Dear Alan (and Vumani),
Thanks for letting me know. I am on the road, but here is my
quick "answer". The test Don and I developed was based on the usual
large-sample argument, namely, the log-likelihood is approximately
quadratic. When that assumption fails, negative values can occur.
But then that serves as a useful warning that the usual likelihood
test based on chi^2 reference distribution should not be trusted. As
should be clear from the derivations given in our paper, the accuracy
of our approximation depends on parametrization, because the normal
approximation depends on it. So one thing could be done is to try
different parametrizations -- anything that leads to negative value
should not be adopted (but of course positive values themselves
do not imply good approximation!).
Hope this is useful -- Don may have more to add.
Cheers to all,
Xiao-Li
On Wed, 23 Jun 2004, Alan Zaslavsky wrote:
> XL,
>
> In case you don't follow this list, here is a request for information that
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>
> Message: 1
> Date: Wed, 23 Jun 2004 11:22:30 +0000
> From: "Vumani Dlamini" <[EMAIL PROTECTED]>
> Subject: [Impute] negative pooled likelihood
> To: [EMAIL PROTECTED]
> Message-ID: <[EMAIL PROTECTED]>
>
> I am using multiple imputation for a logistic regression problem I have. The
> response and one of my varbale is fully observed and am trying select the
> set of model which best describe the data. I am using the likelihood ratio
> test statistics proposed by Meng & Rubin (1992), and am getting negative
> differences in the pooled likelihood for some of the models.
>
> If I fit the different models to each of the data sets, the most complex
> model has the lowest deviance, but when I use the pooled coefficients this
> is not necessarily the case. This leads for some model to a negative value
> in the mean of d_{L} resulting in a negative value in D_{L}. Is this common?
>
> An example of my output is given below.
> d'0(1) = 427.0232
> d'1(1) = 518.6282
>
> d'0(2) = 425.6645
> d'1(2) = 518.6282
>
> d'0(3) = 436.4400
> d'1(3) = 518.6282
> d'0(g) is the deviance of the most complex model for imputation g. d'1(g) is
> the deviance of the model incorporating only the fully observed variable in
> imputation g.
>
> Below is the likelihood from the pooled coefficients:
> d_L(1) = 521.2215
> d_L(1) = 518.6282
>
> d_L(2) = 638.0552
> d_L(2) = 518.6282
>
> d_L(3) = 494.4705
> d_L(3) = 518.6282
> Notice that for the simpler model the likelihood is always the same given
> that the variables is fully observed, but for the pooled data the most
> complex model sometimes has a higher likelihood.
>
> Thanks for your help.
>
> Vumani Dlamini
> Central Statistical Office
> Swaziland
>
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