Dear imputers,
I originally started just to reply to Paul, but
thinking about my reply raised a question that I thought it could be interesting
to put round. VIZ:
How big should an imputation model be? and what
are the costs of including too many variables?
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I am a bit puzzled by your
query. Some questions
A) Do you assume Z is completely
observed?
B) You say that you will assume (X,Y) are ignorable
missing. What do you mean by this?
Ignorable means that missingness does not depend on
any unobserved values.
Do you mean that missingness
is ignorable given all the observations of X,Y and anything else that might have
been observed including Z ( of course this is an assumption, since you
can't test for it)?
If B is assumed then there are two
possibilities.
1) Missingness would have been ignorable even without
information on Z. That is the distribution of Y|X would be
the same for observed and missing
Ys integrating over Z.
2) Missingness is only ignorable given
Z
I would guess that if 1) is true, then
including Z in the imputation process will have no obvious benefit and I would
have thought that it might hurt.
But this is really an academic question since we can
never know if MAR is valid and we always put Zs in to try get as close to MAR as
possible.
It would however have a bearing on how big an
imputation model should be, since if it hurts to have fairly useless stuff in
your imputation then it would be good to make it smaller. Even if the theory
says it doesn't hurt there is a good practical case for making the model small
since the programs will have problems and asymptotics for things like estimation
of variance covariance matrices is likely to mess up.
See the example on the web site that I circulated to
the list yesterday
and the section on problems with imputation software in praticular
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