Logodds, the Basis of Logistic Regression

Abstract

Logistic regression is much similar to linear regression (see Chap. 8). The difference is the type of outcome variable, which is continuous with linear regression and binary with logistic regression. In order for logistic regression to work, we need to transform the binary outcome into the odds of responding, or rather the logodds of responding. In a population

$$ \begin{array}{l}\mathrm{the}\ \mathrm{odds}\ \mathrm{of}\ \mathrm{an}\ \mathrm{infarct}\mathrm{ion}\kern0.5em =\kern0.37em \frac{\mathrm{the}\ \mathrm{number}\ \mathrm{of}\ \mathrm{patients}\ \mathrm{with}\ \mathrm{infarct}}{\mathrm{number}\ \mathrm{of}\ \mathrm{patients}\ \mathrm{with}\mathrm{out}}\\ {}\end{array} $$

The easiest way to understand the term odds is to think of it as though it is the risk.

The odds or risk of an infarction is correlated with age: the older, the larger the odds. The correlation is curvilinear, but if we transform the underneath linear model

$$ \mathrm{y}\kern0.37em =\kern0.37em \mathrm{a} + \mathrm{b}\mathrm{x} $$

into a log linear model

$$ \ln\ \mathrm{odds}\kern0.37em =\kern0.37em \mathrm{a} + \mathrm{b}\ \mathrm{x}\ \left(\mathrm{x}\kern0.37em =\kern0.37em \mathrm{a}\mathrm{ge}\right), $$

then, all of a sudden, we will observe a close to linear relationship (the above right graph). This present from heaven can be used for statistical testing. The current chapter assesses how ln odds, often called logodds, can be used for testing studies with binary outcome data, like numbers of responders to treatment yes or no. We should add that logistic regression is a magnificent methodology with plenty applications.