Ordinal Outcomes Across Dropout Patterns

Why model ordinal outcomes by dropout pattern?

Repeated outcomes such as symptom severity, functional status, or disease stage often have a meaningful order but not a meaningful numerical distance between categories. A transition from “none” to “mild” need not represent the same change as a transition from “moderate” to “severe.” Treating these responses as continuous can therefore impose an inappropriate scale. A cumulative-logit model instead describes probabilities of being at or below each ordered threshold.

In a pattern-mixture analysis, the distribution of an outcome is described separately, or partly separately, for each dropout pattern. This makes it possible to compare ordinal response distributions among pattern groups at a particular visit. A pattern-specific model describes outcomes conditional on pattern membership; by itself, it does not determine the unobserved outcomes after dropout.

The minus sign in the model is a useful convention for interpretation. Increasing a covariate with a positive coefficient lowers the cumulative probability of being at or below any cutpoint, shifting probability toward higher categories. Equivalently, the odds of being above rather than at or below any cutpoint multiply by \(\exp(\beta)\) for a one-unit increase in a scalar covariate with coefficient \(\beta\). Whether “higher” means worse depends on how the outcome categories were ordered.

From cumulative logits to category probabilities

Write \(F(z)=1/(1+\exp(-z))\) for the logistic cumulative distribution function. For fixed \(g,t,x\), define \(C_k=P(Y_{it}\leq k\mid G_i=g,X_{it}=x)\). The model gives \(C_k=F(\alpha_{gtk}-x^\mathsf{T}\beta_{gt})\). To obtain the probability of an individual category, subtract adjacent cumulative probabilities. Set \(C_0=0\) and \(C_K=1\); then \(P(Y_{it}=j\mid g,x)=C_j-C_{j-1}\) for \(j=1,\ldots,K\).

This subtraction also explains why ordered cutpoints matter: because \(F\) is increasing, ordered cutpoints produce nondecreasing cumulative probabilities. If a fitted model violates the cutpoint order, it cannot represent a valid cumulative distribution with the intended category ordering.

Comparing fitted distributions across patterns

Pattern comparisons should hold the visit and covariate values fixed. Otherwise, differences in predicted probabilities can reflect different covariate values rather than different pattern-specific distributions. A model can allow both cutpoints and covariate effects to vary by pattern, or impose constraints such as a common covariate effect while retaining pattern-specific cutpoints. The choice is substantive and should be supported by the data; a more flexible model also requires more information within each pattern.

A pattern-specific model can be fitted separately within each pattern, or estimated in a combined model with pattern indicators and interactions that reproduce the desired pattern-specific cutpoints or effects. In either case, the modeled probabilities are conditional on the pattern and covariates. If pattern weights are available for the target population, pattern-specific probabilities can also be averaged to obtain a population mixture probability at a common covariate value.

Assumptions, estimation, and diagnostics

The model requires genuinely ordered categories and a consistent definition of their order across patterns and visits. Within each pattern and visit, the cumulative logits are assumed linear in the specified covariates unless nonlinear terms are included. The proportional-odds restriction says that a covariate has the same log-odds effect at every cutpoint. Pattern-specific coefficients relax the assumption that this effect is identical across patterns; they do not relax proportional odds across cutpoints within a pattern.

For an observed response \(Y_{it}=j\), its modeled probability is \(C_j-C_{j-1}\). Under a likelihood that treats response records as conditionally independent, the likelihood is the product of these category probabilities over records. Repeated responses from the same subject are generally dependent, so treating every visit record as independent can give incorrect standard errors. The inferential procedure must account for within-subject dependence, for example through an appropriate repeated-measures likelihood or subject-clustered inference. A set of visit-specific cumulative-logit models describes visit-specific distributions, but does not by itself specify the joint distribution of a subject’s repeated outcomes.

Important diagnostics include checking that estimated cutpoints remain ordered, that predicted category probabilities are nonnegative and sum to one, and that observed and fitted category frequencies are reasonably aligned within patterns and covariate groups. Assess whether the proportional-odds restriction is plausible: strong differences in covariate associations across thresholds indicate that a single slope may be inadequate. Also inspect category counts within every pattern. Sparse cells can produce imprecise estimates, very large coefficients, or separation, especially when many pattern-by-covariate parameters are fitted.

A common mistake is to interpret \(\exp(\beta)\) as the odds ratio for a particular category, such as “severe” versus all other categories. In a proportional-odds model it is the common odds ratio for being above rather than at or below each cutpoint. Another mistake is to treat a pattern-specific fit as a model for outcomes after dropout. The model describes the outcome distribution where the relevant outcomes are observed; estimating unobserved post-dropout outcomes requires additional assumptions or modeling choices.

How this approach differs from alternatives

A single ordinal model with common cutpoints and effects pools information across patterns but assumes no pattern-specific distributional differences of the modeled kind. Pattern-specific cumulative-logit models allow such differences, at the cost of more parameters and potentially less precision. Separate binary logistic regressions at each threshold can allow threshold-specific effects, but do not automatically ensure that the resulting cumulative probabilities are ordered. A multinomial model treats categories as nominal and does not use their ordering. The cumulative-logit approach is attractive when the order is meaningful and proportional odds is a defensible simplification; it is limited when that restriction is seriously inconsistent with the data or when patterns are too small for stable estimation.