Multinomial Models for Nominal Responses
The outcome of a response variable might be one of a restricted set of possible values. If there are only two possible outcomes, such as a yes or no answer to a question, these responses are called binary responses. If there are multiple outcomes, then they are called polytomous responses. Some examples include the degree of a disease (mild, medium, severe), preferred districts to live in a city, and so on. When the response variable is nominal, there is no natural order among the response variable categories. Nominal response models explain and predict the probability that an observation is in each category of a categorical response variable.
A nominal response model, also called a multinomial logit model, is one
of several natural extensions of the binary logit model. A multinomial logit model
explains the relative risk of an observation being in one category versus being in the
reference category k, using a linear combination of predictor
variables. Consequently, the probability of each outcome is expressed as a nonlinear
function of p predictor variables. The
function specifies this multinomial model for nominal responses. By default,
fitmnr specifies separate intercepts and slopes among
categories, and uses the
logit link function to create a
MultinomialRegression model object. You cannot specify a different link
function for nominal responses.
The multinomial logit model is
= P(y = j) is the probability of an
outcome being in category j, k is the number of
response categories, and p is the number of predictor variables.
Theoretically, any category can be the reference category, but
fitmnr selects the final category k as the
reference category. Therefore,
fitmnr assumes that the coefficients
of the kth category are zero. The total of j – 1
equations are solved simultaneously to estimate the coefficients.
fitmnr uses the iteratively weighted least-squares algorithm to
find the maximum likelihood estimates.
The coefficients in the model express the effects of the predictor variables on the relative risk or the log odds of being in category j versus the reference category, here k. For example, the coefficient β23 indicates that the probability of the response variable being in category 2 compared to the probability of being in category k increases exp(β23) times for each unit increase in X3, given all else is held constant. Or it indicates that the relative log odds of the response variable being category 2 versus in category k increases β23 times with a one-unit increase in X3, given all else equal.
Based on the nominal response model, and the assumption that the coefficients for the last category are zero, the probability of being in each category is
The probability of the kth category becomes
which is simply equal to 1 – π1 – π2 – ... – πk–1.
After estimating the model coefficients by using
fitmnr to create
MultinomialRegression model object, you can estimate the category
probabilities by using
This function accepts the
MultinomialRegression model object returned
fitmnr, and estimates the category labels, categorical
probabilities, and confidence bounds for each categorical probability. You can specify
predict returns category, cumulative, or conditional
probabilities using the
ProbabilityType name-value argument.
 McCullagh, P., and J. A. Nelder. Generalized Linear Models. New York: Chapman & Hall, 1990.
 Long, J. S. Regression Models for Categorical and Limited Dependent Variables. Sage Publications, 1997.
 Dobson, A. J., and A. G. Barnett. An Introduction to Generalized Linear Models. Chapman and Hall/CRC. Taylor & Francis Group, 2008.