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

R-to-1 matching

unconditional is overestimate of
(correct) conditional estimate

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

L¼L(u)¼likelihood function

u¼(y 1 ,y 2 ,...,yq)

E, V, Wmodel:

logit PðXÞ¼aþbEþ~

p 1

i¼ 1

giVi

þE~

p 2

j¼ 1

djWj

u¼ða;b;g 1 ;g 2 ;...;d 1 ;d 2 ;...Þ

If theonlyvariables being controlled are those

involved in the matching, then the estimate of

the odds ratio obtained by using unconditional

ML estimation, which we denote bydORU, is the

square of the estimate obtained by using con-

ditional ML estimation, which we denote by

dORC. Statisticians have shown that the correct

estimate of this OR is given by the conditional

method, whereas a biased estimate is given by

the unconditional method.

Thus, for example, if the conditional ML esti-

mate yields an estimated odds ratio of 3, then

the unconditional ML method will yield a very

large overestimate of 3 squared, or 9.

More generally, whenever matching is used,

evenR-to-1 matching, whereRis greater than 1,

the unconditional estimate of the odds ratio

that adjusts for covariables will give an overes-

timate, though not necessarily the square, of

the conditional estimate.

Having now distinguished between the two

alternative ML procedures, we are ready to

describe the ML procedure in more detail and

to give a brief overview of how statistical infer-

ences are made using ML techniques.

To describe the ML procedure, we introduce the

likelihood function,L. This is a function of the

unknown parameters in one’s model and, thus,

can alternatively be denoted asL(u), whereu

denotes the collection of unknown parameters

being estimated in the model. In matrix termi-

nology, the collectionuis referred to as avector;

its components are the individual parameters

being estimated in the model, denoted here as

y 1 ,y 2 ,upthroughyq, whereqis the number of

individual components.

For example, using theE, V, Wlogistic model

previously described and shown here again,

the unknown parameters area,b, thegis, and

thedjs. Thus, the vector of parametersuhasa,

b, thegis, and thedjs as its components.

EXAMPLE: (continued)

Assume only variables controlled are

matched

Then

dORU¼ðdORCÞ^2

""

biased correct

e.g.,

dORC¼ 3 )dORU¼ð 3 Þ^2 ¼ 9

Presentation: IV. The Likelihood Function and Its Use in the ML Procedure 111