Latent Growth Models 1

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2 We will use the dataset bp3, which has diastolic blood pressure measurements at four time points for 256 patients undergoing three types of blood pressure medication. These are our observed variables: 2

3 We're going to focus on linear change 3

4 We want to be able to capture two aspects of the subjects' scores their overall level and the way that their blood pressure changes over time 4

5 Unobserved, latent variables The elements of a latent growth model 5

6 The intercept part of the model 6

7 The linear slope part of the model 7

8 The full latent growth model (bp LG-lin.amw) Estimating: The variance of the latent variables The variance of the errors The covariance of the latent variables 8

9 1. Open (bp LG-lin.amw) 2. Click on - Analysis properties 9

10 Click on Calculate estimates 10

11 These results show that there is significant variation between patients in terms intercepts and slopes. There is also a nearly significant (negative) correlation between intercepts and slopes. 11

12 We would now like to know the average intercept and slope for the subjects. To estimate means: Ask AMOS to estimate means and intercepts Name the means (and variances) Fix intercepts for the observed variables at zero (run bp LG-lin5.amw) Constrain all residuals to have the same variance 12

13 From the LGM: From a multilevel/mixed model analysis of the same data: 13

14 Create the individual trajectories 1. Open the model in which the means are estimated 2. Ask AMOS to 'impute' a dataset 3. Use SPSS to create the individual values at the different time points from the intercept and slope values saved from AMOS 4. Stack the SPSS data to graph the individual lines 14

15 Create the individual trajectories the steps (1) 1. Open bp lg-lin5.amw 2. Click on Analyze => Data Imputation Click on Impute 5. Click on OK when the imputation process finishes 15

16 Create the individual trajectories the steps (2) 6. Open bp_c.sav 7. Use the syntax in bp estimates from LGM.sps to create the predicted values and stack the data 16

17 Extending the LG model antecedent and consequential variables 17

18 Extending the LG model antecedent and consequential variables The latent variables become endogenous and are allowed to covary The BP treatment variable is dummy coded Run bp lg-lin-treat.amw 18

19 Extending the LG model antecedent and consequential variables According to this, the mean BP of the participants in the Atenolol group starts 2.8 points lower than that for the participants in the Carvedilol group. The actual difference is

20 Extending the LG model antecedent and consequential variables We can test whether there is an overall effect of treatment on the linear change in blood pressure by setting the linear paths to zero and comparing the fit of the model to the full model. run bp lg-lin-treat-test-linear.amw Full model Reduced model Chi^2 difference = =.437 with 2 df Not significant (Exercise Test whether there is a difference between the groups at Time 1 ) 20

21 Extending the LG model antecedent and consequential variables hr2 is a numeric 'health result' measure run bp lg-lin-treat-outcome.amw 21

22 Extending the LG model antecedent and consequential variables 22

23 Further extensions The effect of different coefficients Modelling curvilinear change Latent variables for indicators Parallel-process LG models 23

24 The effect of different coefficients T1 T2 T3 T4 The intercept will show the mean at: Intercept: Linear: T1 Intercept: Linear: T2 Intercept: Linear: T3 24

25 Curvilinear change Latent Growth Models I L Q

26 Curvilinear change (2) Sue Greig Note constraints to identify model which is estimating quadratic slope from only three time points 26

27 Latent variables as indicators - Sue Greig Note constraints to identify model which is estimating quadratic slope from only three time points 27

28 Parallel Process Models - Sue Greig 28

29 Readings for latent growth models 29