Statistical Methods for Psychology

at substantially higher rates than females. However, when we break the data down by de-

partments, we see that in three of those departments women were admitted at a higher rate,

and in the remaining two the differences in favor of men were quite small.

The Mantel-Haenszel statistic (Mantel and Mantel and Haenszel (1959)) is designed to

deal with the data from each department separately (i.e., we condition on departments). We

then sum the results across departments. Although the statistic is not a sum of the chi-

square statistics for each department separately, you might think of it as roughly that. It is

more powerful than simply combining individual chi-squares and is less susceptible to the

problem of small expected frequencies in the individual 2 3 2 tables (Cochran, 1954).

The computation of the Mantel-Haenszel statistic is based on the fact that for any 2 3 2

table, the entry in any one cell, given the marginal totals, determines the entry in every

other cell. This means that we can create a statistic using only the data in cell 11 of the table

for each department. There are several variations of the Mantel-Haenszel statistic, but the

most common one is

where O 11 kand E 11 kare the observed and expected frequencies in the upper left cell of each

of the k 2 3 2 tables and the entries in the denominator are the marginal totals and grand

total of each of the k 2 3 2 tables. The denominator represents the variance of the numera-

tor. The entry of 21 ⁄ 2 in the numerator is the same Yates’ correction for continuity that I

passed over earlier. These values are shown in the calculations that follow (Table 6.9).

This statistic can be evaluated as a chi-square on 1 df, and its probability under H 0 is .76.

We certainly cannot reject the null hypothesis that admission is independent of gender, in

direct contradiction to the result we found when we collapsed across departments.

In the calculation of the Mantel-Haenszel statistic I left out the data from Department

A, and you are probably wondering why. The explanation is based on odds ratios, which I

won’t discuss until the next section. The short answer is that Department A had a different

=

Aƒ^6862 681.93ƒ^212 B^2

132.777

=

(4.07 2 .5)^2

132.777

=0.096

M^2 =

Aƒ©O 11 k^2 ©E 11 kƒ^212 B^2

gn 11 k n 21 k n 11 k n 12 k>n^211 k(n 11 k 2 1)

M^2 =

AƒgO 11 k 2 ©E 11 kƒ 212 B^2

gn 11 kn 21 kn 11 kn 12 k>n^211 k(n 11 k 2 1)

158 Chapter 6 Categorical Data and Chi-Square

Table 6.8 Admissions data for graduate departments at Berkeley (1973)

Major Males Females

Admit Reject Admit Reject

A 512 313 89 19

B 353 207 17 8

C 120 205 202 391

D 138 279 131 244

E 53 138 94 299

F 22 351 24 317

Total B-F 686 1180 508 1259

% of Total B-F 36.8% 63.2% 28.8% 71.2%