mike zyphur posted on Saturday, January 21, 2006 - 10:14 am
Hi Bengt/Linda, I was wondering your thoughts on something that struck me today: In the Mplus manual, when a single observed variable is used at multiple levels of analysis, the within-groups portion of the multilevel models is shown as observed (rectangles) and the between-groups portion is shown as latent (circles). However, are not the within-groups variables latent too? The within-groups variance is not directly observed, we must remove the between-groups variance to estimate it.
I think this idea is made clear when considering the reticular action model espoused in Mehta & Neale (2005), where observed variables are modeled as a function of a latent group factor, and the residuals are the within-groups variance. Such a model is like a CFA model, except each case is a group, each variable is a person, and the factor loadings are set to unity to act as an equal-weighting linking function. In such a model (if a CFA), the residuals are considered latent because they are not observed. Ergo, is not within-groups variance also latent?
Thanks for your time,
bmuthen posted on Saturday, January 21, 2006 - 1:54 pm
It is a little tricky to connect drawings and formulas here. We have to have observed variables somewhere in the picture so that the parameters are identified. Perhaps an easy way to think about it is 2-level regression with say a random intercept. Level 1 is within and level 2 is between. Level 1 estimates a within variance - namely the residual variance, while level 2 estimates between variance. Still, the level 1 regression equation has an observed y on the LHS and an observed x on the RHS, and the drawing using squares reflects that. The unobserved within variation is the residual which is drawn just as a residual arrow, but could be drawn as a latent within-level variable. Between has latent outcomes, here the random intercept, just like in the level 2 regression equation. So the Within drawing for the observed variables can be said to refer to the total variation in the variables, but the term Within refers to what parameters are estimated and that can be said to be from a latent variable (the within residual).
mike zyphur posted on Sunday, January 22, 2006 - 12:24 am
Thanks, Bengt, That makes sense. I forgot to include the within-groups residual as a latent term but, of course, once one does then the "problem" I describe disappears. Perhaps I shouldn't post at 4:00 AM :-)