Introduction to Structural Equation Modeling (SEM) Day 4: November 29, 2012

Transcription

1 Introduction to Structural Equation Modeling (SEM) Day 4: November 29, 202 ROB CRIBBIE QUANTITATIVE METHODS PROGRAM DEPARTMENT OF PSYCHOLOGY COORDINATOR - STATISTICAL CONSULTING SERVICE COURSE MATERIALS AVAILABLE AT:

2 Longitudinal Data Analysis with SEM Latent growth curve models A simple and informative strategy for modeling longitudinal data also referred to as latent trajectory models Can provide answers to interesting questions associated with longitudinal data, such as: ) What is the overall mean trajectory over time? 2) Do we need distinct trajectories for each individual? 3) Can we find variables to predict individual trajectories?

3 Growth Curve Models Growth modeling is a random-effects analysis of longitudinal data Advantages of a random-effects model Change over time (slopes) are free to vary across individuals Unexplained variability is captured in a residual term Main purpose is the analysis of change on an outcome variable Older methods for longitudinal data, such as repeated measures ANOVA and MANOVA are increasingly less popular and outdated.

4 Unconditional (no predictors) Growth Model Level model: yit = Level 2 model: α + β λ + i i t ε it α i = µ α + ς α i β i = µ β + ς β i

5 Unconditional LCM Level y it = The value of outcome for person i at time t α i = the intercept term; predicted value of y when λ t is equal to 0 (the initial status) λ t = coefficient for time t β i = the slope term; the predicted change in the outcome (y t ) per one unit change in time (λ t λ t- ) ε it = the residual for individual i at time t; the difference between the observed value of y t and the predicted value of y t

6 Unconditional LCM level 2 µ α = mean of the intercept factor; the predicted mean of y when λ t = 0 (usually average initial status, but could be any reference condition) ζ αi = the variability around the intercept mean; individual heterogeneity in the intercept µ β = mean of the slope factor; the predicted mean change in y associated with a one unit increase in time (λ) ζ βi = the variability around the average slope; individual heterogeneity in the slope

7 Unconditional LCM level 2 We are primarily interested in the elements of the level 2 model: The overall trajectory for the sample (mean change per unit of time) And possibly whether the sample trajectory significantly differs from 0 The average initial status (intercept, typically time ) The amount of variability in the trajectories or intercept Significant variability in the trajectories or intercept may indicate the possibility of finding predictors of change or initial staus, or that the individual trajectories or intercept may predict other endogenous variables

8 Unconditional LCM level 2 An important new element to growth models, over previously discussed models, is the usual inclusion of means/intercepts for the variables in the model The means of the observed longitudinal variables are fixed to 0, in order to be able to properly assess the latent means (i.e., mean initial status and mean growth) Residual means are fixed to zero, and other exogenous/endogenous means are freely estimated

9 Unconditional Linear Growth Model

10 Growth Models Time can be measured in many ways: Years, months, days, hours, etc. Time points DO NOT need to be evenly spaced! Simply change the λ values to reflect the spacing of the time points (or in some cases unevenly spaced time points may be necessary to estimate linear growth, e.g., growth in height from birth to two years) The difference between any two consecutive λ values for the slope factor reflects the amount of time elapsed between the two corresponding observed measurements

11 Example LCM A researcher collected data on children s externalizing behaviour over 6 years (ages 5 to 0 years old) We are interested in knowing: ) whether children differ significantly in their externalizing behaviours at age 5 2) whether externalizing behaviours tend to increase or decrease over time 3) is there significant individual variability in the children s externalizing behaviours over time?

12 Unconditional Model for Growth in Externalizing Behaviours

13 AMOS Output: Regression Coefficients Estimate S.E. C.R. P EXT_ <--- ICEPT.000 EXT_ <--- SLOPE.0000 EXT_2 <--- ICEPT.000 EXT_2 <--- SLOPE.000 EXT_3 <--- ICEPT.000 EXT_3 <--- SLOPE EXT_4 <--- ICEPT.000 EXT_4 <--- SLOPE EXT_5 <--- ICEPT.000 EXT_5 <--- SLOPE EXT_6 <--- ICEPT.000 EXT_6 <--- SLOPE 5.000

14 AMOS Output: Means Estimate S.E. C.R. P ICEPT *** SLOPE

15 AMOS Output: Variances Estimate S.E. C.R. P ICEPT *** SLOPE E E *** E *** E *** E E ***

16 AMOS Output: Covariances/Correlations Estimate S.E. C.R. P ICEPT <--> SLOPE Estimate ICEPT <--> SLOPE Relationships between the slope and intercept can sometimes be difficult to interpret

17 Growth Models Quadratic Growth It is sometimes possible that there is non-linear change over time For example, there could curvature in the pattern of change over time If we ignore potential non-linearity, biased estimates of the linear trajectory are likely, and we will probably have a poor model fit Quadratic patterns can be detected by adding a new latent variable whose coefficients are the square of the linear coefficients

18 0, 0, 0, 0, 0, 0, E E2 E3 E4 E5 E EXT_ EXT_2 EXT_3 EXT_4 EXT_5 EXT_ ICEPT SLOPE QUAD

19 Non-linear LCM However, including a quadratic factor may not always be the best way to model non-linear change over time There are two other common methods for investigating nonlinear growth over time ) Freely estimating the coefficients for growth over time a) fix the first loadings, but allow the later loadings to be free b) fix the first loading to 0 and the last loading to and the middle loadings are freely estimated 2) Using a piecewise model, where multiple latent variables are used to measure different patterns of growth

20 0, 0, 0, 0, 0, 0, E E2 E3 E4 E5 E EXT_ EXT_2 EXT_3 EXT_4 EXT_5 EXT_6 0 ICEPT SLOPE

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22 Conditional Latent Growth Models Modeling Predictors and Consequences of Change One of the advantages of latent growth curve models is the ability to test for variables that may covary with, predict, or be affected by initial status or change A researcher might want to describe, explain, or predict individual differences in the slope or intercept with one or more predictors/covariates.

23 Conditional LCM

24 Interpreting Conditional LCMs If the residual variance of the factor is significant; this indicates that there is substantial residual heterogeneity in the linear change in our outcome that was not explained by the predictor Likewise, significant residual variance of the intercept factor indicates that there is substantial residual heterogeneity in initial status that was not explained by the predictor

25 Extensions of Growth Models What else can we do with latent growth curve models? Evaluate whether slopes differ across subpopulations (multiple group latent growth curve models) Evaluate whether the relationship between change and a third variable (e.g., predictor of change) is equivalent across subpopulations Model interactions among two variables, each of which is measured at multiple time points Can also model time-varying covariates

26 Setting up Growth Models in AMOS Check Off the box estimate means and intercepts under View Analysis Properties Estimation Fix each observed variable s intercept to 0 Freely estimate the means of the intercept latent variable and slope latent variable Fix each parameter value for the arrows leading from the intercept latent variable to the observed variables to Fix the parameter value for the arrow leading from the slope latent variable to the reference time point (usually the first) to 0, and increase the parameter value by for each consecutive time point (if you are interested in linear growth and the time points are equally spaced)

27 Setting up Growth Models in AMOS All of these tasks can be done with a shortcut in AMOS for creating growth models Click on Plugins, then Growth Curve Model State the number of time points and most of the work is done Now just copy your observed variables onto the model and you should be ready to go Finally, BE SURE TO CHECK THAT EVERYTHING WAS SET UP PROPERLY This is mostly just my anxiety because the Growth Curve Model shortcut in older versions of AMOS did not set the models up properly

28 Introduction to LAVAAN LAVAAN is free, open-source, software for conducting structural equation modeling within the R software environment More specifically, LAVAAN is a package within R that is able to conduct a broad range of structural equation modeling analyses The fact that LAVAAN is free and open-source makes it especially appealing as both a teaching tool and as a tool for research into SEM LAVAAN is an acronym for LAtent VAriable ANalysis

29 Very, Very, Skimpy Introduction to R R is a free, open-source, software environment for statistical programming and analysis In addition to a core group of individuals responsible for developing R, R is also being built by interested researchers through what are entitled user-contributed packages LAVAAN is an example of a user contributed package LAVAAN was created by Yves Rosseel at Ghent University in Belium R has a steep learning curve, but after that it is a wonderful resource for conducting data analyses with unlimited possibilities E.g., there are currently more than 4000 packages available for conducting a wide range of statistical analyses

30 Introduction to LAVAAN Programming LAVAAN is text-driven, meaning that you specify your models without a graphical user interface I know.. booooo!!!!! There are four primary symbols used for specifying models =~ is used to indicate with indicators load on which latent variables E.g., f =~ item + item2 ~ is used as a regression operator (one-headed arrow in AMOS) E.g., y ~ x ~~ is used a variance/covariance operator (two-headed arrow in AMOS) E.g., y ~~ y2 (covariance) or y~~y (variance) ~ defines an intercept E.g., y ~

31 Introduction to LAVAAN Programming After you have specified your model, it is necessary to run the model using the function lavaan() This will produce output very similar to what we obtain when we run models in AMOS There are other ways to specify models that invoke defaults (e.g., automatically estimating observed variable variances), but we want the full experience Since LAVAAN was just recently developed, some tools are not yet available (e.g., categorical outcomes), but many other tools that are not available with other SEM software packages (e.g., robust estimation methods) are available with LAVAAN

32 SEM Example to try with LAVAAN

33 LAVAAN Example Opening our SPSS Data Set Activate the foreign package that contains the read.spss function library(foreign) Activate a data set called newdat, by reading an SPSS data set from my windows directory newdat<-read.spss(file.choose()) Make newdat a 'data frame' this is a technicality that is only necessary for reading SPSS data sets newdat<-data.frame(newdat) See the first few lines of the dataset head(newdat) See the names of the variables names(newdat)

34 LAVAAN Example Setting Up the Model Activate the package for conducting the SEM analyses library(lavaan) Note: The first time you use LAVAAN you will need to install the package using install.packages( LAVAAN ) Set up the Quant Attitudes and Performance Model Mod<- QuantAtt =~ *HINDR + ANX + SEFF EXAMAVG ~ QuantAtt HINDR ~~ HINDR ANX ~~ ANX SEFF ~~ SEFF EXAMAVG ~~ EXAMAVG QuantAtt ~~ QuantAtt Fit<-lavaan(mod, data=newdat) summary(fit, fit.measures=true) # Latent Variable # Regression # Observed Var Variance # Observed Var Variance # Observed Var Variance # Observed Var Variance # Latent Var Variance # Run the Model # Summary Statistics

35 LAVAAN Example Output on Fit Note: Looks a lot nicer outside of PowerPoint lavaan (0.5-) converged normally after 4 iterations Number of observations 29 Estimator ML Minimum Function Test Statistic.782 Degrees of freedom 2 P-value (Chi-square) 0.40 Full model versus baseline model: Comparative Fit Index (CFI).000 Tucker-Lewis Index (TLI).009 RMSEA Percent Confidence Interval P-value (RMSEA <= 0.05) 0.58 SRMR 0.027

36 LAVAAN Example Regressions Parameter estimates: Latent variables: Est Std.err Z P(> z ) QuantAtt =~ HINDR.000 QuantAtt =~ ANX QuantAtt =~ SEFF Regressions: EXAMAVG ~ QuantAtt

37 LAVAAN Example Output on Variances Variances: Est Std.err HINDR ANX SEFF EXAMAVG QuantAtt Note: These estimate match almost identically to those obtained with AMOS