Profs. Muthen, I am working with a mulitlevel factor model where items indicate factors at both levels. In considering some literature (Skrondal & Laake, 2001) on the appropriateness of modeling 1st-level factor scores as a random coefficient at a higher level, it seems to me that allowing item loadings to vary across levels (e.g., item 1 having a different loading at each level) is only appropriate for reflective constructs at both levels of analysis. In the case of a formative construct at the 2nd-level (e.g., employee-created climate), one would want cross-level item loadings to be equal, so that the 2nd-level factor scores represent an "intercept" of sorts. Is this true?
What are the costs/benefits of specifying cross-level item loading invariance as opposed to simply making the latent variable at the 1st level a random coefficient at the second level?
I'm not sure if you are thinking about loadings that are the same value on the within and between levels, that is, held equal, or if you are thinking about a loading that is random on the within level and has its mean and variance estimated on the between level. I don't see an obvious reason why there would be a distinction between formative and reflective constructs in either case.
I am referencing loadings that are the same value on the within and between levels. In this case, the group-level factor represents a value which is analogous to the lower-level variables' intercept in an HLM-style model, true? If so, then this value is not appropriate for reflective constructs, as discussed by Skrondal & Laake, 2001. I am likely thinking about this issue incorrectly, but according to these authors, using intercepts to represent a group's standing upon a reflective variable is innapropriate because the intercept itself is created formatively. So, if by constraining item loadings to equality across between and within levels creates the between-level factor as analogous to an HLM-style intercept, then this means this technique is appropriate only for formative constructs.
Is this logic fallacious? If so, where? Thank you.
BMuthen posted on Tuesday, June 28, 2005 - 8:23 am
Please repost this after July 1 and give the full Skrondal and Laake reference. I will take a look at it then.
Bengt, here's the repost: I am referencing loadings that are the same value on the within and between levels. In this case, the group-level factor represents a value which is analogous to the lower-level variables' intercept in an HLM-style model, true? If so, then this value is not appropriate for reflective constructs, as discussed by Skrondal & Laake, 2001. I am likely thinking about this issue incorrectly, but these authors suggest that using intercepts (from factor scores) to represent a group's standing upon a reflective variable is innapropriate because the intercept itself is created formatively. So, if by constraining item loadings to equality across between and within levels creates the between-level factor as analogous to an HLM-style intercept, then this means this technique is appropriate only for formative constructs.
Is this logic fallacious? If so, where? Thank you.
Skrondal, A., & Laake, P. (2001). Regression among factor scores. Psychometrika, 66, 563–576.
The Skrondal-Laake article appears to elaborate on the old truth that factor score estimates obtained by the standard regression method (which is used in Mplus) behave badly when they are used as proxies for dependent variables, but perform well when they are used for independent variables. I don't see any statements about reflective versus formative LVs. I assume that reflective means a regular factor model where arrows go from the LV to the indicators and that formative has the arrows in the other direction.
Now, the article only pertains to using estimated factor scores. Doing the analysis in one step without the intermediate step of factor score estimation avoids the problem - and for both the formative and reflective case it would seem. But perhaps I am misunderstanding the question.
To some degree, I am trying to understand what happens when a within-level latent factor (with multiple indicators) is made random at the between-level. In this case, the group-level variable represents the mean of the factor scores for each group, correct? In this case (perhaps I read too much into the S & L article), this group-level variable has been formed by averaging lower-level factor scores. This is ok if the group-level variable is formed as a function of its lower-level parts (i.e., is additively constituted, such as group-level "climate", where the group mean is a group's climate). However, in cases where the group-level variable is reflected in the items in question, then one would need to form a measurement model at the group level to represent the group-level variable, not just aggregate lower-level scores.
So, my question pertains to the practice of setting to equality the L1 and L2 path coefficients for the same items. If this technique is functionally similar to creating a random coefficient of a L1 factor at L2, then it would seem to be innapropriate for reflective constructs at L2. What do you think?
As a side, why does the between-group var/covar matrix have both between and within components in it (i.e., why, as you state in some articles, is it problematic to do EFA on the between-group var/covar matrix and assume you're getting only between-level variance)? I have read many of your articles on the matter but am having difficulty wrapping my mind around this issue.
Thank you very much for your help.
bmuthen posted on Tuesday, July 05, 2005 - 5:19 pm
Re the first and second paragraphs of your post, when a factor has both within and between variation, it can be handled by having equal loadings across the two levels to give a counterpart to random intercept modeling for an observed outcome. But just as with random intercept modeling for an observed outcome, this does not mean, however, that the level 2 scores are actual level 1 scores averaged for each level 2 cluster (group), but should be seen as just a way to decompose the factor variance. For both reflective and formative situations you have a measeurement model, so there is no difference there I think.
The expectation of the between-group sample covariance matrix S_B is not Sigma_B but Sigma_W + s Sigma_B, so S_B is not necessarily a good estimate of Sigma_B, where Sigma_B is what we are interested in. S_B has both within and between components in it because it is a covariance matrix for cluster means - taking the expectation of that covariance matrix involves both within and between variance.
Re the S_B matrix, an older article on this topic (I think by Hox) suggested possibly replicating the within-group model in the between model to "wash out" the Sigma_W in the S_B matrix. Then, after this, fit a hypothesized between-groups model. Does this make sense to you? The technique sounds somewhat like your comments on the possibility of trying to subtracting S_PW from S_B, but I recall you writing that you had some reservations about doing this.
This stuff is fantastically interesting (I should've been a psychometrician). Thanks for all of your help!
bmuthen posted on Thursday, July 07, 2005 - 1:19 pm
My "MUML" estimator is a 2-group approach where the within group has the Sigma_W structure fitted to S_PW and the between group has Sigma_W + s Sigma_B fitted to S_B. Perhaps that's what you refer to. One could make Sigma_W unrestricted and estimate it as S_PW, which amounts to fitting Sigma_B to (S_B - S_PW)/s, but this sample matrix is not necessarily positive definite.