the recent discussion on this phenomenon has been enlightening (to me at 
least, as I had not appreciated how easily this could happen).

I have added an extra FAQ  on    
http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/dicpage.shtml#q17
which shows Dbar can easily be made to increase by fitting a covariate 
with no explanatory power.

*

If Dbar measures 'lack of fit', why can it increase when I add a 
covariate?*

Suppose Yi is assumed to be N(0,1) under model 1, and consider a 
covariate xi with mean 0 and which is uncorrelated with Y. Then it is 
straightforward to show that fitting a more complex model 2: Yi ~ N(b 
xi,1) leads to Dbar increasing by 1. The crucial idea is that Dbar 
should perhaps not really be considered a measure of fit (in spite of 
the title of Spiegelhalter et al (2002)!). Fit might better be measured 
by Dhat. As emphasised by van der Linde (2005) 
<http://www.blackwell-synergy.com/links/doi/10.1111/j.1467-9574.2005.00278.x/abs/> 
(also available from here 
<http://www.math.uni-bremen.de/%7Eavdl/download/papers/varsel2.pdf> ), 
Dbar is more a measure of model 'adequacy', and already incorporates a 
degree of penalty for complexity.


David Spiegelhalter




Joachim Büschken wrote:
> Hello Bugs List:
>
> recently, I posted this email:
>
> I am running a hierarchical logistic regression in Winbugs and have 
> this problem:  When I add a specific explanatory variable to the 
> model, the deviance more than doubles, which implies that adding this 
> variable reduces the fit of the model. This does not happen when I run 
> the same model (without the random effect, i.e. "non-hierarchical") in 
> SPSS. 
>
> I received a number of very helpful comments (many thanks to all who 
> responded!) which I copied below:
>
> "In your case, I wonder if the explanatory variable explains the 
> hierarchy?  I don't know the specifics of your situation, but let's 
> say your random effect was gender and your explanatory variable was 
> hair length.  There might be very good correspondence between gender 
> and hair length such that you do not require both in the model.  One 
> or the other would be sufficient.  Might this describe your case?" 
>
> This may  happen. I checked by running a correlation between my theta 
> vector (random effect) and the explanatory variable. The correlation 
> is 0.11 and insignificant. Thus, I do not think that this is the 
> problem here. However, how do we prevent this from happening (adding a 
> random effect which correlates with an explanatory variable in the model)?
>
> "David Lindely in a discussion (JRSSB) of Aitkens' paper showed that 
> the posterior mean of Log L can have very bad behavior because it 
> counts the data twice: once in log L and once in the posterior 
> distribution."
> Not being a statistician, I cannot really comment on that. 
>
> "One problem is that adding a variable also increases the size of your 
> parameter space a lot in HB models. If you had the same prior 
> variances for each beta for the two models, then the prior in the 
> second model is much flatter/smaller than the prior in the first 
> model.  In fact, the rate of "flattness" decreases exponentialy with # 
> of dimensions.  In a situation were there are a large number of 
> parameters to observations, the flattening of the "hat" will lead to a 
> flattening of the posterior distribution, which means small values of 
> the logL get relatively greater weight in the deviance.  So your 
> E(LogL) can decrease even though the additional covariate is 
> good. Model selection using deviance, or even better, Bayes factors or 
> posterior probs of models, can get tricky for large model spaces and 
> flat likelhoods.  In addition to being hard to compute, seemingly 
> small changes in the prior can have a big impact on these measures. 
> Asymptotically where the likelihood dominates the prior, everything 
> works well.
>
> I find this last point rather troubling. If this can happen, how do we 
> evaluate the fit of a hierarchical model if LogL is "bad"?
>
> Cheers
> Joachim   
>
> Dr. Joachim Büschken,
> Professor of Marketing
>
> Catholic University of Eichstätt-Ingolstadt, Germany
>
> [log in to unmask] <mailto:[log in to unmask]> or [log in to unmask] 
> <mailto:[log in to unmask]>
>
>
> Phone: (49) 841.937.1976
> Fax: (49) 841.937.2976
>
>
>
>
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