---
title: "Adjacent category logit models using **brglm2**"
author: "[Ioannis Kosmidis](http://www.ikosmidis.com)"
date: "05 February 2019"
output: rmarkdown::html_vignette
bibliography: brglm2.bib
nocite: |
  @kosmidis:2019
vignette: >
  %\VignetteIndexEntry{Adjacent category logit models using brglm2}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 6,
  fig.height = 6
)
```

# **bracl**
The [**brglm2**](https://github.com/ikosmidis/brglm2) R package provides `bracl()` which is a wrapper of `brglmFit()` for fitting adjacent category models for ordinal responses using either maximum likelihood or maximum penalized likelihood or any of the various bias reduction methods described in `brglmFit()`. There is a formal equivalence between adjacent category logit models for ordinal responses and multinomial logistic regression models (see, e.g. the [Multinomial logistic regression using brglm2](https://cran.r-project.org/package=brglm2/vignettes/multinomial.html) vignette and the `brmultinom()` function). `bracl()` utilizes that equivalence and fits the corresponding Poisson log-linear model, by appropriately re-scaling the Poisson means to match the multinomial totals (a.k.a. the "Poisson trick"). The mathematical details and algorithm on using the Poisson trick for mean-bias reduction are given in @kosmidis:11.



# Citation
If you found this vignette or **brglm2**, in general, useful, please consider citing **brglm2** and the associated paper. You can find information on how to do this by typing `citation("brglm2")`.

# Opinion on stem cell research and religious fundamentalism

The `stemcell` data set ships with **brglm2**. @agresti:15[, Section 4.1] provides a detailed description of the variables recorded in this data set (see also `?stemcell`).
```{r, echo = TRUE}
library("brglm2")
data("stemcell", package = "brglm2")
stem <- within(stemcell, religion <- as.numeric(religion))
```
## Maximum likelihood estimation
The following chunk of code fits an adjacent category logit model with proportional odds and reproduces @agresti:10[, Table 4.2]. Note that the intercept parameters are different because @agresti:10[, Table 4.2] uses different contrasts for the intercept parameters.
```{r, echo = TRUE}
stem_formula <- research ~ religion + gender
stemcells_ml <- bracl(stem_formula, weights = frequency, data = stem,
                      parallel = TRUE, type = "ML")
summary(stemcells_ml)
```
`stemcells_ml` is an object inheriting from
```{r, echo = TRUE}
class(stemcells_ml)
```
**brglm2** implements `print`, `coef`, `fitted`, `predict`, `summary`, `vcov` and `logLik` methods for


We can check if a model with non-proportional odds fits the data equally well by fitting it and carrying out a likelihood ration test.
```{r, echo = TRUE}
stemcells_ml_full <- bracl(stem_formula, weights = frequency, data = stemcell,
                           parallel = FALSE, type = "ML")
summary(stemcells_ml_full)
```
The value of the log likelihood ratio statistic here is
```{r, echo = TRUE}
(lrt <- deviance(stemcells_ml) - deviance(stemcells_ml_full))
```
and has an asymptotic chi-squared distribution with
```{r, echo = TRUE}
(df1 <- df.residual(stemcells_ml) - df.residual(stemcells_ml_full))
```
The p-value from testing the hypothesis that `stemcells_ml_full` is an as good fit as `stemcells_ml` is
```{r, echo = TRUE}
pchisq(lrt, df1, lower.tail = FALSE)
```
hence, the simpler model is found to be as adequate as the full model is.

# Mean and median bias reduction
We can use `bracl()` to fit the adjacent category model using estimators with smaller mean or median bias. For mean bias reduction we do
```{r, echo = TRUE}
summary(update(stemcells_ml, type = "AS_mean"))
```
and for median
```{r, echo = TRUE}
summary(update(stemcells_ml, type = "AS_median"))
```
The estimates from mean and median bias reduction are similar to the maximum likelihood ones, indicating that estimation bias is not a major issue here.

# Prediction
We can predict the category probabilities using the `predict()` method
```{r, echo = TRUE}
predict(stemcells_ml, type = "probs")
```

# Relevant resources
`?brglmFit` and `?brglm_control` provide descriptions of the various bias reduction methods supported in **brglm2**. The [`iteration`](https://cran.r-project.org/package=brglm2/brglm2.pdf) vignette describes the iteration and gives the mathematical details for the bias-reducing adjustments to the score functions for generalized linear models.


# Citation
If you found this vignette or **brglm2**, in general, useful, please consider citing **brglm2** and the associated paper. You can find information on how to do this by typing `citation("brglm2")`.


# References
