Ordinal Treatment Models

Why model treatment as ordinal?

Some treatments have a natural order but are not well represented by a single binary indicator: for example, no, low, and high dose, or mild, moderate, and intensive treatment. An ordinal treatment model uses that ordering to estimate probabilities conditional on pre-treatment covariates, while still assigning a separate probability to each treatment category. Those category-specific probabilities are the quantities needed when constructing inverse probability weights.

Let \(A\) be a treatment taking one of \(K\) ordered categories \(1,\ldots,K\), with category 1 the lowest. Let \(H\) denote the covariates available before treatment assignment. The model concerns \(P(A=k\mid H)\), not the probability of a treatment level conditional on outcomes or on covariates measured after assignment. The ordering is substantive: it must be defensible that one category is below another, even if the distances between adjacent categories are not meaningful.

The minus sign is a convention: a positive coefficient for a covariate shifts probability toward higher categories. The same coefficient appears at every cutpoint, which is the proportional-odds restriction. The cutpoints are estimated along with the coefficients; they are not ordinary treatment-category intercepts.

From cumulative probabilities to treatment probabilities

A cumulative probability combines several categories. Weighting requires the probability of the category actually received. Write \(F_j(H)=P(A\le j\mid H)\), with \(F_0(H)=0\) and \(F_K(H)=1\). Then the category-specific probabilities are

For the cumulative-logit model, calculate the fitted cumulative probabilities first:

The endpoint conventions mean \(\widehat P(A=1\mid H)=\widehat F_1(H)\) and \(\widehat P(A=K\mid H)=1-\widehat F_{K-1}(H)\). Do not use a cumulative probability as though it were the probability of one category.

Proof. For finite arguments, a strictly increasing distribution function with values strictly between zero and one gives \(0<F_1(H)<\cdots<F_{K-1}(H)<1\). Therefore \(F_1(H)>0\), each difference \(F_k(H)-F_{k-1}(H)>0\) for \(2\le k\le K-1\), and \(1-F_{K-1}(H)>0\). Summing the category probabilities telescopes:

Thus the model produces a valid probability distribution over the categories. This algebraic property does not guarantee that the probabilities are well estimated or that the model fits the data. \(\square\)

Estimation and the proportional-odds restriction

For independent observations \(i=1,\ldots,n\), let \(A_i\) be the observed category and \(H_i\) the pre-treatment covariate vector. Define \(p_{ik}=P(A_i=k\mid H_i)\) using the differences of cumulative probabilities above. Under the model, the conditional likelihood and log-likelihood are

The estimates maximize this likelihood subject to \(\alpha_1<\cdots<\alpha_{K-1}\). In practice, software parameterizes or constrains the cutpoints to preserve their order during optimization. After fitting, obtain each person’s full vector of category probabilities by differencing the fitted cumulative probabilities. Check that the values are finite, lie between zero and one, and sum to one up to numerical rounding.

The model’s substantive restriction can be derived directly. At cutpoint \(j\), the odds of being at or below \(j\), rather than above \(j\), are

where the last equality uses the logistic link. Increasing one covariate by one unit, with the others fixed, multiplies these cumulative odds by \(\exp(-\beta_r)\) at every cutpoint \(j\). This shared effect is why the model is called proportional odds. It does not say that the probabilities of every individual category have the same odds ratio or change by the same amount.

Derivation. At \(H\), the odds are \(\exp(\alpha_j-\beta^\mathsf{T}H)\). At \(H+\delta e_r\), they are \(\exp(\alpha_j-\beta^\mathsf{T}H-\beta_r\delta)\). Dividing the second expression by the first cancels the cutpoint and the other covariates, leaving \(\exp(-\beta_r\delta)\). Since this ratio does not depend on \(j\), the cumulative odds ratios are proportional across cutpoints. \(\square\)

Assumptions, diagnostics, and alternatives

The treatment categories must be correctly ordered, and the covariates in \(H\) must be measured before treatment assignment. The cumulative-logit model further assumes that the log-odds effects are common across all cutpoints. For weighting, estimated probabilities of the categories people actually received must not be so small that weights become extreme. This empirical overlap requirement is distinct from the model’s algebraic guarantee that its fitted probabilities are positive.

Useful checks include comparing observed and predicted category frequencies across important covariate groups, examining calibration of each cumulative probability, and inspecting the distribution of probabilities for the observed treatment category. Sparse categories can produce unstable cutpoints or large standard errors; near-empty covariate-by-category combinations can also yield very small fitted probabilities. A monotone fitted probability vector that sums to one is necessary but not evidence of adequate fit.

The common-slope restriction deserves particular attention. If covariate effects differ across cutpoints, a single proportional-odds coefficient can conceal lack of fit. Compare observed and predicted cumulative patterns, and consider a less restrictive ordinal model if the substantive data support it. The earlier material on Multinomial Treatment Probabilities provides a useful alternative: a multinomial model estimates category probabilities without imposing a common ordered effect, but it also does not use the order in the same way and may require more parameters. Treating category labels as a numeric dose and fitting a linear model is not a substitute: it imposes meaningful equal spacing and a different outcome structure.

No treatment-probability model by itself establishes causal exchangeability, correct measurement, or positivity in the target population. Model-based weights can be unstable even when the model converges. Inspect their range and upper tail, identify which observed categories and covariate patterns drive large values, and assess sensitivity to defensible modeling choices. Weight truncation changes the estimator and should be reported as a sensitivity analysis, not as a correction that makes a misspecified model valid.