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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 is one of several natural extensions
of the binary logit model and is also called a *multinomial
logit* model. The multinomial logit model explains the relative
risk of 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 `'interactions','on'`

name-value pair
argument in `mnrfit`

corresponds to this multinomial
model with separate intercept and slopes among categories. `mnrfit`

uses
the default logit link function for multinomial models. You cannot
specify a different link function for multinomial responses.

The multinomial logit model is

$$\begin{array}{l}\mathrm{ln}\left(\frac{{\pi}_{1}}{{\pi}_{k}}\right)={\alpha}_{1}+{\beta}_{11}{X}_{1}+{\beta}_{12}{X}_{2}+\cdots +{\beta}_{1p}{X}_{p},\\ \mathrm{ln}\left(\frac{{\pi}_{2}}{{\pi}_{k}}\right)={\alpha}_{2}+{\beta}_{21}{X}_{1}+{\beta}_{22}{X}_{2}+\cdots +{\beta}_{2p}{X}_{p},\\ \text{\hspace{1em}}\text{\hspace{0.17em}}\vdots \\ \mathrm{ln}\left(\frac{{\pi}_{k-1}}{{\pi}_{k}}\right)={\alpha}_{(k-1)}+{\beta}_{(k-1)1}{X}_{1}+{\beta}_{(k-1)2}{X}_{2}+\cdots +{\beta}_{(k-1)p}{X}_{p},\end{array}$$

where *π*_{j } =
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 `mnrfit`

chooses the
last one, *k*, as the reference category. Thus, `mnrfit`

assumes
the coefficients of the *k*th category are zero.
The total of *j* – 1 equations are solved
simultaneously to estimate the coefficients. `mnrfit`

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 *β*_{23} indicates
that the probability of the response variable being in category 2
compared to the probability of being in category *k* increases
exp(*β*_{23}) times for
each unit increase in *X*_{3},
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 *β*_{23} times
with a one-unit increase in *X*_{3},
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

$${\pi}_{j}=P\left(y=j\right)=\frac{{e}^{{\alpha}_{j}+{\displaystyle \sum _{l=1}^{p}{\beta}_{jl}{x}_{l}}}}{1+{\displaystyle \sum _{j=1}^{k-1}{e}^{{\alpha}_{j}+{\displaystyle \sum _{l=1}^{p}{\beta}_{jl}{x}_{l}}}}},\text{\hspace{1em}}j=1,\cdots ,k-1.$$

The probability of the *k*th category becomes

$${\pi}_{k}=P\left(y=k\right)=\frac{1}{1+{\displaystyle \sum _{j=1}^{k-1}{e}^{{\alpha}_{j}+{\displaystyle \sum _{l=1}^{p}{\beta}_{jl}{x}_{l}}}}},$$

which is simply equal to 1 – *π*_{1} – *π*_{2} –
... – *π*_{k–1}.

After estimating the model coefficients using `mnrfit`

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

(the default name-value pair
is `'type','category'`

). This function 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 also specify the cumulative
or conditional probabilities or numbers to estimate using the `'type'`

name-value
pair argument in `mnrval`

.

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

[2] Long, J. S. *Regression Models for Categorical
and Limited Dependent Variables*. Sage Publications, 1997.

[3] Dobson, A. J., and A. G. Barnett. *An Introduction
to Generalized Linear Models*. Chapman and Hall/CRC. Taylor
& Francis Group, 2008.

`fitglm`

| `glmfit`

| `glmval`

| `mnrfit`

| `mnrval`