[R-sig-ME] models with no fixed effects

[Steven McKinney]

Not including an intercept term can indeed induce
spurious correlation in linear regression, this
issue is reviewed in Richard Kronmal's excellent paper

 Spurious Correlation and the Fallacy of the Ratio Standard Revisited
 Richard A. Kronmal
 Journal of the Royal Statistical Society. Series A 
  (Statistics in Society), Vol. 156, No. 3 (1993), pp. 379-392 

which could no doubt be readily extended to cover
mixed effect models by A Real Statistician.

The cost of including an intercept term is small
relative to the havoc that can be reaped by not
including one.

Steven McKinney




> 
> -----Original Message-----
> From: r-sig-mixed-models-bounces at r-project.org on behalf of Andy Fugard
> Sent: Thu 9/11/2008 3:57 PM
> To: Peter Dixon
> Cc: r-sig-mixed-models at r-project.org
> Subject: Re: [R-sig-ME] models with no fixed effects
>  
> 
> On 11 Sep 2008, at 22:06, Peter Dixon wrote:
> 
> >
> > On Sep 11, 2008, at 2:15 PM, Andy Fugard wrote:
> >
> >> Peter Dixon wrote:
> >>> On Sep 11, 2008, at 1:15 PM, Douglas Bates wrote:
> >>>> I should definitely add a check on p to the validate method.  (In
> >>>> some
> >>>> ways I'm surprised that it got as far as mer_finalize before  
> >>>> kicking
> >>>> an error).  I suppose that p = 0 could be allowed and I could add
> >>>> some
> >>>> conditional code in the appropriate places but does it really make
> >>>> sense to have p = 0?  The random effects are defined to have mean
> >>>> zero.  If you have p = 0 that means that E[Y] = 0.  I would have
> >>>> difficulty imagining when I would want to make that restriction.
> >>>>
> >>>> Let me make this offer - if someone could suggest circumstances in
> >>>> which such a model would make sense, I will add the appropriate
> >>>> conditional code to allow for p = 0. For the time being I will just
> >>>> add a requirement of  p >  0 to the validate method.
> >>> I think it would make sense to consider a model in which E[Y] = 0
> >>> when  the data are (either explicitly or implicitly) difference
> >>> scores. (In  fact, I tried to fit such a model with lmer a few
> >>> months ago and ran  into exactly this problem.)
> >>
> >> Wouldn't you still need the intercept?  The fixed effect tells you
> >> whether on average the difference differs from zero.  The random
> >> effect estimates tell you by how much each individual's difference
> >> differs from the mean difference.
> >>
> >> A
> >>
> >> -- 
> >> Andy Fugard, Postgraduate Research Student
> >> Psychology (Room S6), The University of Edinburgh,
> >> 7 George Square, Edinburgh EH8 9JZ, UK
> >> +44 (0)78 123 87190   http://figuraleffect.googlepages.com/
> >>
> >> The University of Edinburgh is a charitable body, registered in
> >> Scotland, with registration number SC005336.
> >>
> >>
> >
> >
> > In the context in which this arose, I was interested in assessing the
> > evidence for an overall positive difference score (i.e., that E(Y)>0),
> > and my strategy was to compare the fit of two models, essentially,  
> > D~0+
> > (1|Subject) and D~1+(1|Subject), using AIC values. To get a sensible
> > assessment of the evidence for the fixed effect, it seemed to me that
> > one would want to have the same random effects in the two models being
> > compared. The second model is clearly more obvious, but the
> > interpretation of D~0+(1|Subject) could be that subjects differ
> > randomly in their response to the treatment, but that there is no
> > consistent effect in the population.
> 
> I /think/ I get this, by analogy with how I use AIC/BIC/LRTs to test  
> predictors.  But still a bit confused.  The two models are:
> 
>    y_ij = a + b_j + e_ij     (1)
>    y_ij = c_j + e_ij         (2)
> 
> Suppose a != 0 in model 1.  Then in model 2:
> 
>     c_j = b_j + a.
> 
> (Maybe it's not as simple as this!)  But I'm not sure what effect that  
> would have on the e_ij's - and my intuition says that's what's going  
> to affect the fit.  Also I would have thought model 2 would give a  
> better fit since having one fewer predictor is going to have less of a  
> penalising effect in the AIC and BIC.
> 
> Andy
> 
> -- 
> The University of Edinburgh is a charitable body, registered in
> Scotland, with registration number SC005336.
> 
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