Testing slopes and intercepts in multi group SEM

[Mark Marshall]

I have an unbalanced panel and have performed a simple FIML SEM multigroup modeling exercise.

********************************************************************************
* SIMULATED DATA FOR MULTI-GROUP LATENT GROWTH CURVE MODEL
********************************************************************************

clear all
set seed 12345

* Create 400 observations (200 per group)
set obs 400
gen id = _n

* Create two groups (mode: 1=HD, 2=PD)
gen mode = cond(_n <= 200, 1, 2)

********************************************************************************
* SIMULATE LATENT GROWTH TRAJECTORIES
********************************************************************************

* Group 1 (HD): Baseline ~15, slight increase over time
* Group 2 (PD): Baseline ~16, slight decrease over time

* Individual-level random effects
gen intercept_i = rnormal(0, 2)
gen slope_i = rnormal(0, 0.1)

* Generate observed scores at each timepoint with measurement error
* Group 1 (HD): Mean intercept = 15, Mean slope = 0.02
* Group 2 (PD): Mean intercept = 16, Mean slope = -0.05

gen RAWGH0 = cond(mode==1, 15, 16) + intercept_i + 0*slope_i + rnormal(0, 2)
gen RAWGH6 = cond(mode==1, 15, 16) + intercept_i + 6*cond(mode==1, 0.02, -0.05) + 6*slope_i + rnormal(0, 2)
gen RAWGH12 = cond(mode==1, 15, 16) + intercept_i + 12*cond(mode==1, 0.02, -0.05) + 12*slope_i + rnormal(0, 2)
gen RAWGH18 = cond(mode==1, 15, 16) + intercept_i + 18*cond(mode==1, 0.02, -0.05) + 18*slope_i + rnormal(0, 2)

* Add some missing data to make it realistic
replace RAWGH6 = . if uniform() < 0.10
replace RAWGH12 = . if uniform() < 0.15
replace RAWGH18 = . if uniform() < 0.20

* Clean up intermediate variables
drop intercept_i slope_i

* Label variables
label variable id "Patient ID"
label variable mode "Dialysis modality"
label define mode_label 1 "HD" 2 "PD"
label values mode mode_label
label variable RAWGH0 "General Health at baseline"
label variable RAWGH6 "General Health at 6 months"
label variable RAWGH12 "General Health at 12 months"
label variable RAWGH18 "General Health at 18 months"

********************************************************************************
* DESCRIPTIVE STATISTICS
********************************************************************************

tabulate mode
summarize RAWGH0 RAWGH6 RAWGH12 RAWGH18

bysort mode: summarize RAWGH0 RAWGH6 RAWGH12 RAWGH18

********************************************************************************
* MULTI-GROUP LATENT GROWTH CURVE MODEL
********************************************************************************

sem (Intercept@1 Slope@0 -> RAWGH0) ///
    (Intercept@1 Slope@6 -> RAWGH6) ///
    (Intercept@1 Slope@12 -> RAWGH12) ///
    (Intercept@1 Slope@18 -> RAWGH18), ///
    latent(Intercept Slope) ///
    method(mlmv) ///
    group(mode)

estimates store model_multigroup

* Display the parameter estimates
matrix list e(b)

* Show the model framework
estat framework

.

QUESTION FOR STATALIST: How do I test whether the latent growth trajectories differ between groups?

Specifically, I want to test:

1. Do the mean intercepts differ between mode=1 and mode=2? (i.e., is baseline General Health different between groups?)

2. Do the mean slopes differ between mode=1 and mode=2? (i.e., is the rate of change different between groups?)

3. Joint test: Do trajectories differ overall?

I can see from matrix list e(b) that there are parameters like:

* mean(Intercept) 1.mode
* mean(Intercept) 2.mode
* mean(Slope) 1.mode
* mean(Slope) 2.mode

But I cannot figure out the correct test syntax. I've tried:

test [mean(Intercept)]1.mode = [mean(Intercept)]2.mode
test mean(Intercept):1.mode = mean(Intercept):2.mode
test _b[mean(Intercept):1.mode] = _b[mean(Intercept):2.mode]

All give errors. What is the correct syntax?