Introductory Biostatistics

actions). Therefore, to provide a more comprehensive analysis, it is very desir-
able to consider a large number of factors and sort out which ones are most
closely related to the dependent variable. In this section we discuss a multi-
variate method for risk determination. This method, which is multiple logistic
regression analysis, involves a linear combination of the explanatory or inde-
pendent variables; the variables must be quantitative with particular numerical
values for each patient. A covariate or independent variable, such as a patient
characteristic, may be dichotomous, polytomous, or continuous (categorical
factors will be represented by dummy variables). Examples of dichotomous
covariates are gender and presence/absence of certain comorbidity. Polytomous
covariates include race and di¤erent grades of symptoms; these can be covered
by the use ofdummy variables. Continuous covariates include patient age and
blood pressure. In many cases, data transformations (e.g., taking the loga-
rithm) may be desirable to satisfy the linearity assumption.

9.2.1 Logistic Regression Model with Several Covariates

Suppose that we want to considerkcovariates simultaneously; the simple
logistic model of Section 9.1 can easily be generalized and expressed as

pi¼

1

1 þexp½
ðb 0 þ

Pk

j¼ 1 bjxjiފ

i¼ 1 ; 2 ;...;n

or, equivalently,

ln

pi

1 
pi

¼b 0 þ

Xk

j¼ 1

bjxji

This leads to the likelihood function

L¼

Yn

i¼ 1

½expðb 0 þ

Pk

i¼ 1 bjxjiފ

yi

1 þexpðb 0 þ

Pk

j¼ 1 bjxjiÞ

yi¼ 0 ; 1

from which parameters can be estimated iteratively using a computer-packaged
program such as SAS.
Also similar to the univariate case, expðbiÞrepresents one of the following:

1. The odds ratio associated with an exposure if Xi is binary (exposed
   Xi¼1 versus unexposedXi¼0); or


2. The odds ratio due to a 1-unit increase ifXiis continuous (Xi¼xþ 1
   versusXi¼x).

Afterbb^iand its standard error have been obtained, a 95% confidence interval
for the odds ratio above is given by

326 LOGISTIC REGRESSION