Statistical Methods for Psychology

With 25 df, the difference between the two groups is significant. We now calculate

which, with the exception of the arbitrary sign of the coefficient, agrees with the more

direct calculation.

What is important about the equation linking and t is that it demonstrates that the

distinction between relationships and differences is not as definitive as you might at first

think. More important, we can use and t together to obtain a rough estimate of the prac-

tical, as well as the statistical, significance of a difference. Thus a t 5 3.42 is evidence in

favor of the experimental hypothesis that the two sexes differ in weight. At the same time,

(which is a function of t) tells us that gender accounts for 32% of the variation in

weight. Finally, the equation shows us how to calculate rfrom the research literature when

only t is given, and vice versa.

Testing the Significance of

A test of against the null hypothesis : 5 0 is simple to construct. Since is a Pearson

product-moment coefficient, it can be tested in the same way What is important about the

equation linking and t is that it demonstrates that the distinction between relationships

and differences is not as definitive as you might at first think. More important, we can use

and t together to obtain a rough estimate of the practical, as well as the statistical, signif-

icance of a difference. Thus a t 5 3.42 is evidence in favor of the experimental hypothesis

that the two sexes differ in weight. At the same time, (which is a function of t) tells us

that gender accounts for 32% of the variation in weight. Finally, the equation shows us how

to calculate rfrom the research literature when only t is given, and vice versa.

Testing the Significance of

A test of against the null hypothesis is simple to construct. Since is a

Pearson product-moment coefficient, it can be tested in the same way as r. Namely,

on N 2 2 df. Furthermore, since this equation can be derived directly from the definition of ,

the t 5 3.42 obtained here is the same (except possibly for the sign) as a t test between the two

levels of the dichotomous variable. This makes sense when you realize that a statement that

males and females differ in weight is the same as the statement that weight varies with sex.

and Effect Size

There is one more important step that we can take. Elsewhere we have considered a meas-

ure of effect size put forth by Cohen (1988), who defined

as a measure of the effect of one treatment compared to another. We have to be a bit careful

here, because Cohen originally expressed effect size in terms of parameters (i.e., in terms of

d=

m 1 2m 2

s

rpb^2

r^2 pb

t=

rpb 2 N 22

312 r^2 pb

rpb H 0 : r= 0 rpb

rpb^2

r^2 pb

r^2 pb

r^2 pb

rpb H 0 r rpb

rpb^2

r^2 pb

r^2 pb

r^2 pb

rpb= 1 .319=.565

r^2 pb=

t^2

t^21 df

=

3.42^2

3.42^2125

=.319

298 Chapter 10 Alternative Correlational Techniques