David Rein posted on Monday, August 18, 2003 - 6:09 pm
I recently performed a simple growth model analysis of aggregate state level disease rates over time (23 years), in which I looked at the effect of grand mean centered poverty rates on the intercept and the slope coefficients.
I am struggling to integrate these findings into a more standard econometric fixed effects model framework which was the foundation for this work.
Specifically, my economist colleague thinks doing a growth model like this using state level aggregate data violates a basic assumption of random effects models in which the slope and intercept may not be correlated with the state effects, without biasing random effects results. I'm not sure if this is the actual concern, but the concern seems to be that SOME assumption is violated.
I think the problem lies in different understandings of random effects models verses growth models, but I am not able to put it into words.
1. What is the difference between growth models such as these and fixed and random effects models used in econometrics? Are the differences important, alternatively, what is the correct econometric model to compare this to?
2. Are there assumptions of growth modeling that would be particularly suspect given repeated time masurements on aggregate (state, city, zipcode) data instead of measures from individuals?
4. Are you familiar with a helpful source that used HLGMs with aggregate data?
Many thanks for your time and attention.
bmuthen posted on Monday, August 18, 2003 - 7:31 pm
If your growth model instead was formulated for individuals and you ignored the hierarchical structure of the data, i.e. that the individuals were observed within states (ignored clustering), then I understand that there is a concern that the state effects on individuals' growth intercept and slope, which are ignored, will bias the results. This is the standard concern with ignoring clustered data. I am not familiar with the concern when, as you do, an analysis of the states is done (without concern for individual data). Your analysis has as number of observations the number of states if I understand you correctly. But I may be ignorant regarding this topic and its econometric literature. There is a certainly a big literature on "aggregation bias" but I don't know how it relates to your situation. Chapter 5's Intro in the Raudenbush-Bryk book briefly touches on this. I don't think there is a difference between econometricians' use of the term random effects and growth modelers' use. Perhaps other Mplus Discussion readers have input on this?
David Rein posted on Monday, August 18, 2003 - 7:59 pm
I guess the more general question (and I open this to anyone) is what is the econometric equivilent of a growth model that allows the estimation of time invariant varible effects on intercept and time slope?
David Rein posted on Tuesday, August 19, 2003 - 2:10 pm
After some thought, I've come to the conclusion that the HL Growth Model is a framework to help specify variables and interaction terms within a standard econometric fixed effects model so that their impact on the intercept and slope terms are better understood.
In other words, a HL Growth model is an ML estimated fixed effects model with vairables and interactions specified such that their impact on within and between estimates can be discerned.
Programs like Mplus and Proc mixed aid in identifying the impact of these terms on different components (within and between) of variation in a clustered model. The Hierarchical component obviates the need to specify individual dummies for each group.
bmuthen posted on Thursday, August 21, 2003 - 4:30 pm
I think in part you are referring to how the multilevel model specification (e.g. HLM) can be equivalently expressed as a mixed linear model (e.g. SAS PROC MIXED), where if you have a predictor of a slope in the former specification you end up with a cross-level interaction in the latter formulation (cross-level interaction being a product of a level 1 and level 2 variable). We discuss such equivalences in the Mplus Short Courses. But I don't think a standard fixed effect (econometric) framework is sufficient to represent the multilevel specification because the residuals do not follow the regular assumptions. I think you have our Short course notes from the growth model day related to this. And you will also find good discussions of it in the Raudenbush-Bryk multilevel book.
D. Rein posted on Friday, August 22, 2003 - 3:21 pm
Thanks, I will review those notes. This is exactly the specification I am thinking about. Looking into it more, the model is much more like an econometric random coefficients model than a random effects model. I will review the short course notes and compare them to my econometric textbook to see if I can find any differences.
bmuthen posted on Friday, August 22, 2003 - 3:26 pm
Econometricians often call it random coefficients, while in other stat branches random effects is a more common wording - they are equivalent. One difference is that econometricians also use these random coefficients (or effects) when data are not hierarchical, so for example with single-level regression analysis with a random slope coefficient - this handles modeling of heteroscedastic residuals. Mplus can do that.
Do you know of a reference that uses these models with aggregate (for example state level data)? Is there a quick way to sumarize the main limitations of these models? I can probably figure out the Mplus coding from chapter 22 of the Mplus User's guide, right?
bmuthen posted on Monday, August 25, 2003 - 2:10 pm
I can't think of such a reference; maybe other Mplus Discussion readers can. As far as I can see, there are no limitations of this type of modeling if you are not making inference to the individual level. You are simply using state as the unit of analysis with n=#states. Yes, chapter 22 should guide you.