some aberrant results in terms of IRT estimation. Study results ar e presented in Tables 41 to 46 Table 41 illustrates the MSEb and MSEa values under the first out lier condition that is, when there were no outliers present in the common items. The MSEb and MSEa values for both methods were small when the two groups were equivalent. On the other hand, when the two groups differed in ability, both methods also had sm all MSEa values but large MSEb values. These results indicate that the SL and HA methods performed equally well in terms of estimating the a parameters under equivalent and nonequivalent groups, yet they seemed to be more sensitive to the presence of noneq uivalent groups when estimating the b parameters. Table 41. Mean square errors under the no outliers condition Stocking Lord Haebara Criteria Equivalent Groups Nonequivalent Groups Equivalent Groups Nonequivalent Groups MSE b 0.0111 1.6074 0.0156 1.4139 MSE a 0.0071 0.0381 0.0079 0.0083 Table 42 shows the values for the condition in which the outliers were two common items with extreme b parameter. Similar to the condition with no outliers, the MSEb and MSEa values for both SL and HA methods were small under the equivalent groups condition. As expected, the MSEb values for both characteristic curve methods PAGE 45 45 were very large under the nonequivalent groups condition but their MSEa values remained within a small range. Moreover, the MSEb values for the two methods under the nonequivalent groups condition were larger than they were when there were no outliers in the common items ( Table 41). Therefore, these results suggest that the presence of two outlier common items with outlier b parameter had a specific negative effect on the calibration of the b and a parameter s when the groups were not equivalent. Table 42. Mean square errors under the second outlier condition Stocking Lord Haebara Criteria Equivalent Groups Nonequivalent Groups Equivalent Groups Nonequivalent Groups MSE b 0.0675 2.1942 0.0528 1.8502 MSE a 0.0213 0.1039 0.0306 0.0376 The MSEb and MSEa values for the outlier condition in which there were two outlier common items with extreme a parameter are shown in Table 43. For equivalent groups, the MSEa value for the SL method was moderate and for the HA method was small. For nonequivalent groups, the MSEa value for the SL method was large and for the HA method was small. Thus, the HA method seemed to perform equally well in either group ability condition under the presence of two outlier common items with extreme a parameter. On the other hand, the MSEb values for the SL method were moderate under the equivalent groups condition and very large under the nonequivalent condition. For the HA method, the MSEb values were large in both the equivalent and nonequivalent groups conditions. Not only the char acteristic curve methods were again sensitive to group equivalence, but they were also affected by the presence of two common items with outlier a parameter towards the estimation of the discrimination parameters and towards the calibration of the difficul ty parameters as well. The MSEb PAGE 46 46 and MSEa values significantly increased for all combinations of factors in relation to the previous two conditions. Table 43. Mean square errors under the third outlier condition Stocking Lord Haebara Criteria Equivalent Groups Nonequivalent Groups Equivalent Groups Nonequivalent Groups MSE b 0.1827 1.8205 0.3136 1.7806 MSE a 0.3778 0.7192 0.1230 0.1294 The MSEb and MSEa values for the outlier condition in which there were two outlier common items with an interaction of extreme b and a parameters are shown in Table 44. The MSEa values for the SL method were moderate under both the equivalent and nonequivalent groups con ditions. T he MSEa values for the HA method were small under the equivalent groups condition and moderate under the nonequivalent groups condition. The MSEb values for the SL and HA methods were very large under the nonequivalent groups condition but moderate under the equivalent groups condition. In comparison to the effects caused by having no outliers at all, the presence of two common items with an interaction of outlier a and b parameters did have an overall impact on the errors of both parameter estimations. However, the interesting finding is that the MSEb and MSEa values were less sensitive to the presence of two common items with an interaction of outlier a and b parameters than they had been when two common items with outlier b or a parame ter were introduced separately. Table 44. Mean square errors under the fourth outlier condition Stocking Lord Haebara Criteria Equivalent Groups Nonequivalent Groups Equivalent Groups Nonequivalent Groups MSE b 0.1412 1.4644 0.2117 1.2808 MSE a 0.2335 0.4216 0.0839 0.0765 PAGE 47 47 Table 45 shows the MSEb and MSEa values for the outlier condition in which there were four outlier common items with extreme b parameter. Under the equivalent groups condition, the MSEa value for the SL method was small and for the HA method was moderate. Under the nonequivalent groups condition, the MSEa value for both the SL and HA methods was small. With respect to the MSEb values, under the equivalent groups condition such values were moderate for both SL and HA methods. However, the MSEb values under the nonequivalent groups condition were very large for the two methods As expected, the SL and HA methods performed worse under four outlier items with extreme b parameter than under two outlier items Interestingly, varying from two outlier b parameters to four outlier b parameters did not have a clear impact on the MSEa values. Table 45. Mean square errors under the fifth outlier condition Stocking Lord Haebara Criteria Equivalent Groups Nonequivalent Groups Equivalent Groups Nonequivalent Groups MSE b 0.2206 2.8661 0.1702 2.3255 MSE a 0.0395 0.1704 0.0898 0.1029 Table 46 shows the MSEb and MSEa values for the outlier condition in which there were four outlier common items with extreme a parameter. For both equivalent and nonequivalent groups, the SL method had large MSEa values. In contrast, the HA method had moderate MSEa values under both the equivalent and nonequivalent groups conditions. Similar to the outlier condition in which there were two outlier common items with extreme a parameters, the HA method performed relatively well in either group ability condition under the presence of four outlier common items. With respect to the MSEb values, the SL and HA methods exhibited large values under the two group ability conditions. These results may i ndicate that when there were more PAGE 48 48 common items with outlier a parameter, they not only affect ed the discrimination parameter estimation but also tended to drag the difficulty parameter calibr ation into estimation errors. Table 46. Mean square errors under the sixth outlier condition Stocking Lord Haebara Criteria Equivalent Groups Nonequivalent Groups Equivalent Groups Nonequivalent Groups MSE b 0.4182 1.3100 0.5594 1.7256 MSE a 1.6095 1.9773 0.2081 0.2040 PAGE 49 49 CHAPTER 5 DISCUSSION The purpose of this study was to examine the effects of having common item with outlier b and a parameter estimates on two widely used IRT characteristic curve linking methods: the Stocking Lord and Haebara methods By conducting a series of Monte Carlo simulations, this study compared the performance of both methods under different outlier conditions based on the types of outliers, the number of outliers, and group ability distributions Several studies had previously compared the Stocking Lord and Haebara methods under the commonitem nonequivalent groups design, although they did not show consistent conclusions about which method is more preferable across different testing conditions (Baker & Al Karni, 1991 ; Bguin & Hanson, 2001; Bguin et al., 2000; Hanson & Bguin, 2002; Kim & Cohen, 1998; Kim & Kolen, 2006; Lee & Ban). On the other hand, there have been very few studies examining the effects of estimation errors and outlier common items on the Stocking Lord and Haebara methods (Hu et al., 2008; Kaskowitz & de Ayala, 2001; Michaelides, 2003; Sukin & Keller, 2008) Moreover, no previous studies have addressed the specific effects of common items with outlier a parameter or with an interaction of o utlier b and a parameters. Overall this study accomplished its main purpose. The first research question addressed whether the Stocking Lord and Haebara linking methods differed with respect to bias in the linking process when there were common item s wi th outlier b parameter M ean square errors for the b parameters were very large for both characteristic curve methods under the nonequivalent groups condition. Although the Stocking Lord and Haebara methods were already sensitive to group nonequivalence PAGE 50 50 wh en there were no outliers in the common items both methods became more biased in the calibration of the b parameters when there were two commonitem outliers with extreme b parameters. When the number of common items with outlier b parameter increased from two to four, mean square errors for the b parameters slightly increased under the equivalent groups condition, but significantly increased under the nonequivalent groups condi tion. When comparing the Stocking Lord and Haebara linking methods to one another in the presence of extreme commonitem outliers with extreme b parameter the Haebara method performed slightly better, yet still poorly, than the Stocking Lord method when there were two and four outlier common items. T hese results tend to confirm Lee and Bans (2010) and Michaelides (2003) findings on the Haebara method over the Stocking Lord method. The second research questions addressed whether the Stocking Lord and Haebara linking methods differed with respect to bias in the linking process when there were common items with outlier a parameter. Under the equivalent and nonequivalent groups conditions, the Haebara method consistently outperformed the Stocking Lord method by generating smaller mean square errors for the a parameters. In terestingly both methods performed poorly in terms of the mean square errors for the b parameters, which indicates that linking estimation errors in the a parameters also affect the calibration of the b parameters to a great extent. When the number of com mon items with outlier a parameter increased from two to four, the Haebara method again outperformed the Stocking Lord method by showing low to moderate mean square errors for the a parameters. Similar to when there were two outlier common items with extre me a parameter the mean square errors for the b parameters were very large for PAGE 51 51 both methods. Kolen and Brennan (2004) pointed out that the characteristic curve methods do not explicitly account for the error in estimating item parameters and ignoring the error in parameter estimates might lead to problems in the estimation of the A and B coefficients, and therefore, in the estimation of the linking relationship between different test forms. The third r esearch questions addressed whether the S tocking Lord and Haebara linking methods differed with respect to bias in the linking process when there were commo n items with an interaction of outlier b and a parameters. Although two common item scenarios for this condition were part of the original l inking design, only the condition with two common items with outlier both b and a parameters was examined in this study. When there were four outlier common items with extreme b and a parameters, the Stocking Lord and Haebara methods had serious convergence problems when estimating the A and B coefficients, as well as the posterior truescore equating process. Therefore, those results had to be discarded from any type of further analysis. Instead, when there were two common items with outlier b and a parameters, both the Stocking Lord and Haebara methods performed poorly in terms of mean square errors for the b and a parameters under the nonequivalent groups condition, though they were less sensitive than in the previous conditions where both item parameters were analyzed independently. This difference between the two scenarios is small and is likely due to sampling variability of the common items. The last research question addressed whether the performance of the Stocking Lord and Haebara linking methods was affected by changes in the ability distribution of PAGE 52 52 number of outliers. Based on the findings, the Stocking Lor d and Haebara methods tended to perform better under t he equivalent groups condition than under the nonequivalent groups condition. These results confirm what had also been found in previous studies (Hanson & Bguin, 2002; Hu et al., 2008; Lee & Ban, 2010; Sukin & Keller, 2008). However, special attention must be given to the nonequivalent groups conditions because in real testing situations it is more common to find groups with large ability differences (i.e., retesters vs. first time testers or private vs public elementary school students). In addition, as Kolen and Brennan (2004), a large ability difference can better show which equating methods are most sensitive to group differences. With respect to the number of outliers, both the Stocking Lord and H aebara linking methods performed worse in terms of mean square errors for the b and a parameters when having four outlier common items than when having two outlier common items, except when there were four common items with outlier a parameters because the Haebara method performed equally well under both ability distribution levels. 5.2 Conclusion In conclusion, the Stocking Lord method is not a robust IRT based method to the effects of the presence of common items with outlier b and a parameters under the common item nonequivalent design. On the other hand, the Haebara method is not a robust method to the effects of the presence of commonitems with outlier b parameter under the commonitem nonequivalent design, although it is more robust method to the effects of common items with outlier a parameters. Finally, neither the Stocking Lord nor the Haebara method is a robust IRT based linking method to the effects of the presence of common items with an interaction of outlier b and a parameters. PAGE 53 53 5.1 Limita tions and Suggestions for Future Research There are multiple limitations in this Monte Carlo simulation study involving the Stocking Lord and Haebara linking methods. As with any simulation study, some restraints need to be taken before drawing conclusions because of the small number of conditions investigated. In this simulation study, the results are appropriate to only the use of two test forms, equivalent and nonequivalent groups, two and four outlier common items, two IRT characteristic curve linking m ethods, and one IRT model (the 3PL model). Future simulation research including more test forms (and more items) and different IRT models is necessary. Another limitation of this simulation study has to do with how realistic the simulated data were. Altho ugh this study employed real item parameters from the 2005 and 2006 FCAT 4th Grade M athematics test administration and cautions were taken to ensure that the simulated responses would match real test responses, the simulation of data under the 3PL model is not an easy task because problems in estimating the cparameter can influence the estimation of the items other parameters (de Ayala, 2009). Furthermore the cparameter is considered to be more reflective of a person characteristic rather than a genuine item characteristic, which may have an impact when simulating data under the 3PL model (de Ayala, 2009; Hambleton & Swaminathan, 1985). Therefore, future work using actual student responses is needed in order to determine the real effects of outlier items on the Stocking Lord and Haebara methods. The current study was also limited to analyzing the effects of outliers located on the left side of the straight line of the b and a parameters, where the parameters from Year 2 (Form Y ) were two points lower than in Year 1 (Form X ) Future studies should PAGE 54 54 also focus on other types of outlier characteristics in order to establish more conclusive remarks about the performance of characteristic curve linking methods. Likewise, this study was only limited to examining the effects of two and four common items (out of 22 common items in total). Thus it would be interesting to determine the effects of having a larger percent of common items with outlier b and a parameters. On the other hand, this study did not investigate linking or equating methods that do consider the presence of outlier items as o ther research studies have done in the past (Hu et al., 2008; Michaelides, 2003) Therefore, it would be of interest to investigate the performance of other methods, such as the deltaplot method (Michaelides, 2010), under the presence of outlier common it ems with extreme a parameters. Another limitation of the present study is strictly related with the comparability of tests scores across test for ms. This study did not examine the effect s of common items with outlier b or a parameters on the true score of the examinees, and therefore, it was not possible to quantify the effect of such type s of common items on the actual estimation of the ability levels of the examinees. Further studies ought to consider the estimation of mean square errors for the true scores in order to determine the magnitude of the bias on the estimation of the examinees ability levels Similarly, this study only considered the presence of outlier common items as part of an external anchor test, which does not consider the performanc e of the examinees on such items as part of their to tal observed score. Therefore, future research should also analyze the effects of having outlier common items in an internal anchor test on both the test calibration process and true score estimation. PAGE 55 55 REFERENCES Angoff, W. H. (1982). Summary and derivation of equating methods used at ETS. In P. W. Holland & D. B. Rubin (Eds.), Test equating (pp. 5570). New York: Academic Press. Baker, F. B., & Al Karni, A. (1991). A comparison of two procedures for computing IRT equating coefficients. Journal of Educational Measurement 28 (2), 147 162. Bguin, A. A., & Hanson, B. A. (2001, April). 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PAGE 58 58 BIOGRAPHICAL SKETCH Francisco Andres Jimenez was born in Santiago, Chile in 1982 He received a Bachelor of Science degree in Psychology from the University of Chile in 2005 obtained his professional title of Psychologist from the University of Chile in 2007, and received his Master of Arts degree in Education from the University of Florida in 2011. He has been an A ssistant Professor in the Department of Psychology at the University of Chile since 2008. He is currently a Fulbright doctoral student in the Research and E valuation M ethodology Program in the College of Education at the University of Florida. 