Statistical Methods for Psychology

one is larger. Here we don’t have enough evidence to conclude that Delayed is different

from Nonsmoking, but we dohave enough evidence (i.e., power) to conclude that there is a

significant difference between Active and Nonsmoking. This kind of result occurs fre-

quently with multiple-comparison procedures, and we just have to learn to live with a bit

of uncertainty.

13.7 Power Analysis for Factorial Experiments

Calculating power for fixed-variable factorial designs is basically the same as it was for

one-way designs. In the one-way design we defined

and

where , k 5 the number of treatments, and n 5 the number of observa-

tions in each treatment. In the two-way and higher-order designs, we have more than one “treat-

ment,” but this does not alter the procedure in any important way. If we let , and

, where represents the parametric mean of Treatment (across all levels of

B) and represents the parametric mean of Treatment (across all levels of A), then we

can define the following terms:

and

Examination of these formulae reveals that to calculate the power against a null hypothesis

concerning A, we act as if variable Bdid not exist. To calculate the power of the test against

a null hypothesis concerning B, we similarly act as if variable Adid not exist.

Calculating the power against the null hypothesis concerning the interaction follows

the same logic. We define

where is defined as for the underlying structural model ( ).

Given we can simply obtain the power of the test just as we did for the one-way

design.

Calculating power for the random model is more complicated, and for the mixed model

requires a set of rather unrealistic assumptions. To learn how to obtain estimates of power

with these models, see Winer (1971, p. 334).

In certain situations a two-way factorial is more powerful than are two separate one-

way designs, in addition to the other advantages that accrue to factorial designs. Consider

two hypothetical studies, where the number of participants per treatment is held constant

across both designs.

fab

abij abij=m2mi.2m.j1mij

fab=f¿ab 1 n

f¿ab=

B

aab

(^2) ij
abs^2 e
fb=f¿b 2 na
f¿b=
B
ab
(^2) j
bs^2 e
fa=f¿a 2 nb
f¿a=
B
aa
2
j
as^2 e
m.j Bj
bj=m.j2m mi. Ai
ai=mi.2m
gt^2 j =g(mj2m)^2
f=f¿ 2 n
f¿=
B
at
2
j
ks^2 e
Section 13.7 Power Analysis for Factorial Experiments 429