Introductory Biostatistics

m¼Mþ1 [similar toð;Þandðþ;þÞcells in the 1:1 matching design] will
be ignored.
If we fix the numbermof exposed persons in each stratum, 1amaM,
then

Prðcases exposed=mexposed in a stratumÞ¼

mðORÞ

mðORÞþMmþ 1

where OR is the odds ratio representing the e¤ect of exposure. The result for
pair-matched design in Section 11.5 is a special case whereM¼m¼1.
For the strata, or matched sets with exactlymðm¼ 1 ; 2 ;...;MÞexposed
persons, let

n 1 ;m¼number of sets with an exposed case

n 0 ;m 1 ¼number of sets with an unexposed case

nm¼n 1 ;mþn 0 ;m 1

Then givennmfixed,n 1 ;mhasBðnm;pmÞ, where

pm¼

mðORÞ

mðORÞþMmþ 1

In the special case of 1:1 matching,M¼1, and the result is reduced to the
probabilitypof Section 11.5.1.

11.6.2 Estimation of the Odds Ratio

From the joint (conditional) likelihood function

LðORÞ¼

YM

m¼ 1

mðORÞ

mðORÞþMmþ 1

n 1 ;m 1

Mmþ 1

mðORÞþMmþ 1

n 0 ;m

one can obtain the estimate OR, but such a solution requires an iterative
procedure and a computer algorithm. We will return to this topic in the next
section.
Another simple method for estimating the odds ratio would be to treat a
matched set consisting of one case andMmatched controls as a stratum. We
then present the data from this stratum in the form of a 22 table (Table
11.10) and obtain the Mantel–Haenszel estimate for the odds ratio:

dORMH¼

P

ðad=ðMþ 1 ÞÞ

P

ðbc=ðMþ 1 ÞÞ

410 ANALYSIS OF SURVIVAL DATA