Statistical Methods for Psychology

It should be apparent that in calculating and , we have been asking the same ques-

tion in two different ways. Not surprisingly, we have come to the same conclusion. When

we calculated and tested it for significance, we were asking whether there was any cor-

relation (relationship) between Xand Y. When we ran a chi-square test on Table 10.3, we

were also asking whether the variables are related (correlated). Since these questions are

the same, we would hope that we would come to the same answer, which we did. On the

one hand, relates to the statistical significance of a relationship. On the other, meas-

ures the degree or magnitude of that relationship.

It will come as no great surprise that there is a linear relationship between^2 and.

From the fact that , we can deduce that

For our example,

(again, with a bit of correction for rounding) which agrees with our previous calculation.

f^2 as a Measure of the Practical Significance of x^2

The fact that we can go from to means that we have one way of evaluating the practi-

cal significance (importance) of the relationship between two dichotomous variables. We

have already seen that for Gibson’s data the conversion from to^2 showed that our

of 9.79 accounted for about 1.2% of the variation. As I said, that does not look very

impressive, even if it is significant.

Rosenthal and Rubin (1982) have argued that psychologists and others in the “softer

sciences” are too ready to look at a small value of or , and label an effect as unimpor-

tant. They maintain that very small values of can in fact be associated with important

effects. It is easiest to state their case with respect to , which is why their work is dis-

cussed here.

Rosenthal and Rubin pointed to a large-scale evaluation (called a meta-analysis) of over

400 studies of the efficacy of psychotherapy. The authors, Smith and Glass (1977), reported

f

r^2

r^2 f^2

x^2 f x^2

x^2 f

f=

B

9.79

818

= 1 0.0120=.1095

f=

B

x^2

N

x^2 =N

f^2

1 N

f x^2

x^2 f

f

f x^2

Section 10.1 Point-Biserial Correlation and Phi: Pearson Correlations by Another Name 301

Table 10.3 Calculation of for Gibson’s data on sexual abuse ( is shown as

“approximate” simply because of the effect of rounding error in the table)

Training No Training

Abused 43 (56.85) 50 (36.15) 93

Not Abused 457 (443.15) 268 (281.85) 725

500 318 818

=9.79 (approx.)

x^2 =

(43 2 56.85)^2

56.85

1

(50 2 36.15)^2

36.15

1

(457 2 443.15)^2

443.15

1

(268 2 281.85)^2

281.85

x^2 x^2