Why the response process matters
Ordinal categories have an order, but that order does not tell us how responses arise. A four-level symptom rating may represent crossing progressively higher severity thresholds. Four choices may instead be compared locally, one category with its neighbor. And a four-stage outcome may arise by passing through successive decision points. These response processes suggest different likelihoods and different interpretations of predictor effects.
The goal is not to assign a likelihood merely because one sounds plausible. A model should match the scientific meaning of the categories, the way responses are generated or recorded, and the question of interest. In a generalized linear mixed model, this choice is made for the conditional response distribution given the random effects; integrating over random effects then gives the marginal distribution.
Three likelihoods, three interpretations
A cumulative likelihood asks whether a response falls at or below each cut point. It is suitable when the categories are viewed as ordered levels separated by thresholds, as with ratings of increasing severity. A common cumulative logit mixed model is
where \(\theta_1<\cdots<\theta_{K-1}\) are cut points. Under this sign convention, a positive coefficient along a predictor's direction decreases the probability of being at or below a cut point, and therefore shifts probability toward higher categories. If the proportional-odds form is used, the predictor and random-effect contribution is the same across cut points; this is a substantive restriction, not a consequence of ordering. The earlier material on proportional odds develops that restriction.
An adjacent-category likelihood compares each category with the category immediately above it. A common adjacent-category logit is
Here a positive coefficient means higher odds of category \(k+1\) rather than \(k\), conditional on being in that adjacent pair. It does not directly describe the odds of being above a cut point versus at or below it. This form is natural when neighboring categories are meaningfully compared, such as adjacent response options in a rating task.
A continuation-ratio likelihood represents sequential progression through categories. Define \(q_{ijk}=P(Y_{ij}>k\mid Y_{ij}\geq k,b_j,x_{ij})\). A continuation-ratio logit can be written
At stage \(k\), this compares proceeding beyond category \(k\) with stopping at \(k\), among responses that have reached at least \(k\). A positive coefficient increases the conditional odds of proceeding at each modeled stage. This is appropriate only when the sequential interpretation is sensible; the numerical order alone does not make an outcome a sequence of decisions.
How the category probabilities are built
Each formulation leads to a valid set of category probabilities, but the quantities it models directly differ. For a cumulative model, let \(F_k=P(Y\leq k\mid b,x)\), with \(F_0=0\) and \(F_K=1\). Then \(\pi_k=F_k-F_{k-1}\). For the adjacent-category model, let \(r_k=\exp(\alpha_k+x^{\mathsf T}\beta+z^{\mathsf T}b)\), the odds of category \(k+1\) relative to category \(k\). The probabilities are recovered from the ratios and normalized to sum to one. For the continuation-ratio model, the sequential probabilities give
These equations also show why the models are not interchangeable just because they each describe \(K\) ordered categories. They condition on and compare different events. They can all represent a baseline distribution when category-specific parameters are sufficiently flexible, but their predictor restrictions and coefficient interpretations generally differ.
Worked Example: Ratings as Threshold Crossings
Suppose, hypothetically, that a four-category symptom rating is modeled with a cumulative likelihood. For one patient at a specified covariate and random-effect value, suppose the fitted cumulative probabilities are \(F_1=0.20\), \(F_2=0.55\), and \(F_3=0.85\). The categories mean none, mild, moderate, and severe.
The category probabilities are successive differences:
They are nonnegative and sum to \(0.20+0.35+0.30+0.15=1.00\). If this patient's observed response is moderate (category 3), its conditional likelihood contribution is \(\pi_3=0.30\). The model's interpretation concerns probabilities on either side of each severity threshold, not a separate decision to move from mild to moderate.
The cumulative probabilities are increasing, as required: \(0.20\leq0.55\leq0.85\). In a fitted cumulative-link model, the ordered cut points enforce this structure. If the model allows cut points to cross, the resulting differences could be negative and would not define valid category probabilities.
Worked Example: Neighboring Categories as Comparisons
Suppose, hypothetically, that an adjacent-category model for four satisfaction responses gives the relative category weights \(1,2,1,0.5\) for a specified covariate and random-effect value. These weights encode adjacent ratios: category 2 versus 1 has ratio \(2\), category 3 versus 2 has ratio \(0.5\), and category 4 versus 3 has ratio \(0.5\).
The weights must be normalized. Their total is \(1+2+1+0.5=4.5\), so
The displayed probabilities sum to \(0.9999\) because of rounding; the unrounded fractions sum to one. For a response in category 3, the likelihood contribution is \(1/4.5=0.2222\). The ratio \(P(Y=2)/P(Y=1)=2\) is a local comparison. It would be incorrect to interpret it as saying that the odds of a response above category 1 are twice as high.
Worked Example: Categories as Sequential Stages
Suppose, hypothetically, that a four-level pathway outcome records the furthest stage reached. At each stage, a person either proceeds to the next stage or stops. For a specified covariate and cluster effect, suppose the probabilities of proceeding are \(q_1=0.80\), \(q_2=0.60\), and \(q_3=0.50\).
The category probabilities follow from the probability of reaching a stage and stopping there:
The total is \(0.20+0.32+0.24+0.24=1.00\). An observation in category 3 contributes \(0.24\) to the conditional likelihood. For example, the \(0.24\) is the probability of proceeding past stage 1, proceeding past stage 2, and then stopping at stage 3. The conditional probability of proceeding beyond stage 2 among those who reached stage 2 is \(q_2=0.60\), not the unconditional probability of being in category 3 or higher.
Likelihoods with cluster-level random effects
For all three choices, the model specifies category probabilities conditional on \(b_j\). A standard mixed-model assumption is that observations are independent conditional on their covariates and cluster random effects, and that the random effects follow a specified distribution, commonly \(b_j\sim N(0,D)\). Clusters are taken to be independent, and the random effects are modeled as independent of the observed covariates unless the design or model says otherwise.
For observed responses \(y_{ij}\), the conditional likelihood for a cluster is the product of the relevant category probabilities. The marginal likelihood integrates that product over the random-effects distribution:
The product is inside the integral because observations in the same cluster share \(b_j\). In general, integrating a product of conditional probabilities does not equal the product of marginal probabilities. Thus the likelihood choice concerns the conditional response structure; the random effects also induce marginal dependence among observations from the same cluster.
A short justification: sequential factorization
The continuation-ratio probabilities follow from the chain rule for the sequential events. To be in category \(k<K\), a response must proceed beyond every earlier stage \(h<k\), then stop at \(k\). By the definition \(q_h=P(Y>h\mid Y\geq h,b,x)\), the probability of proceeding at stage \(h\), conditional on reaching it, is \(q_h\), while the probability of stopping there is \(1-q_h\). Multiplying these successive conditional probabilities gives
For the final category \(K\), there is no further stopping stage in the modeled sequence; the response must proceed through all \(K-1\) stages, giving \(P(Y=K\mid b,x)=\prod_{h=1}^{K-1}q_h\). This factorization explains the sequential meaning of the likelihood rather than merely supplying a convenient algebraic formula.
Choosing and checking a formulation
Start by asking what the categories mean, not which model is easiest to fit. A cumulative formulation is a natural starting point when the scientific question concerns being above or below severity or attainment thresholds. An adjacent-category formulation is appropriate when neighboring levels are meaningful contrasts. A continuation-ratio formulation is appropriate when categories correspond to stages in a sequence and progression is defined among those who reached each stage.
| Likelihood | Direct comparison | Response-process interpretation |
|---|---|---|
| Cumulative | At or below versus above each cut point | Crossing ordered thresholds |
| Adjacent-category | Category \(k+1\) versus category \(k\) | Local comparison of neighboring levels |
| Continuation-ratio | Proceed versus stop at stage \(k\), among those reaching it | Sequential progression or stopping |
Diagnostics should reflect both the probability model and its interpretation. Check that fitted probabilities are valid and that the category order and coding match the intended meaning. Compare observed and fitted category frequencies across important predictor values or clusters; inspect whether discrepancies concentrate at particular cut points or stages. For a cumulative proportional-odds specification, assess whether a common predictor effect across cut points is credible. For adjacent-category or continuation-ratio specifications, examine whether local contrasts or stage-specific effects require more flexibility. Also check whether the random-effects structure captures important clustering rather than attributing within-cluster similarity to fixed effects.
A poor fit does not automatically prove that a different ordinal likelihood is correct. It may signal a restrictive predictor structure, an unsuitable random-effects structure, sparse categories, or a mismatch between the assumed response process and the data. Likewise, better numerical fit alone cannot establish that a sequential story is scientifically defensible. If several formulations are plausible, compare fitted category probabilities and the estimands they answer, and explain the chosen interpretation.
Common mistake and limitations
Wrong reasoning: “The categories are ordered, so a continuation-ratio model is the obvious choice because respondents move through the categories in order.” Ordering does not imply that responses pass through successive stages. A person selecting “severe” on a symptom scale need not have made a sequence of decisions to move through “none,” “mild,” and “moderate.”
Correction: Use a continuation-ratio likelihood when a proceed-or-stop interpretation among those reaching a stage is defensible. Use a cumulative likelihood when the relevant idea is crossing thresholds, and an adjacent-category likelihood when neighboring levels are the meaningful contrasts. In every case, explain the conditional event represented by the coefficient.
These likelihoods do not make category distances equal, and their coefficients are not interchangeable measures of a single universal “ordinal effect.” Their interpretations depend on the chosen link, predictor restrictions, and random-effects specification. The conditional interpretation also differs from the marginal association after integrating over random effects. If the response process is uncertain, report that uncertainty and avoid giving a coefficient a stronger process interpretation than the design supports.
Check Your Understanding
Choose the most defensible likelihood and state the probability comparison it models. Solutions follow the questions.
- A clinician records four ordered levels of symptom severity. The main question is whether a patient is above each severity threshold. Which formulation fits that question?
- A task asks respondents to choose among adjacent ordered options, and the study specifically compares each option with its immediate neighbor. Which formulation directly represents those comparisons?
- An outcome is the highest of four treatment stages reached, and each stage is a genuine proceed-or-stop decision. If \(q_1=0.7\) and \(q_2=0.5\), what is the probability of stopping at category 2?
- Why is a positive coefficient in a cumulative logit written as \(\theta_k-x^{\mathsf T}\beta\) interpreted as shifting probability toward higher categories?
- Why can responses in a cluster remain dependent after the random effect is integrated out, even if they are conditionally independent given that effect?
Solutions. (1) Use a cumulative formulation: it models \(P(Y\leq k)\) against \(P(Y>k)\) at each cut point. (2) Use an adjacent-category formulation, which directly models \(P(Y=k+1)/P(Y=k)\). (3) To stop at category 2, proceed past stage 1 and stop at stage 2, so the probability is \(q_1(1-q_2)=0.7(0.5)=0.35\). (4) Increasing the predictor decreases \(\theta_k-x^{\mathsf T}\beta\), and therefore decreases \(P(Y\leq k)\); probability shifts above the cut point, toward higher categories. (5) Observations share the same random effect. Averaging their joint conditional probabilities over its distribution induces association, so marginal independence generally does not follow from conditional independence.