1. G89.2247 Lecture 111 SEM analogue of General Linear Model Fitting structure of mean vector in SEM Numerical Example Growth models in SEM Willett and Sayer (1994) Examples

2. G89.2247 Lecture 112 SEM as Analogue of General Linear Model Regression models can be used to estimate t tests and ANOVAs  Groups are coded with dummy variables (0,1) or effect variables (-.5,+.5)  Regression parameters can be interpreted in terms of group means and differences between means  Continuous covariates as well as interactions can be added to the model

3. G89.2247 Lecture 113 Numerical Example of GLM: Dummy coded X

4. G89.2247 Lecture 114 Taking Means into Account in SEM So far we have analyzed Variance Covariance Matrices, (First moments around the mean) Now we will analyze the general First moment matrix

5. G89.2247 Lecture 115 New Information, New Parameters The general sums of squares matrix has p new pieces of information: the variable means The new models will either  Account for the means in a saturated model  Or they will represent the means in a more parsimonious SEM model New ideas can be explored  Harmony of covariance and means patterns  Variance and covariance of contrasts

6. G89.2247 Lecture 116 Path Representation of Means Models To fit the numerical example we need a constant term  Y = 32.48 + 12.86X + e(Y)  X =.61 + e(X) X Y e(X) e(Y) 1

7. G89.2247 Lecture 117 Means in SEM Software In EQS the mean is the coefficient associated with a system variable called V999  V999 represents the triangle In LISREL there are new Greek constant terms  X =  X  X   Y =  Y  Y    = 

8. G89.2247 Lecture 118 Examples in Handout EQS Examples  GLM version of t test  GLM version of t test with covariate (ANCOVA) Covariate W is strongly related with group indicator X  GLM version of t test with centered covariate  Two group analysis with separate slopes

9. G89.2247 Lecture 119 Means Models with Latent Variables: Saturated Means Structure To date we have thought implicitly about latent variables as having mean zero. Let's be explicit. We have adjusted for manifest variable means. Depression Sadness ~ Energy Despair  1 1 

10. G89.2247 Lecture 1110 Means Models with Latent Variables: Inferred Means Structure If the latent variable drives the means as well as the covariances, we get a different stronger model for the means of the manifest variables. Depression Sadness ~ Energy Despair  1 1 

11. G89.2247 Lecture 1111 Continuing with Examples in Handout EQS Examples  Latent variable as covariate with mean zero Comparable to earlier example with centered covariate  Latent variable as covariate with nonzero mean Comparable to earlier example with noncentered covariate

12. G89.2247 Lecture 1112 Latent Growth Models via SEM Suppose we had five repeated measures, spaced equally over time. An analysis of Y 1, Y 2, Y 3, Y 4, Y 5 that uses only variance/covariances ignores trajectories. Willett and Sayer review SEM models that allow us to think about systematic linear growth. These models use mean structures.

13. G89.2247 Lecture 1113 Example of Trajectories

14. G89.2247 Lecture 1114 Latent Growth Models "Level 1" model: Represents how Y changes over time points (Willett and Sayer notation)  Y ip =  0p +  1p t i +  ip Suppose t 1 = 0. Then  0p is the subject-specific intercept for the trajectory (the value of Y at time 1) The value  1p is the subject-specific slope of Y with a unit change of time. We will be able to study the covariation of the intercept and slopes in "Level 2" parts of the model  Level 1 is between time, Level 2 is between person

15. G89.2247 Lecture 1115 Level 1 Models in SEM Diagram looks like confirmatory factor analysis, but the "loading" are fixed, not estimated. Within person processes are inferred from between person covariance patterns. Y1Y2 Y4 Y3 00 11 D2  D1 Y5  1 11 1 1 01234 1

16. G89.2247 Lecture 1116 Level 2 and Level 1 Models 1 Group Y1Y2 Y4 Y3 00 11 D2  D1 Y5  1 11 1 1 01234

17. G89.2247 Lecture 1117 Willett and Sayer Example 168 adolescents are measured at five points in time (ages 11, 12, 13, 14, 15)  Outcome is Tolerance of Deviant Behavior Transformed with log function for analysis Questions:  Is there evidence that TDB is going up on average?  Do youth vary in their slopes?  Are there individual differences By gender? By early exposure to deviance?

18. G89.2247 Lecture 1118 Three models Fitting random and average trajectories assuming that variances within person are stable (WS Model 1) Fitting random and average trajectories assuming that variances within person are variable (WS Model 2) Fitting random and average trajectories and checking associations of slope and intercept with gender and exposure to deviance (WS Model 4)