For an \(I \times J\) contingency table with ordinal column variable \(y(=1, \ldots, J)\) and ordinal row variable \(x(=1, \ldots, I)\), consider the model

\[\operatorname{logit}[\operatorname{Pr}(y \leq j \mid x)]=\alpha_{j}+\beta x, j=1, \ldots, J-1\]

(a) Show that logit \([\operatorname{Pr}(y \leq j \mid x=i+1)]-\operatorname{logit}[\operatorname{Pr}(y \leq j \mid x=i)]=\beta\).

(b) Show that this difference in logit is log of the odds ratio (cumulative odds ratio) for the \(2 \times 2\) contingency table consisting of rows \(i\) and \(i+1\) and the binary response having cut-point following category \(j\).

(c) Show that independence of \(x\) and \(y\) is the special case when \(\beta=0\).

(d) A generalization of the model replaces \(\beta x\) by unordered parameters \(\left\{\mu_{i}\right\}_{i=1}^{I-1}\), i.e., treat \(x\) as nominal and consider the model \[\operatorname{logit}[\operatorname{Pr}(y \leq j \mid x=i)]=\alpha_{j}+\mu_{i}, i=1, \ldots, I-1\]
For \(i\) th and \(I\) th rows, show that the log cumulative odds ratio equals \(\mu_{i}\) for all \(J-1\) cut-points.