errata8.4.Rmd

---
title: "Errata 8.4"
author: "Michael D. Ward & John S. Ahlquist"
date: "11/1/2018"
output:
  pdf_document: default
  html_document: default
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## Section 8.4

The equation describing parallel regression assumptions in section 8.4 (middle of the page) is incorrect. Rather, for the case of a logit, the partial derivative for the \textbf{log odds} that $Y$ is in any category with respect to the covariate information should be equal. We illustrate this below for a model with one covariate, $x$:
\begin{eqnarray*}
\frac{\partial}{\partial x} 
	\log
	\left(
	\frac
	{\textrm{Pr}(y \leq m | x)}
	{\textrm{Pr}(y > m | x) }
  \right) 
&=&\\ 
\frac{\partial}{\partial x} 
	\log
	\left(
	\frac
	{\textrm{Pr}(y \leq m - 1 | x)}
	{\textrm{Pr}(y > m - 1 | x) }
  \right)  
&=&\\ 
\frac{\partial}{\partial x} 
	\log
	\left(
	\frac
	{\textrm{Pr}(y \leq m - 2 | x)}
	{\textrm{Pr}(y > m - 2 | x) }
  \right) 
&=& 
\beta  
\end{eqnarray*}     

Thanks to Jeff Lewis for alerting us to this error.