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

The conditional formula:

LC¼

Prðobserved dataÞ

Prðall possible configurationsÞ

m 1 cases: (X 1 ,X 2 ,...,Xm 1 )
nm 1 noncases:
(Xm 1 +1,Xm 1 +2,...,Xn)

LC¼Pr(firstm 1 Xs are cases | all
possible configurations ofXs)

Possible configurations
¼combinations of n things
takenm 1 at a time
¼Cmn 1

LC¼

Qm^1

l¼ 1

PðXlÞ

Qn

l¼m 1 þ 1

½ 1 PðXlފ

~

u

Qm^1

l¼ 1

PðXulÞ

Qn

l¼m 1 þ 1

½ 1 PðXulފ

()

vs:

LU¼

Ym^1

l¼ 1

PðXlÞ

Yn

l¼m 1 þ 1

½ 1 PðXlފ

The conditional likelihood formula (LC) reflects

the probability of the observed data configura-

tion relative to the probability of all possible

configurations of the given data. To understand

this, we describe the observed data configura-

tion as a collection ofm 1 cases and nm 1

noncases. We denote the cases by theXvectors

X 1 ,X 2 , and so on throughXm 1 and the non-

cases byXm 1 +1,Xm 1 +2,throughXn.

The above configuration assumes that we have

rearranged the observed data so that them 1

cases are listed first and are then followed

in listing by thenm 1 noncases. Using this

configuration, the conditional likelihood func-

tion gives the probability that the firstm 1 of

the observations actually go with the cases,

given all possible configurations of the above

nobservations into a set ofm 1 cases and a set

ofnm 1 noncases.

The termconfigurationhere refers to one of the

possible ways that the observed set ofXvectors

can be partitioned intom 1 cases andnm 1

noncases. In example 1 here, for instance, the

lastm 1 Xvectors are the cases and the remain-

ingXs are noncases. In example 2, however,

them 1 cases are in the middle of the listing of

allXvectors.

The number of possible configurations is given

by the number of combinations ofnthings

takenm 1 at a time, which is denoted mathe-

matically by the expression shown here, where

theCin the expression denotes combinations.

The formula for the conditional likelihood is

then given by the expression shown here. The

numerator is exactly the same as the likelihood

for the unconditional method. The denomina-

tor is what makes the conditional likelihood

different from the unconditional likelihood.

Basically, the denominator sums the joint pro-

babilities for all possible configurations of the

mobservations intom 1 cases andnm 1 non-

cases. Each configuration is indicated by theu

in theLCformula.

EXAMPLE: Configurations

(1) Lastm 1 Xs are cases

(X 1 ,X 2 ,...,Xn)

—— cases

(2) Cases ofXs are in middle of listing

(X 1 ,X 2 ,...,Xn)

—— cases

Presentation: IV. The Likelihood Function and Its Use in the ML Procedure 115