Why the Response Structure Matters
A categorical response can represent very different kinds of information. “Treatment A, B, or C” may be a set of alternatives with no natural ranking. “Mild, moderate, or severe” has an order. “Did not enroll, enrolled, or graduated” may describe stages reached through a process in which a person must pass earlier stages before later ones. These distinctions affect which category comparisons a model should express.
A logit is not chosen just because the response has several categories. The choice should reflect the meaning of the categories and the question the analysis is intended to answer. A model that compares every category with a reference is natural for nominal alternatives; one that compares neighbors is meaningful when nearby levels matter; and one that models progression through stages requires a genuinely sequential interpretation.
Three Ways to Structure Category Comparisons
Let \(Y\) take one of \(K\) categories, with \(p_k=P(Y=k)\), where \(p_k>0\) and \(\sum_{k=1}^{K}p_k=1\). For simplicity, the definitions below describe probabilities at a fixed covariate value. A regression model can then express these logits as functions of covariates.
These definitions identify different comparisons, not merely different algebraic notations. A baseline-category logit asks how the probability of one named outcome compares with a chosen reference. An adjacent-category logit asks how probability is distributed between neighboring levels. A forward continuation-ratio logit asks, at each stage, whether an individual progresses beyond that stage or stops there.
The response structure and the logit structure should both be considered. An outcome can be ordered without having a sequential generation process. Conversely, a sequence of decisions may matter even when the stage labels are not naturally interpreted as a simple severity ranking. The labels alone do not determine the model.
Choosing by the Scientific Question
| Response structure | Comparison of interest | Natural logit structure |
|---|---|---|
| Unordered nominal outcomes | Each outcome versus a chosen reference | Baseline-category |
| Ordered categories where neighboring levels are the focus | Category \(k\) versus category \(k+1\) | Adjacent-category |
| Stages reached through a sequential process | Progress beyond stage \(k\) versus stop at stage \(k\), among those at risk | Forward continuation-ratio |
This is a guide to the comparison each structure encodes, not a rule that automatically selects a unique model. For instance, ordinal categories may also have a scientifically meaningful reference category, and a sequential process may be analyzed using a different summary if that better answers the research question. The key is to specify what the logit compares and why that comparison is relevant.
Worked Example: Unordered Treatment Preference
Suppose a survey records a person’s preferred treatment: A, B, or C. The outcomes are alternatives, not levels of a common ranking. At a fixed covariate value, let \(P(Y=A)=0.50\), \(P(Y=B)=0.30\), and \(P(Y=C)=0.20\). Take A as the baseline.
The baseline-category logits are
Both logits compare a named alternative with A, matching questions such as whether a covariate is associated with choosing B rather than A. Treating these labels as ordered and forming adjacent logits would imply that B is between A and C in a meaningful sequence. Nothing in the survey description supports that interpretation, so the baseline-category structure is the appropriate starting point.
Worked Example: Ordered Disease Stages
Suppose patients are classified as stage 1, stage 2, or stage 3, with larger numbers denoting more advanced disease. Let the probabilities at a fixed covariate value be \(0.20\), \(0.50\), and \(0.30\), respectively. If the scientific question concerns how probability shifts between neighboring stages, adjacent-category logits directly express those comparisons.
For stages 1 and 2, the odds of stage 1 rather than stage 2 are \(0.20/0.50=0.40\), giving \(\log(0.40)\approx -0.916\). For stages 2 and 3, the odds of stage 2 rather than stage 3 are \(0.50/0.30\approx1.667\), giving \(\log(1.667)\approx0.511\).
For contrast, using stage 3 as a baseline gives \(\log(0.20/0.30)\approx-0.405\) for stage 1 and \(\log(0.50/0.30)\approx0.511\) for stage 2. That structure answers a different set of comparisons: each earlier stage versus stage 3. The categories are ordered, but the scientific question determines whether local comparisons or a reference-stage comparison is more useful.
Worked Example: Progress Through Educational Stages
Consider four outcomes: did not enroll (\(Y=1\)), enrolled but did not complete (\(Y=2\)), completed an undergraduate degree (\(Y=3\)), and completed graduate study (\(Y=4\)). Suppose the process is genuinely sequential: a person must enroll before completing an undergraduate degree and must complete that degree before completing graduate study. At a fixed covariate value, suppose the probabilities of progressing beyond each stage among those who reached it are \(q_1=0.60\), \(q_2=0.50\), and \(q_3=0.25\).
The probability of stopping at stage 1 is \(1-q_1=0.40\). To stop at stage 2, a person must first progress beyond stage 1 and then stop at stage 2, giving \(q_1(1-q_2)=0.60(0.50)=0.30\). To stop at stage 3, the person must pass the first two stages and then stop at the third, giving \(q_1q_2(1-q_3)=0.60(0.50)(0.75)=0.225\). Finally, reaching stage 4 requires progressing at all three stages, giving \(q_1q_2q_3=0.60(0.50)(0.25)=0.075\).
The resulting category probabilities are \(0.40,0.30,0.225,0.075\), which are nonnegative and sum to \(1.00\). The corresponding forward continuation-ratio logits are \(\log(0.60/0.40)\approx0.405\), \(\log(0.50/0.50)=0\), and \(\log(0.25/0.75)\approx-1.099\). Each describes progression versus stopping among people who reached that stage. This conditional interpretation is justified by the process described, not merely by the ordering of the outcome labels.
Two Structural Results
The following results clarify why baseline-category and continuation-ratio logits have the interpretations just described. They are also useful checks that the chosen logits encode a coherent probability distribution.
Proof. Write \(\eta_k=\log(p_k/p_b)\) for \(k\ne b\), and set \(\eta_b=0\). Exponentiating gives \(p_k=p_b e^{\eta_k}\) for every \(k\ne b\), while the same expression holds for \(k=b\) because \(e^{\eta_b}=1\). Summing over all categories yields \(1=p_b\sum_{j=1}^{K}e^{\eta_j}\), so
Thus the logits recover exactly one probability vector. Conversely, every strictly positive probability vector gives finite logits by taking the stated ratios. The reference category changes the contrasts, but not the underlying probability vector. \(\square\)
Proof. At stage \(k\), the event \(Y\ge k\) divides into the disjoint events \(Y=k\) and \(Y>k\). Therefore \(P(Y=k\mid Y\ge k)=1-q_k\). To have outcome \(k\), an individual must progress beyond each of stages \(1,\ldots,k-1\), then stop at \(k\). Applying the multiplication rule for conditional probabilities gives
For outcome \(K\), the individual progresses beyond every earlier stage, so \(P(Y=K)=\prod_{j=1}^{K-1}q_j\). These expressions are the probabilities of the disjoint ways to stop at stages \(1,\ldots,K-1\) or to reach stage \(K\); together they sum to one. Hence the continuation probabilities encode the distribution by successive conditional decisions. \(\square\)
Common Mistake: Treating Order as a Sequential Process
Wrong reasoning: “The categories have a natural order, so a continuation-ratio model must be appropriate.” Order alone does not establish that individuals move through stages or that the relevant comparison is progression versus stopping among those at risk. Disease severity, for example, may be recorded as ordered levels without being generated by the particular sequential decision process represented by a continuation ratio.
Correction: Ask what the categories mean in the study and what comparison the research question requires. Use adjacent-category logits when neighboring levels are the intended comparison. Use continuation-ratio logits when the sequential, conditional progression interpretation is substantively appropriate. If the outcomes are simply distinct alternatives, use baseline-category logits with a scientifically interpretable reference. If more than one structure is plausible, explain the estimand and assumptions for each rather than treating the coding order as decisive.
A second practical check is to state the comparison in words before fitting the model. “B versus A,” “stage 2 versus stage 3,” and “progress beyond stage 2 versus stop at stage 2 among those reaching it” are different scientific questions. A model is well matched when its logits express the desired comparison, not merely when software accepts the category codes.
Check Your Understanding
For each question, identify the response structure and the logit comparison that best matches the stated scientific aim.
- A patient selects one of four equally available treatment plans, with no accepted ranking among plans. Which logit structure is a natural starting point, and what comparison does it express?
- A researcher studies ordered pain levels and specifically wants to compare the probability of moderate pain with the probability of severe pain. Which structure directly makes this neighboring comparison?
- Why is an ordered list of categories not, by itself, enough to justify a continuation-ratio model?
- In a sequential model, what population is represented by the conditional probability \(P(Y>k\mid Y\ge k)\)?
- What does changing the reference category in a baseline-category logit change, and what does it leave unchanged?