Specifying Latent Curve and Other Growth Models Using Mplus. (Revised )

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1 Ronald H. Heck 1 University of Hawai i at Mānoa Handout #20 Specifying Latent Curve and Other Growth Models Using Mplus (Revised ) The SEM approach offers a contrasting framework for use in analyzing various kinds of longitudinal data. In this framework, the distinction between level 1 (within individuals) and level 2 (between individuals) in a two-level random coefficients growth model is not made because the outcomes are considered as multivariate data in a T-dimensional vector (y 1,y 2,,y T ). Because of this difference in conceptualizing the data structure in SEM, the repeated measures over time can be expressed as a type of single-level confirmatory factor analysis (or measurement model), where the intercept and growth latent factors are measured by the multiple indicators of y T. The structural part of the SEM analysis can then be used to investigate the effects of covariates or other latent variables on the latent growth curve factors. Once the overall latent curve model has been specified through relating the observed repeated measures variables to the latent factors that represent the change process, it can be further divided into its within- and between-group components, similar to other types of multilevel models. This implies that the three-level multilevel univariate growth model (with repeated measures nested within individuals and individuals nested within groups) can be expressed two level latent growth model using the SEM specification. Until recently, full information ML for multilevel latent curve models was limited by the complexity of specifying models within existing software and the computational demands in trying to estimate the models. The availability of full information ML for missing data analysis has enhanced the usefulness of SEM methods (e.g., alleviating the need for listwise deletion, allowing unequal intervals between measurement occasions) for examining individual change (Mehta & Neale, 2005). As Mehta and Neal note, this has allowed the fitting of SEM to individual data vectors by making available model estimation methods based on individual likelihood that could accommodate unbalanced data structures (e.g., individuals within clusters). Consider an example where we were interested in examining individual development in math over four years (y 1 - y 4 ) in a random sample of elementary school students. In conducting a latent growth curve analysis, the researcher s first concern is typically to establish the level of initial status (or end status) and the rate of change. Fitting a two-factor latent growth curve model is accomplished first by fixing the intercept factor loadings to 1.0. Because the repeated measures are considered as factor loadings, there are several choices regarding how to parameterize the growth rate factor. In this case, we can specify a linear growth trajectory by using 0, 1, 2, 3 to represent fixed successive linear measurements. Since the first factor loading is fixed at 0, it is shown as a dotted line in Figure 1. Alternatively, researchers may decide not indicate the loading from the factor to the observed indicator in their path diagrams since it is 0.

2 Ronald H. Heck 2 Figure 1. Proposed latent curve model. The typical two-level, random-components growth model with a vector of repeated measures at level 1 nested within individuals at level 2 can be expressed in SEM terms as a single-level multivariate model by utilizing the general measurement and structural models (Muthén, 2002). The measurement model specifies a vector of repeated observed measures which define a smaller set of latent growth factors (often an initial status factor and a linear growth factor) as follows: y x, (1) it t i i it where y it is a vector of math outcomes for individual i at time t ( yi1, yi2,..., yit), t is a vector of measurement intercepts set to 0 at the individual level, t is a p x m design matrix representing the change process, i is an m-dimensional vector of latent variables ( 0, 1), Κ is a p x q parameter matrix of regression slopes relating x i ( x1 i, x2i,..., xpi) to the latent factors, and ε it represents time-specific errors, which are contained in a covariance matrix designated as. The factor loadings for the latent factors (i.e., in this case, η 0i and η 1i ) are contained in the factor-loading matrix, as in the following: t t (2) Fixing the loadings of the four measurement occasions on the first factor (η 0i ) to a value of 1.0 ensures that it is interpreted as a true (i.e., error-free) initial status factor, that is, representing a baseline point of the underlying developmental process under investigation. Because the repeated measures are specified as parameters (i.e., factor loadings) in a latent curve model (i.e., instead of data points in a multilevel model), the researcher can hypothesize that they follow particular shapes or patterns and test the hypothesized growth pattern against the data. The slope factor (η 1i ) can be defined as linear with successive time points of 0, 1, 2, 3. This scaling indicates the

3 Ronald H. Heck 3 slope coefficient will represent successive yearly intervals, starting with initial status as time score 0 (i.e., the loading from the growth rate factor to y 1 is shown as a dotted line in Figure 1). All parameters are fixed, therefore, in the Λ t matrix in this type of specification (which in some cases may be an undesirable restriction). i Below is the lambda matarix for specifying the latent intercept (I) and growth factors (S) as specified in the Mplus TECH1 output. Because the loadings are fixed (at 1.0 for the intercept factor, and from 0-3 for the growth factor), in this growth model specification there are no freely-estimated growth parameters specified. LAMBDA I S GRADE1 0 0 GRADE2 0 0 GRADE3 0 0 GRADE4 0 0 As noted, defining the shape of the change ahead of testing has been referred to as an intercept and slope (IS) approach (Raykov & Marcoulides, 2006). By fixing the factor loadings to a particular pattern (t = 0, 1, 2, 3), the hypothesized growth shape may then be tested against the actual data and its fit determined by examining various SEM fit indices. If the linear model does not fit the data well, one might add a quadratic factor to the factor loading matrix (i.e., which requires an intercept factor, a linear slope factor, and a quadratic slope factor with loadings 0, 1, 4, 9 as a third column). The psi covariance matrix (Ψ) contains latent factor variances and covariances. ii There are also other possible ways to define the factor loadings in the Λ t matrix to indicate different patterns of individual change. The residuals are specified in the theta covariance matrix (Θ). THETA GRADE1 GRADE2 GRADE3 GRADE4 GRADE1 1 GRADE2 0 2 GRADE GRADE Alternatively, we may actually know little about the shape of the individual trajectories. Various types of nonlinear growth can also be captured within the basic two-factor LCA formulation by using different patterns of free (estimated) and fixed factor loadings within the basic level and shape (LS) formulation. In Mplus model statements, the asterisk (*) option is used to designate a free parameter, such as a factor loading. For the LS formulation, we fix the first time score to zero and the last time score to 1.0, while allowing the middle time scores to be freely estimated (0, *, *, 1). This can allow for valleys and peaks within the overall growth interval being measured (e.g., 0, , 1). The factor-loading matrix would be the following:

4 Ronald H. Heck * t 1 * 1 1 (3) We could also capture other types of nonlinear growth, using combinations of free and fixed factor loadings within the factor-loading matrix. For example, we could allow the last time score, or the last two time scores, to be freely estimated [(0, 1, 2,*) or (0, 1, *, *)]. In addition to describing possible peaks or valleys, this type of formulation would allow for the estimation of various trajectories where the change might accelerate (e.g., 0, 1, 3, 6) or decelerate (e.g., 0, 1, 1.5, 1.75) over time. As these models suggest, the basic two-factor LCA formulation is quite flexible in allowing the researcher to consider both the levels of the growth or decline over time as well as the shape of the trend, which summarizes the growth rate over a specified time interval. The analyst can observe whether the change is positive or negative and whether there is heterogeneity in the level and rate of change across the individuals in the sample (Willett & Sayer, 1996). The Structural Model After the measurement model is used to relate the successive observed measures to the initial status and growth rate factors (similar to factor analysis), the second SEM model (i.e., the structural model) is used to relate one or more time-invariant covariates to the random effects (i.e., initial status and growth trend factors). The general structural model for individual i is then x, (4) i i i i where is vector of intercepts for the equations, is an m x m parameter matrix of slopes for the regressions among the latent variables, Γ is an m x q slope parameter matrix relating x covariates to latent variables, and i is a vector of residuals with zero means and are contained in covariance matrix Ψ. Consistent with the two-level growth modeling approach, i contains intercept ( 0i ) and slope ( 1i ) latent factors whose variability can be explained by one or more covariates. For example, here would be a simple model to explain latent growth in math using two covariates (e.g., gender, motivation) to explain differences in the initial status and linear growth factors over time. (5) (6), 0i 0 01x1i 02x2i 0 i, 1i 1 11x1i 12x2i 1 i where 0 and 1 are equation intercepts and coefficients are structural parameters describing

5 Ronald H. Heck 5 the regressions of latent variables on the covariates. As noted, the equation intercepts are contained in alpha. ALPHA I S FEMALE MOTIVATION In practice, it is convenient in Mplus to specify the model such that the regression of the latent intercept (I) and slope (S) factors on the covariates are contained in the B structural matrix (instead of a gamma matrix). iii BETA I S FEMALE MOTIVATION I S FEMALE MOTIVATION Each growth equation has its own residual ( 0i and 1i ) which permits the quality of measurement associated with each individual s growth trajectory to differ from those of other individuals. Residual variances and covariances for the latent factors are contained in Ψ. PSI I S GENDER MOTIVATION I 11 S FEMALE MOTIVATION It is important to keep in mind that the random-coefficients and the latent-curve approaches describe individual development in a slightly different manner. First, as noted, the SEM formulation treats the individual s change trajectory as single-level, multivariate data. The approach accounts for the correlation across time by the same random effects influencing each of the variables in the outcome vector (Muthén & Asparouhov, 2003). In contrast, the randomcoefficients approach treats the trajectories as univariate data and accounts for correlation across time by having two levels in the model. A second difference is the manner in which the time scores (i.e., the scaling of the observations) are considered. For random-coefficients growth modeling, the time scores are data; that is, they are pieces of information entered in the level-1 (within individual) data set to represent what the individual s math score was at each particular point in time. For LCA, however, the time scores are considered as model parameters; therefore, they are defined as factor loadings on the growth factors. The individual scores at each time interval are therefore not represented as data points within the data set. Finally, latent growth

6 Ronald H. Heck 6 curve modeling typically proceeds from an analysis of means and covariance structures. In contrast, random coefficients (multilevel) growth modeling creates a separate growth trajectory with intercept and slope for each individual. Random-coefficients multilevel models are typically not described in terms of means and covariance structures (Raudenbush, 2001). The SEM submodels for the single-level, multivariate specification of latent change can be easily expanded to facilitate specifying multilevel latent change models that decompose observed and latent variables into their uncorrelated within- and between-cluster components. This type of two-level formulation is consistent with a three-level univariate random coefficients growth model, except that the scores are assumed to vary across individuals and clusters in this formulation and there are not time-varying covariates present (Muthen & Asparouhov, 2011). The general measurement model for individual i in group j at time t can be expressed as Y, (7) where tw tij tw ij tij is the within-group factor loading matrix for the latent growth factors and tij are timerelated residuals are assumed to have no means and are contained in Θ W. The measurement intercepts for the repeated measures defining the growth factors are considered to be random intercepts that are referred to in the between part of the model, where in Mplus diagrams they are shown in circles since they are continuous latent variables that vary across clusters (i.e., they have between-cluster variances). For example, for a simple multilevel latent curve model defining three repeated math measures within groups, we have the following specification for the time-related lambda matrix within groups. LAMBDA LW SW LB SB MATH MATH MATH Once again, within groups the latent level factor (LW) has factor loadings fixed to 1.0, so there are no parameters estimated. The latent shape factor (SW) has the first two occasions fixed (0.1) and the third occasion is freely estimated. The between-group latent factors are not specified within groups (with parameters fixed to 0). The time-related residuals are defined in theta as follows: THETA MATH1 MATH2 MATH3 MATH1 0 MATH2 0 2 MATH

7 Ronald H. Heck 7 The matrix specification indicates that the residual for MATH1 was fixed to 0 (which was necessary for model convergence. It is useful to note that in the multilevel specification parameter matrices such as Λ, Β, Γ, Θ, and Ψ can be specified both within clusters and between clusters. The corresponding within-groups structural model can be specified as x, (8) ij j j ij j ij ij where the cluster (j) subscripts on the parameter matrices suggest the possibility that some elements of these matrices vary over clusters. The residuals in equations are assumed to have zero means and are contained in Ψ W. Between groups, a corresponding measurement model can be defined specifying the relationships of the repeated measures to the intercept and growth latent factors:, (9) j tb j tj where ν is a vector measurement intercepts (which are generally fixed to zero because the latent factor means are instead estimated) and tj are time-related residuals that are contained in Θ B. It is possible to specify a different latent growth model between groups from the within-school model describing individual change. Once again, the growth factors are specified in lambda (Between) as follows: LAMBDA LW SW LB SB MATH MATH MATH Time related residuals are contained in Theta (Between), where the first time-related error was fixed to 0: THETA MATH1 MATH2 MATH3 MATH1 0 MATH2 0 8 MATH Factor intercepts, which vary across groups, are contained in alpha (and consequently are not estimated within groups.) ALPHA LW SW LB SB

8 Ronald H. Heck 8 Finally, the factor variances and covariances between groups are contained in PSI (between): PSI SL SW LB SB SL 0 SW 0 0 LB SB It is also possible to specify randomly-varying intercepts and slopes across clusters for other parts of the growth model. For example, in the example we are developing, we might specify that the effect of student SES on the latent growth factors within groups varies across schools. In this latter formulation, multilevel measurement and structural models with random effects can be investigated by permitting elements of some coefficient matrices (e.g., Β and Γ) to vary at the cluster level, as in Eq. 8 (Preacher et al., 2010). Because the slopes vary randomly across groups, they are not estimated in the matrix of structural relationships within groups. In this case, between groups, the intercepts for the random slopes are specified in the alpha vector: ALPHA LW SW S1 S2 LB SB Between schools, we can then specify a structural model to examine variability in the withingroup random intercept and slope effects. A general between-groups structural model can be defined as x, (10) j j j j where j is a vector of the between-group latent growth factors and random effects (e.g., random slopes) from the within-groups model, is a vector of intercepts for the between-groups equations, x j represent between-groups covariates, Β and Γ represent structural relationships regarding latent and observed variables, and j are errors in equations that have zero means and are contained in Ψ B. We might now add a school-level predictor (school quality) to explain variation in the two latent growth parameters (LB and SB) and the random within-groups slopes. We can specify the final model in Figure 2. The freely-estimated (*) math3 parameters within and between groups suggest there might be acceleration or deceleration over the last time period.

9 Ronald H. Heck 9 Figure 2. Proposed multilevel latent curve model. The specification of the Beta matrix suggests that the four latent-variable outcomes are regressed on the quality of the school s educational processes. BETA LW SW S1 S2 LB SB LowSES Quality LW SW S S LB SB LOWSES QUALITY Errors in the between-group equations are once again contained in Ψ B. Of course, we can introduce more complex relationships between groups including direct and indirect effects as well as school-level change processes that might explain growth in student leaning in math. The Mplus input file for the two-level latent curve model is next summarized.

10 Ronald H. Heck 10 TITLE: DATA: VARIABLE: ANALYSIS: Model: OUTPUT: Two-level latent curve model of math growth (M1); FILE IS C:\program files\mplus\ch7 ex2.txt; Format is 10f8.0,2f8.2; Names are schcode math1 math2 math3 female lowses slep sped middle notrans cses quality; Usevariables schcode math1 math2 math3 lowses quality; within = lowses; between = quality; CLUSTER IS schcode; TYPE = twolevel random; Estimator is MLR; %BETWEEN% lb sb math1@0 math2@1 math3*; math1@0; lb sb S1 S2 on quality; %WITHIN% lw sw math1@0 math2@1 math3*; math1@0; S1 lw on lowses; S2 sw on lowses; Sampstat TECH1; Defining A Random-Coefficients Growth Model in Mplus We can also specify a random-coefficients model in Mplus, where the repeated observations on Y T nested within individuals at level 1 and differences between individuals are examined at level 2. Consider, for example, where we wish to look at difference in initial status graduation rates for a sample of four-year institutions and their change in graduation rates over time. We have four years of graduation rates nested within institutions (with four data lines per individual institution within the data set). We also have covariates related to the type of institution (public = 1, private = 0) and prestige (1 = selective student admissions standards, 0 = else). Since there are no latent growth parameters, at level 1 we can write a simple growth model as follows: Y X (11) ti i i ti ti where i is a vector of intercepts for the equations between individuals, Β i is a regression matrix of relationships between growth parameters and covariates, X ti is a vector of time-related variables (e.g., linear and possible quadratic components, time-varying covariates), and ti is a vector of time-related errors which are contained in Ψ W. We will specify the initial status intercept and the growth rate to vary randomly across individual institutions at level 2. In this

11 Ronald H. Heck 11 simple growth model, there is only one parameter specified within groups (i.e., the residual variance for the initial status graduation rate). PSI S GRADUATE GROWRATE PRIVATE PRESTIGE S 0 GRADUATE 0 1 GROWRATE PRIVATE PRESTIGE Between-individuals, we can specify a structural model to explain the random effects (i.e., the initial status intercept and random growth rate) as follows: X (12) i i i i where is a vector of intercepts for the initial status and growth rate random effects, is a matrix of regression coefficients specifying relationships between the random components, Γ describes relationships between level-2 covariates ( X i ) and random growth parameters, and i is a vector of errors in equations, which are contained in Ψ B. The corresponding between-institution matrix specification follows. ALPHA S GRADUATE GROWRATE PRIVATE PRESTIGE BETA S GRADUATE GROWRATE PRIVATE PRESTIGE S GRADUATE GROWRATE PRIVATE PRESTIGE PSI S GRADUATE GROWRATE PRIVATE PRESTIGE S 8 GRADUATE 9 10 GROWRATE PRIVATE PRESTIGE

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13 Ronald H. Heck 13 The Mplus input file for the random-coefficients growth model is summarized below. TITLE: DATA: VARIABLE: define: ANALYSIS: Model: Two-Level Random-Coefficients Growth Model; File is C:\mplus\ch7ex1.dat; Format is 3f8.0, f4.0,2f8.0; Names are id private prestige index1 graduate growrate; Usevariables are id graduate growrate private prestige; cluster = id; within= growrate; between = private prestige; center private prestige(grand); Type = twolevel random; Estimator is MLR; %between% graduate on private prestige; S on private prestige; graduate with S; %within% S graduate on growrate; OUTPUT: sampstat tech1; The model results are presented next in Table 1. The results suggest the adjusted initial status graduation level (grand-mean centered) was percent (in six years). The average linear growth rate was Moreover, private schools had higher initial status graduation rates (6.529) than their public counterparts. Prestige did not affect initial status, however. Regarding growth, private schools made more growth over time (6.059) than public institutions. Table 1. Variables explaining institutional growth over time. Estimate S.E. Est./S.E. P-Value Within Level Residual Variances GRADUATE Between Level S ON PRIVATE PRESTIGE GRADUATE ON PRIVATE PRESTIGE GRADUATE WITH S Intercepts GRADUATE S Residual Variances GRADUATE

14 Ronald H. Heck 14 S References Mehta, P.D. & Neale, M. C. (2005). People are variables too. Multilevel structural equations modeling. Psychological Methods, 10(3), Muthén, B. O. & Asparouhov, T. (2003). Advances in latent variable modeling. Part I: Integrating multilevel and structural equation modeling using Mplus. Unpublished paper. Muthen & Asparouhov, T. (2011). Beyond multilevel regression modeling: Multilevel analysis in a general latent variable framework. In J. Hox and J. K. Roberts (Eds.), Handbook of advanced multilevel analysis (pp ). New York: Taylor and Francis. Preacher, K. J., Zyphur, M. J., & Zhang, Z. (2010). A general multilevel SEM framework for assessing multilevel mediation. Psychological Methods, 15, Raudenbush, S. W. (2001). Comparing personal tranectories and drawing causal inferences from longitudinal data. Annual Review of Psychology, 52, Raykov, T. & Marcoulides, G. A. (2006). A first course in structural equation modeling (2 nd edition). Mahwah, NJ: Lawrence Erlbaum. Willett, J. & Sayer, A. (1996). Cross-domain analysis of change over time: Combining growth modeling and covariance structure analysis. In G. Marcoulides and R. Schumacker (Eds.), Advanced structural equation modeling: Issues and techniques (pp ). Mahwah, NJ: Lawrence Erlbaum. Endnotes i The measurement part of the model can therefore be specified as follows: y1 y 2 y3 y (A1) ii The psi matrix corresponds to: Ψ = , (A2)

15 Ronald H. Heck 15 iii The structural model can be referred to as follows: 0 0 1,3 1, ,3 2, (A3)