Statistical Methods for Psychology

the average sample variance of the two groups in question (perhaps weighted if the sam-

ple sizes were unequal). In this case it would be (26.32 1 37.95) 2 5 32.135 and

5 5.669. This would make sense if I were worried about heterogeneity of vari-

ance. Alternatively, I could consider one of the groups to be a control group and use its

standard deviation as my error term. Here I might argue that M-M is like a control group

because the conditions don’t change on trial 4. In this case I would let serror 5 5.13. I think

that my preference in general would be to base my estimate on the average of the vari-

ances of the groups in question. If the variances are homogeneous across all five groups,

then the average of the groups in question won’t deviate much from the average of the

variances of all five groups, so I haven’t lost much. Others might take a different view.

Effect Size

We have just seen that the confidence interval on the difference between Mc-M and

M-M is 13.26 # (mMc-M– mM-M) #24.74. Both limits are on the same side of 0, reflecting

the fact that the difference was statistically significant. However, the dependent variable

here is the length of time before the animal starts to lick its paws, and I don’t suppose that

any of us have a strong intuitive understanding of what a long or short interval is for this

case. A difference of at least thirteen seconds seems pretty long, but I would like some bet-

ter understanding of what is happening. One way to compute that would be to calculate an

effect size on the difference between these means.

Our effect size measure will be essentially the same as it was in the case for ttests for

independent samples. However, I will write it slightly differently because doing so will

generalize to more complex comparisons. We have just seen that crepresents a contrast

between two means or sets of means, so it is really just a difference in means. We will take

this difference and standardize it, which simply says that we want to represent the differ-

ence in group means in standard deviation units. (That is what we did in Chapter 7 as well.)

In Chapter 7 we defined

where spis the square root of our pooled variance estimate and is a measure of the average

standard deviation within the groups. We are going to calculate essentially the same thing

here, but I will write its expression as

The numerator is a simple linear contrast, while the denominator is some estimate of the

within groups standard deviation.

The preceding formula raises two points. In the first place, the coefficients must form

what we have called a “standard set.” This simply means that the absolute values of the co-

efficients must sum to 2. For example, if we want to compare the mean of two groups with

the mean of a third, we could use coefficients of (^1 ⁄ 2 1 ⁄ 22 1) to form our contrast. Alterna-

tively, we would get to the same place as far as our test of significance is concerned by us-

ing (1 1 –2) or (3 3 –6). The resulting Fwould be the same. But only the first would give

us a numerical answer for the contrast that is the difference between the mean of the first

two groups and the mean of the third. This is easily seen when you write

=

X 11 X 2

2

2 X 3

c=A^12 B(X 1 ) (^1) A^12 BX 21 (-1)X 3
dN=c
se
= a
(ai Xi)
se
dN=
Xi 2 Xj
Sp

1 32.135

>

386 Chapter 12 Multiple Comparisons Among Treatment Means