Logistic Regression

Logistic regression is a statistical technique for analyzing the relationship of an outcome or dependent variable to one or more predictors or independent variables when the dependent variable is (1) dichotomous, having only two categories, for example, the presence or absence of symptoms, or the use or nonuse of tobacco (2) unordered polytomous, a nominal scale variable with three or more categories, for example, type of contraception (none, pill, condom, intrauterine device) used in response to services provided by a family planning clinic or (3) ordered polytomous, an ordinal scale variable with three or more categories, for example, whether a patient's condition deteriorates, remains the same, or improves in response to a cancer treatment. Here, the basic logistic regression model for dichotomous outcomes is examined, noting its extension to polytomous outcomes and its conceptual roots in both log-linear analysis and the general linear model. Next, consideration is given to methods for assessing the goodness of fit and predictive utility of the overall model, and calculation [Page 611]and interpretation of logistic regression coefficients and associated inferential statistics to evaluate the importance of individual predictors in the model. Throughout, the discussion assumes an interest in prediction, regardless of whether causality is implied; hence the language of “outcomes” and “predictors” is preferred to the language of “dependent” and “independent” variables.

The equation for the logistic regression model with a dichotomous outcome is logit(Y) = a + b1X1 + b2X2 + + bKXK, where Y is the dichotomous outcome; logit(Y) is the natural logarithm of the odds of Y, a transformation of Y to be discussed in more detail momentarily; and there are k = 1, 2, … K predictors Xk with associated coefficients bk, plusaconstant or intercept, a, which represents the value of logit(Y)whenallthe Xk are equal to 0. If the two categories of the outcome are coded 1 and 0, respectively, and P1 is the probability of being in the category coded as 1, and P0 is the probability of being in the category coded as 0, then the odds of being in Category 1 is P1=P0 = P1=(1 − P1)(sincetheprobability of being in one category is one minus the probability of being in the other category). Logit(Y)isthe natural logarithm of the odds, ln[P1=(1 − P1)], where ln represents the natural logarithm transformation.