Statistical Methods for Psychology

close to 1.00 for reasonable sample sizes. (See footnote, p. 328). If is false, this ratio

becomes

where

is called lambda ( )or the noncentrality parameter (ncp).^9

You can see that the noncentrality parameter simply displaces the Fdistribution in a

positive direction away from one, with the amount of displacement depending on the true

differences among the population means.

The above formulae may not convey a lot of meaning, but recall that ,

which is the deviation of a group mean from the grand mean. As such, it is a measure of

how much the means differ. Similarly, stis the standard deviation of group means, and, as

such, is an excellent measure of differences between groups. One way of calculating power

is to define a standardized measure^10 of effect size

This statistic ( ) is the same as Cohen’s (1988) measure of effect size, which he labels f.

(If we had two groups it would be numerically equal to half of what we have previously

called d.) You should recall that when we were calculating power for a ttest on two inde-

pendent groups, we took an effect size measure (d) and incorporated the sample size. That

is just what we will do here. We define

This way we can estimate without regard to n, and then include the sample size

when we come to estimating. This just makes our life a bit easier. We can then look up

in the tables of the noncentral Fdistribution, given the level of and the degrees of free-

dom for the numerator and denominator in F. (It is useful to note that is , which is

simply another way to see as a function of the noncentrality parameter.)

An Example

Before we proceed, let’s work with an example that will illustrate several of the points

made here and lead to some further elaboration. Suppose that we take the original data

from the Conti and Musty (1984) experiment referred to earlier. We wish to replicate their

study and want to estimate the power of our experiment. In their paper they analyzed

postinjection activity as a percentage of preinjection activity, rather than the raw activity

measures themselves. We will treat their sample means and the average sample variance

( ) as if they were the actual population values. For this dependent variable, their

sample means were

MSerror

f

f 2 l>k

a

f f

f¿

f=f¿ 2 n

f¿

f¿=

st

se

=

B

aAmj2mB

(^2) >k
se^2
t=Xj 2 X..
l
l=
nat^2 j
s^2 e
E(MStreat)
E(MSerror)

= 11

nat^2 j>(k 2 1)

s^2 e

= 11

l

k 21

H 0

Section 11.12 Power 349

(^9) There are a number of different quantities labeled “noncentrality parameter,” but this is one of the more
common. It is also common to see written with a divisor of (k 2 1).
(^10) To say a measure is a “standardized measure” is just to say that we have divided a quantity by a standard devia-
tion, thus scaling the result in standard deviation units. (This is analogous to dividing 87 inchesby 12, getting 7.25,
and declaring the result to be 7.25feet.)
l
noncentrality
parameter (ncp)