Negative residual covariance between categorical endogenous variables in a cross-lagged SEM

[Blain Waan]

I have two categorical (binary) endogenous variables emp and SA at three time points, 1, 2, and 5. I have run the following cross-lagged SEM with fixed effects age and sex:

full_clpm1 <- '

# synchronous covariances

SA1 ~~ emp1

SA2 ~~ emp2

SA5 ~~ emp5

# autoregressive + cross-lagged paths

emp1 ~ AGE + sex 

SA1 ~ AGE + sex 

emp2 ~ AGE + sex + emp1 + SA1 

SA2 ~ AGE + sex + emp1 + SA1

emp5 ~ AGE + sex + emp1 + SA1 + emp2 + SA2

SA5 ~ AGE + sex + emp1 + SA1 + emp2 + SA2

'

# fit the model 

fit1 <- sem(full_clpm1, data=dp)

summary(fit1)

I got the following results for the residual covariances and wondering why the first covariance is positive and the second one is negative (both statistically significant): 

	Estimate	Std. Err	z-value	P(>|z|)
Covariance at year 1	0.167	0.054	3.112	0.002
Covariance at year 2	-0.077	0.037	-2.069	0.039
Covariance at year 5	-0.009	0.044	-0.212	0.832

I want to know what this means and how I can show descriptively why the second time point had negative residual covariance between emp2 and SA2. I already tried separate probit regressions on emp2 and SA2 and found the correlations of the residuals from these two non-simultaneous regression models. But that correlation appears to be positive (and very small). So, could you suggest me a way to show the reason of this negative residual covariance descriptively for the sake of discussion? I am wondering there must be some way of explaining this negative residual covariance descriptively. 

[Terrence Jorgensen]

This is not a lavaan issue, so a more general forum like SEMNET or Stack Overflow would be a better place to post a general statistical question like this.  But you can also start by reading about what (semi)partial correlations are, and how their interpretation differs from zero-order correlations.

Terrence D. Jorgensen

Postdoctoral Researcher, Methods and Statistics

Research Institute for Child Development and Education, the University of Amsterdam

UvA web page: http://www.uva.nl/profile/t.d.jorgensen

[Blain Waan]

Thanks for your reply. I read about partial correlations. But I'm still not clear how I can descriptively show using my data the reason for this negative partial correlation. Is it even possible to show that descriptively?  

[Blain Waan]

One more thing, does lavaan give me partial "correlation" or "covariance" as the result? So, for year 1, is 0.167 is a "correlation" or a "covariance" given that I didn't standardize age? 

[Terrence Jorgensen]

One more thing, does lavaan give me partial "correlation" or "covariance" as the result? 

correlations are not estimated model parameters (unless by coincidence or design, the corresponding variances are both 1).  The covariances are estimated, and you can request a standardized estimate, which will be in a correlation metric.