Introductory Biostatistics

sured from patients; findings may lead to important decisions in patient man-
agement (or public health interventions in other examples). In epidemiological
studies, such e¤ects are usually measured by therelative risk orodds ratio;
when the logistic model is used, the measure is the odds ratio.
For the case of the logistic regression model, the logistic function for the
probabilitypican also be expressed as a linear model in the log scale (of the
odds):

ln

pi

1 
pi

¼b 0 þb 1 xi

We first consider the case of a binary covariate with the conventional coding:

Xi¼

0 if the patient is not exposed

1 if the patient is exposed



Here, the term exposedmay refer to a risk factor such as smoking, or a
patient’s characteristic such as race (white/nonwhite) or gender (male/female).
It can be seen that from the log-linear form of the logistic regression model,

lnðodds;nonexposedÞ¼b 0

lnðodds;exposedÞ¼b 0 þb 1

So that after exponentiating, the di¤erence leads to

eb^1 ¼

ðodds;exposedÞ

ðodds;nonexposedÞ

represents the odds ratio (OR) associated with the exposure, exposed versus
nonexposed. In other words, the primary regression coe‰cientb 1 is the value of
the odds ratio on the log scale.
Similarly, we have for a continuous covariateXand any valuexofX,

lnðodds;X¼xÞ¼b 0 þb 1 ðxÞ

lnðodds;X¼xþ 1 Þ¼b 0 þb 1 ðxþ 1 Þ

So that after exponentiating, the di¤erence leads to

eb^1 ¼

ðodds;X¼xþ 1 Þ

ðodds;X¼xÞ

represents the odds ratio (OR) associated with a 1-unit increase in the value of
X,X¼xþ1 versusX¼x. For example, a systolic blood pressure of 114
mmHg versus 113 mmHg. For an m-unit increase in the value ofX, say
X¼xþmversusX¼x, the corresponding odds ratio isemb^1.

SIMPLE REGRESSION ANALYSIS 319