A SPSS summary table for a factorial design differs somewhat from others you have
seen in that it contains additional information. The line labeled “Corrected model” is the
sum of the main effects and the interaction. As such its sum of squares is what we earlier
called SScells. The line labeled “Intercept” is a test on the grand mean, here showing that the
grand mean is significantly different from 0.00, which is hardly a surprise. Near the bottom
the line labeled “Corrected total” is what we normally label “Total,” and the line that they
label “Total” is These extra lines rarely add anything of interest.
The summary table reveals that there are significant effects due to Task and to the in-
teraction of Task and SmokeGrp, but there is no significant effect due to the SmokeGrp
variable. The Task effect is of no interest, because it simply says that people make more er-
rors on some kinds of tasks than others. This is like saying that your basketball team scored
more points in yesterday’s game than did your soccer team. You can see the effects graphi-
cally in the interaction plot, which is self-explanatory.13.6 Multiple Comparisons
All of the multiple-comparison procedures discussed in Chapter 12 are applicable to the
analysis of factorial designs. Thus we can test the differences among the five Condition
means in the Eysenck example, or the three SmokeGrp means in the Spilich example using
the Bonferroni t test, the Tukey test, Ryan’s REGWQ, or any other procedure. Keep in
mind, however, that we must interpret the “n” that appears in the formulae in Chapter 12 to
be the number of observations on which each treatment mean was based. Since the Condi-
tion means are based on (a 3 n) observations, that is the value that you would enter into
the formula, not n.
In the Spilich smoking example, there is no significant effect due to SmokeGrp, so you
would probably not wish to run contrasts among the three levels of that variable. Because
the dependent variable (errors) is not directly comparable across groups, it makes no sense
to look for specific group differences there. We could do so, but no one would be likely to
care. (Remember the basketball and soccer teams referred to above.) However, the interac-
tion suggests that you might wish to run multiple comparisons on simple effects. In partic-
ular, you might wish to examine the effect of smoking on cognitive tasks. You could run
these tests by restricting yourself just to the data from the Cognitive task. However, I would
suggest making these contrasts using from the overall analysis, assuming that you
have no reason to think that you have heterogeneity of variance. If you run your analysis
using standard computer software, you will have to recalculate your effects by substituting
from the main summary table.
The analysis of SmokeGrp differences on the Cognitive task gives a frequent, but un-
welcome, result. Whether you use standard contrasts, Ryan’s procedure, or Tukey’s proce-
dure, you will find that the Nonsmoking group performs significantly better than the Active
group, but not significantly better than the Delayed group. The Delayed group is also not
significantly different from the Active group. Representing this graphically, we have
Nonsmoking Delayed Activewith the groups that did not differ significantly underlined.
If you just came from your class in Logic 132, you know that it does not make sense to
say A 5 B, B 5 C, but But, don’t confuse Logic, which is in some sense exact, with
Statistics, which is probabilistic. Don’t forget that a failure to reject does not mean that
the means are equal. It just means that they are not sufficiently different for us to know whichH 0
AZC.
MSerrorMSerror(gX^2 >N).428 Chapter 13 Factorial Analysis of Variance