Statistical Methods for Psychology

As an illustration of the use of the likelihood ratio test for contingency tables, consider

the data found in the death sentence study. The cell and marginal frequencies follow:

This answer agrees with the likelihood ratio statistic found in Exhibit 6.1b. It is a on

1 df, and since it exceeds , it will lead to rejection of.

6.10 Mantel-Haenszel Statistic

We have been dealing with two-dimensional tables where the interpretation is relatively

straightforward. But often we have a 2 3 2 table that is replicated over some other vari-

able. There are many situations in which we wish to control for (often called “condition

on”) a third variable. We might look at the relationship between (X) stress (high/low) and

(Y) mental status (normal/disturbed) when we have data collected across several different

environments (Z). Or we might look at the relationship between the race of the defendant

(X) and the severity of the sentence (Y) conditioned on the severity of the offense (Z)—see

Exercise 6.41. The Mantel-Haenszel statistic(often referred to as the Cochran-Mantel-

Haenszelstatistic because of Cochran’s (1954) early work on it) is designed to deal with

just these situations. For our example here we will take a well-known example involving a

study of sex discrimination in graduate admissions at Berkeley in the early1970s. This ex-

ample will serve two purposes because it will also illustrate a phenomenon known as

Simpson’s paradox.This paradox was described by Simpson in the early 1950s, but was

known to Yule nearly half a century earlier. (It should probably be called the Yule-Simpson

paradox.) It refers to the situation in which the relationship between two variables, seen at

individual levels of a third variable, reverses direction when you collapse over the third

variable. The Mantel-Haenszel statistic is meaningful whenever you simply want to con-

trol the analysis of a 2 3 2 table for a third variable, but it is particularly interesting in the

examination of the Yule-Simpson paradox.

The University of California at Berkeley investigated racial discrimination in graduate

admissions in 1973 (Bickel, Hammel, and O’Connell (1975)). A superficial examination of

admissions for that year revealed that approximately 45% of male applicants were admit-

ted compared with only about 30% of female applicants. On the surface this would appear

to be a clear case of gender discrimination. However, graduate admissions are made by de-

partments, not by a University admissions office, and it is appropriate and necessary to

look at admissions data at the departmental level. The data in Table 6.8 show the break-

down by gender in six large departments at Berkeley. (They are reflective of data from all

101 graduate departments.) For reasons that will become clear shortly, we will set aside for

now the data from the largest department (Department A), which is why that department is

shaded in Table 6.8.

Looking at the bottom row of Table 6.8, which does not include Department A, you can

see that 36.8% of males and 28.8% of females were admitted by the five departments. A

chi-square test on the data produces which has a probability under H 0 that is

0.00 to the 9th decimal place. This seems to be convincing evidence that males are admitted

x^2 =37.98,

x^2 .05(1)=3.84 H 0

x^2

=2[3.6790]=7.358

=2[33(.3733) 1 251(-.0401) 1 33(-0.2172) 1 508(0.0204)]

= 2 c 33 lna

33

22.72

b 1 251 lna

251

261.28

b 1 33 lna

33

43.28

b 1 508 lna

508

497.72

bd

x^2 = (^2) aOijlna
Oij
Eij
b
Section 6.10 Mantel-Haenszel Statistic 157
The Mantel-
Haenszel
statistic
Cochran-Mantel-
Haenszel
Simpson’s
paradox