Rasch with lme4

Dear all,

What happens in practice when you compare the two approaches of item as
a fixed versus as a random effect?

Consider:

  M1 = lmer(Reaction ~ Days + (1|Subject), sleepstudy)
  M2 = lm(Reaction ~ Days + factor(Subject), sleepstudy)

The slope estimates for Days for are practically identical, the mean
intercepts differ:

For M1:

  ...
  Fixed effects:
              Estimate Std. Error t value
  (Intercept) 251.4051     9.7459   25.80
  Days         10.4673     0.8042   13.02
  ...

For M2:

                      Estimate Std. Error t value Pr(>|t|)
  (Intercept)         295.0310    10.4471  28.240  < 2e-16 ***
  Days                 10.4673     0.8042  13.015  < 2e-16 ***
  ...

I didn't look at the estimators for Subject, e.g., for M2 the predictors:

  factor(Subject)309 -126.9008    13.8597  -9.156 2.35e-16 ***
  factor(Subject)310 -111.1326    13.8597  -8.018 2.07e-13 ***
  factor(Subject)330  -38.9124    13.8597  -2.808 0.005609 **
  ...

But it could be done...

Is there a paper on these sorts of comparisons?  How does the mixed
effects approach differ from a standard regression model with a heap of
categorical predictors for representing, e.g., deviations from the mean
intercept?

Presumably this could be done too for estimates for items, e.g., for
binary logistic models and beyond.

Cheers,

Andy


Ken Beath wrote:

On 09/06/2009, at 8:58 AM, Stuart Luppescu wrote:

On ?, 2009-06-09 at 08:04 +1000, Ken Beath wrote:

The model treats item as a random effect and should be a fixed effect.

Hmm. In Doran, Bates, Bliese and Dowling (2007), the authors treat the
item as random.

It can be argued that the items are a sample from a population of
items which is possibly reasonable for educational testing where there
might be a population of questions which can be asked. Even so,
assumptions about the distribution are optimistic and most items are
used because they test something obvious. Maybe others have a
different philosophy. A more pedantic argument is that this isn't the
model Rasch used.

[snip]

Another question to ask is whether the Rasch model is appropriate. If
an IRT is more sensible it would cause some problems with the second
model.

Sorry, but I don't understand this at all.

By an IRT I mean the 2 parameter version where there is a discriminant
parameter which varies among items, in contrast to the Rasch where it
is constant. It probably gives problems with the other model as well
but the second model should have more problems.

I don't like the idea of assuming a Rasch model at all, its popularity
seems to derive from an era when fitting anything else was difficult.
Modern software offers proper solutions, unfortunately at a cost but
that shouldn't be a consideration.

Ken

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