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

[Paul Johnson]

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