Modeling Ordinal Joint Outcomes

Why Model the Grades Jointly?

Efficacy and toxicity are often recorded in ordered grades rather than as simple yes-or-no outcomes. An efficacy scale might distinguish no response, partial response, and major response; a toxicity scale might distinguish none, moderate toxicity, and severe toxicity. Reducing each scale to a binary indicator discards distinctions that may matter clinically. Modeling the two ordinal outcomes jointly also preserves the fact that both are observed for the same patient.

The goal here is to specify a probability model for the pair of grades at each dose. Such a model can estimate probabilities of clinically meaningful combinations, such as achieving at least a partial response without severe toxicity. Those probabilities can inform dose assessment, but they do not by themselves define a complete dose-selection rule.

Cumulative-Link Marginal Models

A cumulative-link model represents probabilities of being at or above each grade threshold. This is convenient because the thresholds have direct interpretations: \(P(E\geq r\mid d)\) is the probability of attaining efficacy grade \(r\) or better, while \(P(T\geq s\mid d)\) is the probability of experiencing toxicity grade \(s\) or worse in the severity ordering.

For example, a simple dose predictor is \(\eta_E(d)=\beta_{E,0}+\beta_{E,1}x_d\), where \(x_d\) is a chosen numerical coding of dose. A corresponding predictor can be used for toxicity. The proportional-odds assumption is that the dose predictor has the same coefficient across all grade thresholds within an outcome; the cutpoints, not the dose coefficient, vary by threshold. If larger doses are expected to improve efficacy and increase toxicity, positive dose coefficients are natural, but the expected direction should be justified clinically and checked rather than imposed automatically.

The category probabilities follow by differencing the cumulative probabilities. Define \(S_{E,r}(d)=P(E\geq r\mid d)\), and set \(S_{E,0}(d)=1\), \(S_{E,J+1}(d)=0\). Then \(P(E=j\mid d)=S_{E,j}(d)-S_{E,j+1}(d)\). The same construction applies to toxicity. Ordered cutpoints ensure the survival probabilities decrease as the threshold rises, so these category probabilities are nonnegative and sum to one.

Combining the Marginals Into a Joint Model

Separate cumulative-link models specify the two marginal distributions, but they do not determine how efficacy and toxicity grades occur together. A dependence structure is needed to assign probabilities to the \((J+1)(K+1)\) possible grade pairs. One option is a copula. In a Gaussian-copula specification, the fitted marginal cumulative probabilities are linked through a bivariate normal distribution with correlation parameter \(\rho\). The parameter represents dependence on the copula’s latent scale; it is not generally equal to the ordinary correlation between the observed grades.

For interior grades, the cell probability is

where \(C(u,v;\rho)\) denotes the Gaussian-copula cumulative probability. This inclusion–exclusion calculation is the probability of the rectangle between the two adjacent efficacy and toxicity cumulative boundaries. It also explains why the resulting cells sum to one: summing over adjacent rectangles cancels their shared boundaries, leaving the total cumulative probability at the upper corner, which is one.

When \(\rho=0\), the Gaussian copula gives independence, so each joint cell is the product of its marginal probabilities. For nonzero \(\rho\), the cell probabilities change while the marginal probabilities remain fixed. The dependence specification should be treated as a modeling choice: observed ordinal margins alone do not determine a unique dependence structure.

Clinically Meaningful Categories and Dose Assessment

The grade pair is the basic modeled outcome, but decisions are often discussed using broader categories. These are defined by grouping grade pairs according to clinical priorities. For example, a favorable category might be \(E\geq1,T\leq1\); an unfavorable category might be \(T=2\) regardless of efficacy; and a highly favorable category might require \(E=2,T=0\). Each category is a set of cells in the joint probability table, so its probability is the sum of the probabilities of those cells.

The grouping should reflect the clinical meaning of the grades, not merely simplify computation. In particular, “no severe toxicity” is not equivalent to “no toxicity,” and “any response” is not equivalent to “major response.” State each criterion in terms of the actual grade thresholds and retain the units and clinical definitions used to assign grades. If a category combines cells with different clinical consequences, report its component probabilities as well rather than concealing that variation in one total.

Assumptions, Diagnostics, and Limitations

The model requires several assumptions to be made explicit. Grades must be ordered consistently, defined in a way that can be applied across dose groups, and mutually exclusive and exhaustive for each outcome. The cumulative-logit form assumes proportional odds within each marginal outcome. A linear dose predictor assumes the chosen dose coding adequately describes the dose trend; it may be unsuitable if the response is nonlinear or plateaus. The joint model adds a dependence assumption, such as the Gaussian copula, which cannot be checked from marginal probabilities alone.

Useful diagnostics include comparing observed and fitted proportions above each grade threshold across doses, examining whether discrepancies are concentrated at particular thresholds, and checking whether observed dose trends are compatible with the chosen predictor. Marked differences in dose effects across thresholds challenge proportional odds. For instance, a dose could substantially increase the probability of partial response while barely changing the probability of major response; a single common dose coefficient may then be inadequate. Sparse grade-pair counts also make dependence estimates unstable, especially for rare severe outcomes. These checks should be interpreted in light of sample size and uncertainty rather than as mechanical pass-or-fail tests.

Alternatives include fitting separate binary models at each threshold, using a partial proportional-odds model that allows selected threshold effects to differ, or specifying a different joint dependence structure. Separate threshold models can be more flexible but may yield cumulative probabilities that are not ordered, producing invalid negative category probabilities unless constraints are imposed. A different copula may represent dependence differently, but changing the copula does not remove the need to assess sensitivity to dependence assumptions. No model repairs inconsistent grading, unmeasured differences between dose groups, or a clinically poorly chosen category definition.

From Model Output to a Defensible Interpretation

A dose-specific joint probability should be reported with its category definition and model assumptions. For example, “the estimated probability of at least partial response without severe toxicity at dose \(d\) is 0.583 under the fitted marginal models and an independence assumption” is more informative than “the dose has a 58% success rate.” The former states what counts as success and identifies the dependence assumption; the latter hides both.

Ordinal modeling preserves information, but it does not make every grade distinction equally important. The model estimates probabilities for the recorded outcomes; clinical judgment determines which combinations matter. Dose comparisons should therefore keep the grade-scale definitions, dependence assumptions, and intended clinical category visible. A probability for one category is one component of dose assessment, not by itself a recommendation to treat at that dose.