Ordinal Outcomes and Proportional-Odds Selection

Why Model the Current Ordinal Category?

A longitudinal outcome may be recorded on an ordered scale such as none, mild, moderate, and severe. Treating these categories as numerical measurements assumes more than their ordering: it assigns distances between categories. A cumulative-logit model uses the ordering without requiring equal category spacing. In a selection model, dropout may also depend on the current category. For example, participants with more severe symptoms may be more likely to miss the next assessment, even after accounting for their recorded history.

The key timing convention matters. Let \(Y_{it}\) be subject \(i\)'s ordinal outcome at visit \(t\), whether or not it is observed, and let \(R_{it}=1\) mean that the outcome is observed. Let \(R_{i0}=1\). Among subjects observed through the previous visit, the current outcome is generated and then observation at visit \(t\) is determined. Thus dropout at \(t\) can depend on \(Y_{it}\), which is unobserved when \(R_{it}=0\). After dropout, subsequent outcomes are not observed.

The Proportional-Odds Outcome Model

Suppose \(Y_{it}\) takes one of the ordered categories \(1,\ldots,K\). A cumulative-logit model specifies the probability of being at or below each cutpoint. One longitudinal version conditions on baseline covariates \(x_i\), visit \(t\), and the observed or latent prior outcome history \(\mathcal H_{i,t-1}\). Its linear predictor can also include a subject effect \(b_i\), if such heterogeneity is part of the chosen outcome model.

The thresholds must be ordered, \(\alpha_1<\alpha_2<\cdots<\alpha_{K-1}\), so the cumulative probabilities increase with \(k\). The predictor \(\eta_{it}\) may include covariate, visit, and history terms. With the minus sign shown, increasing \(\eta_{it}\) lowers the cumulative probabilities and shifts probability toward higher categories. The defining proportional-odds restriction is that a predictor has the same slope at every cumulative cutpoint; the thresholds differ, but the slopes do not.

Write \(F_{it}(k)\) for the modeled cumulative probability at cutpoint \(k\), with \(F_{it}(0)=0\) and \(F_{it}(K)=1\). Then the probability of category \(k\) is \(p_{itk}=F_{it}(k)-F_{it}(k-1)\). This difference is essential: cumulative probabilities are not themselves category probabilities.

A Current-Category Dropout Model

For a subject still observed through visit \(t-1\), define the conditional dropout hazard \(h_{it}(y)\) as the probability of not being observed at visit \(t\), given the current outcome category \(y\), the covariates, and relevant history. A useful ordinal specification assigns a score \(s(y)\) to each category and uses a logistic hazard:

For example, \(s(y)=y-1\) gives a one-unit increase for each step up the ordinal scale. Under this specification, the dropout odds ratio for a one-category increase is \(\exp(\delta)\), holding the other terms fixed. A positive \(\delta\) means higher categories are associated with higher conditional dropout odds. This is a model assumption about the selection association, not a causal effect.

The phrase “proportional odds” describes the cumulative-logit outcome model, not necessarily the selection equation. A score-based logistic hazard imposes a constant change in log dropout odds per score unit. If that restriction is unsuitable, category indicators can replace \(\delta s(y)\), allowing a separate selection effect for each category. The outcome and selection models are distinct parts of the specification.

Worked Probabilities and Selection Consequences

Why Missing Current Categories Require Integration

When \(R_{it}=1\), the current category is observed. Conditional on the history and covariates, its joint probability with retention is the outcome probability multiplied by the retention probability. When \(R_{it}=0\), the current category is unknown, so its possible values must be averaged over rather than replaced by a single category. For the following identities, write \(p_{ity}=P(Y_{it}=y\mid \mathcal H_{i,t-1},x_i,b_i,z_{it},R_{i,t-1}=1)\) and \(h_{it}(y)=P(R_{it}=0\mid Y_{it}=y,\mathcal H_{i,t-1},x_i,b_i,z_{it},R_{i,t-1}=1)\), so both probabilities condition on the same history, covariates, and latent effect.

The second expression follows by partitioning the event \(R_{it}=0\) according to the mutually exclusive possible values of \(Y_{it}\), then summing the conditional joint probabilities \(p_{ity}h_{it}(y)\). If \(b_i\) is marginalized, integrate these joint products over its conditional distribution given the observed conditioning variables rather than multiplying separately marginalized probabilities. This also shows why the missing current category cannot simply be omitted from the selection equation. Across visits, the same conditional reasoning is applied among subjects still at risk; any latent subject effects or unobserved histories must be handled according to the specified model.

Diagnostics, Alternatives, and Limitations

The model should be checked in both components. For the outcome model, compare observed cumulative category frequencies across visits and relevant covariate groups with fitted probabilities; examine whether the common-slope proportional-odds restriction is plausible. Sparse categories can make thresholds and category probabilities unstable. For selection, inspect implied dropout probabilities across all categories and visits, and check that fitted probabilities remain between 0 and 1 and are not implausibly close to the extremes. Compare observed dropout rates across recorded histories with model-based rates where such comparisons are available.

A useful diagnostic is sensitivity to the selection specification: compare a linear category score with category-specific selection effects, and assess whether conclusions change under plausible values of \(\delta\). A score model can be too restrictive if dropout risk changes sharply between two categories but little between others. Conversely, a saturated category model may be unstable when few participants occupy some categories. The outcome model and selection equation should not be treated as interchangeable ways to impose the same assumption.

Under this parameterization, setting \(\delta=0\) removes the direct current-category term from the selection hazard, conditional on the other specified variables. That is a conditional independence restriction, not proof that the data are missing at random under every possible conditioning set. A pattern-mixture model instead parameterizes outcome distributions by missingness pattern; it provides a different route to sensitivity analysis, not an automatic correction for a misspecified selection model.

A central limitation is that the current category is unavailable precisely for participants whose dropout may depend on it. The observed data alone may therefore provide little information about the strength of this dependence. Estimates can rely heavily on modeling assumptions, external information, or sensitivity parameters. The earlier material on Missing Data gives the broader terminology for such assumptions; here, the practical implication is to report the selection specification and assess how conclusions vary over credible alternatives. A fitted selection coefficient should not be given a causal interpretation.