Statistical Methods for Psychology

are looking at comparisons of individual means, or sets of means, it generally makes more

sense to calculate confidence limits on our differences and or to use a d-family measure of

the effect size.

There are several ways that we could approach d-family measures. One very simple

way is to go back to Chapter 7, which discussed ttests, and apply the measures that were

discussed there. We will come out at the same place, however, if we approach the problem

through linear contrasts. Remember that when you are looking at two groups, it makes no

difference whether you run a ttest between those groups, or compute a linear contrast and

then an F, and take the square root of that F. The advantage of going with linear contrasts

is that they are more general, allowing us to compare means of sets of groups rather than

just two individual groups.

We will take an example from our morphine study by Siegel. One contrast that really in-

terests me is the contrast between Group M-M and Group Mc-M. If their means are statisti-

cally significantly different, then that tells us that there is something important about changing

the physical context in which the morphine is given. The group statistics are given below.

Condition M-M Mc-M

Mean 10.00 29.00

St. Dev 5.13 6.06

Variance 26.32 37.95

MSerror 32.00

The coefficients for the linear contrast of these two groups would be “ 2 1” for M-M, “ 1 1”

for Mc-M, and “0” for the other three conditions.

Confidence Interval

Let us first compute a confidence interval on the difference between conditions. The gen-

eral formula for a confidence interval on a contrast of two means is

or, if we let “ j” represent the value of the contrast, where , then

whereserror is the standard error of the contrast, which is

For our confidence interval on the difference between the two conditions of interest I have

The probability is .95 that an interval formed as I have formed this one will include the true

difference between the population means.

When it comes time to form our effect size measure, we have a choice of what we will

use as the error term—the standard deviation in the equation. I could choose to use the

square root of MSerrorfrom the overall analysis, because that represents the square root of

the average variance within each groups. Kline (2004) recommends this approach. I have

two other perfectly reasonable alternatives, however. First I could take the square root of

13.26...mM-M2mMc-M...24.74

= 196 2.03(2.828)= 19 6 5.74

CI.95=(-1(10) 1 1(29)) 6 2.03 1 8.00

B

2 MSerror

n

CI.95=(cj) 6 t.025serror

c cj=©aiXi

CI.95=(Xi 2 Xj) 6 t.025sXi 2 Xj

>

12.4 Confidence Intervals and Effect Sizes for Contrasts 385