Statistical Methods for Psychology

This is a chi-square on 1 dfand is significant because it exceeds the critical value of 3.84.

There is reason to conclude that the intervention was successful.

One Further Step

The question that Dr Freedenthal asked was actually more complicated than the one that I

just answered, because she also had a control group that did not receive the intervention

but was evaluated at both times as well. She wanted to test whether the change in the

intervention group was greater than the change in the control group. This actually turns out

to be an easier test than you might suspect. The test is attributable to Marascuilo and Serlin

(1979). The data are independent because we have different children in the two treatments

and because those who change in one direction are different from those who change in the

other direction. So all that we need to do is create a 2 3 2 contingency table with Treat-

ment Condition on the columns and Increase versus Decrease on the rows and enter data

only from those children in each group who changed their behavior from fall to spring. The

chi-square test on this contingency table tests the null hypothesis that there was an equal

degree of change in the two groups. (A more extensive discussion of the whole issue of

testing non-independent frequency data can be found at http://www.uvm.edu/~dhowell/

StatPages/More_Stuff/Chi-square/Testing Dependent Proportions.pdf.)

6.8 One- and Two-Tailed Tests

People are often confused as to whether chi-square is a one- or a two-tailed test. This confu-

sion results from the fact that there are different ways of defining what we mean by a one-

or a two-tailed test. If we think of the sampling distribution of , we can argue that is a

one-tailed test because we reject only when our value of lies in the extreme right tail

of the distribution. On the other hand, if we think of the underlying data on which our ob-

tained is based, we could argue that we have a two-tailed test. If, for example, we were

using chi-square to test the fairness of a coin, we would reject if it produced too many

heads orif it produced too many tails, since either event would lead to a large value of.

The preceding discussion is not intended to start an argument over semantics (it does

not really matter whether you think of the test as one-tailed or two); rather, it is intended to

point out one of the weaknesses of the chi-square test, so that you can take this into ac-

count. The weakness is that the test, as normally applied, is nondirectional. To take a sim-

ple example, consider the situation in which you wish to show that increasing amounts of

quinine added to an animal’s food make it less appealing. You take 90 rats and offer them a

choice of three bowls of food that differ in the amount of quinine that has been added. You

then count the number of animals selecting each bowl of food. Suppose the data are

Amount of Quinine

Small Medium Large

39 30 21

The computed value of is 5.4, which, on 2 df, is not significant at p ,.05.

The important fact about the data is that any of the six possible configurations of the

same frequencies (such as 21, 30, 39) would produce the same value of , and you receive

no credit for the fact that the configuration you obtained is precisely the one that you pre-

dicted. Thus, you have made a multi-tailedtest when in fact you have a specific prediction

x^2

x^2

x^2

H 0

x^2

H 0 x^2

x^2 x^2

x^2 =

©(O 2 E)^2

E

=

(4 2 8.0)^2

8.0

1

(12 2 8.0)^2

8.0

=4.00

Section 6.8 One- and Two-Tailed Tests 155