# R Programming/Multinomial Models

## Multinomial Logit

• mlogit package.
• mnlogit package
• Bayesm package
• multinom() nnet
• multinomial(), which is used by vglm() VGAM

## Conditional Logit

• clogit() in the survival package
• mclogit package.

## Multinomial Probit

• mprobit package 
• MNP package to fit a multinomial probit.

## Multinomial ordered logit model

We consider a multinomial ordered logit model with unknown thresholds. First, we simulate fake data. We draw the residuals in a logistic distribution. Then we draw some explanatory variable x and we define ys the latent variable as a linear function of x. Note that we set the constant to 0 because the constant and the thresholds cannot be identified simultaneously in this model. So we need to fix one of the parameters. Then, we define thresholds (-1,0,1) and we define our observed variable y using the `cut()` function. So y is an ordered multinomial variable.

```N <- 10000
u <- rlogis(N)
x <- rnorm(N)
ys <- x + u
mu <- c(-Inf,-1,0,1, Inf)
y <- cut(ys, mu)
plot(y,ys)
df <- data.frame(y,x)
```

### Maximum likelihood estimation

This model can be estimated by maximum likelihood using the `polr()` function in the MASS package. Since it is not possible to achieve identification of the constant and the thresholds, R assumes by default that the constant is equal to 0.

```library(MASS)
fit <- polr(y  ~ x, method = "logistic", data = df)
summary(fit)
```

### Bayesian estimation

• bayespolr() (arm) performs a bayesian estimation of the multinomial ordered logit
```library("arm")
fit <- bayespolr(y ~ x, method = "logistic", data = df)
summary(fit)
```

## Multinomial ordered probit model

We generate fake data by drawing an error term in normal distribution and cutting the latent variables in 4 categories.

```N <- 1000
u <- rnorm(N)
x <- rnorm(N)
ys <- x + u
mu <- c(-Inf,-1,0,1, Inf)
y <- cut(ys, mu)
plot(y,ys)
df <- data.frame(x,y)
```

### Maximum likelihood estimation

The model can be fitted using maximum likelihood method. This can be done using the `polr()` function in the MASS package with the `probit` method.

```library(MASS)
fit <- polr(y  ~ x, method = "probit", data = df)
summary(fit)
```

### Bayesian estimation

• bayespolr() (arm) performs a bayesian estimation of the multinomial ordered probit

## Rank Ordered Logit Model

This model was introduced in econometrics by Beggs, Cardell and Hausman in 1981. One application is the Combes et alii paper explaining the ranking of candidates to become professor. Is is also known as Plackett–Luce model in biomedical literature or as exploded logit model in marketing.

## Conditionally Ordered Hierarchical Probit

• The Conditionally Ordered Hierarchical Probit can be estimated using the anchors package developped by Gary King and his coauthors.