# In SEM, why would adding covariance for construct C's items cause the path between construct A and B to be significant?

Suppose I have the following model (using lavaan's syntax).

# Four latent variables
A =~ a1 + a2 + a3
B =~ b1 + b2 + b3
C =~ c1 + c2 + c3
D =~ d1 + d2 + d3

# Regression paths
B ~ A
D ~ C + B


When I run the model, I find that the regression from A -> B is not significant (p=.17). However, I add the following covariance as suggested by modification indices.

c1 ~~ c2


When I run the model again, the regression from A -> B is significant at p=.02. Why would specifying the covariance between two measurement items cause such a difference in the path between two other constructs?

When you change things in a model, and parameters flip from significant to not-significant, two things might have happened:

• Parameter estimates changed
• Standard errors changed

(Or both).

Without knowing which of these happened, it's hard to say.

Each of your latent variables has three indicators - that means each latent essentially has one df to add to the model. This means that the model is only identified because it is 'borrowing' from other parts of the model. When this is the case, the model is pretty unstable, and changes can 'flow' through the model - a small change in one place can change things in any other part of the model. (If you have more indicators, this is less likely).

When you add the correlation of c1~~c2, the nature of the C latent changes. (Now C can be unrelated to c1 and c2, and only have something to do with c3). The correlation of B and C changes, so the path from B to D can change. There is no direct relationship between A and D, it has to be estimated via B, so the path from A to B is dependent on the path from B to D.

In this sort of situation, it's worth taking what's sometimes called the 2-step (or, confusingly, 4-step) approach. First, let your latent variables correlate and sort out the measurement model. When you have the measurement model correct, sort out your structural model. If the measurement model isn't correct, your structural model also won't be correct.