[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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