Selecting a Logit Structure from the Response Mechanism

Start With the Response Mechanism

A multinomial response can be described through several sets of logits. The choice is not merely a matter of which formula is easiest to fit: each structure emphasizes a different comparison. A useful choice begins by asking what the categories mean, which comparisons answer the scientific question, and whether responses arise through a sequence of stages.

Let \(Y\) take one of \(K\) categories, with probabilities \(p_j=P(Y=j)\), where \(p_j>0\) and \(\sum_{j=1}^K p_j=1\). If the outcome is modeled conditional on covariates \(x\), these probabilities may depend on \(x\); the comparisons below then apply at a fixed value of \(x\). The three structures considered here are baseline-category logits, adjacent-category logits, and continuation-ratio logits.

These are different sets of contrasts. Baseline-category logits need a designated reference but do not require an order. Adjacent-category logits use the immediate neighbors in a specified ordering. Continuation-ratio logits use an ordering and treat each category as a possible stage at which the response process stops or proceeds.

What Scientific Question Does Each Structure Answer?

For a nominal outcome, categories are distinct labels without a scientifically meaningful order. If the question is how each outcome compares with a particular control, usual-care, or reference outcome, the baseline-category structure directly encodes that question. The reference is a choice of comparison, not a claim that the other categories form a sequence around it.

An ordering alone does not automatically identify the right model. If neighboring categories are meaningfully distinct and the question concerns the local change from one level to the next, adjacent-category logits may be appropriate. For example, a researcher might ask how the odds of moderate rather than mild disease differ from the odds of severe rather than moderate disease. These are separate local comparisons.

A continuation-ratio structure is appropriate when the ordering also reflects a sequential process. At each stage, some responses are in the current category and the others continue to a later category. The relevant comparison is therefore the probability of stopping at stage \(j\) against proceeding beyond it, conditional on having reached the decision at \(j\). This can describe educational progression, treatment escalation, or a sequence of eligibility decisions, provided that such a process is scientifically defensible.

In short, ask whether the outcome is unordered, ordered with important neighboring contrasts, or ordered because responses pass through sequential decision stages. These are different properties: an ordered scale does not by itself mean that a sequential process generated the responses.

Worked Examples: Matching Contrasts to Questions

Why One Category Table Cannot Decide the Structure

A single vector of positive category probabilities can be expressed through all three sets of logits. For a fixed table, each structure is simply a different way of describing the same probabilities. A fitted table alone therefore cannot establish which response mechanism is scientifically appropriate. The distinction matters when a model relates logits to covariates, because the structures impose different patterns on those covariate effects and answer different questions.

Proof. Because every \(p_j>0\), both \(p_j\) and \(\sum_{r=j+1}^K p_r\) are positive for \(j<K\). Thus \(0<h_j<1\). Write \(S_j=\sum_{r=j}^K p_r\). Then \(h_j=p_j/S_j\), so

Since \(S_1=1\), repeated substitution gives \(S_j=\prod_{m=1}^{j-1}(1-h_m)\), with an empty product equal to \(1\). Consequently,

These equations reconstruct every category probability from the continuation probabilities, so the representation is unique. They also show why a positive category table can always be represented by forward continuation logits: that fact alone does not show that the data arose through sequential stages. \(\square\)

The Sequential Likelihood Makes the Risk Sets Explicit

The interpretation of continuation logits is especially clear for grouped counts. Let \(n_j\) be the number of responses in category \(j\), with \(N=\sum_{j=1}^K n_j\), and define the number at risk of being in category \(j\) or later by \(R_j=\sum_{r=j}^K n_r\). At stage \(j\), \(n_j\) stop and \(R_{j+1}\) continue. The stage-specific stopping probability is \(h_j\).

Proof. From the probability reconstruction above, \(p_j=h_j\prod_{m<j}(1-h_m)\) for \(j<K\), and \(p_K=\prod_{m<K}(1-h_m)\). In the product \(\prod_j p_j^{n_j}\), the factor \(h_j\) appears with exponent \(n_j\). For each \(j<K\), the factor \(1-h_j\) appears for categories \(j+1,\ldots,K\), so its total exponent is \(R_{j+1}\). Hence

For the combinatorial factors, \(R_j=n_j+R_{j+1}\) and \(R_1=N\). Therefore

where \(R_K=n_K\) cancels the final remaining factorial in the denominator. Combining the probability and combinatorial identities proves the factorization. It shows precisely which people are in each stage-specific risk set; using this interpretation requires that those risk sets make sense for the scientific process. \(\square\)

Practical Checks and a Common Mistake

Before fitting a structure, state in words what each logit compares. If that sentence does not match the scientific question, the model is not made appropriate by a good numerical fit. Also distinguish the outcome’s order from the mechanism generating it: a severity scale can be ordered without being sequential, while a sequence of eligibility decisions has a process structure beyond its ordered labels.

• Choose baseline-category logits when the outcomes are nominal or when the central question is comparison with a chosen reference category.
• Choose adjacent-category logits when neighboring categories have substantive meaning and local comparisons are the target.
• Choose continuation-ratio logits when responses proceed through ordered stages and the question is whether a response stops at each stage or continues.

Common mistake: “The response has three ordered categories, so a continuation-ratio model must be the right choice.” The ordering makes such logits definable, but does not establish that responses pass through stage-specific risk sets. The correction is to check how the response is generated and what comparison is of interest. If the scientific question is about neighboring severity levels, adjacent-category logits may be more direct; if there is no meaningful order, a baseline-category comparison may be preferable.