# Log-Link and Logistic Regression

• Bryan Kestenbaum
Chapter

## Abstract

The previous chapter described linear regression models, which assume the general form:
$$\mathrm{Mean}\left(\mathrm{outcome}\right)={\beta}_0+{\beta}_1\times \left(\mathrm{predictor}\ 1\right)+{\beta}_2\times \left(\mathrm{predictor}\ 2\right)+{\beta}_3\times \left(\mathrm{predictor}\ 3\right)\dots$$

By definition, linear regression models specify a linear relationship between the predictor variables in the model and the outcome variable under study. In linear regression, each one-unit difference in a predictor variable is associated with some constant difference in the mean value of the outcome variable. In many instances, the assumption of a linear relationship between two characteristics is reasonable. However, there are circumstances in which nonlinear relationships might be expected. For example, the HIV viral load, a measure of disease severity, grows exponentially over time among untreated patients, such that each week the viral load may be 10% greater than it was the previous week. This growth pattern motivates evaluation of the relative change (or percent change) in the HIV viral load.