[R] some help interpreting ANOVA results, please?

[Prof Brian Ripley]

On Sun, 10 Oct 2004, RenE J.V. Bertin wrote:

> Could I ask some hints/help in interpreting the following ANOVA results,
> please? This concerns an experiment where I study the incidence and
> severity of motion sickness. I have Sickness.norm, a subjective
> discomfort/sickness estimate, normalised to 0..1, the session time T
> (normalised to 0..1 and binned in 0.2 wide bins) and a qualitive
> indicator if a given Subject was sick or not. For instance, to see if
> there is an effect of time, I do
> 
> 
> > summary( aov.SS1( Sickness.norm~WasSick*T.norm.Class + Error(Subject/(WasSick*T.norm.Class)), na.action=na.exclude ) )
> Factor "WasSick", levels: 0 1 
> Factor "T.norm.Class", levels: 0 0.2 0.4 0.6 0.8 1 
> Sun Oct 10 13:45:45 2004 
> WasSick data: 377
> Warning in aov(Sickness.norm ~ WasSick * T.norm.Class + Error(Subject/(WasSick*T.norm.Class)), na.action=naa,  :
> 	 Error model is singular
> 
> Error: Subject
>                       Df Sum Sq Mean Sq F value Pr(>F)    
> WasSick               1 11.170  11.170  21.037 0.0001 ***
> T.norm.Class          5  5.058   1.012   1.905 0.1278    
> WasSick:T.norm.Class  5  1.680   0.336   0.633 0.6765    
> Residuals 26 13.805   0.531                   
> ---
> Signif. codes:  0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 
> 
> Error: Subject:T.norm.Class
>                        Df Sum Sq Mean Sq F value   Pr(>F)    
> T.norm.Class           5 10.916   2.183  13.673 3.29e-11 ***
> WasSick:T.norm.Class   5  2.318   0.464   2.903   0.0153 *  
> Residuals 169 26.984   0.160                     
> 
> 
> 
> There is a very nice effect of time (the line T.norm.Class under
> Subject:T.norm.Class). There is no interaction between the two factors
> (that would have been given on a line WasSick:T.norm.Class under the
> Subject:WasSick:T.norm.Class caption, right?!). 

Not right.

> That means that the
> effect of time is not different for the two groups, which is true:
> sickness increases for both, only less so for the non-sick group. The
> time effect is significant only for the sick group, when tested
> seperately. But what I do not manage to grasp is the interpretation of
> the other significant effects:


> *1 What does the very significant WasSick effect (under the Error:
> Subject caption) mean? That there is a sign. difference between the
> subjects who were sick and who were not, which is indeed very true?

See next point.

> *2 What is the meaning of the interaction WasSick:T.norm.Class
> (p=0.0153)? Can I interpret this as meaning that sickness increases
> significantly less in the non-sick group?

Here you have a problem, as that term appears in two strata, so presumably 
the nesting was unbalanced.  You either need to use Yates' `recovery of 
inter-block information' or, simpler, use lme.

You have evidence that the effect of WasSick differs by T.norm.Class.
If that is substantiated by a more refined analysis, it does not make 
sense to say there is a `very significant WasSick effect', as it is likely 
there is an effect for some session times and not others, or at least that 
the size of the effect differs by session time.

Trying to interpret main effects in the presence of interactions depends 
on the (unstated) coding of factors used.  If this is treatment coding,
WasSick is the effect at T=0 (significant) and the effect appears to 
change with time.

> *3 What importance should I give to the warning that the error model is
> singular? Does this stem from the fact that Subject and WasSick are not
> independent? R 1.8 generally refused to calculate this sort of test: can
> I rely on the results R 1.9.1 gives me??

I think that means the correct error model is Error(Subject/T.norm.Class):
my guess is that WasSick is a subject-level observation and so each 
subject only has one level of it.  Certainly that is the model which was 
fitted.

-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595