G89 2247 Lecture 11 SEM analogue of General Linear Model Fitting structure of mean vector in SEM Numerical Example Growth models in SEM Willett and Sayer 1994 Examples G89 2247 Lecture 11 1 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 G89 2247 Lecture 11 2 Numerical Example of GLM Dummy coded X t TEST Variables are selected from file LEC12 ESS y on x 0 00 N MEAN STND DEV y on x 32 480 15 891 1 00 N MEAN STND DEV METHOD POOLED t SEPARATE t 39 MEDIAN 34 171 MEDIAN 47 816 61 45 345 15 409 TEST STATISTICS 4 0229 3 9955 DF 98 0 79 3 P VALUE 0 0001 0 0001 MULTIPLE REGRESSIONS Dependent variable Number of obs Multiple R R square y 100 0 3765 0 1417 REGRESSION TEST STATISTICS VARIABLES Intercept x COEFFICIENTS 32 48041 12 86502 STD ERROR t P VALUE 3 1979 4 023 0 0001 G89 2247 Lecture 11 3 Taking Means into Account in SEM So far we have analyzed Variance Covariance Matrices First moments around the mean S X X nXX n 1 Now we will analyze the general First moment matrix X X n 1 S n XX G89 2247 Lecture 11 4 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 G89 2247 Lecture 11 5 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 e Y e X Y X 1 G89 2247 Lecture 11 6 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 G89 2247 Lecture 11 7 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 G89 2247 Lecture 11 8 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 Sadness 1 Energy 1 Depression Despair G89 2247 Lecture 11 9 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 1 Sadness Energy 1 Depression Despair G89 2247 Lecture 11 10 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 G89 2247 Lecture 11 11 Latent Growth Models via SEM Suppose we had five repeated measures spaced equally over time An analysis of Y1 Y2 Y3 Y4 Y5 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 G89 2247 Lecture 11 12 Example of Trajectories Ten Trajectories 7 00 Response 6 00 5 00 4 00 3 00 2 00 1 00 0 00 1 2 3 4 5 Time Point G89 2247 Lecture 11 13 Latent Growth Models Level 1 model Represents how Y changes over time points Willett and Sayer notation Yip 0p 1pti ip Suppose t1 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 G89 2247 Lecture 11 14 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 1 0 D1 1 1 0 1 1 1 1 1 2 D2 3 4 Y1 Y2 Y3 Y4 Y5 G89 2247 Lecture 11 15 Level 2 and Level 1 Models Group 1 0 D1 1 1 0 1 1 1 1 1 2 3 D2 4 Y1 Y2 Y3 Y4 Y5 G89 2247 Lecture 11 16 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 G89 2247 Lecture 11 17 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 G89 2247 Lecture 11 18