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, *y*_{1} of
them are not smokers, then at level two, the risk group is the number
of all smoking individuals, *m* – *y*_{1}.
If *y*_{2} of these *m* – *y*_{1} individuals
smoke from one to five cigarettes a day, then at level three, the
risk group is *m* – *y*_{1} – *y*_{2}.
So, at each level, the number of people in that category becomes a
conditional binomial observation.

The hierarchical multinomial regression models are extensions
of binary regression models based on conditional binary observations.
The default is a model with different intercept and slopes (coefficients)
among categories, in which case `mnrfit`

fits a
sequence of conditional binomial models. The `'interactions','on'`

name-value
pair specifies this in `mnrfit`

. The default link
function is logit and the `'link','logit'`

name-value
pair specifies this model in `mnrfit`

.

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

$$\begin{array}{l}\mathrm{ln}\left(\frac{{\pi}_{1}}{1-P\left(y\le {c}_{1}\right)}\right)=\mathrm{ln}\left(\frac{{\pi}_{1}}{1-{\pi}_{1}}\right)={\alpha}_{1}+{\beta}_{11}{X}_{1}+{\beta}_{12}{X}_{2}+\cdots +{\beta}_{1p}{X}_{p},\\ \mathrm{ln}\left(\frac{{\pi}_{2}}{1-P\left(y\le {c}_{2}\right)}\right)=\mathrm{ln}\left(\frac{{\pi}_{2}}{1-\left({\pi}_{1}+{\pi}_{2}\right)}\right)={\alpha}_{2}+{\beta}_{21}{X}_{2}+{\beta}_{22}{X}_{2}+\cdots +{\beta}_{2p}{X}_{p},\\ \text{\hspace{1em}}\text{\hspace{1em}}\vdots \\ \mathrm{ln}\left(\frac{{\pi}_{k-1}}{1-P\left(y\le {c}_{k-1}\right)}\right)=\mathrm{ln}\left(\frac{{\pi}_{k-1}}{1-\left({\pi}_{1}+\cdots +{\pi}_{k-1}\right)}\right)={\alpha}_{k-1}+{\beta}_{(k-1)1}{X}_{1}+{\beta}_{(k-1)2}{X}_{2}+\cdots +{\beta}_{(k-1)p}{X}_{p}.\end{array}$$

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

$$\begin{array}{l}\mathrm{ln}\left(\frac{\pi {}_{1}}{\pi {}_{2}+\pi {}_{3}+\pi {}_{4}}\right)={\alpha}_{1}+{\beta}_{11}{X}_{1}+{\beta}_{12}{X}_{2}+\cdots +{\beta}_{1p}{X}_{p},\\ \mathrm{ln}\left(\frac{\pi {}_{2}}{\pi {}_{3}+\pi {}_{4}}\right)={\alpha}_{2}+{\beta}_{21}{X}_{1}+{\beta}_{22}{X}_{2}+\cdots +{\beta}_{2p}{X}_{p},\\ \mathrm{ln}\left(\frac{\pi {}_{3}}{\pi {}_{4}}\right)={\alpha}_{3}+{\beta}_{31}{X}_{1}+{\beta}_{32}{X}_{2}+\cdots +{\beta}_{3p}{X}_{p}.\end{array}$$

The coefficients *β*_{ij} are
interpreted within each level. For example, for the previous smoking
example, *β*_{12} shows
the impact of *X*_{2} on the
log odds of a person being a smoker versus a nonsmoker, provided that
everything else is held constant. Alternatively, *β*_{22} shows
the impact of *X*_{2} 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, *β*_{23},
shows the effect of *X*_{2} 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

$$\begin{array}{l}{\Phi}^{-1}\left({\pi}_{1}\right)={\alpha}_{1}+{\beta}_{11}{X}_{1}+\cdots +{\beta}_{1p}{X}_{p},\text{\hspace{1em}}\\ {\Phi}^{-1}\left({\pi}_{2}\right)={\alpha}_{2}+{\beta}_{21}{X}_{1}+\cdots +{\beta}_{2p}{X}_{p},\\ \text{\hspace{1em}}\text{\hspace{1em}}\vdots \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\vdots \\ {\Phi}^{-1}\left({\pi}_{k}\right)={\alpha}_{k}+{\beta}_{k1}{X}_{1}+\cdots +{\beta}_{kp}{X}_{p},\end{array}$$

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 using `mnrfit`

,
you can estimate the cumulative probabilities or the cumulative number
in each category using `mnrval`

with the `'type','conditional'`

name-value
pair argument. The function `mnrval`

accepts the
coefficient estimates and the model statistics `mnrfit`

returns,
and estimates the categorical probabilities or the number in each
category and their confidence bounds. You can specify which category
or cumulative probabilities or numbers to estimate by changing the
value of the `'type'`

name-value pair argument in `mnrval`

.

[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.

`fitglm`

| `glmfit`

| `glmval`

| `mnrfit`

| `mnrval`