Analysis of dependent nominal variable Y on independent variables X

Nominal logistic regression

Logistic regression analysis studies the association between a categorical dependent variable and a set of independent (explanatory) variables. The name logistic regression is often used when the dependent variable has only two values. The name multiple-group logistic regression (MGLR) is usually reserved for the case when the dependent variable has three or more unique values. Multiple-group logistic regression is sometimes called multinomial, polytomous, polychotomous, or nominal logistic regression. Although the data structure is different from that of multiple regression, the practical use of the procedure is similar.

Logistic regression competes with discriminant analysis as a method for analyzing discrete response variables. In fact, the current feeling among many statisticians is that logistic regression is more versatile and better suited for most situations than is discriminant analysis because it does not assume that the independent variables are normally distributed, as discriminant analysis does.

If discrete dependent variable Y has G unique values (G ≥ 2), The logistic regression model is given by the G equations:

where:
p_g - the probability that an individual with values X₁, X₂, X_p is in group g.
P_g - the prior probabilities of group membership
β_gi - population regression coefficients that are to be estimated from the data

Group one is called the reference group. The regression coefficients β for the reference group are set to zero. The choice of the reference group is arbitrary. Usually, it is the largest group or a control group to which the other groups are to be compared. This leaves G-1 logistic regression equations in the multinomial logistic model.

If Y has three unique values: A, B, C, where C is reference group, logistic regression model consists of two equations: