Three Logit Structures for One Category Table

One Table, Three Questions

For an ordered response, the same set of category probabilities can be summarized through different odds comparisons. A baseline-category logit compares each category with a chosen reference category. An adjacent-category logit compares neighboring categories. A forward continuation-ratio logit compares stopping at a category with proceeding to a higher one. These choices do not change the observed table; they change the conditional or relative question represented by each coefficient.

We will use one two-group table throughout. Its four response categories are ordered from lowest to highest. The group label is \(A\) or \(B\); it can be treated as a binary predictor, without implying that group membership was randomized or caused any difference.

Write \(p_{gj}=P(Y=j\mid G=g)\), where \(g\) denotes a group and \(j\) a category. The observed proportions are \((0.12,0.18,0.30,0.40)\) in group \(A\), and \((0.24,0.24,0.30,0.22)\) in group \(B\). In the calculations below, all three model forms are fitted with a separate parameter for each group and logit. This is a saturated model for these two group-specific category distributions.

Definitions and Probability Reconstructions

The direction in the continuation-ratio definition matters. Some conventions instead put the continuation event in the numerator. We use the stated “stop versus proceed” convention throughout, so a positive logit means greater odds of category \(j\) rather than a higher category at that step.

Here are the reconstructions. For baseline logits \(\eta_j=\log(p_j/p_K)\), let \(r_j=e^{\eta_j}\). Then \(p_K=1/(1+\sum_{j=1}^{K-1}r_j)\), and \(p_j=r_jp_K\). For adjacent logits \(\alpha_j=\log(p_j/p_{j+1})\), set \(w_K=1\) and \(w_j=\prod_{\ell=j}^{K-1}e^{\alpha_\ell}\). Then \(p_j=w_j/\sum_{r=1}^{K}w_r\). The product works because \(p_j/p_K\) is the product of the intervening adjacent ratios.

For forward continuation logits \(\gamma_j\), put \(h_j=e^{\gamma_j}=p_j/S_{j+1}\), where \(S_j=\sum_{r=j}^Kp_r\). Since \(S_j=p_j+S_{j+1}=(1+h_j)S_{j+1}\), the tail probabilities are recovered successively, starting with \(S_1=1\). In particular,

Each reconstruction uses positive weights and normalizes them to sum to one. Conversely, applying the corresponding logit definition to the reconstructed probabilities returns the original logits. This proves both existence and uniqueness for each parameterization. \(\square\)

Why the Saturated Fits Give the Same Probabilities

The logit values just calculated are transformations of the proportions. To connect them to model fitting, consider a group with counts \(n_1,\ldots,n_K\), total \(n=\sum_j n_j\), and probabilities \(p_1,\ldots,p_K\). Its multinomial likelihood, apart from a factor not involving the probabilities, is \(\prod_jp_j^{n_j}\). When every category count is positive, the maximum-likelihood probabilities are \(n_j/n\).

To prove the first claim, maximize \(\ell(p)=\sum_j n_j\log p_j\) subject to \(\sum_jp_j=1\) and \(p_j>0\). The Lagrange equations are \(n_j/p_j=\lambda\) for every \(j\). Thus \(p_j=n_j/\lambda\); summing over \(j\) gives \(\lambda=n\), so \(p_j=n_j/n\). The log-likelihood is strictly concave on the positive simplex because its second derivative in each coordinate is \(-n_j/p_j^2<0\). The stationary point is therefore the unique maximum. For the second claim, the probability reconstructions above show that each unrestricted logit set represents every strictly positive probability vector. Each model has the same probability space and likelihood, so each attains that same maximum. \(\square\)

The positivity condition avoids boundary cases in this proof. If a cell count is zero, the multinomial maximum can lie on the boundary, with a fitted probability of zero; the finite-logit reconstructions above cover strictly positive probabilities, not such boundary points. The table here has no zero counts.

Different Coefficients, Different Comparisons

Although each parameterization gives the same fitted proportions in this saturated example, its group coefficients answer a different question. For a binary group indicator, define a logit in group \(B\) minus the corresponding logit in group \(A\). This difference is the coefficient of the group indicator in a model with a separate intercept and group effect for that logit. It is a contrast in odds, not a direct difference in category probabilities.

The values need not agree across structures, even though the group-specific fitted probabilities do. For example, the first group contrast is \(1.2910\) under the baseline comparison, \(0.4055\) under the adjacent comparison, and \(0.8390\) under the continuation comparison. They are not competing estimates of one common odds ratio: their denominators describe different outcomes.

When the Equality of Fits Does Not Apply

The shared fitted probabilities here depend on fitting a saturated distribution separately in each group. In regression, one often imposes restrictions—for example, a common slope across several covariate values or a particular pattern among category-specific effects. Such restrictions generally define different sets of allowable probability distributions for the three logit structures. Their fitted probabilities can then differ. The result above is not a claim that different restricted regression models always fit identically.

The structures are useful when their comparisons match the scientific question. A baseline logit is appropriate when a particular category is a meaningful reference. Adjacent logits focus on local comparisons between neighboring outcomes. A forward continuation ratio is natural when the response process can be described as a sequence of opportunities to stop or proceed. These interpretations should be based on the response mechanism, not selected merely because one coefficient pattern is convenient.

Common Mistake

A common error is to see the same fitted category probabilities and conclude that all three models have identical coefficients or scientific interpretations. They do not. In the table, category 1 versus category 4 yields a group contrast of \(1.2910\), while category 1 versus category 2 yields \(0.4055\); the forward comparison of category 1 versus all higher categories yields \(0.8390\). The correction is to name the numerator and denominator for every odds contrast before interpreting its coefficient.

A related error is to read a positive logit coefficient as meaning that the probability of a particular category must increase. A positive group effect says that the specified odds increase. Because category probabilities must sum to one, their changes depend on the full probability vector and on the other category contrasts.