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 n11k n12k n1+k
2 n21k n22k n2+k
Total n+1k n+2k Nk
k=1,…,K
2 2× ORs
ORk=n n11k 22k/n n12k 21k.
ORs
ORs, ORk OR
ORs ORs ORs
wk=n n21k 12k/Nk.
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
(
ORMH)
±z / VarRBG⎡⎣⎢ln(
ORMH)
⎤ ,⎡ ⎦⎥
⎣⎢
⎤
− ⎦⎥
1 α2
0000 101058 GRMAT #1C41220.book Page 83 Thursday, November 1, 2012 2:45 PM
