My colleagues and I are having a disagreement about what is being predicted in Example 9.6 of the User Guide. Let me, however, contextualize my question in terms of our particular model.
Imagine we have a two level CFA model. Lets say we have 4 continuous indicators, and we have one factor at both within and between -- the factor loadings are allowed to be different between the levels. We have no predictors at level 1. Imagine we have one predictor at level 2.
The disagreement is whether that level 2 predictor is predicting (a) the factor mean, or (b) the combined factor mean and the measurement intercept.
Here is the code we are using.
VARIABLE: NAMES ARE country y1 - y4; USEVARIABLES ARE country y1-y4; CLUSTER = country; MISSING = y1-y4 (999);
ANALYSIS: TYPE = TWOLEVEL; MODEL: %WITHIN% Fw BY y1-y4; %BETWEEN% Fb BY y1-y4; Fb on w;
Your very helpful response actually triggered more questions. Please note that we have read several of Bengt's papers and we are still debating your reply.
So, when you write that "The regression of fb on w explains the variability of fb. As a consequence, the mean of fb and the intercepts of y1-y4 may change." The first sentence seems obvious to us, but the second sentence is causing some consternation.
What we are debating is what is the variation in Fb representing? And, in particular, what is the variation in Fb representing when one thinks in terms of the mixed effects models language of Level 1 and and Level 2, etc..
We are struggling with whether there are measurement intercepts at the "within" level. We suspect that the observed variables are centered so that the within intercepts are zero.
So following equation (4) in Muthen (1991, JEM) "nu(j)" is not indexed by "g" so obviously it does not vary by g. Is this nu(j) therefore the intercepts of y1-y4 you refer to above? And if so, what could does it mean that the intercepts of y1-y4 may change?
.. please see second post, below, for the rest of this question.
My group is split with two possible interpretations of the phrase "regression of fb on w explains the variability of fb":
Interpretation 1. The w is explaining the variation in factor means and intercepts from the within model. Therefore, variation in Fb (the between latent variable) reflects both the variation in within groups factor means and within groups (measurement) intercepts of y1-y4.
Interpretation 2. The w is explaining the variation in factor means from the within model. Therefore, variation in Fb (the between latent variable) reflects the variation in within groups factor means.
In Raudenbush and Bryk language of levels, we are discussing the "level 2" model and the debate seems to center about whether the dependent variable at level 2 is either (a) the factor means from level 1, or (b) some composite of factor means and measurement intercepts both from level 1.
I hope that this is helpful. Again, my apologies for this long question but this seems very subtle to us.
Hi Linda and Bengt, Sorry for sending so many notes about this.
I suppose the simplest way to ask our question is what model is being fit in by our code. Is it (a) the Muthen-Satorra model for varying factor means, (b) the varying factor means and measurement intercepts model, or (c) some other model?
You may want to take a look at the handout for Topic 7 that we have on our web site (see version of 8/17/09). See slide 86 and on. You can watch the video there too. A good starting point is slide 87 where you see a random effect ANOVA juxtaposed with a 2-level factor model like the one you specified (except you added a between-level factor covariate w).
As in the ANOVA we decompose the observed outcomes into between and within components. They are latent variables representing variation in the observed outcomes on the two levels. As in ANOVA you can think of the between level component as approximately the sample cluster mean of the outcome and you want to know the variance across clusters of that. The extension to the factor model is to simply look at a decomposition of several outcomes and describe their covariance matrix on between and within in terms of a factor model, one for each level. Your model then further decomposes fb into variation due to w and due to a residual.
So in multilevel regression terms you can think of this as a random intercept model (although we don't even have a level 1 covariate) where both level 1 and level 2 use a factor model.