[R-sig-ME] 4 binary DVs, subjects nested within schools

[Chris Howden]

You've just described a classic market research problem and method.
It's called choice modes.

They used to be modelled using aggregate multinomial logit models.

But these days they are more commonly modelled using Bayesian
multinomial logit, this can allow us to get individual level
parameters and since a lot of the variance is at the individual level
we model it that way.

Sawtooth software are experts on this. You'll find all types of good
reference material on their web site. Plus they have a Bayesian
software for multinomial logit.

Chris Howden
Founding Partner
Tricky Solutions
Tricky Solutions 4 Tricky Problems
Evidence Based Strategic Development, IP Commercialisation and
Innovation, Data Analysis, Modelling and Training

(mobile) 0410 689 945
(fax / office)
chris at trickysolutions.com.au

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On 23/11/2011, at 4:25, Paul Johnson <pauljohn32 at gmail.com> wrote:

> Greetings
>
> I'm trying to get my footing under a researcher's request for
> statistical support. I need your advice.
>
> The gist of this is that there are 4 dichotomous outputs that can be
> modeled separately with logistic or probit models, and lme4 works fine
> treating each one separately.  There is a random effect at the school
> level.
>
> However, a reviewer says a multivariate model is needed to fully model
> this problem.
>
> The data is like selections from a menu, where all of the above is
> possible.  This actual project is about student behaviors in the class
> room, but it seems more understandable to me to think of it as a
> person's taste for ice cream. Respondents are asked "do you like
> chocolate ice cream" or "do you like vanilla ice cream" or "strawberry
> ice cream".  So the dependent variable is multivariate like this (yes,
> no, yes, no).
>
> Where can I learn more about the multivariate approach to this?
>
> And why are multivariate approaches not making the same mistake that
> is described in this literature on comparison of coefficients across
> logit models fitted for separate groups. I mean, if the variance
> parameter is not identified, how can I meaningfully put together 4
> logit models?
>
> Allison, Paul. 1999. “Comparing Logit and Probit Coefficients Across
> Groups.” Sociological Methods and Research 28(2): 186-208
>
> Richard Williams, 2008, "Using Heterogeneous Choice Models To Compare
> Logit and Probit Coefficients Across Groups"
> http://nd.edu/~rwilliam/oglm/RW_Hetero_Choice.pdf
>
> Mood, C. (2010). Logistic Regression: Why We Cannot Do What We Think
> We Can Do, and What We Can Do About It. European Sociological Review,
> 26(1), 67 -82. doi:10.1093/esr/jcp006
>
> Well, anyway, this looks like a project to me.  I (probably) first
> need to understand how to fit this model without any distractions due
> to nested effects or sampling weights, and then I need to take into
> account the fact that students are nested in classrooms.
>
> I've been digging about for models of more-than-one dichotomy.  VGAM
> has bivariate logit and probit.   The brand new package mvProbit has
> "experimental" support for several dichotomous DVs.   But I don't
> think it is going to help with the classroom random effect.
>
> I'm trying to find the simplest way to write all this down as a model
> so I can see where the correlations come in across questions and
> across units. For each outcome,  yj, j=1,2,3,4, there is a coefficient
> vector Bj and an error term ej and the model states:
>
> y1 = 1 if XB1 + e1 > 0; 0 otherwise
> y2 = 1 if XB2 + e2 > 0; 0 otherwise
> y3 = 1 if XB3 + e3 > 0; 0 otherwise
> y4 = 1 if XB4 + e4 > 0; 0 otherwise
>
> Suppose (e1,e2,e3,e4) is multivariate (normal or logistic?).  Because
> of the "you can't compare logistic regressions across groups" problem,
> it appears problematic to assert that the variances of ej = 1.
>
> Pj
>
>
>
> --
> Paul E. Johnson
> Professor, Political Science
> 1541 Lilac Lane, Room 504
> University of Kansas
>
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