actions). Therefore, to provide a more comprehensive analysis, it is very desir-
able to consider a large number of factors and sort out which are most closely
related to the dependent variable. In this section we discuss a multivariate
method for such a risk determination. This method, multiple conditional logis-
tic regression analysis, involves a linear combination of the explanatory or
independent variables; the variables must be quantitative with particular nu-
merical values for each patient. A covariate or independent variable such as a
patient characteristic may be dichotomous, polytomous, or continuous (cate-
gorical 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 trans-
formations (e.g., taking the logarithm) may be desirable to satisfy the linearity
assumption. We illustrate this process for a very general design in which
matched setið 1 aNÞhasnicases matched tomicontrols; the numbers of
casesniand controlsmivary from matched set to matched set.
Likelihood Function For the general case ofnicases matched tomicontrols in
a set, we have the conditional probability of the observed outcome (that a spe-
cific set ofnisubjects are cases) given that the number of cases isni(anyni
subjects could be cases):
exp½Pni
j¼ 1 ðbTx
jÞ
P
Rðni;miÞexp½Pni
j¼ 1 ðbTxjÞwhere the sum in the denominator ranges over the collectionsRðni;miÞof all
partitions of theniþmisubjects into two: one of sizeniand one of sizemi. The
full conditional likelihood is the product over all matched sets, one probability
for each set:
L¼
YN
i¼ 1exp½Pni
j¼ 1 ðbTx
jÞ
P
Rðni;miÞexp½Pni
j¼ 1 ðbTx
jÞwhich can be implemented using the same SAS program.
Similar to the univariate case, expðbiÞrepresents one of the following:
- The odds ratio associated with an exposure if Xi is binary (exposed
Xi¼1 versus unexposedXi¼0); or - 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
CONDITIONAL LOGISTIC REGRESSION 419