Multilevel Structural Equation Modeling - model identification

[Cecilia Raab]

Hi everyone,

I have a more general, conceptual question about multilevel SEM and would be grateful for any input.

I am estimating a multilevel SEM with a latent outcome variable that is predicted by both within-level and between-level predictors. Accordingly, the measurement model is specified on both levels. I assume measurement invariance across levels, and therefore constrain the factor loadings of the indicators to be equal at the within and between levels. For metric identification, I fix the loading of the first indicator to 1.

Following the latent within–between decomposition, I conceptualize the latent outcome as consisting of a within-level component, representing individual deviations from the cluster-specific latent mean, and a between-level component, representing the cluster-level latent mean (random intercept). Based on this decomposition, I assume that the expected value of the within-level latent variable conditional on the cluster is zero, because it captures deviations from the cluster mean.

What I am not entirely sure about is the source of this zero expectation:
Is the zero mean of the within-level latent variable a direct consequence of the latent within–between decomposition itself, or does it only arise if additional constraints are imposed (e.g., fixing an intercept or the overall latent mean to zero)?

Ultimately, by answering this question you will help me immensively with my model identification.

Any confirmation or clarification would be very welcome and highly appreciated. Thanks in advance!

[Clyde Schechter]

In any set of clustered data, it will always be true that the expected value of the deviation from the within-cluster mean is zero.

Suppose x_ij is the observed value of the j'th observation in the i'th cluster. Let m_i be the mean of the i'th cluster. By definition, m_i = (1/n_i)*SUM(x_ij), where n_i is the number of observations in the i'th cluster. Now consider the deviation of the j'th observation in the i'th cluster, d_ij. We have, by definition, d_ij = x_ij - m_i. So:

d_ij = x_ij - m_i = x_ij - (1/n_i)*SUM(x_ij)

And taking expectations over j we get:

E(d_ij)= E(x_ij - (1/n_i)*SUM(x_ij)) = E(x_ij) - (1/n_i)*SUM(E(x_ij))
    = m_i - (1/n_i)*SUM(m_i) = m_i - (1/n_i)*n_i*m_i = m_i - m_i = 0

Aside from your question, and with the caveat that I don't know what your variables represent in the real world, the assumption of measurement invariance across levels is a very strong assumption and I strain to think of examples where this is true.

[Cecilia Raab]

Dear Clyde,

thank you for your response and help! It is totally clear now.
I am very much aware that it cross level variance is indeed a strong assumption. I am (trying) to test it via two-level CFA prior to my MSEM. Still, constraining equal factor loadings between levels is in my model specification a necessary condition to reach model estimation. In any case it is mainly explorative and comes with a bunch of limitations. But thank you for raising the issue!