Editing Logistic regression (section)

====Case-control sampling====
Suppose cases are rare. Then we might wish to sample them more frequently than their prevalence in the population. For example, suppose there is a disease that affects 1 person in 10,000 and to collect our data we need to do a complete physical. It may be too expensive to do thousands of physicals of healthy people in order to obtain data for only a few diseased individuals. Thus, we may evaluate more diseased individuals, perhaps all of the rare outcomes. This is also retrospective sampling, or equivalently it is called unbalanced data. As a rule of thumb, sampling controls at a rate of five times the number of cases will produce sufficient control data.<ref name="islr">https://class.stanford.edu/c4x/HumanitiesScience/StatLearning/asset/classification.pdf slide 16</ref>

Logistic regression is unique in that it may be estimated on unbalanced data, rather than randomly sampled data, and still yield correct coefficient estimates of the effects of each independent variable on the outcome.  That is to say, if we form a logistic model from such data, if the model is correct in the general population, the <math>\beta_j</math> parameters are all correct except for <math>\beta_0</math>. We can correct <math>\beta_0</math> if we know the true prevalence as follows:<ref name="islr"/>

: <math>\widehat{\beta}_0^* = \widehat{\beta}_0+\log \frac \pi {1 - \pi} - \log{ \tilde{\pi} \over {1 - \tilde{\pi}} } </math>

where <math>\pi</math> is the true prevalence and <math>\tilde{\pi}</math> is the prevalence in the sample.