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

The study involves 117 persons in 39 matched

sets, or strata, each strata containing 3 per-

sons, 1 of whom is a case and the other 2 are

matched controls.

The logistic model for the above situation can

be defined as follows: logit P(X) equalsaplusb

times SMK plus the sum of 38 terms of the form

g 1 itimesV 1 i, whereV 1 is are dummy variables

for the 39 matched sets, plusg 21 times SBP plus

g 22 times ECG plus SMK times the sum ofd 1

times SBP plusd 2 times ECG.

Here, we are considering two potential con-

founders involving the two variables (SBP and

ECG) not involved in the matching and also

two interaction variables involving these same

two variables.

The odds ratio for the above logistic model is

given by the formula e to the quantitybplus the

sum ofd 1 times SBP andd 2 times ECG.

Note that this odds ratio expression involves

the coefficientsb,d 1 , andd 2 , which are coeffi-

cients of variables involving the exposure vari-

able. In particular,d 1 andd 2 are coefficients of

the interaction termsESBP andEECG.

The model we have just described is the start-

ing model for the analysis of the dataset on 117

subjects. We now address how to carry out an

analysis strategy for obtaining a final model

that includes only the most relevant of the cov-

ariates being considered initially.

The first important issue in the analysis con-

cerns the choice of estimation method for

obtaining ML estimates. Because matching is

being used, the appropriate method is condi-

tional ML estimation. Nevertheless, we also

show the results of unconditional ML estima-

tion to illustrate the type of bias that can result

from using the wrong estimation method.

The next issue to be considered is the assess-

ment of interaction. Based on our starting

model, we, therefore, determine whether or

not either or both of the product terms SMK

SBP and SMKECG are retained in the model.

EXAMPLE (continued)

n¼117 (39 matched sets)

The model:

logit P(X) = a + bSMK + Σ g 1 iV 1 i

38

i= 1

= g 21 SBP + g 22 ECG

+ SMK (d 1 SBP + d 2 ECG)

modifiers

confounders

ROR¼expðbþd 1 SBPþd 2 ECGÞ

b¼coefficient ofE
d 1 ¼coefficient ofESBP
d 2 ¼coefficient ofEECG

Starting model

analysis strategy

Final model

Estimation method:
üConditional ML estimation
(also, we illustrate unconditional
ML estimation)

Interaction:
SMKSBP and SMKECG?

Presentation: V. An Application 401