The Mantel-Haenszel Statistic.

In biological and medical applications, it is not uncommon to be confronted with a series of 2 × 2 tables that examine the same effect under different conditions. If there are K such tables, the data can be combined to form a 2 × 2 × K table. Because each 2 × 2 table examines the same effect, it is often assumed that the odds ratio for the effect is constant over tables. This is equivalent to assuming the no three-factor interaction model. To test for the existence of the effect, one tests whether the common log odds ratio is zero while adjusting for the various circumstances under which data were collected. In terms of log-linear models, this is a one degree of freedom test of conditional independence given the layer k. Prior to the development of log-linear model theory, Mantel and Haenszel (1959) proposed a statistic for testing this hypothesis. The statistic, apart from a continuity correction factor, is [

k(n11k − mˆ 11k)]2 
k [ ˆm11kmˆ 22k]

[n··k − 1]
, where the ˆm’s are obtained from the conditional independence model. This statistic has an asymptotic χ2(1) distribution under the conditional independence model.
The Berkeley graduate admission data of Exercise 3.8.4 and Table 3.3 is a set of six 2 × 2 tables. In each table we are interested in the effect of sex on admission; the six departments constitute various conditions under which this effect is being investigated.

a) Give a justification for whether or not use of the Mantel-Haenszel statistic is appropriate for these data.

b) If appropriate, use both G2 and the Mantel-Haenszel statistic to test whether there is an effect of sex on admission.

c) Show that the denominator of the Mantel-Haenszel statistic can be written as 
k [ ˆm12kmˆ 21k]


[n··k − 1].