Multinomial Models for Hierarchical Responses

The outcome of a response variable might sometimes be one of a restricted set of possible values. If there are only two possible outcomes, such as male and female for gender, these responses are called binary responses. If there are multiple outcomes, then they are called polytomous responses. These responses are usually qualitative rather than quantitative, such as preferred districts to live in a city, the severity level of a disease, the species for a certain flower type, and so on. Polytomous responses might also have categories which are not independent of each other. Instead the response happens in a sequential manner, or one category is nested in the previous one. These types of responses are called hierarchical, or sequential, or nested multinomial responses.

For example, if the response is the number of cigarettes a person smokes in a given day, the first level is whether the person is a smoker or not. Given that he or she is a smoker, the number of cigarettes he or she smokes can be from one to five or more than five a day. Given that it is more than 5, this person might be smoking from 6 to 10 or more than 10 cigarettes a day, and so on. The risk group at each level changes accordingly. At level one, the risk group is all of the individuals of interest (smoker or not), say m. If out of m individuals, y₁ of them are not smokers, then at level two, the risk group is the number of all smoking individuals, m – y₁. If y₂ of these m – y₁ individuals smoke from one to five cigarettes a day, then at level three, the risk group is m – y₁ – y₂. So, at each level, the number of people in that category becomes a conditional binomial observation.

A hierarchical multinomial regression model is an extension of a binary regression model based on conditional binary observations. The default is a model with separate intercepts and slopes (coefficients) among categories. In this case, the fitmnr function creates a MultinomialRegression model object with a sequence of conditional binomial models. The name-value argument IncludeClassInteractions=true in fitmnr specifies the default multinomial model. By default, fitmnr uses the logit link function to create a MultinomialRegression model object. You can specify a different link function using the Link name-value argument.

Suppose the probability that an individual is in category j given that he or she is not in the previous categories is π_j, and the cumulative probability that a response belongs to a category j or a previous category is P(y ≤ c_j). Then the hierarchical model with a logit link function and different slopes assumption is

$$\begin{array}{l}\ln \left(\frac{{\pi }_1}{1-P\left(y\le c_1\right)}\right)=\ln \left(\frac{{\pi }_1}{1-{\pi }_1}\right)={\alpha }_1+{\beta }_{11}{X}_1+{\beta }_{12}{X}_2+\cdots +{\beta }_{1p}{X}_p,\\ \ln \left(\frac{{\pi }_2}{1-P\left(y\le c_2\right)}\right)=\ln \left(\frac{{\pi }_2}{1-\left({\pi }_1+{\pi }_2\right)}\right)={\alpha }_2+{\beta }_{21}{X}_2+{\beta }_{22}{X}_2+\cdots +{\beta }_{2p}{X}_p,\\ ⋮\\ \ln \left(\frac{{\pi }_{k-1}}{1-P\left(y\le c_{k-1}\right)}\right)=\ln \left(\frac{{\pi }_{k-1}}{1-\left({\pi }_1+\cdots +{\pi }_{k-1}\right)}\right)={\alpha }_{k-1}+{\beta }_{(k-1)1}{X}_1+{\beta }_{(k-1)2}{X}_2+\cdots +{\beta }_{(k-1)p}{X}_p.\end{array}$$

For example, for a response variable with four sequential categories, there are 4 – 1 = 3 equations as follows:

$$\begin{array}{l}\ln \left(\frac{\pi {}_1}{\pi {}_2+\pi {}_3+\pi {}_4}\right)={\alpha }_1+{\beta }_{11}{X}_1+{\beta }_{12}{X}_2+\cdots +{\beta }_{1p}{X}_p,\\ \ln \left(\frac{\pi {}_2}{\pi {}_3+\pi {}_4}\right)={\alpha }_2+{\beta }_{21}{X}_1+{\beta }_{22}{X}_2+\cdots +{\beta }_{2p}{X}_p,\\ \ln \left(\frac{\pi {}_3}{\pi {}_4}\right)={\alpha }_3+{\beta }_{31}{X}_1+{\beta }_{32}{X}_2+\cdots +{\beta }_{3p}{X}_p.\end{array}$$

The coefficients β_ij are interpreted within each level. For example, for the previous smoking example, β₁₂ shows the impact of X₂ on the log odds of a person being a smoker versus a nonsmoker, provided that everything else is held constant. Alternatively, β₂₂ shows the impact of X₂ on the log odds of a person smoking one to five cigarettes versus more than five cigarettes a day, given that he or she is a smoker, provided that everything else is held constant. Similarly, β₂₃, shows the effect of X₂ on the log odds of a person smoking 6 to 10 cigarettes versus more than 10 cigarettes a day, given that he or she smokes more than 5 cigarettes a day, provided that everything else is held constant.

You can specify other link functions for hierarchical models. The 'link','probit' name-value pair argument uses the probit link function. With the separate slopes assumption, the model becomes

$$\begin{array}{l}{\Phi }^{-1}\left({\pi }_1\right)={\alpha }_1+{\beta }_{11}{X}_1+\cdots +{\beta }_{1p}{X}_p,\\ {\Phi }^{-1}\left({\pi }_2\right)={\alpha }_2+{\beta }_{21}{X}_1+\cdots +{\beta }_{2p}{X}_p,\\ ⋮⋮\\ {\Phi }^{-1}\left({\pi }_k\right)={\alpha }_k+{\beta }_{k1}{X}_1+\cdots +{\beta }_{kp}{X}_p,\end{array}$$

where π_j is the conditional probability of being in category j, given that it is not in categories previous to category j. And Φ^-1(.) is the inverse of the standard normal cumulative distribution function.

After estimating the model coefficients by using fitmnr to create a MultinomialRegression model object, you can estimate the cumulative probabilities by using predict with the name-value argument ProbabilityType="conditional". predict accepts the MultinomialRegression model object returned by fitmnr, and estimates the category labels, categorical probabilities, and confidence bounds for each categorical probability. You can specify whether predict returns category, cumulative, or conditional probabilities using the ProbabilityType name-value argument.

References

[1] McCullagh, P., and J. A. Nelder. Generalized Linear Models. New York: Chapman & Hall, 1990.

[2] Liao, T. F. Interpreting Probability Models: Logit, Probit, and Other Generalized Linear Models Series: Quantitative Applications in the Social Sciences. Sage Publications, 1994.