Statistical Methods for Psychology

relationship between gender and admissions than did the other five departments, which

were largely homogeneous in that respect. The Mantel-Haenszel statistic is based on the

assumption that departments are homogeneous with respect to the pattern of admissions.

The obvious question following the result of our analysis of these data concerns why it

should happen. How is it that there is a clear bias toward men in the aggregated data, but

no such bias when we break the results down by department. If you calculate the percent-

age of applicants admitted by each department, you will find that Departments A, B, and D

admit over 50% of their applicants, and those are also the departments to which males ap-

ply in large numbers. On the other hand, women predominate in applying to Departments

C and E, which are among the departments who reject two-thirds of their applicants. In

other words, women are admitted at a lower rate overall because they predominately apply

to departments with low admittance rates (for both males and females). This is obscured

when you sum across departments.

6.11 Effect Sizes

The fact that a relationship is “statistically significant” does not tell us very much about

whether it is of practical significance. The fact that two independent variables are not sta-

tistically independent does not necessarily mean that the lack of independence is important

or worthy of our attention. In fact, if you allow the sample size to grow large enough, al-

most any two variables would likely show a statistically significant lack of independence.

What we need, then, are ways to go beyond a simple test of significance to present one

or more statistics that reflect the size of the effect we are looking at. There are two differ-

ent types of measures designed to represent the size of an effect. One type, called the

d-familyby Rosenthal (1994), is based on one or more measures of the differencesbe-

tween groups or levels of the independent variable. For example, as we will see shortly, the

probability of receiving a death sentence is about 5% points higher for defendants who are

nonwhite. The other type of measure, called the r-family,represents some sort of correla-

tion coefficient between the two independent variables. We will discuss correlation thor-

oughly in Chapter 9, but I will discuss these measures here because they are appropriate at

this time. Measures in the r-family are often called “measures of association.”

An Example

An important study of the beneficial effects of small daily doses of aspirin on reducing

heart attacks in men was reported in 1988. Over 22,000 physicians were administered as-

pirin or a placebo over a number of years, and the incidence of later heart attacks was

recorded. The data follow in Table 6.10. Notice that this design is a prospective study

Section 6.11 Effect Sizes 159

Table 6.9 Observed and expected frequencies for Berkeley data

Department O 11 E 11 Variance

A 512 531.43 21.913

B 353 354.19 5.572

C 120 114.00 47.861

D 138 141.63 44.340

E 53 48.08 24.251

F 22 24.03 10.753

Total B-F 686 681.93 132.777

d-family

r-family

measures of
association

prospective
study