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

Two alternatives:

Unconditional algorithm (LU)

vs.

Conditional algorithm (LC)

likelihoods

Formula forLis built into
computer algorithms

User inputs data and
computer does calculations

Lformulae are different for
unconditional and conditional
methods

The unconditional formula:
(a joint probability)
cases noncases

LU¼

Ym^1

l¼ 1

PðXlÞ

Yn

l¼m 1 þ 1

½ 1 PðXlފ

PðXÞ¼logistic model

¼

1

1 þeðaþSbiXiÞ

LU¼

Qn

l¼ 1

exp aþ~

k

i¼ 1

biXil



Qn

l¼ 1

1 þexp aþ~

k

i¼ 1

biXil



As described earlier, if the model is logistic,

there are two alternative typesof computer

algorithms to choose from, anunconditional

vs. aconditionalalgorithm. These algorithms

use different likelihood functions, namely,LU

for the unconditional method andLCfor the

conditional method.

The formulae for the likelihood functions for

both the unconditional and conditional ML

approaches are quite complex mathematically.

The applied user of logistic regression, however,

never has to see the formulae forLin practice

because they are built into their respective com-

puter algorithms. All the user has to do is learn

how to input the data and to state the form of

the logistic model being fit. Then the computer

does the heavy calculations of forming the like-

lihood function internally and maximizing this

function to obtain the ML solutions.

Although we do not want to emphasize the

particular likelihood formulae for the uncondi-

tional vs. conditional methods, we do want

to describe how these formulae are different.

Thus, we briefly show these formulae for this

purpose.

Theunconditional formulais given first and

directly describes the joint probability of the

study data as theproduct of the joint probability

for the cases(diseased persons)and the joint

probability for the noncases(nondiseased per-

sons). These two products are indicated by the

largePsigns in the formula. We can use these

products here by assuming that we have inde-

pendent observations on all subjects. The prob-

ability of obtaining the data for thelth case is

given by P(Xl), where P(X) is the logistic model

formula for individualX. The probability of the

data for thelth noncase is given by 1 – P(Xl).

When the logistic model formula involving the

parameters is substituted into the likelihood

expression above, the formula shown here is

obtained after a certain amount of algebra is

done. Note that this expression for the likeli-

hood functionLis a function of the unknown

parametersaand thebi.

114 4. Maximum Likelihood Techniques: An Overview