Statistical Methods for Psychology

where and represent the largest and smallest of a set of treatment means and ris the

number of treatments in the set. You probably have noticed that the formula for qis very

similar to the formula for t. In fact

and the only difference is that the formula for t has a “ ” in the denominator. Thus, qis a

linear function of t and we can always go from t to qby the relation. The real dif-

ference between qand t tests comes from the fact that the tables of q(Appendix q) are set

up to allow us to adjust the critical value of qfor the number of means involved, as will be-

come apparent shortly. When there are only two treatments, whether we solve for t or qis

irrelevant as long as we use the corresponding table.

When we have only two means or when we wish to compare two means chosen at ran-

domfrom the set of available means, t is an appropriate test.^10 Suppose, however, that we

looked at a set of means and deliberately selected the largest and smallest means for test-

ing. It is apparent that we have drastically altered the probability of a Type I error. Given

that is true, the largest and smallest means certainly have a greater chance of being

called “significantly different” than do means that are adjacent in an ordered series of

means. This is the point at which the Studentized range statistic becomes useful. It was de-

signed for just this purpose.

To use q, we first rank the means from smallest to largest. We then take into account

the number of steps between the means to be compared. For adjacent means, no change is

made and. For means that are not adjacent, however, the critical value of q

increases, growing in magnitude as the number of intervening steps between means in-

creases.

As an example of the use of q, consider the data on morphine tolerance. The means are

4 10112429

with n 5 8, 5 35, and 5 32.00. The largest mean is 29 and the smallest is 4,

and there are a total (r) of 5 means in the set (in the terminology of most tables, we say that

these means are r 5 5 steps apart).

Notice that ris not involved in the calculation. It is involved, however, when we go to the ta-

bles. From Appendix q, for r 5 5 and 5 35,. Because 12.5 .4.07,

we will reject and conclude that there is a significant difference between the largest and

smallest means.

An alternative to solving for and referring to the sampling distribution of q

would be to solve for the smallest difference that would be significant and then to compare

qobt qobt

H 0

dferror q.05(5,35)=4.07

q 5 =

X 12 Xs

B

MSerror

n

=

2924

B

32.00

8

=

25

14

=12.5

dferror MSerror

X 1 X 2 X 3 X 4 X 5

q.05=t.05 12

H 0

q=t 12

12

t=

Xi 2 Xj

B

2(MSerror)

n

qr=

Xl 2 Xs

B

MSerror

n

Xl Xs

390 Chapter 12 Multiple Comparisons Among Treatment Means

(^10) With only two means, we obtain all of the information we need from the Fin the analysis of variance table and
would have no need to run any contrast.