You only have four cells [(A,B) = {(0,0), (0,1), (1,0), (1,1)}], with
only 3 degrees of freedom among them. If you have three significant
effects (A, B, and A*B) your strategy in post hoc tests ought to be to
simplify the description, not to complicate things further. What you SAY
you expected was that A is toxic (or harmful), so that (on the average)
cell number (1,0) < cell number (0,0);
and that B is helpful (or perhaps anti-toxic?), so that perhaps
cell number (0,1) >= cell number (0,0)
[it is not clear to me whether the effect of B alone is expected to
_increase_ cell numbers, or merely not to decrease them.]
and you apparently expect B to prevent effects of A, so that
cell number (1,1) = cell number (0,0) approximately.
What, in fact, are the means of your four cells? Have you plotted them?
(Ordinate = average cell number; abscissa = the two values of, say, A;
two means are plotted for each value of A and labelled by value of B.)
Until you have looked at the means, you cannot easily judge how best to
carry out post hoc analyses.
There are a limited number of patterns of four means in a 2x2 design that
can produce significant effects for all three formal sources of
variation. One of those is [a, a, a, b] for some ordering of the four
cells, where a represents three means that are nearly equal and b is
a fourth mean that is different from the others. This would be
consistend with your expectations as described above if B alone actually
has no effect on cell number but merely (?!) counteracts the toxic
effects of A, so that mean cell numbers for (0,0), (0,1), and (1,1) are
roughly equal, while average cell number for (1,0) is significantly
smaller than any of these due to the toxicity of treatment A.
This pattern will always produce two significant main effects and a
significant interaction if the average (a+b)/2 is significantly larger
than the average (a+a)/2 = a.
If this is what you see, your post hoc analysis should compare
the mean of (1,0) with the average of the means of the other three cells
(the Scheffe' method would be preferable to the Tukey method).
Subtract the SS for this contrast from the total SS between groups (with
3 d.f.: the total of the SS's for the A main effect, the B main effect,
and the A*B interaction), and ask whether the SS remaining (with 2 d.f.)
is large enough to contain any further interesting effects. 'Twouldn't
surprise me if it weren't; but if it should be, construct another
contrast among the other three cells that would account for the bulk of
the remaining SS, and see if it's significant.
We can in any case be sure that if B actually has a positive effect on
cell number, that effect is different in magnitude (smaller, I'd guess)
than the negative effect of A; because the interaction is significant.
On Sat, 8 Jan 2000, JE wrote:
> I am trying to figure out how to properly perform a "posthoc" analysis
> of an experiment in which I exposed subjects to 2 different treatments
> (drug A and drug B). Treatment A had 2 levels: treatment or no
> treatment. Treatment B had 2 levels: treatment or no treatment.
> We hypothesized that A is a toxic agent, and would reduce cell number.
> We hypothesized that B is a helpful agent, and would prevent toxic
> effects by A.
>
> A univariate anova found a significant interaction A*B : p <0.00001.
> A and B were also highly signficant.
>
> In SPSS, I set up a four column ANOVA. Column 1: Treatment A [0, 1];
> Column 2: Treatment B [0, 1]; Column 3: Cell Number [scale value];
> Column 4: Treatment [no A-noB, A-noB, B-noA, AB]
>
> I ran a regular ANOVA using columns 1, 2, and 3. I performed a "post
> hoc" analysis using column 4 and column 3 with a Tukey's test.
>
> Is there another way?
You have not described your "post hoc" analysis. For openers, which
"Tukey's test" did you use (there are at least three post hoc tests due,
or at any rate attributed, to Tukey)? Did you carry out a "complete"
post hoc analysis, accounting for 3 degrees of freedom all together, and
if so were the comparisons orthogonal? For all you've said to the
contrary, you might equally well have done something as simple-minded as
examining all six possible pairwise contrasts -- that sort of thing
tends unfortunately to get programmed in packages because it's easy to
program, but it is not usually a useful way of detecting interesting
patterns among the means for designs with more than 3 cells.
-- DFB.
------------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 603-535-2597
184 Nashua Road, Bedford, NH 03110 603-471-7128
