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Re: Re: Re: st: Situation where multiple imputation may be of no use?


From   Maarten Buis <maartenlbuis@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   Re: Re: Re: st: Situation where multiple imputation may be of no use?
Date   Wed, 15 Feb 2012 16:07:36 +0100

On Wed, Feb 15, 2012 at 3:40 PM, Stas Kolenikov wrote:
> It is unusual that MAR and MCAR led to the same results (although if
> you generated the outcome as independent of the covariates except for
> treatment, that's how it should be, indeed).

This result does not surprise me. Clyde's problem was missing values
in the outcome variable, so in both the MAR and MCAR case the
probability of missingness is independent of the outcome variable. In
that case only using the observed cases in both the MAR and MCAR cases
should lead to consistent estimates.

Lets say we have an outcome variable y, some treatments and/or
covariates x, and a indicator variable m which is 1 when there are
missing value an 0 when everything is observed. We want to model y
conditional on x, and if we use only the observed cases we get:

f(y | x, m== 0)

Using Bayes' theorem:

f(y | x, m == 0) = f(y, x, m==0) / f(x, m == 0)

= { Pr(m==0 | y, x) * f(y | x) * f(x)  } / { Pr(m==0 | x) f(x) }

In Clyde's case the missing values are in y, and per the MAR
assumption the probability of being missing does not depend on y as
the MAR assumption states that that probability does not depend on the
(unobserved) value of the variable that is missing (otherwise it would
be NMAR). As a consequence  Pr(m==0 | y, x) =  Pr(m==0 | x). So we can
write:

f(y | x, m== 0 ) = { Pr(m==0 |  x) * f(y | x) * f(x)  } / { Pr(m==0 | x) f(x) }
 = f(y | x)

Hope this helps,
Maarten

--------------------------
Maarten L. Buis
Institut fuer Soziologie
Universitaet Tuebingen
Wilhelmstrasse 36
72074 Tuebingen
Germany


http://www.maartenbuis.nl
--------------------------
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