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

Maximizing L(u) is equivalent to
maximizing lnL(u)

Solve:
@lnLðuÞ
@yj

¼0,j¼1, 2,...,q

qequations inqunknowns require
iterativesolution by computer

The standard approach for maximizing an

expression like the likelihood function for the

binomial example here is to use calculus by

setting the derivative dL/dpequal to 0 and solv-

ing for the unknown parameter or parameters.

For the binomial example, when the derivative

dL/dpis set equal to 0, the ML solution obtained

is^pequal to 0.75. Thus, the value 0.75 is the

“most likely” value forp in the sense that it

maximizes the likelihood functionL.

If we substitute into the expression for La

value for pexceeding 0.75, this will yield a

smaller value forLthan obtained when substi-

tutingpequal to 0.75. This is why 0.75 is called

theMLestimator.Forexample,whenpequals1,

the value forLusing the binomial formula is 0,

which is as small asLcan get and is, therefore,

less than the value ofLwhenpequals the ML

value of 0.75.

Note that for the binomial example, the ML

valuep^equal to 0.75 is simply the sample pro-

portion of the 100 trials that are successful. In

other words, for a binomial model,the sample

proportionalways turns out to be the ML esti-

mator of the parameterp. So for this model, it is

not necessary to work through the calculus to

derive this estimate. However, for models more

complicated than the binomial, for example, the

logistic model, calculus computations involving

derivatives are required and are quite complex.

In general, maximizing the likelihood function

L(u) is equivalent to maximizing the natural

log ofL(u), which is computationally easier.

The components ofuare then found as solu-

tions of equations of partial derivatives as

shown here. Each equation is stated as the

partial derivative of the log of the likelihood

function with respect toyjequals 0, whereyj

is thejth individual parameter.

If there areq parameters in total, then the

above set of equations is a set ofqequations

inqunknowns. These equations must then be

solved iteratively, which is no problem with the

right computer program.

EXAMPLE (continued)

Maximum value obtained by solving

dL

dp

¼ 0

forp:

^p¼0.75 “most likely”

maximum

#

p>^p¼0.75)L(p)<L(p¼0.75)

e.g.,

binomial formula

p¼ 1 )L(1)¼c 175 (11)^25

¼ 0 <L(0.75)

^p¼ 0 : 75 ¼

75

100

, a sample proportion

Binomial model

)^p¼

X

nis ML estimator

More complicated models)complex

calculations

Presentation: IV. The Likelihood Function and Its Use in the ML Procedure 113