Introductory Biostatistics

1. Fit a simple logistic linear regression model to each factor, one at a time.


2. Select the most important factor defined as the one with the largest value
   of the measure of goodness of fitC.


3. Compare this value ofCfor the factor selected in step 2 and determine,
   according to a predetermined criterion, whether or not to add this factor
   to the model—say, to see ifCb 0 :53, an increase of 0.03 or 3% over 0.5
   when no factor is considered.


4. Repeat steps 2 and 3 for those variables not yet in the model. At any
   subsequent step, if none meets the criterion in step 3—say, increase the
   separation power by 0.03, no more variables are included in the model
   and the process is terminated.

9.3 BRIEF NOTES ON THE FUNDAMENTALS

Here are a few more remarks on the use of the logistic regression model as well
as a new approach to forming one. The usual approach to regression modeling
is (1) to assume that independent variableXis fixed, not random, and (2) to
assume a functional relationship between a parameter characterizing the distri-
bution of the dependent variableYand the measured value of the independent
variable. For example:

1. In the simple (Gaussian) regression model of Chapter 8, the model
   describes themeanof that normally distributed dependent variableYas a
   function of the predictor or independent variableX,

mi¼b 0 þb 1 xi

2. In the simple logistic regression of this chapter, themodelassumes that
   the dependent variableYof interest is a dichotomous variable taking the
   value 1 with probabilitypand the value 0 with probabilityð 1
   pÞ, and
   that the relationship betweenpiand the covariate valuexiof the same
   person is described by the logistic function

pi¼

1

1 þexp½
ðb 0 þb 1 xiފ

In both cases, thexi’s are treated as fixed values.
A di¤erent approach to the same logistic regression model can be described
as follows. Assume that the independent variableXis also a random variable
following, say, a normal distribution. Then using the Bayes’ theorem of Chap-
ter 3, we can express the ratio of posterior probabilities (after data onXwere
obtained) as the ratio of prior probabilities (before data onXwere obtained)
times the likelihood ratio:

BRIEF NOTES ON THE FUNDAMENTALS 339