Chapter 1 The Unconditional Linear Latent Curve Model

Chapter 1 The Uncondtonal Lnear Latent Curve Model 1.1 Introducton and Organzaton of the Workshop... 1 3 1.2 Defnng a Latent Growth Curve... 1 9 1.3 Latent Growth Curves as a Confrmatory Factor Model... 1 17 1.4 Thnkng More Closely About Tme... 1 35 1.5 Demonstraton: Lnear Trajectores of Negatve Affect... 1 41 Adolescent and Famly Development Project... 1 43 Examnng Descrptve Statstcs... 1 43 Intercept only LCM, Heteroscedastc Resduals... 1 48 Intercept and Lnear Slope LCM, Heteroscedastc Resduals... 1 50 Intercept and Lnear Slope LCM, Homoscedastc Resduals... 1 52

1 2 Chapter 1 The Uncondtonal Lnear Latent Curve Model

1.1 Introducton and Organzaton of the Workshop 1 3 1.1 Introducton and Organzaton of the Workshop Objectves Provde a bref overvew of the latent curve model (LCM) Present recent exemplar uses of the LCM n practce Descrbe the organzaton of the next three days What s the Latent Curve Model? Latent Curve Model (LCM) s a form of structural equaton model (SEM) for the analyss of repeated measures (RMs) data Uses latent factors to nfer exstence of unobserved growth trajectores based on set of observed RMs Model no change, lnear, quadratc, pecewse, etc. Include one or more tme-nvarant predctors e.g., ethncty, bologcal sex, country of orgn, etc. Include one or more tme-varyng predctors e.g., emotons, onset of dagnoss, marrage, arrest, etc. Test nteractons among tme-nvarant, tme-varyng, or both

1 4 Chapter 1 The Uncondtonal Lnear Latent Curve Model What s the Latent Curve Model? Estmate growth n two or more constructs at once multvarate latent curve model Test medatng and moderatng effects medators of predctors of growth growth as medators n predcton of one or more dstal outcomes Test observed group heterogenety multple group analyss for gender, ethncty, treatment group, etc. Explore unobserved group heterogenety growth mxture modelng to dentfy two or more latent subgroups A non exhaustve samplng of ntal readngs about latent curve modelng. Bauer, D.J., & Curran, P.J. (2003). Dstrbutonal assumptons of growth mxture models: Implcatons for over extracton of latent trajectory classes. Psychologcal Methods, 8, 338 363. Bauer, D.J., & Curran, P.J. (2004). The ntegraton of contnuous and dscrete latent latent varable models: Potental problems and promsng opportuntes. Psychologcal Methods, 9, 3 29. Bollen, K.A., & Curran, P.J. (2006). Latent Curve Models: A Structural Equaton Approach. Wley Seres on Probablty and Mathematcal Statstcs. John Wley & Sons: New Jersey. Curran, P. J., & Hussong, A. M. (2003). The use of latent trajectory models n psychopathology research. Journal of Abnormal Psychology, 112, 526 544. Curran, P.J., Obedat, K., & Losardo, D. (2010). Twelve frequently asked questons about growth curve modelng. Journal of Cognton and Development, 11, 121 136. Curran, P. J., & Wlloughby, M. T. (2003). Implcatons of latent trajectory models for the study of developmental psychopathology. Development and Psychopathology, 15, 581 612. Duncan, T. E., Duncan, S. C., & Strycker, L. A. (2006). An ntroducton to latent varable growth curve modelng: Concepts, ssues, and applcatons (2nd ed.). Mahwah, NJ: Lawrence Erlbaum Assocates. McArdle, J. J. (2009). Latent varable modelng of dfferences n changes wth longtudnal data. Annual Revew of Psychology, 60, 577 605. Preacher, K. J., Wchman, A. L., MacCallum, R., & Brggs, N. E. (2008). Latent growth curve modelng. Thousand Oaks, CA: Sage Publcatons. Wllett, J. B., & Sayer, A. G. (1994). Usng covarance structure analyss to detect correlates and predctors of ndvdual change over tme. Psychologcal Bulletn, 116, 363 381.

1.1 Introducton and Organzaton of the Workshop 1 5 Curran, Bauer & Wlloughby (2004) Curran, P.J., Bauer, D.J., & Wlloughby, M.T. (2004). Testng man effects and nteractons n latent curve analyss. Psychologcal Methods, 9, 220 237. Abstract from manuscrpt: A key strength of latent curve analyss (LCA) s the ablty to model ndvdual varablty n rates of change as a functon of 1 or more explanatory varables. The measurement of tme plays a crtcal role because the explanatory varables multplcatvely nteract wth tme n the predcton of the repeated measures. However, ths nteracton s not typcally captalzed on n LCA because the measure of tme s rather subtly ncorporated va the factor loadng matrx. The authors goal s to demonstrate both analytcally and emprcally that classc technques for probng nteractons n multple regresson can be generalzed to LCA. A worked example s presented, and the use of these technques s recommended whenever estmatng condtonal LCAs n practce.

1 6 Chapter 1 The Uncondtonal Lnear Latent Curve Model Cole et al. (2002) Cole. D.A., Tram, J.M., Martn, J.M., Hoffman, K.B., Ruz, M.D., Jacquez, F.M., & Maschman, T.L. (2002). Indvdual dfferences n the emergence of depressve symptoms n chldren and adolescents: A longtudnal nvestgaton of parent and chld reports. Journal of Abnormal Psychology, 111, 156 165. Abstract from manuscrpt: The authors address questons about the rate that depressve symptoms emerge, developmental and gender dfferences n ths rate, and dfferences between parent and chld estmates of ths rate. In a 12 wave, cohort sequental, longtudnal desgn, 1,570 chldren (Grades 4 11) and parents completed reports about chldren s depresson. Cross doman latent growth curve analyss revealed that (a) the rate of symptom growth vared wth developmental level, (b) gender dfferences symptom growth preceded emergence of mean level gender dfferences, (c) the rate of symptom development vared wth age, and (d) parent chld agreement about rate of symptom change was stronger than agreement about tme specfc symptoms. The authors suggest that predctablty of depressve symptoms vares wth age and the dmenson under nvestgaton.

1.1 Introducton and Organzaton of the Workshop 1 7 McCarty et al. (2013) McCarty, C.A., Wymbs, B.T., Mason, W.A., Kng, K.M., McCauley, E., Baer, J., & Vander Stoep, A. (2013). Early adolescent growth n depresson and conduct problem symptoms as predctors of later substance use mparment. Journal of Abnormal Chld Psychology, 41, 1041 1051. Abstract from manuscrpt: Most studes of adolescent substance use and psychologcal comorbdty have examned the contrbutons of conduct problems and depressve symptoms measured only at partcular ponts n tme. Yet, durng adolescence, rsk factors such as conduct problems and depresson exst wthn a developmental context, and vary over tme. Though nternalzng and comorbd pathways to substance use have been theorzed (Hussong et al. Psychology of Addctve Behavors 25:390 404, 2011), the degree to whch developmental ncreases n depressve symptoms and conduct problems elevate rsk for substance use mparment among adolescents, n ether an addtve or potentally a synergstc fashon, s unclear. Usng a school based sample of 521 adolescents, we tested addtve and synergstc nfluences of changes n depressve symptoms and conduct problems from 6th to 9th grade usng parallel process growth curve modelng wth latent nteractons n the predcton of late adolescent (12th grade) substance use mparment, whle examnng gender as a moderator. We found that the nteracton between growth n depresson and conduct dsorder symptoms unquely predcted later substance use problems, n addton to man effects of each, across boys and grls. Results ndcated that adolescents whose parents reported ncreases n both depresson and conduct dsorder symptoms from 6th to 9th grade reported the most substance use related mparment n 12th grade. The current study demonstrates that patterns of depresson and conduct problems (e.g., growth vs. decreasng) are lkely more mportant than the statc levels at any partcular pont n tme n relaton to substance use rsk.

1 8 Chapter 1 The Uncondtonal Lnear Latent Curve Model Structure of the Next Three Days Introduce concept of a latent growth curve Defne latent curve as a confrmatory factor analyss model Dscuss estmaton of lnear and nonlnear forms of growth Test and plot effects of tme-nvarant & tme-varyng predctors Buld LCMs for two constructs at once Ft LCMs to non-normal or dscrete RMs (skewed, bnary, ordnal) Estmate LCM as a functon of two or more observed groups Estmate LCM as a functon of two or more latent groups Intersperse lecture wth software demonstratons n Mplus Assume pror exposure to SEM, but see Appendx A for a revew Although we focus exclusvely on Mplus for our demonstratons, the vast majorty of latent curve models can be equvalently estmated usng any standard software package ncludng Amos, CALIS, EQS, R (lavaan), LISREL, OpenMx, or Stata (sem).

1.2 Defnng a Latent Growth Curve 1 9 1.2 Defnng a Latent Growth Curve Objectves Introduce general concept of a latent growth curve Descrbe a growth curve for a sngle ndvdual Descrbe a set of growth curves for multple ndvduals Dfferentate mean trajectory from ndvdual varablty around the mean trajectory Cross-sectonal Regresson Model Tradtonal models for repeated measures are varants of multple regresson and path analyss Consder two-predctor cross-sectonal regresson gender chld delnquency chld depresson Lmted n that cannot establsh temporal precedence

1 10 Chapter 1 The Uncondtonal Lnear Latent Curve Model Two Tme Pont Regresson Model Regresson model can be extended to nclude two tme ponts The T2 assessment s the dependent varable, and the T1 assessment of the same varable s an addtonal predctor sometmes called resdualzed change model gender T1 chld delnquency T2 chld depresson T1 chld depresson Autoregressve Path Analytc Model Usng path analyss, can expand to more than two assessments T1 chld delnquency T2 chld delnquency T3 chld delnquency T4 chld delnquency gender T1 chld depresson T2 chld depresson T3 chld depresson T4 chld depresson Prmarly captures tme-adjacent relatons among set of RMs Does not allow for estmaton of contnuous trajectory of change Often dsjont between theoretcal model and statstcal model

1.2 Defnng a Latent Growth Curve 1 11 The Latent Growth Curve To capture contnuous trajectory of change, wll approach precsely same data structure from dfferent perspectve Wll buld model for data that estmates change over tme wthn each ndvdual and then compare change across ndvduals e.g., estmate nter-ndvdual varablty n ntra-ndvdual change Ths s core concept behnd a growth curve also sometmes called latent trajectores, latent curves, growth trajectores, or tme paths Although growth models are often descrbed as frst fttng trajectores to each ndvdual observaton and then examnng the set of trajectores across all ndvduals, the models are typcally estmated n a sngle analytc step. Repeated Measures for One Person Consder hypothetcal case where we have fve repeated measures assessng depresson n a sngle adolescent depresson tme

1 12 Chapter 1 The Uncondtonal Lnear Latent Curve Model Repeated Measures for One Person Could connect observatons to see tme-adjacent changes depresson tme A Growth Curve for One Person Could nstead smooth over repeated measures and estmate a lne of best ft for ths ndvdual depresson tme

1.2 Defnng a Latent Growth Curve 1 13 A Growth Curve for One Person Can summarze lne by two peces of nformaton the ntercept ( ) and the slope ( ) unque to ndvdual 1 2 depresson tme We use the 1 and 2 to denote ntercept and slope so that later latent curve models correspond wth the standard notaton used n the general SEM. Growth Curves for Multple Persons Rarely nterested n one ndvdual, but n a sample of ndvduals can extend to 8 trajectores (but would use 100 or more n practce) can also consder the mean and varance of the 8 trajectores depresson mean trajectory ndvdual trajectores tme

1 14 Chapter 1 The Uncondtonal Lnear Latent Curve Model The Latent Growth Curve Characterstcs of the latent trajectores captured n two ways Trajectory means the average value of the parameters that defne the growth trajectory poolng over all ndvduals n the sample e.g., the mean startng pont and mean rate of change for the entre sample Trajectory varances the varablty of ndvdual cases around the mean trajectory parameters e.g., ndvdual varablty n startng pont and rate of change over tme larger varances reflect greater varablty n growth Can consder varous restrctons on these parameters to model dfferent patterns of growth over tme In the multlevel modelng framework, the means are referred to as fxed effects and the varances as random effects. There are many close tes between the multlevel and SEM approaches to growth modelng. For example, see: Wllett, J. B., & Sayer, A. G. (1994). Usng covarance structure analyss to detect correlates and predctors of change. Psychologcal Bulletn, 116, 363 381. No Varance for Intercept or Slope Imples all ndvdual trajectores are stacked precsely on top of one another n a sngle lne depresson tme

1.2 Defnng a Latent Growth Curve 1 15 Mean Slope, Mean & Varance Intercept Imples ndvdual varablty n startng pont but constant rate of change over tme depresson tme Mean & Varance both Intercept & Slope Imples ndvdual varablty n startng pont and rate of change depresson tme

1 16 Chapter 1 The Uncondtonal Lnear Latent Curve Model Intal Motvatng Questons What s the mean course of change over tme? lnear, quadratc, exponental, no change, etc. Are there ndvdual dfferences n the course of change? varablty n startng pont and rate of change Are there tme-nvarant predctors of change? person-level characterstcs lke gender, ethncty dagnostc status Are there tme-varyng predctors of change? tme-specfc characterstcs lke onset of drnkng, arrest, marrage Do two constructs travel together through tme? trajectores of alcohol use lnked wth trajectores of depresson Do trajectores meanngfully vary as functon of observed or latent group membershp? Summary Intal motvaton s that there exsts a true underlyng contnuous trajectory of change that was not drectly observed Want to use the observed repeated measures to nfer exstence of underlyng latent trajectory Means capture overall values of parameters that defne growth trajectory Varances capture ndvdual varablty n parameters that defne growth trajectory Goal s to buld a model that ncorporates predctors of ndvdual varablty n growth Wll buld ths model usng the SEM analytcal framework

1.3 Latent Growth Curves as a Confrmatory Factor Model 1 17 1.3 Latent Growth Curves as a Confrmatory Factor Model Objectves Brefly revew confrmatory factor analyss (CFA) model Defne latent curve as a specal type of CFA Explore mean and covarance structures mpled by the LCM Capturng Growth as a Latent Factor Theory posts exstence of unobserved contnuous trajectory Cannot drectly observe, but can nfer exstence based on set of repeated measures Growth curve thus fts naturally nto latent varable model e.g., depresson, self esteem, worker productvty, etc. Can draw on strengths of confrmatory factor analyss (CFA) to defne latent curve model The LCM s fundamentally a hghly restrcted CFA model t s thus helpful to frst revew the standard CFA so we may then modfy ths to defne the LCM

1 18 Chapter 1 The Uncondtonal Lnear Latent Curve Model Confrmatory Factor Analyss Prmarly theory-drven: test model that specfes the number and nature of the latent factors behnd set of observed measures e.g., latent depresson and anxety underle set of 20 symptom tems Model dentfed through restrctons on parameters Number of latent factors determned by theory Factor pattern matrx s restrcted by analyst to reflect theory e.g., some loadngs freely estmated, others fxed to zero Attenton pad to global and local ft of model to data The Parameters of the General CFA 11 1 22 2 33 3 44 4 55 1 2 3 4 5 y1 y2 y3 y4 5 y 5 y observed tem tem resdual resdual varance 11 21 32 42 52 tem ntercept 1 2 factor loadng factor score 1 2 factor mean 11 1 2 22 factor dsturbance 21 factor co/varance There s no sngle method for creatng path dagrams. Here we use the conventon that rectangles are observed varables and crcles are unobserved (or latent) varables. We do not use shapes to denote means or ntercepts, but nstead use symbols to denote the parameter adjacent to the correspondng varable (.e., ) or factor (.e., ). We wll not always carry ths level of dagrammatc detal forward.

1.3 Latent Growth Curves as a Confrmatory Factor Model 1 19 The Measurement Equaton Measurement equaton expresses observed vector of measures as functon of underlyng latent factors y νλη ε y 1 1 11 k1 1 1 y 2 2 21 k2 2 2 y p p p1 kp k p There are p-observed varables and k-latent factors See Appendx A for a revew of the general structural equaton model. The Measurement Equaton Further, the resduals from the measurement equaton y νλη ε are dstrbuted as ε ~ MVN 0, Θ where 11 0 22 Θ 0 0 0 0 0 pp

1 20 Chapter 1 The Uncondtonal Lnear Latent Curve Model The Structural Equaton Next defne model for latent factors n the structural equaton η αζ 1 1 1 2 2 2 k k k for k-latent factors The Structural Equaton The devatons from the structural model are dstrbuted as η αζ ζ ~ MVN 0, Ψ where 11 21 22 Ψ k1 k2 kk

1.3 Latent Growth Curves as a Confrmatory Factor Model 1 21 The Reduced-Form Equaton Can substtute structural equaton nto measurement equaton to obtan the reduced-form equaton y νλη ε η αζ y νλ αζ ε Because we now express observed measures n solely as a functon of model parameters, can defne mean and covarance structure among observed measures as mpled by the model y In the standard CFA model, the mean structure s typcally saturated. In other words, there are usually as many means estmated as were observed n the sample, and thus the mean structure does not contrbute to the overall ft of the model. In contrast, as we wll see n a moment, the mean structure for the LCM s not typcally saturated and thus fewer means are estmated than were observed. Model-Impled Moment Structures Motvatng goal of CFA s to defne model that reproduces mean and covarance structure of observed data as closely as possble Begn by defnng a vector theta that contans all of the parameters that defne the CFA, say ' θ 1 2 11 22 11 Use ths vector to compute a matrx-valued functon that expresses the means and covarances among our observed measures n y solely as a functon of model parameters model-mpled mean structure denoted θ model-mpled covarance structure denoted θ

1 22 Chapter 1 The Uncondtonal Lnear Latent Curve Model Model-Impled Moment Structures We start by defnng the model-mpled mean structure of as the expected value of our reduced-form expresson of the CFA: θ E y Mean structure mpled by CFA defned by the tem ntercepts, factor loadngs, and factor means y νλ αζ ε E νλ αζ ε νλα y Model-Impled Moment Structures Can smlarly defne model-mpled covarance structure of the varance of our reduced form-expresson of the CFA: y νλ αζ ε y as var y Σθ Covarance structure mpled by the CFA s defned by the factor loadngs, factor varances and covarances, and tem resduals var νλ αζ ε ΛΨΛ ' Θ

1.3 Latent Growth Curves as a Confrmatory Factor Model 1 23 Model-Impled Moment Structures Thus the model-mpled mean and covarance structure s θ νλα Σθ ΛΨΛ ' Θ Goal of model estmaton s to select values of that make mean and covarance structure mpled by model as close as possble to the mean and covarance structure observed n the sample see Appendx A for revew of estmaton n SEM We now have all of the necessary nformaton to defne the LCM as a specal case of the CFA Returnng to the Latent Curve We beleve ndvdual trajectores exst for each case, but these were not observed drectly want to nfer the latent curves based on the data that were observed y t tme

1 24 Chapter 1 The Uncondtonal Lnear Latent Curve Model Equaton for Indvdual Curves Begn by expressng repeated measures for a gven ndvdual as addtve functon of underlyng lnear trajectory weghted by tme y tme t 1 2 t t note that there s an equaton that determnes for each tme pont For example, consder three tme ponts: y y y tme tme tme 1 1 2 1 1 2 1 2 2 2 3 1 2 3 3 Note that the ntercept ( ) s mplctly weghted by 1, and the slope ( ) s weghted by the numercal value of tme Equaton for Indvdual Curves If we place each term n the scalar equatons nto matrces: y y y y 1 2 3 1 tme1 1 1 tme 1 Λ 2 η ε 2 2 1 tme 3 3 then we can equvalently express our set of scalar equatons as: y Λη ε y1 1 tme1 1 1 2tme11 1 y 2 1 tme 2 2 1 2tme2 2 2 y 1 tme 3 3 3 1 2tme3 3 Whereas n the multlevel growth model the numercal value of tme s entered as an exogenous covarate, n the latent curve model the numercal value of tme s entered va the factor loadng matrx.

1.3 Latent Growth Curves as a Confrmatory Factor Model 1 25 The Measurement Equaton Note smlarty of LCM equaton: to CFA equaton from earler: y Λη ε y νλη ε Indeed, measurement equaton for the LCM s the same as the measurement equaton for the CFA wth the restrcton that ν 0 tem-specfc ntercepts are restrcted to zero and LCM must reproduce sample means entrely through means of latent factor We next need to defne a model for the latent curve factors The trple equal sgn (.e., ) means "s equal to by defnton", so ν 0 ndcates that the vector of tem ntercepts are fxed to zero. The Structural Equaton Recall we can wrte the ntercept and slope as a functon of the mean and ndvdual devaton from the mean: 1 11 2 2 2 We can agan compactly summarze these n matrx form η αζ 1 1 1 1 1 2 2 2 2 2 Ths expresson precsely matches that of the CFA

1 26 Chapter 1 The Uncondtonal Lnear Latent Curve Model The LCM Equatons General expresson for the measurement equaton for a lnear trajectory wth t=1,2,...,t repeated measures s: numercal value of tme enters model va factor loadng matrx The structural equaton s η αζ y Λη ε y 1 0 1 1 y 2 1 1 1 2 2 yt 1 T 1 T vectors are same length as number of latent factors that defne trajectory Here we defne the loadng for the slope factor to be equal to one less than the assocated value of tme. Ths results n the frst assessment beng set equal to zero and n turn defnes the ntercept as the mean of the trajectory at the frst occason. We explore the codng of tme n much greater detal n a moment. LCM Model-Impled Moment Structures The model-mpled mean and covarance structure for the LCM: θ Λα Σθ ΛΨΛ ' Θ Precsely same expressons as CFA except no tem ntercepts Also precsely same goal of estmaton to select values of that make mean and covarance structure mpled by model as close as possble to mean and covarance structure observed n sample We can graphcally represent LCM as a path dagram

1.3 Latent Growth Curves as a Confrmatory Factor Model 1 27 A Lnear LCM as a Path Dagram 11 1 22 2 33 3 44 4 55 5 y1 y2 y3 y4 y 5 y Λη ε 1 1 1 1 1 1 2 1 3 4 1 2 2 η α ζ 11 1 2 22 21 We can examne dfferent expressons of ths model... In the standard CFA model descrbed earler, typcally some or all of the factor loadngs are freely estmated. In contrast, n the standard LCM all of the factor loadngs are fxed to pre defned values. Ths s a unque characterstc of the latent curve model relatve to the CFA. Intercept-only LCM Imples outcome does not change as functon of tme all ndvdual trajectores horzontal but at dfferent levels 1 2 3 4 y1 y2 y3 y4 1 1 1 1 1 5 y 5 y Λη ε y1 1 1 y 2 1 2 y 3 1 1 3 y 1 4 4 y 5 1 5 1 1 1 η α ζ 1 1 1

1 28 Chapter 1 The Uncondtonal Lnear Latent Curve Model Intercept-only LCM: Trajectores Intercept-only LCM mples between-person varablty n overall level of outcome, but outcome does not change wth tme depresson tme An ntercept only model mght hold n a daly dary study n whch repeated assessments are obtaned for daly mood; there may be person to person varablty n overall levels of mood, but mood s not systematcally ncreasng or decreasng as a functon of tme. Intercept-only LCM: Factor Mean Just one factor mean because only one factor s defned to represent latent ntercept η αζ E Ths smply reflects the mean level of all repeated measures pooled over all ndvduals η α α 1

1.3 Latent Growth Curves as a Confrmatory Factor Model 1 29 Intercept-only LCM: Factor Varance Just one factor varance agan because only one factor s defned to represent latent ntercept η αζ ζ ~ N 0, Ψ Ψ Ths represents the ndvdual varablty around the overall mean 11 Intercept-only LCM: Resdual Varance Fnally, tme-specfc resduals for RMs allowed to obtan a unque value at each tme pont and are typcally uncorrelated over tme y Λη ε ε ~ N 0, Θ 11 0 22 Θ 0 0 33 0 0 0 0 0 0 0 44 55 Here we estmate a unque resdual varance at each tme perod; ths s called heteroscedastcty. In a moment we wll compare ths to a model n whch we estmate a sngle resdual varance for all tme perods; ths s called homoscedastcty.

1 30 Chapter 1 The Uncondtonal Lnear Latent Curve Model Lnear LCM Can add a second correlated factor to capture lnear change: 11 1 22 2 33 3 44 4 55 5 y1 y2 y3 y4 1 1 1 1 1 1 2 1 3 4 1 2 11 1 2 22 21 2 y 5 y Λη ε y1 1 0 1 y 2 1 1 2 1 y 3 1 2 3 2 y4 1 3 4 y 1 4 5 5 η α ζ 1 1 1 2 2 2 In ths chapter we focus explctly on the lnear growth model. We wll expand ths to a varety of nonlnear functons n Chapter 2. Lnear LCM: Trajectores Intercept and lnear slope model mples ndvdual dfferences n both level and rate of change depresson tme

1.3 Latent Growth Curves as a Confrmatory Factor Model 1 31 Lnear LCM: Mean Structure Two factor means, one for ntercept and one for lnear slope η αζ E η defnes startng pont because we set frst value of tme to be zero we wll explore ths n much greater detal shortly α 1 α 2 Reflects mean startng pont and mean lnear rate of change Lnear LCM: Varance Components Factor varance now expressed as a covarance matrx wth varance of ntercept and slope and covarance between the two η αζ ζ ~ N 0, Ψ Ψ 11 21 22 Represents ndvdual varablty around startng pont and rate of change, and covarance between startng pont and rate of change

1 32 Chapter 1 The Uncondtonal Lnear Latent Curve Model Lnear LCM: Varance Components Interestngly, the covarance structure among the tme-specfc resduals s precsely the same as wth the ntercept-only LCM y Λη ε ε ~ N 0, Θ 11 0 22 Θ 0 0 33 0 0 0 0 0 0 0 44 55 Heteroscedastc Resduals Standard LCM assumes each tme-specfc repeated measure s defned by a unque resdual varance called heteroscedastcty 11 0 22 Θ 0 0 33 0 0 0 0 0 0 0 44 55 A smplfyng condton s to assume error varances are equal called homoscedastcty

1.3 Latent Growth Curves as a Confrmatory Factor Model 1 33 Homoscedastc Resduals Can mpose equalty constrant on resduals over tme 0 Θ 0 0 0 0 0 0 0 0 0 Ths s a testable hypothess homoscedastcty more parsmonous, but may not correspond to characterstcs of the observed data Typcally want to dentfy most parsmonous structure that does not sgnfcantly contrbute to model msft Correlatons Among Resduals Standard LCM typcally assumes condtonal ndependence n whch resduals are uncorrelated net the nfluence of the factors Possble to ntroduce tme-adjacent correlatons f needed many more complex structures also possble 11 21 22 Θ 0 32 33 0 0 43 44 0 0 0 54 55 Correlated resduals usually avoded n LCM, although these play a greater role n the multlevel growth model Correlated resduals are not typcally estmated n many LCMs because the correlaton structure among the repeated measures can be fully reproduced as a functon of the underlyng latent factors. However, n some cases (e.g., daly assessments), correlated resduals may be necessary even n the presence of the latent curve factors.

1 34 Chapter 1 The Uncondtonal Lnear Latent Curve Model Communalty: Tme-specfc R 2 Regardless of homoscedastc vs. heteroscedastc, can use tmespecfc resduals to estmate proporton of observed varance n repeated measure that s accounted for by the latent factors n tradtonal factor analyss, called communalty or h 2 usually called r-squared n SEM Smply complement of rato of resdual varance to total varance R 2 t Values are nterpreted n the usual way hgher values reflect stronger assocatons between tems and factors Good standardzed effect szes to report n practce var t 1 1 var y t tt 2 yt Summary The latent curve model fts logcally wthn the CFA Estmate a latent factor for each trajectory component Enter numercal measure of tme va factor loadng matrx Can buld basc to complex models and test ntervenng steps Can compare homo- vs. heteroscedastcty Can allow for correlated resduals f needed

1.4 Thnkng More Closely About Tme 1 35 1.4 Thnkng More Closely About Tme Objectves More closely consder role of tme n the LCM wth regard to where to place the zero pont the unequal spacng of tme alternatve unts of tme ndvdually-varyng assessments of tme Codng Tme: Zero Pont Up to ths pont we have set numercal value of tme to equal zero at the frst assessment perod: t 0, 1, 2, 3 Mean and varance of ntercept s scaled for ntal tme perod because when tme s zero the slope factor does not contrbute 0 1 2 3 Note that t denotes the numercal scalng of tme and appears n the second column of the factor loadng matrx for a lnear LCM (where the frst column conssts of all 1 s to defne the ntercept factor).

1 36 Chapter 1 The Uncondtonal Lnear Latent Curve Model Codng Tme: Zero Pont Can code tme such that the ntercept defnes the last assessment 3, 2, 1, 0 t 3 2 1 0 Mean and varance of ntercept scaled to reflect the last tme perod; mght be useful f conductng a treatment evaluaton Codng Tme: Zero Pont Can code tme so the ntercept defnes the mddle assessment 1.5,.5,.5,1.5 t 1.5.5.5 1.5 Mean and varance of ntercept scaled to reflect the mddle tme perod; for case of even number of assessments, mddle may not have been drectly observed but s mpled by model

1.4 Thnkng More Closely About Tme 1 37 Codng Tme: Zero Pont Alternatve codng schemes for tme are done va factor loadngs 1 0 1 1 1 2 1 3 1 3 1 2 1 1 1 0 1 1.5 1 0.5 1 0.5 1 1.5 All of these result n same model ft, but several propertes of LCM change dependng on the numercal codng of tme Codng Tme: Zero Pont What potentally does change the mean of the ntercept factor the varance of the ntercept factor the covarance between the ntercept and slope factor effects of predctors of the ntercept factor What does not change model ft the mean of the lnear slope factor the varance of the lnear slope factor effects of predctors of the lnear slope factor The above summary holds for the lnear LCM. Some complcatons arse n codng of tme when consderng hgher order polynomal functons (e.g., quadratc or cubc). For detals, see: Besanz, J.C., Deeb Sossa, N., Aubrecht, A.M., Bollen, K.A., & Curran, P.J. (2004). The role of codng tme n estmatng and nterpretng growth curve models. Psychologcal Methods, 9, 30 52.

1 38 Chapter 1 The Uncondtonal Lnear Latent Curve Model Codng Tme: Spacng Thus far have assumed equally spaced assessments e.g., one year ncrement between each assessment No reason to be lmted to equally spaced assessment perods can have some or even all assessments unequally spaced For example, say assessed subjects at ages 6, 7, 10, 12, & 16 factor loadngs smply scaled relatve to spacng of frst two assessments ' 1 1 1 1 1 Λ 0 1 4 6 10 All of our pror dscusson about placement of the zero pont holds here as well Codng Tme: Unt-of-Tme We also control the unt of tme under study e.g., assessments taken at 6, 12, 18, 30, and 42 months Can scale tme n months, half-years, or full-years Λ month 1 0 1 6 1 12 1 24 1 36 Λ half year 1 0 1 1 1 2 1 4 1 6 Λ full year 1 0 1.5 1 1 1 2 1 3 All models ft dentcally, but mean and varance of slope factor scaled n the gven unt of tme Both the choce of zero pont, spacng, and unt of tme can be controlled wth a sngle lnear transformaton equaton; see Bollen & Curran (2006), pages 115 120. Bollen, K.A., & Curran, P.J. (2006). Latent Curve Models: A Structural Equaton Approach. Wley Seres on Probablty and Mathematcal Statstcs. John Wley & Sons: New Jersey.

1.4 Thnkng More Closely About Tme 1 39 Codng Tme: Unt-of-Tme What potentally does change mean and varance of slope factor covarance of ntercept wth slope factor effects of predctors of slope factor, What does not change model ft mean and varance of ntercept factor effects of predctors of ntercept factor p-values of any parameter estmates assocated wth slope numercal values smply rescaled to dfferent unts-of-tme, but sgnfcance same Indvdually-varyng Measures of Tme LCM allows for dfferent assessment schedules across ndvduals e.g., some subjects assessed at ages 11, 12, and 14 and others assessed at ages 10, 13, and 16, etc. LCM requres each age-specfc outcome be a manfest varable e.g., observed measures of outcome at ages 11, 12, 13, 14, 15 & 16 Fundamentally a mssng data problem subjects assessed at age 10 and 12 are "mssng" at age 11 subjects assessed at age 12 and 14 are "mssng" at age 13 but jontly sample has observatons at all ages sometmes called an accelerated-cohort desgn SEM typcally uses drect maxmum lkelhood estmator that allows for partally mssng data As long as mssngness not nformatve, can be handled n LCM

1 40 Chapter 1 The Uncondtonal Lnear Latent Curve Model Indvdually-varyng Measures of Tme Some longtudnal desgns have hghly varable assessments e.g., daly dary data where subjects randomly pnged throughout day At the extreme, could have no two people provde assessments at same tme pont Standard LCM not well suted for data such as these must be able to compute mean and varance of RM at each tme perod Must nstead use defnton varable methodology n LCM defnes a unque factor loadng matrx for each person based on personspecfc tme scores, then fts LCM aggregatng over ndvdual matrces Places the LCM much closer to a multlevel modelng approach ndeed, MLM may be a better analytc strategy Boker, S., Neale, M., Maes, H., Wlde, M., Spegel, M., Brck, T.,... & Fox, J. (2011). OpenMx: an open source extended structural equaton modelng framework. Psychometrka, 76, 306 317. Neale, M. C., Aggen, S. H., Maes, H. H., Kubarych, T. S., & Schmtt, J. E. (2006). Methodologcal ssues n the assessment of substance use phenotypes. Addctve Behavors, 31, 1010 1034. Summary Can select numercal values of tme to acheve dfferent goals Can defne alternatve zero-ponts for tme e.g., begnnng, mddle, or end of trajectory Can defne unequal assessment ntervals e.g., 6 month, 12 month, 24 month Can alter metrc of change e.g., days, weeks, months, etc. Can ncorporate ndvdually-varyng measures of tme defnton varables All centered around ultmate goal of dentfyng optmal functonal form of change over tme

1.5 Demonstraton: Lnear Trajectores of Negatve Affect 1 41 1.5 Demonstraton: Lnear Trajectores of Negatve Affect Objectves Introduce real data set studyng trajectores of adolescent depresson and delnquency from ages 11 to 16 Ft seres of models to repeated measures of depresson to dentfy optmally fttng LCM Draw ntal conclusons about the developmental course of depresson pror to ncludng predctors of level and change Adolescent & Famly Development Project Data for demonstraton provded by Dr. Laure Chassn, Drector of the Adolescent and Famly Development Project (AFDP) Demonstraton data drawn from much larger sample & measures Sample conssts of n=452 chldren assessed 1, 2, or 3 tmes between ages 11 and 16 54% were chldren of alcoholcs (COAs) 53% were male Outcomes of nterest are IRT-based scores of negatve affect (NA) based on parent-reports of 13 bnary tems e.g., lonely, cres a lot, worres, has to be perfect, feels gulty, etc. full detals n Bauer et al., (2013) Unt of analyss s age-specfc contnuous measure of NA Bauer, D.J., Howard, A.L., Baldasaro, R.E., Curran, P.J., Hussong, A.M., Chassn, L., & Zucker, R.A. (2013). A trfactor model for ntegratng ratngs across multple nformants. Psychologcal Methods, 18, 475 493.

1 42 Chapter 1 The Uncondtonal Lnear Latent Curve Model Adolescent & Famly Development Project We use an accelerated longtudnal cohort desgn such that chronologcal age s our numercal metrc of tme Ths data structure results n mssngness ntroduced by desgn "x" denotes data that are present " " denotes data that are mssng 11 12 13 14 15 16 x x x x x x x x x x x x Ths desgn allows for a 6 year age span even though no ndvdual chld was assessed more than 3 tmes See Appendx A for a bref revew of mssng data estmaton n SEMs. Motvatng Questons Is there evdence of ndvdual varablty n the overall level negatve affect? Does the ncluson of a lnear slope factor sgnfcantly mprove model ft? Is there evdence of ndvdual varablty n startng pont and rate of change over tme? Does the model support the constrant that the tme-specfc resduals be homoscedastc? Does the fnal model adequately ft the observed data?

Adolescent and Famly Development Project 1.5 Demonstraton: Lnear Trajectores of Negatve Affect 1 43 The data for ths demonstraton were drawn from the Adolescent Famly Development Project (AFDP) Drected by Dr. Laure Chassn from Arzona State Unversty. We are ndebted to Dr. Chassn for generously sharng these data wth us. Note that these data were provded for strctly pedagogcal purposes and should not be used for any other purposes beyond ths workshop. These data are stored n the fle na_ext.dat. Brefly, the AFDP s a mult year longtudnal study of a large sample of adolescents and ther parents. Approxmately one half of the orgnal 452 famles were characterzed by at least one parent dagnosed as alcoholc (54%), and the remanng were characterzed by nether parent dagnosed as alcoholc (46%). Data collecton spanned more than two decades, but here we consder just the frst three waves of assessment. Parents and chldren were frst assessed when the chld was between 11 and 15 years of age, and were re assessed up to two more tmes at 12 month ntervals. Of the 452 famles consdered here, 358 (79%) were assessed three tmes, 87 (19%) twce, and 7 (2%) one tme. Two chld specfc predctors are of nterest: chld of an alcoholc (COA) and gender. The parent's report of the chld's negatve affect was obtaned at each age of assessment. Item response theory (IRT) scores were estmated for 13 bnary symptom tems ndcatng the absence or presence of negatve affect symptomatology. Sample tems nclude lonely, cres a lot, worres, has to be perfect, and feels gulty. Fnally, parent's report of the chld's externalzng behavor were obtaned at each age of assessment. Item response theory (IRT) scores were estmated for 20 bnary symptom tems ndcatng the absence or presence of externalzng symptomatology. Sample tems nclude swearng, truancy, cruelty, destroys thngs, fghts, etc. The varables n the data set are: d Unque numercal dentfer for each chld rangng from 1 to 452 coa male base_ext na11-na16 ext11-ext16 0=chld of a non alcoholc, 1=chld of an alcoholc 0=grl, 1=boy Contnuous measure of externalzng behavor at baselne, mean centered Age specfc IRT score for negatve affect Age specfc IRT score for externalzng symptomatology In ths chapter, we demonstrate how to ft a seres of growth models to negatve affectvty between ages 11 and 16. In later chapters wll wll consder the two predctors and externalzng symptomatology. Examnng Descrptve Statstcs The focus of our workshop s on the conceptualzaton and fttng of latent curve models and not on the general use of Mplus. However, there are a varety of powerful features of Mplus that can help us stay as close to our data as possble. We wll demonstrate these features throughout the workshop but refer you to the corpus of Mplus support materal avalable onlne for detals. As a startng pont, we can obtan mportant ntal nformaton about our data pror to fttng the LCMs. For example, consder the followng code that s stored n fle ch01_na_1.np:

1 44 Chapter 1 The Uncondtonal Lnear Latent Curve Model ttle: Negatve Affect: Examnng Descrptve Statstcs data: fle=na_ext.dat; varable: names=d coa male base_ext ext11-ext16 na11-na16; usevarables=na11-na16; mssng=.; analyss: type=basc; coverage=0; plot: type=plot3; Mplus requres that lnes end wth a colon or a semcolon. The equal sgn s nterchangeable wth the phrase s or are. The ttle command s optonal and t provdes a descrptve ttle for the analyss. The data command specfes the name and locaton of the data source. If no drectory path s specfed (as above), the default locaton s the same folder n whch the Mplus nput fle s saved. The varable command names all of the varables ncluded n the data set (names = ) and the usevarables statement specfes whch varables are to be used for ths analyss. In ths example, mssng cases are denoted wth perod; however, ths could be mssng=-9999 or whatever s used to denote mssng n a gven data set. In the analyss secton we defne our model to be type=basc whch wll provde core summary nformaton about the data. Mplus uses the term coverage to refer to the proporton of non mssng data for any gven varable or covarance between two varables. So a coverage value of say.25 for the covarance between two varables reflects that the covarance was observed for 25% of the avalable cases. By default, Mplus wll not estmate models that have coverage values fallng below.10. However, n some desgns (such as the accelerated longtudnal desgn used n the AFDP data here), coverage can go to zero because of the planned mssng desgn. For example, although 107 chldren were assessed at age 11 and 150 chldren were assessed at age 16, no chldren were assessed at both ages 11 and 16; thus the coverage for the covarance between NA11 and NA16 would be zero and the model estmaton would be stopped. However, the command coverage=0 overrdes ths default value of.10 and nstructs Mplus to proceed wth estmaton regardless of how many cases contrbuted to each sample statstc. Fnally, the plot command provdes a varety of avalable plots of data and model results. There are three clusters of plots logcally named plot1, plot2, and plot3. See the onlne documentaton for more nformaton about what each of these clusters contan. We are not fttng a model yet, so the results provde basc characterstcs of our data. We only present subsets of the full output. Negatve Affect: Examnng Descrptve Statstcs SUMMARY OF ANALYSIS Number of groups 1 Number of observatons 452 Number of dependent varables 6 Number of ndependent varables 0 Number of contnuous latent varables 0

1.5 Demonstraton: Lnear Trajectores of Negatve Affect 1 45 Ths ndcates that we are consderng a sngle group wth 452 ndvduals and sx varables. A varety of useful nformaton s provded about mssng data: SUMMARY OF MISSING DATA PATTERNS MISSING DATA PATTERNS (x = not mssng) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 NA11 x x x x x x NA12 x x x x x x x x NA13 x x x x x x x x x x x NA14 x x x x x x x x x x x x x NA15 x x x x x x x NA16 x x x x x x NA11 NA12 NA13 NA14 NA15 NA16 21 x MISSING DATA PATTERN FREQUENCIES Pattern Frequency Pattern Frequency Pattern Frequency 1 0 8 2 15 2 2 73 9 1 16 94 3 3 10 1 17 4 4 29 11 101 18 3 5 1 12 2 19 1 6 1 13 1 20 46 7 82 14 1 21 4 COVARIANCE COVERAGE OF DATA Mnmum covarance coverage value 0.000 PROPORTION OF DATA PRESENT Covarance Coverage NA11 NA12 NA13 NA14 NA15 NA11 0.237 NA12 0.232 0.423 NA13 0.166 0.347 0.588 NA14 0.009 0.192 0.414 0.650 NA15 0.000 0.002 0.226 0.442 0.546 NA16 0.000 0.000 0.007 0.219 0.312 Covarance Coverage NA16 NA16 0.332 Ths s valuable nformaton about the presence and absence of data across all of the measures. For example, there are no cases for pattern 1 (denotng complete data across all sx measures); ths reflects

1 46 Chapter 1 The Uncondtonal Lnear Latent Curve Model the accelerated cohort sequental desgn n whch no gven subject was assessed more than three tmes, yet we have data spannng sx years of age. The hghest frequency s pattern 11 at whch chldren were assessed at ages 13, 14 and 15; ths too reflects the sample desgn n whch the majorty of cases resde at the center of the age span. The covarance coverage matrx also reflects the desgn n whch no cell exceeds.65 and some cells have zero observed cases (e.g., covarance of ages 11 and 15, 11 and 16, and 12 and 16). We can next examne the sample statstcs for our observed data. It s mportant to note that these values are not the smple descrptve statstcs one would obtan from SAS or SPSS. Smple descrptve statstcs gnore mssng data and delete any cases that are not nvolved n the calculaton of the statstc at hand. However, these results use maxmum lkelhood (ML) estmaton to provde optmal estmates of the sample statstcs. Usually these are qute close to those that are obtaned gnorng mssngness, but at tmes they are not. Vsual examnaton of the tme specfc means ndcate that negatve affect s decreasng approxmately lnearly over tme. ESTIMATED SAMPLE STATISTICS Means NA11 NA12 NA13 NA14 NA15 1 0.166 0.151 0.089-0.041-0.055 Means NA16 1-0.025 Further, the sample varances appear to be approxmately comparable over tme, rangng from.48 to.69. And the standardzed correlatons reflect relatons typcally found n repeated measures data; namely, measures are more strongly correlated wth one another the closer they are together n tme (e.g., age 11 and 12 NA s correlated.70, 11 and 13.62, 11 and 14.52, etc.). Covarances NA11 NA12 NA13 NA14 NA15 NA11 0.543 NA12 0.356 0.477 NA13 0.324 0.340 0.510 NA14 0.285 0.303 0.368 0.546 NA15 0.237 0.260 0.315 0.404 0.614 NA16 0.284 0.304 0.414 0.428 0.463 Covarances NA16 NA16 0.692

1.5 Demonstraton: Lnear Trajectores of Negatve Affect 1 47 Correlatons NA11 NA12 NA13 NA14 NA15 NA11 1.000 NA12 0.698 1.000 NA13 0.615 0.689 1.000 NA14 0.524 0.593 0.697 1.000 NA15 0.410 0.481 0.563 0.698 1.000 NA16 0.463 0.528 0.697 0.695 0.710 Correlatons NA16 NA16 1.000 Fnally, we can examne the dstrbutons of our data. Agan, we wll not go nto detal about the plottng utltes that are avalable n Mplus, but there are many optons and they are qute good. If you request plots va the plot command, ths wll actvate the tab "Plot" n the toolbar. For example, select Plot, then Hstograms : As a bref example, we wll select "Hstograms" and here s the dstrbuton for the frst assessment of negatve affect:

1 48 Chapter 1 The Uncondtonal Lnear Latent Curve Model We have done nothng to modfy ths plot, although there are many optons for scalng and labelng these plots. Havng examned our data, we wll now move to fttng our frst LCM. Intercept only LCM, Heteroscedastc Resduals We begn by fttng an ntercept only model to the sx repeated measures of negatve affect. There are many dfferent ways a latent curve model can be defned n Mplus. We wll begn by usng the smplest approach that s based on the vertcal bar opton (or " ") and allows for the effcent use of a number of defaults nvoked by Mplus. We wll demonstrate other approaches to defnng these models as we move nto more complex analyses. The path dagram for ths model s:

1.5 Demonstraton: Lnear Trajectores of Negatve Affect 1 49 The Mplus nput code for estmatng an ntercept only LCM s provded n the fle ch01_na_2.np. ttle: Negatve Affect: Intercept only data: fle=na_ext.dat; varable: names=d coa male base_ext ext11-ext16 na11-na16; usevarables=na11-na16; mssng=.; analyss: estmator=ml; coverage=0; model: na_nt na11@0 na12@1 na13@2 na14@3 na15@4 na16@5; output: sampstat stdyx; In the ANALYSIS secton, we specfy that we are usng maxmum lkelhood (ML) estmaton. Ths s the default estmator but t s sometmes good habt to explcate ths when movng to more complex models. The MODEL command specfes the one factor latent curve model. Here we use the vertcal bar " " to nvoke the defaults assocated wth the LCM. To the left of the vertcal bar are the labels for the ntercept (lsted frst), the lnear (lsted second), the quadratc (lsted thrd), and so on. If only one label s provded, then an ntercept only model s estmated; f two labels are provded, then a lnear model s estmated; etc. To the rght of the vertcal bar are the varables to be used to defne the LCM and the values of tme that are selected for the analyss. For an ntercept only model, the values of tme are superfluous, but we wll use these n the followng model to add a lnear slope component. Fnally, the OUTPUT secton requests sample statstcs (sampstat) and standardzed estmates (stdyx) n addton to the unstandardzed estmates that are provded by default. There are a myrad of other output optons, some of whch we wll use n later models. Let us now turn to the output. Much of the ntal nformaton s precsely the same as before, so we wll not present ths agan. THE MODEL ESTIMATION TERMINATED NORMALLY MODEL FIT INFORMATION Number of Free Parameters 8 Loglkelhood H0 Value -1150.220 H1 Value -1129.519 Informaton Crtera Akake (AIC) 2316.440 Bayesan (BIC) 2349.350 Sample-Sze Adjusted BIC 2323.961

1 50 Chapter 1 The Uncondtonal Lnear Latent Curve Model Ch-Square Test of Model Ft Value 41.402 Degrees of Freedom 16 P-Value 0.0005 RMSEA (Root Mean Square Error Of Approxmaton) CFI/TLI Estmate 0.059 90 Percent C.I. 0.037 0.082 Probablty RMSEA <=.05 0.223 CFI 0.953 TLI 0.965 Ch-Square Test of Model Ft for the Baselne Model Value 554.362 Degrees of Freedom 12 P-Value 0.0000 SRMR (Standardzed Root Mean Square Resdual) Value 0.115 The model estmaton termnated normally (whch s always a good sgn). However, as expected (based on the lnear pattern n the sample means) the ft of the ntercept only model s poor to borderlne as 2 ndcated by a large model ch square ( (16)=41.4, p=.0005) and RMSEA=.06. Interestngly, the CFI and TLI are both equal to.95 ndcatng good ft, but the SRMR s rather large at.12. Taken together, despte the hgh CFI and TLI values, the set of ft statstcs suggest a rather poor ft of the model to the data. Because of lkely model msspecfcaton reflected n the borderlne ft of the model, we wll not yet examne the parameter estmates and nstead attempt to extend the model to mprove ft. Intercept and Lnear Slope LCM, Heteroscedastc Resduals To extend the LCM to nclude a slope factor, we make a mnor adjustment n our model lne. The path dagram for ths model s on the followng page:

1.5 Demonstraton: Lnear Trajectores of Negatve Affect 1 51 The code correspondng code s provded n ch01_na_3.np. ttle: Negatve Affect: Intercept & slope, heteroscedastc errors data: fle=na_ext.dat; varable: names=d coa male base_ext ext11-ext16 na11-na16; usevarables=na11-na16; mssng=.; analyss: estmator=ml; coverage=0; model: na_nt na_slp na11@0 na12@1 na13@2 na14@3 na15@4 na16@5; where we have smply added na_slp followng na_nt. Ths nvokes addtonal defaults to estmate a lnear slope factor wth factor loadngs set to 0, 1, 2, 3, 4, and 5; ths slope factor s defned by a mean and varance and covares wth the ntercept factor. All other code remans precsely the same as before. Examnng more abbrevated model output, we see that the model acheves much mproved ft: Ch-Square Test of Model Ft Value 14.349 Degrees of Freedom 13 P-Value 0.3497 RMSEA (Root Mean Square Error Of Approxmaton) Estmate 0.015 90 Percent C.I. 0.000 0.050 Probablty RMSEA <=.05 0.948

1 52 Chapter 1 The Uncondtonal Lnear Latent Curve Model CFI/TLI CFI 0.998 TLI 0.998 The model ch square s non sgnfcant, the RMSEA s low, and the CFI and TLI are approachng ther maxma. Recall from earler that the ntercept only model s nested wthn the lnear model; that s, the ntercept only model s equvalent to a lnear growth model wth the mean and varance of the latent slope factor, and the covarance between the ntercept and the slope factor, all fxed to zero. As such, we can conduct a lkelhood rato test (LRT) to formally evaluate the mprovement n model ft wth the ncluson of the lnear slope factor (see Appendx A for a revew of ths topc). The LRT s smply the dfference between the model ch squares (41.4 14.35=27.1) that s dstrbuted on the dfference between the model degrees of freedom (16 13=3). A ch square of 27.1 dstrbuted on df=3 s hghly sgnfcant and ndcates that there s a substantal mprovement n model ft wth the ncluson of the lnear slope factor. Pror to nterpretng fnal parameter estmates, we can consder whether the model mght support homoscedastc resduals. The unque tme specfc estmates for the heteroscedastc resduals are Resdual Varances NA11 0.152 0.040 3.804 0.000 NA12 0.136 0.024 5.577 0.000 NA13 0.156 0.021 7.434 0.000 NA14 0.158 0.021 7.401 0.000 NA15 0.193 0.027 7.223 0.000 NA16 0.163 0.041 3.927 0.000 Note that the estmates all seem to vary n the.14 to.19 range, suggestng that a sngle value could stand for all sx tme ponts. We wll next mpose an equalty constrant on the tme specfc resdual varances to ascertan whether the fewer parameters are needed to reproduce the observed data. Intercept and Lnear Slope LCM, Homoscedastc Resduals The pror lnear LCM allowed for a unque tme specfc resdual varance to be estmated at each age (heteroscedastcty). However, a more parsmonous model would use a sngle resdual varance estmate to hold for all ages (homoscedastcty). Because the latter s nested wthn the former, we can conduct a formal LRT to evaluate the change n model ft wth the mposton of equalty constrants on the resduals. To do ths, we wll mpose an equalty constrant on the resduals; ths code s stored n ch02_na_4.np. (We also nclude an addtonal plottng command to whch we wll return shortly). ttle: Negatve Affect: Intercept & slope, homoscedastc errors data: fle=na_ext.dat; varable: names=d coa male base_ext ext11-ext16 na11-na16; usevarables=na11-na16; mssng=.; analyss: estmator=ml; coverage=0; model: na_nt na_slp na11@0 na12@1 na13@2 na14@3 na15@4 na16@5; na11-na16 (1);

1.5 Demonstraton: Lnear Trajectores of Negatve Affect 1 53 plot: type=plot3; seres=na11-na16 (*); output: sampstat stdyx; In Mplus, lstng a varable n the model statement wthout any brackets or parenthess references the varance of that varable. In the pror models, the resduals of the repeated measures have been freely estmated by default (.e., usng the " " command). We would now lke to estmate these wth the constrant that they be equal. To accomplsh ths, we do two thngs. Frst, we lst the repeated measures separately to reference the resdual varance; second, we assgn a number n parenthess to set those values to be equal. In Mplus, all parameters lsted on a gven lne that share the same number n parenthess are set to be equal. Thus, to set all of the resdual varances to be equal, we add the lne na11-na16 (1); Instead of estmatng sx separate resdual varances (one for each unque age), we wll obtan a sngle estmate of the resdual varances that holds for all sx ages. Ths model s nested wthn the pror model, so we can conduct an LRT to formally evaluate whether one resdual varance estmate s suffcent to hold for all sx tme ponts. The ch square for the lnear LCM wth homoscedastcty s Ch-Square Test of Model Ft Value 17.395 Degrees of Freedom 18 P-Value 0.4961 Recall that the ch square for the prevous model wth heteroscedastcty was Ch-Square Test of Model Ft Value 14.349 Degrees of Freedom 13 P-Value 0.3497 As expected, the LCM wth homoscedastcty has 5 fewer parameters than the LCM wth heteroscedastcty (because one parameter s needed for the resduals n the former and sx parameters are needed n the latter). The dfference n ch square statstcs s 17.395 14.349=3.046 that s dstrbuted on df=18 13=5 and s non sgnfcant (p=.31; note that we must determne ths probablty value based on the ch square dfference and df outsde of Mplus). We can conclude that ncreasng the model parsmony by estmatng just a sngle resdual varance for all sx ages does not lead to a sgnfcant reducton n model ft; that s, the model fts no worse wth one resdual varance compared to sx resdual varances. We thus retan ths model smplfcaton and can examne our fnal results. To begn, the model fts the data qute well, wth all ndces ndcatng excellent ft: Ch-Square Test of Model Ft Value 17.395 Degrees of Freedom 18 P-Value 0.4961

1 54 Chapter 1 The Uncondtonal Lnear Latent Curve Model RMSEA (Root Mean Square Error Of Approxmaton) CFI/TLI Estmate 0.000 90 Percent C.I. 0.000 0.040 Probablty RMSEA <=.05 0.989 CFI 1.000 TLI 1.001 SRMR (Standardzed Root Mean Square Resdual) Value 0.050 We can next examne the parameter estmates and standard errors (we only present a subset of the output here): MODEL RESULTS Two-Taled Estmate S.E. Est./S.E. P-Value NA_INT NA11 1.000 0.000 999.000 999.000 NA12 1.000 0.000 999.000 999.000 NA13 1.000 0.000 999.000 999.000 NA14 1.000 0.000 999.000 999.000 NA15 1.000 0.000 999.000 999.000 NA16 1.000 0.000 999.000 999.000 NA_SLP NA11 0.000 0.000 999.000 999.000 NA12 1.000 0.000 999.000 999.000 NA13 2.000 0.000 999.000 999.000 NA14 3.000 0.000 999.000 999.000 NA15 4.000 0.000 999.000 999.000 NA16 5.000 0.000 999.000 999.000 Because the ntercept and slope factor loadngs are set to predetermned values, these have no standard errors and Mplus reports 999.000 for crtcal ratos and p values; ths s perfectly fne and s exactly how we've set up the model. NA_SLP WITH NA_INT -0.035 0.019-1.842 0.065 The covarance of the ntercept and slope s negatve and margnally sgnfcant ( ˆ 21.035 ); we'll see n the standardzed output later n the program that ths translates nto a correlaton of r.39. The substantve nterpretaton s that adolescents who reporter hgher ntal levels of negatve affect to tend decrease more rapdly over tme (and vce versa). Care should be taken when nterpretng these covarances because they are drectly nfluenced by where the zero pont s placed when codng tme. As we would fully expect based on how the ntercept s defned, ths covarance would lkely be qute dfferent f the zero pont were placed at the mddle or at the end of the growth process.

1.5 Demonstraton: Lnear Trajectores of Negatve Affect 1 55 Means NA_INT 0.158 0.046 3.392 0.001 NA_SLP -0.048 0.014-3.449 0.001 Both factor means are sgnfcantly dfferent from zero and mply that, on average, the trajectory of negatve affects s ˆ 1.158 at age 11 and changes ˆ 2.048 unts per one year ncrease n age. Intercepts NA11 0.000 0.000 999.000 999.000 NA12 0.000 0.000 999.000 999.000 NA13 0.000 0.000 999.000 999.000 NA14 0.000 0.000 999.000 999.000 NA15 0.000 0.000 999.000 999.000 NA16 0.000 0.000 999.000 999.000 The tem ntercepts are fxed at zero ( 0 ) and also have no standard errors. Varances NA_INT 0.403 0.063 6.411 0.000 NA_SLP 0.020 0.007 2.869 0.004 The varances of both the ntercept ( ˆ 11.403) and the slope ( ˆ 22.02 ) sgnfcantly dffer from zero ndcatng potentally mportant ndvdually varablty n both startng pont and rate of change over tme. Ths would suggest that we mght be able to nclude one or more predctors of ntercept and slope to partally model ths varablty/ We wll address ths topc n detal n the next chapter. Resdual Varances NA11 0.163 0.010 15.780 0.000 NA12 0.163 0.010 15.780 0.000 NA13 0.163 0.010 15.780 0.000 NA14 0.163 0.010 15.780 0.000 NA15 0.163 0.010 15.780 0.000 NA16 0.163 0.010 15.780 0.000 The homoscedastcty condton s easly seen n that all sx tme specfc resduals take on precsely the same value ( ˆ.163 ). Ths s nterpreted as that part of the observed varance of the repeated measures that s not explaned by the underlyng latent factors. As such, we can standardze these nto measures of r squared: R-SQUARE Observed Two-Taled Varable Estmate S.E. Est./S.E. P-Value NA11 0.712 0.039 18.341 0.000 NA12 0.684 0.030 22.463 0.000 NA13 0.677 0.025 27.553 0.000 NA14 0.696 0.023 30.558 0.000 NA15 0.731 0.024 30.168 0.000 NA16 0.773 0.027 28.677 0.000 Note that although the resdual varances are equal over tme, the r squared values are not. Ths s because the observed varance dffers at each age, and thus the proporton of the observed varance explaned by the latent factors (as reflected n the multple r squared values) also dffers at each age. These values ndcate that the underlyng latent factors jontly explan about 70% 75% of the observed varance n the tme specfc measures; ths s actually a farly strong predcton for a context such as

1 56 Chapter 1 The Uncondtonal Lnear Latent Curve Model chld development. Ths could be n part ndcatve of the rather hgh relablty of the repeated measures that were scored usng a rgorous tem response theory modelng approach. Fnally, recall that we ncluded a plot functon that allows us to examne the observed vs. ftted mean trajectory. To revew, the new code was plot: type=plot3; seres=na11-na16 (*); The key lne here s seres=na11-na16 (*) whch requests that means be plotted for negatve affect from ages 11 and 16, and the astersk nvokes the default that tme be coded from 0 to 5 by 1. Agan, there are many varants of these plots; here we smply want to see the model mpled mean trajectory. From the "Plot" tab we select "Vew plots" and we are presented wth: We wll select "Sample and estmated means". The unedted plot we get s

1.5 Demonstraton: Lnear Trajectores of Negatve Affect 1 57 n whch the squares are the sample means and the trangles are the model mpled means. Much can be done to rescale and label ths plot, but we don't pursue ths further here. Fnally, we can request a plot of ndvdual trajectores for all or a subset of cases. These are sometmes called "spaghett plots" for obvous reasons. To obtan these, we select the "Plot" menu and then "Estmated ndvdual values": We next selected cases n consecutve order and arbtrarly asked for the frst 25:

1 58 Chapter 1 The Uncondtonal Lnear Latent Curve Model The resultng unedted plot s on the followng page: The plot s consstent wth the sgnfcant varablty n both startng pont and rate of change over tme. Plots such as these are often excellent ways to communcate model results n a research report and to assst n nterpretng varances and covarances of the latent factors. For example, a negatve correlaton between ntercept and slope mght ndcate hgher ntercepts are assocated wth less postve slopes or that hgher ntercepts are assocated wth more negatve slopes; movng from.50 to.25 (less postve) s equvalent to movng from.50 to.75 (more negatve) as both are smaller n value by.25. Augmentng numercal output wth graphcal plots can provde valuable nsghts nto the nature of these relatons. Agan, much could be done to modfy labels, axes, legends, etc., but we don't pursue ths further here.