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Let us suppose that the response categories 1, 2, . . . , r are ordered, (e.g., nine responses in cheese data). Rather than considering the probability of each category versus a baseline, it now makes sense to consider the probability of

outcome 1 versus 2,
outcome 2 versus 3,
outcome 3 versus 4,
...
outcome r − 1 versus r.

This comparison of adjacent-categories will make more sense for the mortality data example. For the mortality data, consider the logits of "alive vs. dead",  "cancer death vs. non-cancer death", etc.

The adjacent-category logits are defined as:

\begin{array}{rcl}
L_1 &=& \text{log} \left(\dfrac{\pi_1}{\pi_2}\right)\\
L_2 &=& \text{log} \left(\dfrac{\pi_2}{\pi_3}\right)\\
& \vdots &  \\
L_{r-1} &=& \text{log} \left(\dfrac{\pi_{r-1}}{\pi_r}\right)
\end{array}

This is similar to a baseline-category logit model, but the baseline changes from one category to the next. Suppose we introduce covariates to the model:

\begin{array}{rcl}
L_1 &=& \beta_{10}+\beta_{11}X_1+\cdots+\beta_{1p}X_p\\
L_2 &=& \beta_{20}+\beta_{21}X_1+\cdots+\beta_{2p}X_p\\
& \vdots &  \\
L_{r-1} &=& \beta_{r-1,0}+\beta_{r-1,1}X_1+\cdots+\beta_{r-1,p}X_p\\
\end{array}

It is easy to see that the β-coefficients from this model are linear transformations of the β's from the baseline-category model. To see this, suppose that we create a model in which category 1 is the baseline.

Then

\begin{array}{rcl}
\text{log} \left(\dfrac{\pi_2}{\pi_1}\right)&=& -L_1,\\
\text{log} \left(\dfrac{\pi_3}{\pi_1}\right)&=& -L_2-L_1,\\
& \vdots &  \\
\text{log} \left(\dfrac{\pi_r}{\pi_1}\right)&=& -L_{r-1}-\cdots-L_2-L_1
\end{array}

Without further structure, the adjacent-category model is just a reparametrization of the baseline-category model. But now, let's suppose that the effect of a covariate in each of the adjacent-category equations is the same:

\begin{array}{rcl}
L_1 &=& \alpha_1+\beta_1X_1+\cdots+\beta_p X_p\\
L_2 &=& \alpha_2+\beta_1X_1+\cdots+\beta_p X_p\\
& \vdots &  \\
L_{r-1} &=& \alpha_{r-1}+\beta_1X_1+\cdots+\beta_p X_p
\end{array}

What does this model mean? Let us consider the interpretation of β1, the coefficient for X1. Suppose that we hold all the other X's constant and change the value of X1. Think about the 2 × r table that shows the probabilities for the outcomes 1, 2, . . . , r at a given value of X1 = x, and at a new value X1 = x + 1:

The relationship between X1 and the response, holding all the other X-variables constant, can be described by a set of r − 1 odds ratios for each pair of adjacent response categories. The adjacent-category logit model says that each of these adjacent-category odds ratios is equal to exp(β1). That is, β1 is the change in the log-odds of falling into category j + 1 versus category j when X1 increases by one unit, holding all the other X-variables constant.

For example, for the cheese data, j + 1 = 7, and j = 6, for x = cheese A, and x + 1 = cheese B, log-odds ratio = (19 × 6 / 8 × 1)

This adjacent-category logit model can be fit using software for Poisson loglinear regression using a specially coded design matrix, or a log-linear model when all data are categorical. In this model, the association between the r-category response variable and X1 would be included by

• the product of r − 1 dummy indicators for the response variable with
• a linear contrast for X1.

This model can be fit in SAS using PROC CATMOD or PROC GENMOD and in R using the vgam() package, for example. However, we will not discuss this model further, because it is not nearly as popular as the proportional-odds cumulative-logit model, for an ordinal response, which we discuss next.