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

42 views

### Blain Waan

Jul 1, 2018, 2:55:14 PM7/1/18
to lavaan
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

Jul 2, 2018, 6:26:22 AM7/2/18
to lavaan
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

### Blain Waan

Jul 2, 2018, 12:57:31 PM7/2/18
to lavaan
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

Jul 3, 2018, 1:38:41 AM7/3/18
to lavaan
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?