Specifying Latent Curve and Other Growth Models Using Mplus. (Revised )
|
|
- Rachel Cross
- 6 years ago
- Views:
Transcription
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
12 Ronald H. Heck 12
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)
Multilevel Structural Equation Modeling
Multilevel Structural Equation Modeling Joop Hox Utrecht University j.hox@uu.nl http://www.joophox.net 14_15_mlevsem Multilevel Regression Three level data structure Groups at different levels may have
More informationModel Estimation Example
Ronald H. Heck 1 EDEP 606: Multivariate Methods (S2013) April 7, 2013 Model Estimation Example As we have moved through the course this semester, we have encountered the concept of model estimation. Discussions
More informationCHAPTER 9 EXAMPLES: MULTILEVEL MODELING WITH COMPLEX SURVEY DATA
Examples: Multilevel Modeling With Complex Survey Data CHAPTER 9 EXAMPLES: MULTILEVEL MODELING WITH COMPLEX SURVEY DATA Complex survey data refers to data obtained by stratification, cluster sampling and/or
More informationRon Heck, Fall Week 3: Notes Building a Two-Level Model
Ron Heck, Fall 2011 1 EDEP 768E: Seminar on Multilevel Modeling rev. 9/6/2011@11:27pm Week 3: Notes Building a Two-Level Model We will build a model to explain student math achievement using student-level
More informationAdditional Notes: Investigating a Random Slope. When we have fixed level-1 predictors at level 2 we show them like this:
Ron Heck, Summer 01 Seminars 1 Multilevel Regression Models and Their Applications Seminar Additional Notes: Investigating a Random Slope We can begin with Model 3 and add a Random slope parameter. If
More informationIntroducing Generalized Linear Models: Logistic Regression
Ron Heck, Summer 2012 Seminars 1 Multilevel Regression Models and Their Applications Seminar Introducing Generalized Linear Models: Logistic Regression The generalized linear model (GLM) represents and
More informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Today s Class (or 3): Summary of steps in building unconditional models for time What happens to missing predictors Effects of time-invariant predictors
More informationReview of CLDP 944: Multilevel Models for Longitudinal Data
Review of CLDP 944: Multilevel Models for Longitudinal Data Topics: Review of general MLM concepts and terminology Model comparisons and significance testing Fixed and random effects of time Significance
More informationRon Heck, Fall Week 8: Introducing Generalized Linear Models: Logistic Regression 1 (Replaces prior revision dated October 20, 2011)
Ron Heck, Fall 2011 1 EDEP 768E: Seminar in Multilevel Modeling rev. January 3, 2012 (see footnote) Week 8: Introducing Generalized Linear Models: Logistic Regression 1 (Replaces prior revision dated October
More informationComputationally Efficient Estimation of Multilevel High-Dimensional Latent Variable Models
Computationally Efficient Estimation of Multilevel High-Dimensional Latent Variable Models Tihomir Asparouhov 1, Bengt Muthen 2 Muthen & Muthen 1 UCLA 2 Abstract Multilevel analysis often leads to modeling
More informationAn Introduction to Multilevel Models. PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 25: December 7, 2012
An Introduction to Multilevel Models PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 25: December 7, 2012 Today s Class Concepts in Longitudinal Modeling Between-Person vs. +Within-Person
More informationMultilevel Analysis of Grouped and Longitudinal Data
Multilevel Analysis of Grouped and Longitudinal Data Joop J. Hox Utrecht University Second draft, to appear in: T.D. Little, K.U. Schnabel, & J. Baumert (Eds.). Modeling longitudinal and multiple-group
More informationDescribing Change over Time: Adding Linear Trends
Describing Change over Time: Adding Linear Trends Longitudinal Data Analysis Workshop Section 7 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Section
More informationMultilevel regression mixture analysis
J. R. Statist. Soc. A (2009) 172, Part 3, pp. 639 657 Multilevel regression mixture analysis Bengt Muthén University of California, Los Angeles, USA and Tihomir Asparouhov Muthén & Muthén, Los Angeles,
More informationEstimating a Piecewise Growth Model with Longitudinal Data that Contains Individual Mobility across Clusters
Estimating a Piecewise Growth Model with Longitudinal Data that Contains Individual Mobility across Clusters Audrey J. Leroux Georgia State University Piecewise Growth Model (PGM) PGMs are beneficial for
More informationGoals for the Morning
Introduction to Growth Curve Modeling: An Overview and Recommendations for Practice Patrick J. Curran & Daniel J. Bauer University of North Carolina at Chapel Hill Goals for the Morning Brief review of
More informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Today s Topics: What happens to missing predictors Effects of time-invariant predictors Fixed vs. systematically varying vs. random effects Model building
More informationSEM Day 3 Lab Exercises SPIDA 2007 Dave Flora
SEM Day 3 Lab Exercises SPIDA 2007 Dave Flora 1 Today we will see how to estimate SEM conditional latent trajectory models and interpret output using both SAS and LISREL. Exercise 1 Using SAS PROC CALIS,
More informationReview of Multiple Regression
Ronald H. Heck 1 Let s begin with a little review of multiple regression this week. Linear models [e.g., correlation, t-tests, analysis of variance (ANOVA), multiple regression, path analysis, multivariate
More informationNesting and Equivalence Testing
Nesting and Equivalence Testing Tihomir Asparouhov and Bengt Muthén August 13, 2018 Abstract In this note, we discuss the nesting and equivalence testing (NET) methodology developed in Bentler and Satorra
More informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Topics: Summary of building unconditional models for time Missing predictors in MLM Effects of time-invariant predictors Fixed, systematically varying,
More informationLogistic And Probit Regression
Further Readings On Multilevel Regression Analysis Ludtke Marsh, Robitzsch, Trautwein, Asparouhov, Muthen (27). Analysis of group level effects using multilevel modeling: Probing a latent covariate approach.
More informationCentering Predictor and Mediator Variables in Multilevel and Time-Series Models
Centering Predictor and Mediator Variables in Multilevel and Time-Series Models Tihomir Asparouhov and Bengt Muthén Part 2 May 7, 2018 Tihomir Asparouhov and Bengt Muthén Part 2 Muthén & Muthén 1/ 42 Overview
More informationMeasurement Invariance (MI) in CFA and Differential Item Functioning (DIF) in IRT/IFA
Topics: Measurement Invariance (MI) in CFA and Differential Item Functioning (DIF) in IRT/IFA What are MI and DIF? Testing measurement invariance in CFA Testing differential item functioning in IRT/IFA
More informationCHAPTER 3. SPECIALIZED EXTENSIONS
03-Preacher-45609:03-Preacher-45609.qxd 6/3/2008 3:36 PM Page 57 CHAPTER 3. SPECIALIZED EXTENSIONS We have by no means exhausted the possibilities of LGM with the examples presented thus far. As scientific
More informationGrowth Mixture Modeling and Causal Inference. Booil Jo Stanford University
Growth Mixture Modeling and Causal Inference Booil Jo Stanford University booil@stanford.edu Conference on Advances in Longitudinal Methods inthe Socialand and Behavioral Sciences June 17 18, 2010 Center
More informationInvestigating Models with Two or Three Categories
Ronald H. Heck and Lynn N. Tabata 1 Investigating Models with Two or Three Categories For the past few weeks we have been working with discriminant analysis. Let s now see what the same sort of model might
More informationAn Efficient State Space Approach to Estimate Univariate and Multivariate Multilevel Regression Models
An Efficient State Space Approach to Estimate Univariate and Multivariate Multilevel Regression Models Fei Gu Kristopher J. Preacher Wei Wu 05/21/2013 Overview Introduction: estimate MLM as SEM (Bauer,
More informationTime Invariant Predictors in Longitudinal Models
Time Invariant Predictors in Longitudinal Models Longitudinal Data Analysis Workshop Section 9 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Section
More informationRonald Heck Week 14 1 EDEP 768E: Seminar in Categorical Data Modeling (F2012) Nov. 17, 2012
Ronald Heck Week 14 1 From Single Level to Multilevel Categorical Models This week we develop a two-level model to examine the event probability for an ordinal response variable with three categories (persist
More informationMultilevel Regression Mixture Analysis
Multilevel Regression Mixture Analysis Bengt Muthén and Tihomir Asparouhov Forthcoming in Journal of the Royal Statistical Society, Series A October 3, 2008 1 Abstract A two-level regression mixture model
More informationCategorical and Zero Inflated Growth Models
Categorical and Zero Inflated Growth Models Alan C. Acock* Summer, 2009 *Alan C. Acock, Department of Human Development and Family Sciences, Oregon State University, Corvallis OR 97331 (alan.acock@oregonstate.edu).
More informationLongitudinal Invariance CFA (using MLR) Example in Mplus v. 7.4 (N = 151; 6 items over 3 occasions)
Longitudinal Invariance CFA (using MLR) Example in Mplus v. 7.4 (N = 151; 6 items over 3 occasions) CLP 948 Example 7b page 1 These data measuring a latent trait of social functioning were collected at
More informationModeling Heterogeneity in Indirect Effects: Multilevel Structural Equation Modeling Strategies. Emily Fall
Modeling Heterogeneity in Indirect Effects: Multilevel Structural Equation Modeling Strategies By Emily Fall Submitted to the Psychology and the Faculty of the Graduate School of the University of Kansas
More informationIntroduction to Within-Person Analysis and RM ANOVA
Introduction to Within-Person Analysis and RM ANOVA Today s Class: From between-person to within-person ANOVAs for longitudinal data Variance model comparisons using 2 LL CLP 944: Lecture 3 1 The Two Sides
More informationTime-Invariant Predictors in Longitudinal Models
Time-Invariant Predictors in Longitudinal Models Topics: What happens to missing predictors Effects of time-invariant predictors Fixed vs. systematically varying vs. random effects Model building strategies
More informationEstimation of Curvilinear Effects in SEM. Rex B. Kline, September 2009
Estimation of Curvilinear Effects in SEM Supplement to Principles and Practice of Structural Equation Modeling (3rd ed.) Rex B. Kline, September 009 Curvlinear Effects of Observed Variables Consider the
More informationUsing Mplus individual residual plots for. diagnostics and model evaluation in SEM
Using Mplus individual residual plots for diagnostics and model evaluation in SEM Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 20 October 31, 2017 1 Introduction A variety of plots are available
More informationClass Notes: Week 8. Probit versus Logit Link Functions and Count Data
Ronald Heck Class Notes: Week 8 1 Class Notes: Week 8 Probit versus Logit Link Functions and Count Data This week we ll take up a couple of issues. The first is working with a probit link function. While
More informationGeneralized Linear Models for Non-Normal Data
Generalized Linear Models for Non-Normal Data Today s Class: 3 parts of a generalized model Models for binary outcomes Complications for generalized multivariate or multilevel models SPLH 861: Lecture
More informationHow to run the RI CLPM with Mplus By Ellen Hamaker March 21, 2018
How to run the RI CLPM with Mplus By Ellen Hamaker March 21, 2018 The random intercept cross lagged panel model (RI CLPM) as proposed by Hamaker, Kuiper and Grasman (2015, Psychological Methods) is a model
More informationDESIGNING EXPERIMENTS AND ANALYZING DATA A Model Comparison Perspective
DESIGNING EXPERIMENTS AND ANALYZING DATA A Model Comparison Perspective Second Edition Scott E. Maxwell Uniuersity of Notre Dame Harold D. Delaney Uniuersity of New Mexico J,t{,.?; LAWRENCE ERLBAUM ASSOCIATES,
More informationHypothesis Testing for Var-Cov Components
Hypothesis Testing for Var-Cov Components When the specification of coefficients as fixed, random or non-randomly varying is considered, a null hypothesis of the form is considered, where Additional output
More informationNELS 88. Latent Response Variable Formulation Versus Probability Curve Formulation
NELS 88 Table 2.3 Adjusted odds ratios of eighth-grade students in 988 performing below basic levels of reading and mathematics in 988 and dropping out of school, 988 to 990, by basic demographics Variable
More informationVariable-Specific Entropy Contribution
Variable-Specific Entropy Contribution Tihomir Asparouhov and Bengt Muthén June 19, 2018 In latent class analysis it is useful to evaluate a measurement instrument in terms of how well it identifies the
More informationMplus Short Courses Day 2. Growth Modeling With Latent Variables Using Mplus
Mplus Short Courses Day 2 Growth Modeling With Latent Variables Using Mplus Linda K. Muthén Bengt Muthén Copyright 2007 Muthén & Muthén www.statmodel.com 1 Table Of Contents General Latent Variable Modeling
More informationMLMED. User Guide. Nicholas J. Rockwood The Ohio State University Beta Version May, 2017
MLMED User Guide Nicholas J. Rockwood The Ohio State University rockwood.19@osu.edu Beta Version May, 2017 MLmed is a computational macro for SPSS that simplifies the fitting of multilevel mediation and
More informationStructural Equation Modeling and Confirmatory Factor Analysis. Types of Variables
/4/04 Structural Equation Modeling and Confirmatory Factor Analysis Advanced Statistics for Researchers Session 3 Dr. Chris Rakes Website: http://csrakes.yolasite.com Email: Rakes@umbc.edu Twitter: @RakesChris
More informationIntroduction to Random Effects of Time and Model Estimation
Introduction to Random Effects of Time and Model Estimation Today s Class: The Big Picture Multilevel model notation Fixed vs. random effects of time Random intercept vs. random slope models How MLM =
More informationMplus Short Courses Topic 3. Growth Modeling With Latent Variables Using Mplus: Introductory And Intermediate Growth Models
Mplus Short Courses Topic 3 Growth Modeling With Latent Variables Using Mplus: Introductory And Intermediate Growth Models Linda K. Muthén Bengt Muthén Copyright 2008 Muthén & Muthén www.statmodel.com
More informationFall Homework Chapter 4
Fall 18 1 Homework Chapter 4 1) Starting values do not need to be theoretically driven (unless you do not have data) 2) The final results should not depend on starting values 3) Starting values can be
More informationDescribing Nonlinear Change Over Time
Describing Nonlinear Change Over Time Longitudinal Data Analysis Workshop Section 8 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development Section 8: Describing
More informationIntroduction to mtm: An R Package for Marginalized Transition Models
Introduction to mtm: An R Package for Marginalized Transition Models Bryan A. Comstock and Patrick J. Heagerty Department of Biostatistics University of Washington 1 Introduction Marginalized transition
More informationThe fixed- and random-effects parts of the mixed-model are specified in the MODEL and RANDOM
1 A. Univariate Random-Intercepts Model in Proc MIXED Softare programs for estimating parameters of mixed-effects model such as SAS Proc Mixed and HLM, are designed to accept data in a univariate format
More informationModel Assumptions; Predicting Heterogeneity of Variance
Model Assumptions; Predicting Heterogeneity of Variance Today s topics: Model assumptions Normality Constant variance Predicting heterogeneity of variance CLP 945: Lecture 6 1 Checking for Violations of
More informationPlausible Values for Latent Variables Using Mplus
Plausible Values for Latent Variables Using Mplus Tihomir Asparouhov and Bengt Muthén August 21, 2010 1 1 Introduction Plausible values are imputed values for latent variables. All latent variables can
More informationA Re-Introduction to General Linear Models (GLM)
A Re-Introduction to General Linear Models (GLM) Today s Class: You do know the GLM Estimation (where the numbers in the output come from): From least squares to restricted maximum likelihood (REML) Reviewing
More informationResearch Design - - Topic 19 Multiple regression: Applications 2009 R.C. Gardner, Ph.D.
Research Design - - Topic 19 Multiple regression: Applications 2009 R.C. Gardner, Ph.D. Curve Fitting Mediation analysis Moderation Analysis 1 Curve Fitting The investigation of non-linear functions using
More informationPart 8: GLMs and Hierarchical LMs and GLMs
Part 8: GLMs and Hierarchical LMs and GLMs 1 Example: Song sparrow reproductive success Arcese et al., (1992) provide data on a sample from a population of 52 female song sparrows studied over the course
More informationCourse Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model
Course Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model EPSY 905: Multivariate Analysis Lecture 1 20 January 2016 EPSY 905: Lecture 1 -
More informationIntroduction to lnmle: An R Package for Marginally Specified Logistic-Normal Models for Longitudinal Binary Data
Introduction to lnmle: An R Package for Marginally Specified Logistic-Normal Models for Longitudinal Binary Data Bryan A. Comstock and Patrick J. Heagerty Department of Biostatistics University of Washington
More informationResearch Design: Topic 18 Hierarchical Linear Modeling (Measures within Persons) 2010 R.C. Gardner, Ph.d.
Research Design: Topic 8 Hierarchical Linear Modeling (Measures within Persons) R.C. Gardner, Ph.d. General Rationale, Purpose, and Applications Linear Growth Models HLM can also be used with repeated
More informationSTAT 730 Chapter 9: Factor analysis
STAT 730 Chapter 9: Factor analysis Timothy Hanson Department of Statistics, University of South Carolina Stat 730: Multivariate Data Analysis 1 / 15 Basic idea Factor analysis attempts to explain the
More informationPsychology 282 Lecture #4 Outline Inferences in SLR
Psychology 282 Lecture #4 Outline Inferences in SLR Assumptions To this point we have not had to make any distributional assumptions. Principle of least squares requires no assumptions. Can use correlations
More informationAn Introduction to Path Analysis
An Introduction to Path Analysis PRE 905: Multivariate Analysis Lecture 10: April 15, 2014 PRE 905: Lecture 10 Path Analysis Today s Lecture Path analysis starting with multivariate regression then arriving
More informationWU Weiterbildung. Linear Mixed Models
Linear Mixed Effects Models WU Weiterbildung SLIDE 1 Outline 1 Estimation: ML vs. REML 2 Special Models On Two Levels Mixed ANOVA Or Random ANOVA Random Intercept Model Random Coefficients Model Intercept-and-Slopes-as-Outcomes
More informationCourse Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model
Course Introduction and Overview Descriptive Statistics Conceptualizations of Variance Review of the General Linear Model PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 1: August 22, 2012
More informationAn Introduction to Mplus and Path Analysis
An Introduction to Mplus and Path Analysis PSYC 943: Fundamentals of Multivariate Modeling Lecture 10: October 30, 2013 PSYC 943: Lecture 10 Today s Lecture Path analysis starting with multivariate regression
More informationOnline Appendices for: Modeling Latent Growth With Multiple Indicators: A Comparison of Three Approaches
Online Appendices for: Modeling Latent Growth With Multiple Indicators: A Comparison of Three Approaches Jacob Bishop and Christian Geiser Utah State University David A. Cole Vanderbilt University Contents
More informationIntroduction to Structural Equation Modeling
Introduction to Structural Equation Modeling Notes Prepared by: Lisa Lix, PhD Manitoba Centre for Health Policy Topics Section I: Introduction Section II: Review of Statistical Concepts and Regression
More informationTesting Main Effects and Interactions in Latent Curve Analysis
Psychological Methods 2004, Vol. 9, No. 2, 220 237 Copyright 2004 by the American Psychological Association 1082-989X/04/$12.00 DOI: 10.1037/1082-989X.9.2.220 Testing Main Effects and Interactions in Latent
More informationMultilevel Modeling: A Second Course
Multilevel Modeling: A Second Course Kristopher Preacher, Ph.D. Upcoming Seminar: February 2-3, 2017, Ft. Myers, Florida What this workshop will accomplish I will review the basics of multilevel modeling
More informationChapter 5. Introduction to Path Analysis. Overview. Correlation and causation. Specification of path models. Types of path models
Chapter 5 Introduction to Path Analysis Put simply, the basic dilemma in all sciences is that of how much to oversimplify reality. Overview H. M. Blalock Correlation and causation Specification of path
More informationModel fit evaluation in multilevel structural equation models
Model fit evaluation in multilevel structural equation models Ehri Ryu Journal Name: Frontiers in Psychology ISSN: 1664-1078 Article type: Review Article Received on: 0 Sep 013 Accepted on: 1 Jan 014 Provisional
More informationBasic IRT Concepts, Models, and Assumptions
Basic IRT Concepts, Models, and Assumptions Lecture #2 ICPSR Item Response Theory Workshop Lecture #2: 1of 64 Lecture #2 Overview Background of IRT and how it differs from CFA Creating a scale An introduction
More informationModeration 調節 = 交互作用
Moderation 調節 = 交互作用 Kit-Tai Hau 侯傑泰 JianFang Chang 常建芳 The Chinese University of Hong Kong Based on Marsh, H. W., Hau, K. T., Wen, Z., Nagengast, B., & Morin, A. J. S. (in press). Moderation. In Little,
More informationPath Analysis. PRE 906: Structural Equation Modeling Lecture #5 February 18, PRE 906, SEM: Lecture 5 - Path Analysis
Path Analysis PRE 906: Structural Equation Modeling Lecture #5 February 18, 2015 PRE 906, SEM: Lecture 5 - Path Analysis Key Questions for Today s Lecture What distinguishes path models from multivariate
More informationAdvanced Structural Equations Models I
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. Your use of this material constitutes acceptance of that license and the conditions of use of materials on this
More informationLatent Variable Centering of Predictors and Mediators in Multilevel and Time-Series Models
Latent Variable Centering of Predictors and Mediators in Multilevel and Time-Series Models Tihomir Asparouhov and Bengt Muthén August 5, 2018 Abstract We discuss different methods for centering a predictor
More informationPart 7: Hierarchical Modeling
Part 7: Hierarchical Modeling!1 Nested data It is common for data to be nested: i.e., observations on subjects are organized by a hierarchy Such data are often called hierarchical or multilevel For example,
More informationMULTILEVEL IMPUTATION 1
MULTILEVEL IMPUTATION 1 Supplement B: MCMC Sampling Steps and Distributions for Two-Level Imputation This document gives technical details of the full conditional distributions used to draw regression
More informationA (Brief) Introduction to Crossed Random Effects Models for Repeated Measures Data
A (Brief) Introduction to Crossed Random Effects Models for Repeated Measures Data Today s Class: Review of concepts in multivariate data Introduction to random intercepts Crossed random effects models
More informationTesting and Interpreting Interaction Effects in Multilevel Models
Testing and Interpreting Interaction Effects in Multilevel Models Joseph J. Stevens University of Oregon and Ann C. Schulte Arizona State University Presented at the annual AERA conference, Washington,
More informationTime Metric in Latent Difference Score Models. Holly P. O Rourke
Time Metric in Latent Difference Score Models by Holly P. O Rourke A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved June 2016 by the Graduate
More informationCitation for published version (APA): Jak, S. (2013). Cluster bias: Testing measurement invariance in multilevel data
UvA-DARE (Digital Academic Repository) Cluster bias: Testing measurement invariance in multilevel data Jak, S. Link to publication Citation for published version (APA): Jak, S. (2013). Cluster bias: Testing
More informationApplication of Plausible Values of Latent Variables to Analyzing BSI-18 Factors. Jichuan Wang, Ph.D
Application of Plausible Values of Latent Variables to Analyzing BSI-18 Factors Jichuan Wang, Ph.D Children s National Health System The George Washington University School of Medicine Washington, DC 1
More informationSupplementary materials for: Exploratory Structural Equation Modeling
Supplementary materials for: Exploratory Structural Equation Modeling To appear in Hancock, G. R., & Mueller, R. O. (Eds.). (2013). Structural equation modeling: A second course (2nd ed.). Charlotte, NC:
More informationModeling growth in dyads at the group level
Modeling growth in dyads at the group level Thomas Ledermann University of Basel Siegfried Macho University of Fribourg The Study of Change The study of change is important for the understanding of many
More informationThursday Morning. Growth Modelling in Mplus. Using a set of repeated continuous measures of bodyweight
Thursday Morning Growth Modelling in Mplus Using a set of repeated continuous measures of bodyweight 1 Growth modelling Continuous Data Mplus model syntax refresher ALSPAC Confirmatory Factor Analysis
More informationAssessing the relation between language comprehension and performance in general chemistry. Appendices
Assessing the relation between language comprehension and performance in general chemistry Daniel T. Pyburn a, Samuel Pazicni* a, Victor A. Benassi b, and Elizabeth E. Tappin c a Department of Chemistry,
More informationFrom Micro to Macro: Multilevel modelling with group-level outcomes
From Micro to Macro: Multilevel modelling with group-level outcomes by C.A. (Marloes) Onrust s274 A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Methodology
More informationChapter 4: Factor Analysis
Chapter 4: Factor Analysis In many studies, we may not be able to measure directly the variables of interest. We can merely collect data on other variables which may be related to the variables of interest.
More informationOutline
2559 Outline cvonck@111zeelandnet.nl 1. Review of analysis of variance (ANOVA), simple regression analysis (SRA), and path analysis (PA) 1.1 Similarities and differences between MRA with dummy variables
More informationADVANCED C. MEASUREMENT INVARIANCE SEM REX B KLINE CONCORDIA
ADVANCED SEM C. MEASUREMENT INVARIANCE REX B KLINE CONCORDIA C C2 multiple model 2 data sets simultaneous C3 multiple 2 populations 2 occasions 2 methods C4 multiple unstandardized constrain to equal fit
More informationLongitudinal Data Analysis of Health Outcomes
Longitudinal Data Analysis of Health Outcomes Longitudinal Data Analysis Workshop Running Example: Days 2 and 3 University of Georgia: Institute for Interdisciplinary Research in Education and Human Development
More informationA Study of Statistical Power and Type I Errors in Testing a Factor Analytic. Model for Group Differences in Regression Intercepts
A Study of Statistical Power and Type I Errors in Testing a Factor Analytic Model for Group Differences in Regression Intercepts by Margarita Olivera Aguilar A Thesis Presented in Partial Fulfillment of
More informationUsing Structural Equation Modeling to Conduct Confirmatory Factor Analysis
Using Structural Equation Modeling to Conduct Confirmatory Factor Analysis Advanced Statistics for Researchers Session 3 Dr. Chris Rakes Website: http://csrakes.yolasite.com Email: Rakes@umbc.edu Twitter:
More informationConfirmatory Factor Analysis: Model comparison, respecification, and more. Psychology 588: Covariance structure and factor models
Confirmatory Factor Analysis: Model comparison, respecification, and more Psychology 588: Covariance structure and factor models Model comparison 2 Essentially all goodness of fit indices are descriptive,
More informationHow well do Fit Indices Distinguish Between the Two?
MODELS OF VARIABILITY VS. MODELS OF TRAIT CHANGE How well do Fit Indices Distinguish Between the Two? M Conference University of Connecticut, May 2-22, 2 bkeller2@asu.edu INTRODUCTION More and more researchers
More informationHierarchical Generalized Linear Models. ERSH 8990 REMS Seminar on HLM Last Lecture!
Hierarchical Generalized Linear Models ERSH 8990 REMS Seminar on HLM Last Lecture! Hierarchical Generalized Linear Models Introduction to generalized models Models for binary outcomes Interpreting parameter
More information