Accuracy of Estimates and Statistical Power for Testing Meditation in Latent Growth Curve Modeling

Transcription

1 Structural Equaton Modelng, 18: , 2011 Copyrght Taylor & Francs Group, LLC ISSN: prnt/ onlne DOI: / Accuracy of Estmates and Statstcal Power for Testng Medtaton n Latent Growth Curve Modelng JeeWon Cheong Unversty of Pttsburgh The latent growth curve modelng (LGCM) approach has been ncreasngly utlzed to nvestgate longtudnal medaton. However, lttle s known about the accuracy of the estmates and statstcal power when medaton s evaluated n the LGCM framework. A smulaton study was conducted to address these ssues under varous condtons ncludng sample sze, effect sze of medated effect, number of measurement occasons, and R 2 of measured varables. In general, the results showed that relatvely large samples were needed to accurately estmate the medated effects and to have adequate statstcal power, when testng medaton n the LGCM framework. Gudelnes for desgnng studes to examne longtudnal medaton and ways to mprove the accuracy of the estmates and statstcal power were dscussed. Keywords: accuracy of medated effects, ndrect effects, latent growth curve modelng, longtudnal medaton, statstcal power Assessng medaton s a crtcal part of socal scence research. Researchers hypothesze the underlyng causal mechansm and assess how the effect of the ndependent varable on the dependent varable s carred out through the hypotheszed ntervenng varables (Baron & Kenny, 1986; Sobel, 1990). For example, alcohol researchers hypothesze that drnkng alcohol reduces attenton capacty, whch, n turn, facltates aggressve behavors because ndvduals wth reduced attenton capacty are lkely to attend to more salent cues, such as provocaton from others (Steele & Josephs, 1990; Taylor & Leonard, 1983). For another example, the theory of cogntve dssonance (Festnger, 1957) suggests that behavors that are ncongruent wth an ndvdual s atttudes produce an unpleasant dssonance state, and thus lead to atttude change n an effort to reduce the unpleasantness. Statstcal approaches to testng medaton have been actvely studed n recent years (Baron & Kenny, 1986; Collns, Graham, & Flaherty, 1998; Kenny, 2008; MacKnnon, Farchld, & Frtz, 2007; MacKnnon, Lockwood, Hoffman, West, & Sheets, 2002; Preacher & Hayes, 2004; Sobel, 1990). The majorty of these methods are based on a varable-centered approach, Correspondence should be addressed to JeeWon Cheong, Department of Psychology, Unversty of Pttsburgh, 210 S. Bouquet Street, Pttsburgh, PA 15260, USA. E-mal: jcheong@ptt.edu 195

2 196 CHEONG n whch researchers focus on the relatons among the ndependent varable, the medator, and the outcome varable. MacKnnon et al. (2002) examned Type I error rates and statstcal power of 14 varable-centered methods that were wdely used for testng medaton n varous dscplnes. Other researchers take a person-centered approach and focus on ndvduals response patterns that are consstent or nconsstent wth the medaton hypothess (Collns et al., 1998; Robns & Greenland, 1992). Medaton analyss has been extended and appled to nvestgatng medatonal processes n more complcated study desgns such as longtudnal studes (Cheong, MacKnnon, & Khoo, 2003; Cole & Maxwell, 2003), studes wth multlevel data structure (Bauer, Preacher, & Gl, 2006; Kenny, Korchmaros, & Bolger, 2003; Krull & MacKnnon, 1999, 2001; Raudenbush & Sampson, 1999), and studes nvestgatng the combnaton of medaton and moderaton (Bauer et al., 2006; Edwards & Lambert, 2007; Farchld & MacKnnon, 2009; Morgan-Lopez & MacKnnon, 2006). Although advances have been made regardng the accuracy of estmates and statstcal power to detect medaton effects for basc medaton models (MacKnnon et al., 2002; MacKnnon, Lockwood, & Wllams, 2004; Shrout & Bolger, 2002), our understandng about testng longtudnal medaton s stll lmted. Several studes (Cheong et al., 2003; Cole & Maxwell, 2003; Frtz & MacKnnon, 2008) ntroduced and llustrated dfferent modelng technques for nvestgatng longtudnal medaton, and emprcal studes (Jagers, Morgan-Lopez, Howard, Browne, & Flay, 2007; Roesch et al., 2009; Smons-Morton, Hayne, Saylor, Crump, & Chen, 2005) began to utlze these modelng technques. However, smulaton studes are needed to examne statstcal performance of these technques and to provde researchers wth gudelnes for analyzng exstng data or desgnng future studes that nvolve testng longtudnal medaton. In ths study, a smulaton study was conducted to examne how accurately the medated effects were estmated and how statstcal power changed under dfferent condtons when the methods for testng medaton were utlzed n the latent growth curve modelng (LGCM) framework. An example of a growth curve model for nvestgatng longtudnal medaton s shown n Fgure 1. Based on the repeated measures of the medator and the outcome made on the same ndvduals, the trajectores of the medator and the outcome are modeled as two parallel processes that are nfluenced by the ndependent varable. Fgure 1 presents a sngle medator parallel process latent growth model wth fve measurement occasons, n whch the trajectores of the medator and the outcome varables are modeled as lnear wth equal ntervals between measurement occasons. Estmatng the parallel process medaton model requres several steps. The growth trajectory of the medator and the growth trajectory of the outcome are modeled separately to determne the trajectory shapes and to estmate the overall changes n the medator and the outcome. These two processes are modeled n the measurement models as follows: Medator Process: Outcome Process: M D v.m/ Y D v.y/ C ƒ.m/.m/ C ƒ.y/.y/ C.m/ (1) C.y/ (2) In the precedng equatons, the superscrpts m and y denote the medator and the outcome process, respectvely. M and Y are T 1 vectors of the repeated measures of the medator M and the outcome Y for the ndvdual across T measurement occasons; v.m/ and v.y/ are

3 TESTING MEDIATION IN LATENT GROWTH CURVE MODELING 197 FIGURE 1 A parallel process latent growth model for medaton. X D ndependent varable; Y D outcome; M D medator; 1 D ntercept (ntal status) factor of medator; 3 D ntercept (ntal status) factor of outcome; 2 D slope (growth rate) factor of medator; 4 D slope (growth rate) factor of outcome. T 1 vectors of the ntercepts of the repeated measures; ƒ.m/ and ƒ.y/ are T J matrces of factor loadngs on the latent growth factors (j D 1;2;:::;J,wherej D the number of growth factors);.m/ and.y/ are J 1 vectors of the latent growth factors; and.m/ and.y/ are T 1 vectors of the measurement errors of the repeated measures of the medator and the outcome processes. In Fgure 1, the two latent growth factors n each process are the common ntercept factor and the slope factor. The means of the common ntercept factors estmate the average ntal level of the medator and the average ntal level of the outcome n the study sample. The means of the slope factors estmate the average growth rates of the medator and the outcome processes. The varances of the ntercept and the slope factors estmate the varablty n the ntal levels and the varablty n the growth rates among ndvduals. Once the growth curves of the medator and the outcome are modeled adequately, supportng the hypotheszed trajectory shapes, and the varances of the latent growth factors are statstcally sgnfcant, the latent factors can be regressed on the ndependent varable X to nvestgate the effects of the ndependent varable on the growth trajectory of the medator and the growth trajectory of the outcome. Medatonal process s modeled by combnng the two growth curve models estmated n the prevous steps and relatng the growth factors and the ndependent varable X n the structural model. When the growth trajectores of the medator and the outcome are modeled as lnear,

4 198 CHEONG the medtatonal process can be expressed as follows: 1 D.m/ 0 C 1 X C 1 (3) 2 D.m/ 1 C 2 3 C X C 2 (4) 3 D.y/ 0 C 3 X C 3 (5) 4 D.y/ 1 C 4 1 C 0 X C 2 C 4 (6) In these equatons,.m/ 0,.m/ 1,.y/.y/ 0,and 1 denote the regresson ntercepts of the growth factors, and 1, 2, 3,and 4 denote the resduals. Path coeffcents 1 n Equaton 3 and 3 n Equaton 5 ndcate the effects of X on the ntal level of the medator process and the ntal level of the outcome process, respectvely. These coeffcents can be modeled to be 0, for example, when the ndependent varable represents the groups ndvduals are randomly assgned to. In such cases, the groups are expected to be equvalent on the pretreatment measures and thus, the group membershp s not expected to be related to the baselne measures (ntal levels) of the medator and the outcome. Path coeffcents 2 and 4 n Equatons 4 and 6 represent the effects of the ntal levels on the growth rates. Reflectng the nterrelatedness between the medator and the outcome varables and modelng the temporal sequence of the ntal level at Tme 1 and the growth rate estmated based on the multple measurements from Tme 1 to Tme T, the ntal level can be modeled to have drect effect on the growth rate of the other process. Path coeffcent represents the effect of the ndependent varable X on the growth rate of the medator (.e., change n M per tme unt), controllng for the other predctors n the model. Path coeffcent ndcates the effect of the growth rate of the medator process on the growth rate of the outcome process, controllng for the effects of the ndependent varable and the ntercept factor of the medator. Path coeffcent 0 represents the effect of X on the growth rate of the outcome process that s not medated by the medator process, or the drect effect. In the model descrbed n Fgure 1 and the precedng equatons, medatonal process can be hypotheszed n several ways. Frst, one can hypothesze that the ndependent varable X affects the growth rate of the medator process, 2, and subsequently affects the growth rate of the outcome process, 4 (.e., X! 2! 4). Another potental medatonal pathway s the ndependent varable affectng the growth rate of the outcome process va the ntal level of the outcome process, 3, and the growth rate of the medator process, 2 (.e., X! 3! 2! 4). It s also possble to hypothesze a medatonal process such that the ndependent varable nfluences the ntal level of the medator, whch, n turn, affects the growth rate of the outcome process (.e., X! 1! 4). Ths study focuses on the frst type of medatonal process. The queston of nterest here s whether the ndependent varable nfluences the trajectory of the outcome process through ts effects on the trajectory of the medator process. Thus, the medaton s defned by the pathway from the ndependent varable X through the growth rate of the medator process to the growth rate of the outcome process, X! 2! 4. Among a number of methods for testng the sgnfcance of the medated effects (see MacKnnon et al. (2002) for revew of these methods), some of the most wdely used are

5 TESTING MEDIATION IN LATENT GROWTH CURVE MODELING 199 TABLE 1 Three Methods for Testng Medated Effects Examned n the Current Smulaton Study Method Estmate of Medated Effect Test of Sgnfcance O O Frst-order soluton (Sobel, 1982) O O z D q O 2 O 2 C O 2 O 2 q Asymmetrc confdence nterval (MacKnnon & Lockwood, 2001) O O O O CL O 2 O 2 C O 2 O 2 Jont sgnfcance test (Cohen & Cohen, 1983) None t D O O I t D O O the jont sgnfcance test and the methods classfed as the product of coeffcents methods. Three of these methods were examned n ths study (Table 1) to examne statstcal power to detect sgnfcant medaton effects. In the jont sgnfcance test, the medated effect s evaluated by examnng the extent to whch the ndependent varable X affects the growth rate of the medator process (path coeffcent ) and the extent to whch the growth rate of the medator affects the growth rate of the outcome (path coeffcent ). Sgnfcant medaton s clamed when both path coeffcents are statstcally sgnfcant. The jont sgnfcance test does not estmate the medated effect or calculate confdence ntervals to test the sgnfcance of the medated effect. In the product of coeffcents tests, the medated effects are estmated by the product of the two coeffcents, O O. For the Sobel frst-order test, the sgnfcance of medaton s tested by comparng the rato of the estmated medated effect to ts standard error aganst the standard normal dstrbuton. The assumpton here s that the product of the two coeffcents, O O, s normally dstrbuted. However, the dstrbuton of the product of the two coeffcents s not always normal, often beng asymmetrc wth hgh kurtoss. To take nto account the nonnormalty of the dstrbuton of O O, MacKnnon and Lockwood (2001) proposed the asymmetrc confdence nterval test. In the asymmetrc confdence nterval test, the sgnfcance of medaton s tested by constructng the confdence nterval of the medated effect based on the dstrbuton of the product of two normally dstrbuted random varables (Meeker, Cornwell, & Aroan, 1981). Unlke the typcal cases of confdence ntervals, the confdence ntervals obtaned n ths method are asymmetrc because t accommodates the nonnormalty of the dstrbutons of the product of two random varables. 1 When the confdence nterval does not nclude zero, the medated effect s consdered statstcally sgnfcant. The purpose of ths smulaton study was to examne the statstcal performance of testng medaton n the LGCM framework, focusng on the accuracy of the estmates of the medated effects and statstcal power to detect sgnfcant medaton for the tests prevously descrbed 1 The asymmetrc confdence ntervals of estmated medated effects can be obtaned drectly usng a program called PRODCLIN (MacKnnon, Frtz, Wllams, & Lockwood, 2007), whch can be downloaded on the Web ( publc.asu.edu/davdpm/rpl/prodcln/).

6 200 CHEONG (Table 1). Smulaton data were generated under varous condtons defned by the combnaton of effect sze of medated effect, sample sze, number of measurement occasons, and R 2 of the measured varables; that s, proporton of the varance of the measured medator and the outcome explaned by the growth factors. Effect sze and sample sze were chosen as smulaton factors based on pror smulaton studes carred out n the multple regresson framework, as they (e.g., MacKnnon et al., 2002) found that effect sze of the medated effect and sample sze were crtcal factors that affected the accuracy of the estmates of the medated effects and statstcal power of the methods for testng medaton. Thus, t was expected that the accuracy of estmates and power of medaton tests would be enhanced as sample sze and effect sze ncrease when the longtudnal medaton s tested n the LGCM framework. The other two smulaton condtons, the number of measurement occasons and R 2 of the measured varables, are related to the precson and relablty of growth curves. When medatonal processes are modeled n the LGCM framework, accurate and relable estmaton of growth trajectores s crucal and thought to affect statstcal performance of the methods for testng medaton. Precson of a growth curve (Snger & Wllet, 2003) s the extent to whch the growth rate estmates the true rate of change n the sample and s mproved by ncreasng the number of measurement occasons. As such, t was found that statstcal power to detect group dfferences n the mean of the slope factor n a sngle growth curve model mproved dramatcally as the number of measurements ncreased (Muthen & Curran, 1997). In ths study, t was expected that an ncreased number of measurements, gven the same magntude of growth over tme, would mprove the precson of growth curve and consequently, enhance the accuracy of the estmates of the medated effects and statstcal power to detect the sgnfcant medaton effect. Fnally, R 2 of the measured varable was thought to affect the accuracy of estmates and statstcal power. R 2 of the measured varable s the proporton of the total varance of the measured varable that s accounted for by the growth factors. The R 2 value s calculated at each measurement tme pont and can vary across tme. A large R 2 value ndcates that the varance of the measured varable at tme t s well explaned by the hypotheszed growth trajectory. In a smulaton study examnng statstcal power to detect the sgnfcant covarance between the slope factors of two growth curves (Hertzog, Lndenberger, Ghsletta, & Oertzen, 2006), R 2 at Tme 1 (.e., the proporton of the measured varable at Tme 1 explaned by the ntercept factor), was found to be an mportant factor for statstcal power. In ths study, the R 2 of the measured varable was defned as the proporton of the varance of the measured varable accounted for by both ntercept and slope factors, and t was expected that greater R 2 would mprove the accuracy of estmaton and statstcal power for the medaton tests. METHOD Statstcal Model In ths smulaton study, the longtudnal medaton model was smlar to the model n Fgure 1. All the varables (X, M,andY) n the model were smulated to be contnuous. The growth trajectores of the medator and the outcome were modeled as lnear wth equal ntervals between measurements. The medatonal process was modeled n the relatons among the

7 TESTING MEDIATION IN LATENT GROWTH CURVE MODELING 201 ndependent varable, the slope of the medator trajectory, and the slope of the outcome trajectory. As the man research questons examned n the growth curve models center around the growth rates over tme and the ndvdual dfferences n the growth rates, the medaton effect n ths study was defned as the ndependent varable affectng the growth rate of the outcome va the growth rate of the medator. The parameters n the growth trajectores were selected based on pror smulaton studes on sngle growth trajectory models (Muthén & Curran, 1997) and the author s pror experences n longtudnal data analyses. Specfcally, the means of the ntercept factors ( 1 and 3) were set to be 1.0 wth varances of 0.5. The growth of the medator and the growth of the outcome over the multple measurement occasons were modeled to be the same n magntude, such that the level at the last measurement occason was 1 SD above the mean of the baselne level (.e., M 1 and Y 1 ). In addton, the ntercept factors of the medator and the outcome processes ( 1 and 3) were smulated to be correlated wth each other (r D :30) but unrelated wth the ndependent varable X to mmc the randomzed trals, where ndvduals are randomly assgned to the levels of the ndependent varable, and thus, the medator and the outcome measures at baselne are equvalent across dfferent levels of the ndependent varable. The covarances between the ntercept factors and the dsturbances of the slope factors were set to be 0.1. The relatons between the ntercept factors and the slope factors of the other processes (e.g., the ntercept factor of the medator and the slope factor of the outcome) were also set to be 0.1. These negatve relatons of the ntercept and the slope factors reflected the stuatons frequently observed n the emprcal studes usng the LGCM technque, where the ndvduals wth hgher levels at the ntal measurement tme pont showed smaller changes at the later tme ponts (Duncan, Duncan, & Strycker, 2006; Hancock & Lawrence, 2006). Condtons of Smulaton Study The smulaton study was conducted n a factoral desgn n the combnaton of the followng four factors. All smulatons were carred out usng Mplus (Muthén & Muthén, 2006) software program (verson 4.1). In each of the 72 condtons, a total of 1,000 replcatons were conducted. Out of 1,000 replcatons n each condton, any replcatons n whch estmates of varances were negatve or the model was not converged were excluded, and then the frst 500 replcatons were selected for the subsequent analyses. Thus, a total of 36,000 replcatons were ncluded n the analyses to examne statstcal performance of the medaton tests. Effect sze of medated effect. Effect sze of the medated effect was defned by the proporton medated, that s, the rato of the medated effect to the total effect Œ =. C 0 /. Three dfferent values of the proporton medated were chosen: 0.1, 0.3, and 0.5. The dfferent values of the proporton medated measure were obtaned by varyng the sze of the path coeffcents and, whle settng the sze of the drect effect, 0, to be constant. For the proporton medated of 0.1 (.e., 10% of the total effect s medated), the coeffcent was 0.18 and was For the proporton medated of 0.3, was 0.31 and was For the proporton medated of 0.5, was 0.52 and was In all condtons, the path coeffcent 0, the drect effect, was 0.25.

8 202 CHEONG R 2 of measured varables. Two dfferent values for R 2 of the measured medator and outcome varables were chosen to represent moderate and large proporton of varance accounted for by the latent growth factors: 0.5 and 0.8. The R 2 values were set to be constant across tme to reflect that the growth trajectory explaned the varances of the measured varables equally across the measurement occasons. In addton, the same R 2 values were set for both medator and outcome processes. As the varance explaned by the growth factors at each tme pont was a functon of the tme scores represented by the factor loadngs and the varances and covarances of the growth factors, the total varances of the measured varables ncreased over tme n the current smulatons. Thus, wth the constant R 2 values across the measurement occasons, the error varance ncreased across tme as well. Number of measurement occasons. The number of measurement occasons was chosen to be three and fve tme ponts. The number of measurements was modeled to be the same for both medator and outcome processes. Snce the growth of the medator and the outcome at the last measurement occason was defned as 1 SD above the mean of the ntal level at Tme 1, the level at the last measurement was the same regardless of the number of measurement occasons. In other words, ncreasng the measurement occasons n ths study corresponded to addng more measurement occasons for a gven study perod. Sample sze. Sample szes were vared by sx dfferent szes: 100, 200, 500, 1,000, 2,000, and 5,000. These sample szes were chosen to represent the sample szes used n research studes n socal scences and preventon scence. Smulaton Outcome Measures Measure of accuracy of estmate of medated effect and standard error. Relatve bas was used to assess the accuracy of the pont estmate of the medated effect and the standard error of estmate. As shown next, relatve bas of the pont estmate of the medated effect was calculated by the rato of the dfference between the estmate and the true value of the medated effect to the true value. _ Relatve bas D (7) In Equaton 7, Ö s the pont estmate of the medated effect based on the smulated data and s the true value of the medaton effect. Out of three tests examned n ths study, medated effects are not estmated n the jont sgnfcance test. In the other two methods, the medated effects are estmated by the product of the two path coeffcents, O O. Thus, the relatve bas of the medated effect was obtaned for the product of coeffcents, O. O For the relatve bas of the standard error of the medated effect, the frst-order delta-method standard error (Sobel, 1982) was calculated as shown n Table 1 for each of 500 replcatons. The mean of these 500 standard error estmates was then used as the estmate of the standard error of the medated effect. The true value of the standard error was obtaned by the standard devaton of the estmated medated effects across 500 replcatons (Yang & Robertson, 1986). A relatve bas smaller than.10 was consdered acceptable (Kaplan, 1988).

9 TESTING MEDIATION IN LATENT GROWTH CURVE MODELING 203 Emprcal power. Estmaton of statstcal power n ths study was emprcally based. To estmate power emprcally, the medated effect was calculated and tested for sgnfcance repeatedly for 500 replcatons n each smulaton condton usng each of the three methods n Table 1. The proporton of replcatons, n whch the medated effect was statstcally sgnfcant at the Type I error rate of.05, was computed as an estmate of statstcal power for each condton and for each method for testng medaton. Because the true values of and were not 0 n ths study, the proporton of replcatons that the medated effect was sgnfcant provded the measure of statstcal power. Accuracy of Estmates of Medated Effects RESULTS The relatve bases of the medated effect estmates.o O / are reported n Table 2. The relatve bases were generally larger when R 2 was.50 compared to the condtons wth R 2 of.80, especally when the number of measurement occasons was reduced (.e., three tme ponts). Wth R 2 of.50 across three measurement occasons, the relatve bas was smaller than.10 only when the sample sze was 1,000 or larger. Wth two addtonal measurements on both medator and outcome varables, the relatve bas decreased tremendously: Relatve bas was larger than.10 only wth the sample sze of 100. When R 2 was.80, the estmates of medated effects were qute accurate. Specfcally, only when the sample sze was 100 wth three measurement occasons, the relatve bas was larger than.10. In other condtons wth R 2 of.80, the relatve bas of the estmated medated effect was smaller than.10. TABLE 2 Relatve Bas of Medated Effect across Smulaton Condtons Sample Sze Number of Measurements Effect Sze ,000 2,000 5,000 Three occasons R 2 D :5 Small Medum Large R 2 D :8 Small Medum Large Fve occasons R 2 D :5 Small Medum Large R 2 D :8 Small Medum Large Note. Effect sze was defned as the proporton medated: small D.10, medum D.30, and large D.50. R 2 was the proporton of varances of the measured medator and outcome accounted for by both ntercept and slope factors.

10 204 CHEONG TABLE 3 Relatve Bas of Standard Error of Medated Effect across Smulaton Condtons Sample Sze Number of Measurements Effect Sze ,000 2,000 5,000 Three occasons R 2 D :5 Small Medum Large R 2 D :8 Small Medum Large Fve occasons R 2 D :5 Small Medum Large R 2 D :8 Small Medum Large Note. Effect sze was defned as the proporton medated: small D.10, medum D.30, and large D.50. R 2 was the proporton of varances of the measured medator and outcome accounted for by both ntercept and slope factors. Standard error of medated effect was obtaned by Sobel s (1982) frst-order delta-method. Accuracy of Standard Errors The relatve bases of the standard errors of the medated effects calculated for the frst-order delta-method are reported n Table 3. Smlar to the result patterns found n the relatve bases of the medated effect estmates, the relatve bases of the standard error were larger than.10 when R 2 was.50, wth relatvely smaller sample szes, such as 100 or 200, and wth three measurement tme ponts. Wth two addtonal measurements, the relatve bas of the standard error of estmate decreased. When the medator and the outcome are measured over fve occasons and R 2 of the measures s.80, the relatve bas of the standard error was smaller than.10 across all the condtons. Emprcal Power The results on emprcal power are presented n Fgures 2 and 3: Fgure 2 for power under the three measurement occasons and Fgure 3 for power under the fve measurement occasons. In each fgure, the top row represents the R 2 value of.50 and the bottom row represents the R 2 value of.80. Power curves are represented as a functon of sample sze for each method for testng medaton. As expected, emprcal power was affected by the factors vared across smulaton condtons. In addton, power vared dependng on the methods used for testng medaton. The test usng Sobel s frst-order soluton (Sobel, 1982) had relatvely lower power than the other tests. The jont sgnfcance test (Cohen & Cohen, 1983) and the asymmetrc confdence nterval test (MacKnnon & Lockwood, 2001) had equvalent power. Examnng emprcal power under the condtons wth three measurement occasons n Fgure 2, power greater than.80 requred ex-

11 FIGURE 2 Emprcal power: Three measurement occasons. 205

12 FIGURE 3 Emprcal power: Fve measurement occasons. 206

13 TESTING MEDIATION IN LATENT GROWTH CURVE MODELING 207 tremely large samples when the R 2 value was.50. When usng the jont sgnfcance test and the asymmetrc confdence nterval test wth R 2 of.50 across three measurement occasons, the sample sze requred for.80 power was approxmately 2,000, 1,000, and 500, for small, medum, and large effect szes, respectvely. The Sobel test needed larger samples than the jont sgnfcance test or the asymmetrc confdence nterval test under the same condtons. Wth the ncrease n R 2 (.e.,.80) of the measured varables, power mproved. However, the sample sze needed for.80 power was stll large when the effect sze of medaton was small. For example, the jont sgnfcance test and the asymmetrc confdence nterval test needed a sample sze of approxmately 750 to have.80 power wth R 2 of.80 across three measurement occasons. When the effect sze of the medated effect was medum and the R 2 value was.80 across three measurement occasons, a sample sze of 200 was needed for power greater than.80, except for the Sobel test. The Sobel test needed a sample sze larger than 200 to have.80 power, when the effect sze was medum and the R 2 value was.80 across three measurement occasons. As can be seen n Fgure 3, the addtonal two measurements (.e., fve measurement occasons) ncreased power substantally. However, when the effect sze was small wth the R 2 value of.50 across fve measurements, the sample sze needed for.80 power was stll as large as approxmately 750 for the jont sgnfcance test and the asymmetrc confdence nterval test. For the Sobel test, the sample sze needed for.80 power was 1,000 under the same condton. Wth the R 2 value of.80 over fve measurements and small medated effect, the requred sample sze for.80 power was approxmately 500 for the Sobel test and slghtly smaller for the other two tests. When the effect sze of the medated effect was medum or large, a sample sze of approxmately 200 was enough to have power greater than.80 across dfferent R 2 values for the jont sgnfcance and the asymmetrc confdence nterval tests; however, when the R 2 value was.50 wth the medum effect sze, the Sobel test needed a sample sze greater than 200. DISCUSSION Ths study examned the accuracy of estmated medated effects and statstcal power of dfferent methods for testng medaton when the medaton was nvestgated n the parallel process LGCM framework. As expected, the factors vared across smulaton condtons n ths study affected the accuracy of estmates and statstcal power to detect sgnfcant medated effects. As sample sze, effect sze of the medated effect, number of measurement occasons, and R 2 of the measured varables ncreased, the estmates of the medated effect and ts standard error were more accurate and statstcal power mproved. However, n general, large samples were needed for accurate estmates of medaton and decent statstcal power. The accuracy of the estmates of medated effects suffered, partcularly when R 2 was.50 across three measurement occasons. In such condtons, a sample sze of approxmately 1,000 was needed. Wth the addtonal two measurements on both medator and outcome varables, the accuracy of the estmated medated effects mproved tremendously, except for the small sample sze (.e., 100) wth R 2 of.50. For statstcal power of.80 or hgher, large samples, such as a sample sze of approxmately 1,000, were needed for most of the methods for testng medaton, when the effect sze of medaton s medum and R 2 was.50 across three measurement occasons. Wth the addtonal

14 208 CHEONG two measurements, the sample sze needed for.80 power under the same condton decreased to approxmately 200. Increases n the number of measurements and n the R 2 value of the measured varables mproved statstcal power substantally, although qute large samples were stll needed when the effect sze of medaton was small. Based on the fndngs of ths study, t appears that researchers could mprove statstcal power to detect medaton n the LGCM framework by securng large samples, ncreasng the effect sze of the medated effect, ncreasng the number of measurement occasons, and ncreasng R 2 of the measured varables. Consderng that the resources are often lmted, ncreasng R 2 of the measured varables appears to be an effcent way to mprove statstcal power to detect the medaton effect. Ths fndng s consstent wth pror work by Hertzog et al. (2006), whch examned power to detect the correlatons between the slope factors of two growth curves. Large R 2 can be acheved by reducng the proporton of the resdual varances of the measured varables. One way to reduce the proporton of the resdual varances of the measures s to reduce random measurement errors by choosng relable measures for the medator and the outcome. Alternatvely, one could use multple ndcators n measurng the medator and the outcome and model them as latent varables to separate out the measurement errors from the true estmates of the medator and the outcome at each tme pont. In such cases, the growth of the medator and the growth of the outcome can be modeled as growth of factors usng second-order latent varable models (Hertzog et al., 2006; Sayer & Cumslle, 2001). However, modelng the growth of the latent factors mght requre larger sample szes due to the complexty of the models. Another factor to consder for statstcal power was the choce of method for testng medaton. The method usng Sobel s frst-order soluton, whch assumes that the medated effect.o O / s normally dstrbuted, showed the lowest power among the three methods examned n ths study. The dstrbutons of O O are often nonnormal and thus, testng medaton by usng the crtcal values under the standard normal dstrbuton leads to low statstcal power. Among the three methods examned n ths study, the jont sgnfcance test does not estmate the medated effects and the asymmetrc confdence nterval test s based on the dstrbuton of the product of two normally dstrbuted random varables (see MacKnnon et al., 2002, for more detals). These two tests showed essentally the same statstcal power; however, the asymmetrc confdence nterval test would be preferable because one can obtan the pont estmate of the medated effect and also construct confdence lmts of the medated effect. Another set of tests that were not examned n ths study but need to be noted are those based on resamplng methods such as bootstrappng (Efron & Tbshrn, 1993). These methods create emprcal dstrbutons of the medated effects, adjustng for the nonnormalty of the dstrbuton of the product of the two coeffcents (MacKnnon et al., 2004; Preacher & Hayes, 2004; Shrout & Bolger, 2002). Future study s warranted to examne statstcal power of these resamplng methods when medaton effects are evaluated n the LGCM framework. Although the parallel process LGCM approach s a superor method for nvestgatng longtudnal medaton, there are several ssues researchers should note. The foremost strength of the parallel process LGCM approach s that researchers estmate the longtudnal changes based on the repeated measures across multple occasons. Consequently, one can estmate the changes n the medator and the outcome that are closer to true values and, thus, evaluate medaton more accurately. However, the dffcultes n causal nterpretaton of the medtatonal process reman n ths approach, wthout the desgn aspects and theoretcal ratonale supportng the

15 TESTING MEDIATION IN LATENT GROWTH CURVE MODELING 209 causal drectons. When the ndependent varable represents random assgnment to condtons, the path coeffcent n Fgure 1 can be nterpreted as causal effect. The causal nterpretaton of coeffcent s stll not clear, as the relaton between the two slope factors s correlatonal and t s possble that the outcome actually causes the medator (Holland, 1988; MacCallum, 1986), especally when the medator and the outcome are concurrently measured. The causal nterpretaton of the relaton between the growth rate n the medator and the growth rate n the outcome should depend on other factors, such as tme precedence of the medator or clear theoretcal ratonale. Another ssue s that n the parallel process LGCM approach, the overall, as opposed to tme-specfc, medaton effect s estmated based on the repeated measures across tme. If researchers are nterested n examnng whether the effect of the ndependent varable on the medator and the effect of the medator on the outcome vary across tme or assessng when the medated effect ceases to occur n longtudnal studes, the cross-lagged model approach (Cole & Maxwell, 2003) mght be more sutable. If the medaton effects of nterest are based on the change scores between two measurement occasons, rather than the overall changes across several measurement occasons, latent dfference score models (Ferrer & McArdle, 2003; McArdle, 2001) can be appled. Gven the recent ncreased use of the LGCM approach to medaton, the fndngs reported here provde researchers wth mportant nformaton that can be utlzed n desgnng longtudnal studes or analyzng exstng data; however, there are several lmtatons of ths study. Frst, the LGCM medaton n the current study was modeled n a relatvely smple model. Researchers mght run nto more complcated modelng ssues. For example, the growth mght take nonlnear forms, the number of occasons mght be dfferent for the medator and the outcome processes, the R 2 values mght vary over tme, the two path coeffcents nvolved n the medtatonal process ( and ) mght show dscrepancy n magntude, and the ndependent varable mght be modeled as another growth process. In such cases, the medaton models become more complcated than the current model and the accuracy of estmates and statstcal power could be reduced n the smlar condtons. Second, the ndependent varable n ths study was smulated as a contnuous varable. Although the ndependent varable n ths study was not smulated as a dchotomous varable, t s speculated that the results for the dchotomous ndependent varable mght not be dfferent from the results presented here, as other smulaton studes found the smlartes n the results when comparng contnuous and categorcal ndependent varables, f other condtons are equvalent for the model parameters (Frtz & MacKnnon, 2008; MacKnnon et al., 2002). Thrd, ths study dd not examne Type I error rates. Based on the fndngs from a pror smulaton study on medaton carred out n the multple regresson framework (MacKnnon et al., 2002), t s expected that the Type I error rates would be lower than the conventonal level when usng the jont sgnfcance test, the Sobel s frst-order soluton test, and the asymmetrc confdence nterval test. However, t s not clear to what extent the Type I error rates would be based when medaton s modeled n an LGCM framework. Issues of Type I error rates and other modelng aspects should be nvestgated n future studes. Fnally, effect sze of the medated effect n ths study was defned as the proporton medated. Proporton medated s one of the most frequently used effect sze measures; however, t tends to be stablzed wth relatvely large samples. Future study s needed to examne whether smlar results are obtaned wth dfferent effect sze measures, such as the rato of the medated effect to the drect effect.o O =O 0 / and R 2 measures (Farchld, MacKnnon, Taborga, & Taylor, 2009).

16 210 CHEONG ACKNOWLEDGMENTS Ths project was supported by Grants R01-DA07356 and R01 AA A awarded to Dr. Davd MacKnnon. The author thanks Dr. Davd MacKnnon for hs generous support and valuable comments on ths study. The author also thanks Drs. Sek Toon Khoo and Matthew Frtz for ther helpful comments and acknowledges the members of the Preventon Scence Methodology Group for ther comments on the presentatons related to ths study. REFERENCES Baron, R. M., & Kenny, D. A. (1986). The moderator medator varable dstncton n socal psychologcal research: Conceptual, strategc, and statstcal consderatons. Journal of Personalty and Socal Psychology, 51, Bauer, D. J., Preacher, K. J., & Gl, K. M. (2006). Conceptualzng and testng random ndrect effects and moderated medaton n multlevel models: New procedures and reecommendatons. Psychologcal Methods, 11(2), Cheong, J., MacKnnon, D. P., & Khoo, S. T. (2003). Investgaton of medatonal processes usng parallel process latent growth curve modelng. Structural Equaton Modelng, 10(2), Cohen, J., & Cohen, P. (1983). Appled multple regresson/correlaton analyss for the behavoral scences. Hllsdale, NJ: Lawrence Erlbaum Assocates, Inc. Cole, D. A., & Maxwell, S. E. (2003). Testng medatonal models wth longtudnal data: Questons and tps n the use of structural equaton modelng. Journal of Abnormal Psychology, 112(4), Collns, L. M., Graham, J. W., & Flaherty, B. P. (1998). An alternatve framework for defnng medaton. Multvarate Behavoral Research, 33, Duncan, T. E., Duncan, S. C., & Strycker, L. A. (2006). An ntroducton to latent varable growth curve modelng: Concepts, ssues, and applcatons. Mahwah, NJ: Lawrence Erlbaum Assocates, Inc. Edwards, J. R., & Lambert, L. S. (2007). Methods for ntegratng moderaton and medaton: A general analytcal framework usng moderated path analyss. Psychologcal Methods, 12(1), Efron, B., & Tbshrn, R. J. (1993). An ntroducton to the bootstrap. Boca Raton, FL: Chapman & Hall/CRC Press. Farchld, A. J., & MacKnnon, D. P. (2009). A general model for testng medaton and moderaton effects. Preventon Scence, 10(2), Farchld, A. J., MacKnnon, D. P., Taborga, M. P., & Taylor, A. B. (2009). R-squared effect sze measures for medaton analyss. Behavoral Research Methods, 41, Ferrer, E., & McArdle, J. J. (2003). Alternatve structural models for multvarate longtudnal data analyss. Structural Equaton Modelng, 10, Festnger, L. (1957). A theory of cogntve dssonance. Palo Alto, CA: Stanford Unversty Press. Frtz, M. S., & MacKnnon, D. P. (2008, May 27). An exponental decay model for medaton. Paper presented at the Socety for Preventon Research, San Francsco, CA. Hancock, G. R., & Lawrence, F. R. (2006). Usng latent growth models to evaluate longtudnal change. In G. R. Hancock & R. O. Mueller (Eds.), Structural equaton modelng: A second course (pp ). Greenwch, CT: Informaton Age. Hertzog, C., Lndenberger, U., Ghsletta, P., & von Oertzen, T. (2006). On the power of multvarate latent growth curve models to detect correlated change. Psychologcal Methods, 11(3), Holland, P. W. (1988). Causal nference, path analyss, and recursve structural equatons models. In C. C. Clogg (Ed.), Socologcal methodology (pp ). Washngton, DC: Amercan Socologcal Assocaton. Jagers, R. J., Morgan-Lopez, A. A., Howard, T.-L., Browne, D. C., & Flay, B. R. (2007). Medators of the development and preventon of volent behavor. Preventon Scence, 8(3), Kaplan, D. (1988). The mpact of specfcaton error on the estmaton, testng, and mprovement of structural equaton models. Multvarate Behavoral Research, 23, Kenny, D. A. (2008). Reflectons on medaton. Organzatonal Research Methods, 11(2), Kenny, D. A., Korchmaros, J. D., & Bolger, N. (2003). Lower level medaton n multlevel models. Psychologcal Methods, 8(2),

17 TESTING MEDIATION IN LATENT GROWTH CURVE MODELING 211 Krull, J. L., & MacKnnon, D. P. (1999). Multlevel medaton modelng n group-basednterventon studes. Evaluaton Revew, 23(4), Krull, J. L., & MacKnnon, D. P. (2001). Multlevel modelng of ndvdual and group level medated effects. Multvarate Behavoral Research, 36(2), MacCallum, R. (1986). Specfcaton searches n covarance structure modelng. Psychologcal Bulletn, 100, MacKnnon, D. P., Farchld, A. J., & Frtz, M. S. (2007). Medaton analyss. Annual Revew of Psychology, 58, MacKnnon, D. P., Frtz, M. S., Wllams, J., & Lockwood, C. M. (2007). Dstrbuton of the product confdence lmts for the ndrect effect: Program PRODLIN. Behavor Research Methods, 39(3), MacKnnon, D. P., & Lockwood, C. M. (2001). Dstrbuton of products tests for the medated effect. Unpublshed manuscrpt, Arzona State Unversty, Tempe, AZ. MacKnnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparson of methods to test medaton and other ntervenng varable effects. Psychologcal Methods, 7(1), MacKnnon, D. P., Lockwood, C. M., & Wllams, J. (2004). Confdence lmts for the ndrect effect: Dstrbuton of the product and resamplng methods. Multvarate Behavoral Research, 39(1), McArdle, J. J. (2001). A latent dfference score approach to longtudnal dynamc structural analyss. In R. Cudeck, S. du Tot, & D. Sörbom (Eds.), Structural equaton modelng: Present and future. A festschrft n honor of Karl Jöreskog (pp ). Lncolnwood, IL: Scentfc Software Internatonal. Meeker, W. Q., Cornwell, L. W., & Aroan, L. A. (1981). Selected tables n mathematcal statstcs: Vol. VII. The product of two normally dstrbuted random varables. Provdence, RI: Amercan Mathematcal Socety. Morgan-Lopez, A. A., & MacKnnon, D. P. (2006). Demonstraton and evaluaton of a method for assessng medated moderaton. Behavor Research Methods, 38(1), Muthén, B. O., & Curran, P. J. (1997). General longtudnal modelng of ndvdual dfferences n expermental desgns: A latent varable framework for analyss and power estmaton. Psychologcal Methods, 4(2), Muthén, B. O., & Muthén, L. (2006). Mplus user s gude. Los Angeles, CA: Muthén & Muthén. Preacher, K. J., & Hayes, A. F. (2004). SPSS and SAS procedures for estmatng ndrect effects n smple medaton models. Behavor Research Methods, Instruments, & Computers, 36(4), Raudenbush, S. W., & Sampson, R. (1999). Assessng drect and ndrect effects n multlevel desgns wth latent varables. Socologcal Methods and Research, 28(2), Robns, D. R., & Greenland, S. (1992). Identfablty and exchangeablty for drect and ndrect effects. Epdemology, 3, Roesch, S. C., Norman, G. J., Adams, M. A., Kerr, J., Salls, J. F., Ryan, S., ::: Patrck, K. (2009). Latent growth curve modelng of adolescent physcal actvty testng parallel process and medaton models. Journal of Health Psychology, 14(2), Sayer, A. G., & Cumslle, P. E. (2001). Second-order latent growth models. In L. M. Collns & A. G. Sayer (Eds.), New methods for the analyss of change (pp ). Washngton, DC: Amercan Psychologcal Assocaton. Shrout, P. E., & Bolger, N. (2002). Medaton n expermental and nonexpermental studes: New procedures and recommendatons. Psychologcal Methods, 7(4), Smons-Morton, B., Hayne, D., Saylor, K., Crump, A. D., & Chen, R. A. (2005). Impact analyss and medaton of outcomes: The gong places program. Health Educaton & Behavor, 32(2), Snger, J. D., & Wllet, J. B. (2003). Appled longtudnal data analyss: Modelng change and event occurrence. New York, NY: Oxford Unversty Press. Sobel, M. E. (1982). Asymptotc confdence ntervals for ndrect effects n structural equaton models. In S. Lenhardt (Ed.), Socologcal methodology (pp ). Washngton, DC: Amercan Socologcal Assocaton. Sobel, M. E. (1990). Effect analyss and causaton n lnear structural equaton models. Psychometrka, 55(2), Steele, C., & Josephs, R. (1990). Alcohol myopa: Its przed and dangerous effects. Amercan Psychologst, 45, Taylor, S., & Leonard, K. (1983). Alcohol and human physcal aggresson. In R. Geen & E. Donnersten (Eds.), Aggresson: Theoretcal and emprcal revews (Vol. 2, pp ). New York, NY: Academc Press. Yang, M. C. K., & Robertson, D. H. (1986). Understandng and learnng statstcs by computer. Sngapore: World Scentfc.