Let us suppose that the response categories \(1, 2,\ldots , r\) are ordered, (e.g., nine responses in ice cream rating). 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 &=& \log \left(\dfrac{\pi_1}{\pi_2}\right)\\
L_2 &=& \log \left(\dfrac{\pi_2}{\pi_3}\right)\\
& \vdots & \\
L_{r-1} &=& \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 \(\beta\)-coefficients from this model are linear transformations of the \(\beta\)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}
\log \left(\dfrac{\pi_2}{\pi_1}\right)&=& -L_1,\\
\log \left(\dfrac{\pi_3}{\pi_1}\right)&=& -L_2-L_1,\\
& \vdots & \\
\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 \(\beta_1\), the coefficient for \(x_1\). Suppose that we hold all the other \(x\)'s constant and change the value of \(x_1\). Think about the \(2\times r\) table that shows the probabilities for the outcomes \(1, 2, \ldots , r\) at a given value of \(x_1=x\) and at a new value \(x_1=x+1\):
The relationship between \(x_1\) 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 \(\exp(\beta_1)\). That is, \(\beta_1\) is the change in the log-odds of falling into category \(j + 1\) versus category \(j\) when \(x_1\) increases by one unit, holding all the other \(x\)-variables constant.
This adjacent-category logit model can be fit using software for Poisson log-linear 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 \(x_1\) would be included by
- the product of \(r - 1\) indicators for the response variable with
- a linear contrast for \(x_1\).
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.