Vandenberg & Lance (2000) explain that measurement invariance in SEM is important because the variance-covariance matrix of a group, $\Sigma$, and item scores, $X$, can be decomposed, such that




where $g$ is a group identifier, $k$ is a variable identifier, $\tau$ is an intercept, $\lambda$ is a regression slope relating $X$ to $\xi$, $\delta$ are unique factors, $\Lambda$ is an item's factor loadings on the latent factor $\xi$, $\Phi$ is the variance-covariance among $\xi$, and $\Theta$ is the unique variance to an item.

Is my diagram below correct in representing this relationship? I expect the $\delta$ and $\theta$ parts are incorrect, so I am unsure of how to diagram them.

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1 Answer 1


You've drawn the diagram a little unconventionally, for example you've only drawn one indicator, and $\Phi$ is often irrelevant when you're looking at invariance, so it is not modelled. $\theta$ is usually thought of as being equal to 1, and so can also be ignored / removed.

Apart from that, looks OK to me.


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