Re: st: Regression Across Two Groups

[Richard Williams <richardwilliams.ndu@gmail.com>]

At 01:04 PM 12/13/2011, Maarten Buis wrote:
> On Tue, Dec 13, 2011 at 6:44 PM, Cameron McIntosh wrote:
> > I will note that one of the easiest ways to do this is via the 
> Mplus package (www.statmodel.com), which through a special THETA 
> parameterization allows the difference in residual variance to be 
> directly estimated for the multi-group case in both logit and 
> probit models. Thus, cross-group differences in residual variation 
> will not be absorbed by the model coefficients, and not confound 
> the comparison.
> I find that rather suspect: The residuals we are talking here about
> are the differences between the latent (and thus unobserved) variable
> and the predicted probability. The only information in the data
> concerning any patterns in the variance of these residuals is in the
> form the fit of a model with a more complex functional form for the
> relationship between the explanatory variables on the probability of
> success. So I find it hard to see how one could separate the
> estimation of the parameters from the estimation of patterns in the
> residual variance. As a consequence, these models tend to be very
> (i.e. way too) sensitive to model specification. Moreover, the
> difference between the complex functional form and the "regular"
> functional form are really subtle, which means that there is very
> little information from the data that these models can use. In
> essence, the problem is real and it cannot be solved.
>
> -- Maarten
I go over the Pros and Cons of various methods (Allison, Long, My 
own) in http://www.nd.edu/~rwilliam/stats/Oglm.pdf (starting around 
slide 20.) In logit and probit models that want to examine group 
differences, radically different interpretations can be consistent 
with the same empirical results (which I think is what Maarten means 
when he says "I find it hard to see how one could separate the 
estimation of the parameters from the estimation of patterns in the 
residual variance.") Based on theory, you can offer an 
interpretation, but that interpretation may be wrong.
Long takes a sort of a high-tech descriptive approach that may avoid 
such issues, but is also somewhat unsatisfying because it doesn't 
offer a substantive explanation as to why group differences exist.
My own bias is to offer the interpretation I most believe in while 
conceding the inherent problems that come from working with binary or 
ordinal dependent variables.

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Richard Williams, Notre Dame Dept of Sociology
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