[R-sig-ME] How to use mixed-effects models on multinomial data

Douglas Bates bates at stat.wisc.edu
Thu May 28 20:13:37 CEST 2009

On Thu, May 28, 2009 at 9:24 AM, Jonathan Baron <baron at psych.upenn.edu> wrote:
> I had already replied to Linda Mortensen, but Emmanuel Charpentier's
> reply gives me the courage to say to the whole list roughly what I
> said before, plus a little more.

> The assumption that 0-1, 1-2, ... 4-5 are equally spaced measures of
> the underlying variable of interest may indeed be incorrect, but so
> may the assumption that the difference between 200-300 msec reaction
> time is equivalent to the difference between 300-400 msec (etc.).
> Failure of the assumptions will lead to some additional error, but, as
> argued by Dawes and Corrigan (Psych. Bull., 1974), not much.  (And you
> can look at the residuals as a function of the predictions to see how
> bad the situation is.)  In general, in my experience (for what that is
> worth), you lose far less power by assuming equal spacing than you
> lose by using a more "conservative" model that treats the dependent
> measure as ordinal only.

I'm glad to see you write that, Jonathon.  I don't have a lot of
experience modeling ordinal response data but my impression is that
there is more to lose by resorting to comparatively exotic models for
an ordinal response than by modeling it with a Gaussian "noise" term.
In cases like this where there are six levels, 0 to 5, I think your
suggestion of beginning with a linear mixed-effects model and checking
the residuals for undesirable behavior is a good start.

> Occasionally you may have a theoretical reason for NOT treating the
> dependent measure as equally spaced (e.g., when doing conjoint
> analysis), or for treating it as equally spaced (e.g., when testing
> additive factors in reaction time).
> In the former sort of case, it might be appropriate to fit a model to
> each subject using some other method, then look at the coefficients
> across subjects.  (This is what I did routinely before lmer.)
> Jon
> On 05/28/09 14:35, Emmanuel Charpentier wrote:
>> Le mercredi 27 mai 2009 �  18:08 +0200, Linda Mortensen a écrit :
>> > Dear list members,
>> >
>> > In the past, I have used the lmer function to model data sets with
>> > crossed random effects (i.e., of subjects and items) and with either a
>> > continuous response variable (reaction times) or a binary response
>> > variable (correct vs. incorrect response). For the reaction time data,
>> > I use the formula:
>> > lmer(response ~ predictor1 * predictor2 ....  + (1 + predictor1 *
>> > predictor2 .... | subject) + (1 + predictor1 * predictor2 .... |
>> > item), data)
> I think that the second random effect term should be (0 + ...), since
> there is already an intercept in the first one.

I don't think so.  It is quite legitimate to have random effects of
the form (1|subject) + (1|item) and the formula above is a
generalization of this.  A additive random effect for each subject is
not confounded with an additive random effect for each item.

I would be a more concerned about the number of random effects per
subject and per item when you have a complex formula like 1 +
predictor1 * predictor2 on the left hand side of the random-effects
term.  If predictor1 and predictor2 are both numeric predictors this
might be justified but I would look at it carefully.

> > I'm currently working on a data set for which the response variable is
>> > number of correct items with accuracy ranging from 0 to 5. So, here
>> > the response variable is not binomial but multinomial.
>> This approximation may be too rough with only 5 items, though.
>> Furthermore, depending on your beliefs on the cognitive model involved
>> in giving a "correct" response, the distance between 0 and 1 correct
>> response(s) may be close to or very different from the distance between
>> 4 and 5 correct responses, which is exactly what proportional risks
>> model (polr) tries to explain away.
> --
> Jonathan Baron, Professor of Psychology, University of Pennsylvania
> Home page: http://www.sas.upenn.edu/~baron
> Editor: Judgment and Decision Making (http://journal.sjdm.org)
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