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In-Depth Information
this approach has bias that increases with the number of subjects. If data
include a large number of subjects, the associated bias of the results makes
this a very poor model choice.
Conditional Fixed Effects.
These are applied in logistic regression,
Poisson regression, and negative binomial regression. A sufficient statistic
for the subject effect is used to derive a conditional likelihood such that
the subject level effect is removed from the estimation.
While conditioning out the subject level effect in this manner is alge-
braically attractive, interpretation of model results must continue to be in
terms of the conditional likelihood. This may be difficult and the analyst
must be willing to alter the original scientific questions of interest to ques-
tions in terms of the conditional likelihood.
Questions always arise as to whether some function of the independent
variable might be more appropriate to use than the independent variable
itself. For example, suppose
X
=
Z
2
where
E
(
Y
|
Z
) satisfies the logistic
equation; then
E
(
Y
|
X
) does not.
Random Effects.
The choice of a distribution for the random effect too
often is driven by the need to find an analytic solution to the problem,
rather than by any actual knowledge. If we assume a normally distributed
random effect when the random effect is really Laplace, we will get the
same point estimates (since both distributions have mean zero), but we
will get different standard errors. We will not have any way of checking
the approaches short of fitting both models.
If the true random effects distribution has a nonzero mean, then the
misspecification is more troublesome as the point estimates of the fitted
model are different from those that would be obtained from fitting the
true model. Knowledge of the true random-effects distribution does not
alter the interpretation of fitted model results. Instead, we are limited to
discussing the relationship of the fitted parameters to those parameters we
would obtain if we had access to the entire population of subjects, and we
fit that population to the same fitted model. In other words, even given
the knowledge of the true random effects distribution, we cannot easily
compare fitted results to true parameters.
As discussed in Chapter 5 with respect to group-randomized trials, if
the subjects are not independent (say, they all come from the same class-
room), then the true random effect is actually larger. The attenuation of
our fitted coefficient increases as a function of the number of supergroups
containing our subjects as members; if classrooms are within schools and
there is within school correlation, the attenuation is even greater.
GEE (Generalized Estimating Equation).
Instead of trying to derive the
estimating equation for GLM with correlated observations from a likeli-
