Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

LC¼

Qm^1

l¼ 1

exp ~

k

i¼ 1

biXli



~

u

Qm^1

l¼ 1

exp ~

k

i¼ 1

biXlui



Note:adrops out ofLC

Conditional algorithm:
 Estimatesbs

 Does not estimatea(nuisance
parameter)

Note:OR involves onlybs

Case-control study:
cannot estimatea

direct joint
probability

LU ≠ LC

does not require

estimating nuisance

parameters

many nuisance parameters

Stratified data, e.g., matching,

100 nuisance parameters

⇓

are not estimated

using LC

unnecessarily

estimated using LU

When the logistic model formula involving the

parameters is substituted into the conditional

likelihood expression above, the resulting for-

mula shown here is obtained. This formula

is not the same as the unconditional for-

mula shown earlier. Moreover, in the condi-

tional formula, the intercept parameterahas

dropped out of the likelihood.

The removal of the interceptafrom the condi-

tional likelihood is important because it means

that when a conditional ML algorithm is used,

estimates are obtained only for thebicoeffi-

cients in the model and not fora. Because the

usual focus of a logistic regression analysis is

to estimate an odds ratio, which involves the

bs and nota, we usually do not care about

estimatingaand, therefore, considerato be a

nuisance parameter.

In particular, if the data come from a case-

control study, we cannot estimateabecause

we cannot estimate risk, and the conditional

likelihood function does not allow us to obtain

any such estimate.

Regarding likelihood functions, then, we have

shown that the unconditional and conditional

likelihood functions involve different formu-

lae. The unconditional formula has the theoret-

ical advantage in that it is developed directly

as a joint probability of the observed data.

The conditional formula has the advantage

that it does not require estimating nuisance

parameters likea.

If the data are stratified, as, for example, by

matching, it can be shown that there are as

many nuisance parameters as there are

matched strata. Thus, for example, if there

are 100 matched pairs, then 100 nuisance para-

meters do not have to be estimated when

using conditional estimation, whereas these

100 parameters would be unnecessarily esti-

mated when using unconditional estimation.

116 4. Maximum Likelihood Techniques: An Overview