# [R] ANOVA and TukeyHSD disagrees?

Rolf Turner r.turner at auckland.ac.nz
Fri Mar 20 22:07:52 CET 2009

```On 21/03/2009, at 12:50 AM, Fredrik Karlsson wrote:

> Dear list,
>
> Sorry for posting a borderline statistical question on the list,
> but hte
> SPSS people around me just stares at me blankly when refering to
> tests with
> any term other than ANOVA and post-hoc. I would appreciate any
> insight on
> how this all is possible:
>
> I have a model fitted by aov() stored in "ppdur", which gives this
> result
> when using ANOVA:

<snip>

> As you can see, I don't get a significant p-value for this interaction
> effect  anymore. How could that be?

How?  Well it just could.  That's the way with statistics.  Remember
you're not talking about things being definitely true, you're talking
about there being ``significant'' evidence that they're true.  This

A simple example of such ``paradoxes'' arises in the context of
multiple comparisons.  In a one-way anova these might show you that
level A is ``the same as'' level B, and level B is ``the same as''
level C, but nevertheless level A is *different from* level C.

This is just saying that we have *evidence* that level A is different
from level C.  Ergo it follows, as doth the night follow the day,
that level B must differ either from level A or from level C, or
both.  There just isn't enough information in the data to decide
which of the possibilities is true.  A larger data set would be able
to ``make the decision''.

anova results is not an unheard of phenomenon.  I like to illustrate
what's going on via the diagram shown in the attached pdf file.

Think of the null hypothesis of ``no difference between the levels''
being rejected whenever the sample falls outside of a certain
enclosure.
In the diagram the circle represents the enclosure corresponding to the
anova test; the square represents the enclosure corresponding to the
multiple comparisons test.  If the sample lands outside both the circle
and the square, then both tests reject the null.  But it can happen,
rarely but not too rarely, that the sample lands inside one of the bits
of the circle that stick out beyond the square.  In this case the
multiple
comparisons test will say that there are differences, but the anova
test
will say there are none.  Alternatively, the sample could land in
one of
the corners of the square that stick out beyond the circle.  In this
case
the anova test will say that there are differences, but the multiple
comparisons
test will find none.

That's just All Part of the Rich Tapestry of Life when you do
statistical
hypothesis testing.

BTW don't take the circle and the square too literally.  They are just
illustrative analogies; don't try to interpret them in terms of what's
really going on in the actual hypothesis testing mechanism.

HTH.

cheers,

Rolf Turner

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