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Some of my friends/colleagues have recently taken an interest in structural equation modelling, and I have been having to field an increasing number of questions about SEM. Often times, these questions are about how to interpret the meaning of the estimates of various measurement model parameters (i.e., factor loadings, residual variances, and intercepts), and why it is important for these values to be roughly equivalent between groups (i.e., establishing measurement invariance) before comparing groups on estimated structural parameters (i.e., variances, covariances, and means).

I feel as though I have a pretty good handle on providing accessible definitions of factor loadings and residual variances, and accessible explanations for why it might be important for these estimated parameters to be equivalent between groups before comparing those groups on structural parameters. But for some reason, I've felt like a similarly accessible definition and explanation of intercepts has eluded me.

So, my question is: how best to accessibly explain what an intercept is, and explain why it might be important for intercepts to be invariant between groups before comparing groups' latent means?

For example: a factor loading represents the estimated direction and strength of association between an observed variable and a latent variable. In other words, a factor loading represents how central that observed variable is to the manifestation of its associated latent variable. When comparing groups' structural parameters, it's important to ensure that they are invariant across groups, because otherwise it suggests that the same observed variables aren't equally important to both groups' understanding of a given latent variable--the latent variable means something different to each group.

An intercept is the expected value of a given observed variable when its associated latent variable is equal to zero...what is the In other words... and it's important to ensure they are invariant, because... portions of the explanation of an interpret (in the SEM context)?

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The intercept or mean of a latent variable is arbitrary, like the variance, and is usually fixed to zero if you have a single group model (or a single time point model). The intercept of the measured variable is the expected value when the predictor (the latent variable) is equal to zero.

You anchor the mean of the latent variable to the intercept of the measured variables, and that means that you can compare them over time. But if the intercepts of the measured variables drift apart, you can't anchor the means to them any more, because you don't know where they are anchored.

Enough analogies, let's have a concrete example.

Let's say you want to compare depression symptoms in men and women.

So you ask three questions: How many days in the past week have you:

  1. Felt lonely.
  2. Felt sad
  3. Cried

I create a latent variable based on this, and error and loadings look good. Now I want to compare the means of the latent variables, so I fix the male latent mean to zero. I constrain the intercepts of the three measured variables to be equal across groups.

Women and men do not differ on how much they have felt lonely, how much they have felt sad, but then we find that women say that they have cried more than men.

Does that mean that the women have 'more' depression than the men? If we anchor to crying - yes. If we anchor to the other two variables - no. We don't have intercept invariance, and because of that, we can't compare the means of the latent variables.

Another (only slightly different) way to think about it. The intercept of the measured variable is the expected value of the variable if the mean of the factor is equal to zero. The predicted values for the measured variables should be the same between men and women when the values of the factors are equal (that is, when the value of the factors is zero). But the predicted values of the measured variables are not equal when the factors are equal. Some are equal (in our example, 1 and 2), one is not (3).

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    $\begingroup$ I won't lie; when I posted this question, I thought to myself, "I bet Jeremy Miles is going to take a crack at this..." Glad you did--thanks for a very intuitive example! $\endgroup$ – jsakaluk Apr 8 '16 at 2:52
  • $\begingroup$ I'm finding part of this unclear. If you constrain the intercepts of the three measured variables to be equal across groups then how can you observe that women say that they have cried more than men? I am assuming that the invariance test indicates that this constraint reduces fit significantly, and when you examine the modification indices, they suggest freeing the intercept for crying. $\endgroup$ – John Flournoy Nov 3 '19 at 17:26
  • $\begingroup$ Generally this was very helpful. What might you speculate substantively about the hypothetical observed non-invariance? Something like, "depression manifests differently for men and women, with higher rates of crying for women than men," or perhaps just, "men tend to cry less than women in general, possibly due to social norms, and so have lower levels of crying for the same level of depression?" Contrast to loading-non-invariance on the same item which would indicate that, say, crying is more strongly determined by depression for women than men (or vice versa). $\endgroup$ – John Flournoy Nov 3 '19 at 17:30
  • $\begingroup$ I'm not sure I'd want to speculate as to the reasons - that's outside my area of expertise. :) $\endgroup$ – Jeremy Miles Nov 4 '19 at 18:33

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