The primary distinction between the linear regression and logistic procedures is that the latter is intended for the modeling of dichotomous categorical outcomes (e.g., dead vs. alive, present vs. absent or yes vs. no). Logistic and linear regression are both based on many of the same assumptions and theory. However the outcome is dichotomous and predicting unit change has little or no meaning. For this reason logistic regression focuses instead upon the relative probability (odds) of obtaining a given result category. The natural logarithm of the odds is linear across most of its range, and, therefore, we can continue using many of the methods developed for linear models. We can express logistic regression as following:
Ln [ p / (1-p) ] = b0 + b1*X1 + b2*X2 + b3*X3 . . . bk*Xk + e
where p represents the probability of an event (e.g., present), b0 is the y-intercept, and X1 to Xk represent the independent variables included in the model. As with the linear model, each independent variable’s association with the outcome (log odds) is indicated by the coefficients b1 to bk. Again, an error term is included to account for differences between the observed outcome values and those predicted by the model. In effect, we are trying to model the probability that an event is a result of a linear combination of variables as indicated in the equation above.
Typical outcome parameters of the logistic model are shown in the figure below.