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
Disclaimer: The information in this email and any attachments to it are
confidential and may contain legally privileged information. If you are not
the named or intended recipient, please delete this communication and
contact us immediately. Please note you are not authorised to copy,
use or disclose this communication or any attachments without our
consent. Although this email has been checked by anti-virus software,
there is a risk that email messages may be corrupted or infected by
viruses or other
interferences. No responsibility is accepted for such interference. Unless
expressly stated, the views of the writer are not those of the
company. Tricky Solutions always does our best to provide accurate
forecasts and analyses based on the data supplied, however it is
possible that some important predictors were not included in the data
sent to us. Information provided by us should not be solely relied
upon when making decisions and clients should use their own judgement.
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