If we were to take the average of the strata specific each would contribute equally to the overall

categorical data and contingency tables 83

2.6.1 Estimation

A first step in the analysis of several tables is to calculate strata specific

(2.40)

This will provide us with a first impression of the distribution of ORs across the subgroups defined by the stratification variable. Provided there is a common underlying OR for the association of exposure/treatment and outcome, we want to combine the to obtain an estimate of this common OR. If we were to take the average of the strata specific each would contribute equally to the overall . This approach usually leads to unnecessary loss of precision. To increase precision, we would like strata with many observations to carry more weight than strata with fewer observations. This is because the from large tables are usually estimated with greater precision than the from small tables.

The Mantel-Haenszel estimate of the common OR (Mantel and Haenszel, 1959) is a weighted average of the strata specific with weights equal to

The Mantel-Haenszel estimate is given as

(2.41)

The confidence interval using the variance estimate of Robins et al. (1986) is given as

(2.42) Table 2.20 The general counts of 2 × 2 table number k, tables.

Rows

Columns

Total

1 2

1 n_11k n_12k n_1+k

2 n21k n22k n2+k

Total n_+1k n_+2k N_k

k=1,…,K

2 2× ORs

OR_k=n n_{11k 22k}/n n_{12k 21k}.

ORs

ORs, OR_k OR

ORs ORs ORs

w_k=n n_{21k 12k}/N_k.

OR

n n N n n N

MH

k k k

k K

k k k

k K

= ⁼

=

∑

∑

11 22 1

12 21 1

/ /

.

exp ln

(

)

(

)

⎡ ⎦⎥

⎣⎢

⎤

− ⎦⎥

1 α2

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