## Contents

## Multinomial LogitEdit

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

## Conditional LogitEdit

`clogit()`in the**survival**package**mclogit**package.

## Multinomial ProbitEdit

**mprobit**package^{[1]}**MNP**package to fit a multinomial probit.

## Multinomial ordered logit modelEdit

We consider a multinomial ordered logit model with unkwnown 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 simultanously 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 estimationEdit

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 estimationEdit

`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 modelEdit

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 estimationEdit

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 estimationEdit

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

## Rank Ordered Logit ModelEdit

This model was introduced in econometrics by Beggs, Cardell and Hausman in 1981.^{[2]}^{[3]} One application is the Combes et alii paper explaining the ranking of candidates to become professor.^{[3]} Is is also known as Plackett–Luce model in biomedical literature or as exploded logit model in marketing.^{[3]}

## Conditionally Ordered Hierarchical ProbitEdit

- The Conditionally Ordered Hierarchical Probit can be estimated using the
**anchors**package developped by Gary King and his coauthors^{[4]}.

## ReferencesEdit

- ↑ Harry Joe, Laing Wei Chou and Hongbin Zhang (2006). mprobit: Multivariate probit model for binary/ordinal response. R package version 0.9-2.
- ↑ Beggs, S; Cardell, S; Hausman, J (1981). "Assessing the potential demand for electric cars".
*Journal of Econometrics***17**: 1–19. doi:10.1016/0304-4076(81)90056-7. - ↑
^{a}^{b}^{c}Combes, Pierre-Philippe; Linnemer, Laurent; Visser, Michael (2008). "Publish or peer-rich? The role of skills and networks in hiring economics professors".*Labour Economics***15**(3): 423–41. doi:10.1016/j.labeco.2007.04.003. - ↑ Jonathan Wand, Gary King, Olivia Lau (2009). anchors: Software for Anchoring Vignette Data. Journal of Statistical Software, Forthcoming. URL http://www.jstatsoft.org/.