[R] significant anova but no distinct groups ?

rolf at math.unb.ca rolf at math.unb.ca
Sat Mar 3 15:18:17 CET 2007


Frederic Jean wrote:

> I am studying a dataset using the aov() function.
> 
> The independant variable 'cds' is a factor() with 8 levels and here is  
> the result in studying the dependant variable 'rta' with aov() :
> 
> > summary(aov(rta ~ cds))
>              Df  Sum Sq Mean Sq F value  Pr(>F)
> cds          7 0.34713 0.04959  2.3807 0.02777
> Residuals   92 1.91635 0.02083
> 
> The dependant variable 'rta' is normally distributed and variances are  
> homogeneous.
> But when studying the result with TukeyHSD, no differences in 'rta'  
> are seen among groups of 'cds' :
> 
> > TukeyHSD(aov(rta ~ cds), which="cds")
>    Tukey multiple comparisons of means
>      95% family-wise confidence level
> 
> Fit: aov(formula = rta ~ cds)
> 
> $cds
>               diff        lwr        upr     p adj
> 1-0 -0.1046092796 -0.4331100 0.22389141 0.9751178
> 2-0  0.0359991860 -0.1371359 0.20913425 0.9980970
> 3-0  0.0261665235 -0.1348524 0.18718540 0.9996165
> 4-0  0.0004502442 -0.1805448 0.18144531 1.0000000
> 5-0 -0.1438949939 -0.3104752 0.02268526 0.1422670
> [...]
> 7-5  0.0621598639 -0.1027595 0.22707926 0.9386170
> 7-6  0.0256519274 -0.1757408 0.22704465 0.9999248
> 
> I tried a pairwise.t.test (holm correction) which also was not able to  
> detect differences in 'rta' among groups of 'cds'
> I've never been confronted to such a situation before : is it just a  
> problem of power of the /a posteriori/ tests used ? Do I miss  
> something important in basic stats or in R ?
> How to highlight differences among 'cds' groups seen with aov() ?

        The apparent paradox is only apparent.  This sort of thing
        can and does happen.

	One way of thinking about this situation is to envisage a
	circle (Anova) and a square (multiple comparisons),
	superimposed, with the corners of the square sticking outside
	of the circle, and the extremities of the circle protruding
	beyond the edges of the square.

	You get a ``significant'' result from the Anova if a point
	lands outside the circle; you get a ``significant'' result
	from the multiple comparisons if a point lands outside the
	square.  So if a point lands in the corners of the square
	that stick out beyond the circle, you have a ``significant''
	Anova result, but find no ``significant'' differences in the
	multiple comparisons.  Conversely a point could land in the
	extremities of the circle that protrude beyond the edges of
	the square, in which case you would find ``significant''
	differences in the multiple comparisons but your Anova test
	would not be ``significant''.

	These are rare but not unheard of phenomena.  The essence of
	the situation is that the data are giving you an ambiguous
	message.  There is no real way to resolve the ambiguity
	except by collecting more data.

        Note that if there is a significant Anova result there
        will be at least one ``contrast'' amongst the means that
        is significantly different from zero on an a posteriori
        basis.  This contrast need not however be a pairwise
        difference between means.

                                        cheers,

                                                Rolf Turner
                                                rolf at math.unb.ca



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