# How to explain negative residual covariance in cross-lagged SEM?

Question in short:

If I see a positive (significant) residual covariance between two endogenous variables in a cross-lagged SEM for the first lag and a negative (significant) residual covariance for the second lag, how can I explain that descriptively to the audience? I am particularly interested in describing why we see such positive and negative residual covariances from the data itself. Could you please suggest?

Details of my scenario:

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):

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.

Disclaimer: Cross-posted from https://groups.google.com/forum/#!topic/lavaan/4H38vTlUcL4 as suggested there.

This is not surprising. These covariances are different things. The first covariance is controlling for age and sex, the second is controlling for the t1 measures of the same variables.

It's so uninteresting that I wouldn't even bother to mention it. It might have happened to me. It might not. I'm not sure I would have noticed.

• I'm surprised why such a small partial correlation turns out to be statistically significant. I guess calling this "residual covariance" as "partial correlation" is correct? Jul 2, 2018 at 19:53
• It's a partial covariance (I think), not correlation, so you can't say anything about the size. You can call it a residual covariance or partial covariance. Jul 2, 2018 at 19:55