Ordinal Toxicity Models for Escalation

Why Model Toxicity Grades?

A binary dose-limiting toxicity (DLT) outcome is useful when a decision depends on whether a prespecified event occurred. But it can discard clinically relevant distinctions: a patient with a moderate toxicity and a patient with a severe toxicity may both be coded “DLT,” while two outcomes below the DLT threshold may differ substantially in burden. An ordinal model retains the order of toxicity grades and estimates the probability of each grade at each dose.

The goal is not to treat toxicity grades as precise measurements with equal spacing. Rather, the model uses their ordering to describe cumulative risks such as the probability of grade 2 or worse, grade 3 or worse, and grade 4 or worse. These probabilities can then be examined separately when considering whether a dose is acceptable for escalation.

A Cumulative-Logit Model

A common ordinal model assigns a logistic regression to each cumulative exceedance probability while sharing a dose-effect slope across grade thresholds. For positive dose levels, one convenient dose model uses the logarithm of dose relative to a reference dose \(d_{\mathrm{ref}}\). The model is

Here, \(\operatorname{logit}(u)=\log\{u/(1-u)\}\), and \(\operatorname{expit}(v)=1/(1+\exp(-v))\) is its inverse. The \(\kappa_k\) values are grade thresholds. Their ordering, \(\kappa_1<\kappa_2<\cdots<\kappa_J\), ensures that exceeding a more severe grade is less likely than exceeding a less severe one. If dose is assumed to increase toxicity risk, the model also requires \(\beta_1>0\). In a Bayesian analysis, this can be enforced through a prior supported only on positive slopes.

The shared slope is the proportional-odds assumption in this parameterization: increasing log dose shifts the log odds of exceeding every grade by the same amount. It does not require the probabilities of individual grades to change by the same amount. The model assumes a common, consistently defined toxicity grading system and assessment window; patients’ outcomes are conditionally independent given their doses and the model parameters. The specified dose-response form and proportional-odds structure must also be adequate for the data and intended decision.

From Cumulative Risk to Grade-Specific Risk

The cumulative probabilities overlap. For example, every grade 3 event is also a grade 2-or-worse event. Category probabilities are obtained by taking differences between adjacent cumulative probabilities, not by treating the cumulative probabilities as separate categories.

To justify the middle expression, the event \(Y\geq j\) consists of grades \(j,j+1,\ldots,J\), while \(Y\geq j+1\) consists of grades \(j+1,\ldots,J\). Subtracting their probabilities leaves exactly the probability of grade \(j\). Likewise, the probability of grade 0 is the complement of the probability of any toxicity grade, and grade \(J\) is the most severe cumulative event itself. The threshold ordering makes each difference nonnegative, and the category probabilities sum to 1.

For observed counts \(n_{ij}\) at dose \(d_i\), with \(n_i=\sum_{j=0}^{J}n_{ij}\), the likelihood contribution at that dose is multinomial:

where \(\boldsymbol{\theta}\) denotes the thresholds and dose-effect parameters. In a Bayesian analysis, this likelihood is combined with a prior distribution to obtain a posterior distribution. For every posterior draw, calculate the cumulative and grade-specific probabilities at each dose. The resulting posterior distributions, rather than just a single fitted curve, express uncertainty relevant to a dose decision.

Using Posterior Grade Risks in Escalation

An escalation rule should identify which grades matter clinically and what posterior risk criterion is acceptable. For instance, a protocol might consider the probability of grade 3 or worse relative to a prespecified threshold. A different decision may also constrain grade 2-or-worse toxicity, particularly when moderate toxicity is important to patients or affects treatment delivery. The thresholds and rules are design choices that require clinical justification; the ordinal model does not determine them automatically.

A posterior constraint can be written as the probability, under the posterior distribution, that the toxicity risk at a candidate dose exceeds a specified risk limit. This is distinct from the toxicity risk itself. The risk is a model quantity such as \(q_3(d)\); the posterior probability of exceeding a limit measures uncertainty about that quantity after observing data.

What Is Added Beyond Binary DLT Coding?

Binary coding can be recovered from an ordinal outcome if DLT is defined as reaching or exceeding a particular grade. For example, if grade 3 or worse defines DLT, its risk is \(q_3(d)\). But replacing every outcome with a DLT indicator removes distinctions among grades below that threshold and among the more severe grades above it. An ordinal analysis can retain those distinctions and support multiple grade-specific summaries.

One summary is the expected numeric grade. For grades 0 through \(J\), the identity \(Y=\sum_{k=1}^{J}I(Y\geq k)\) gives

This is a valid mathematical summary of the coded grade, but its clinical interpretation depends on the grade scale. It implicitly treats each one-grade increase as the same numerical increment. It should not replace grade-specific risks unless that summary is appropriate for the decision.

Assumptions, Diagnostics, and Alternatives

The central structural assumption is that a common dose slope describes all cumulative grade thresholds on the log-odds scale. The ordered thresholds ensure coherent category probabilities, but they do not guarantee that the shared slope fits observed outcomes. Before relying on posterior dose risks, examine whether the model is consistent with the observed grade counts and whether its fitted dose-response pattern is clinically plausible.

• Check cumulative ordering. At each dose, the fitted risks must satisfy \(q_1(d)\geq q_2(d)\geq\cdots\geq q_J(d)\). Category probabilities should be nonnegative and sum to 1.
• Compare observed and replicated grade patterns. Posterior predictive simulations can be compared with observed counts by dose and grade. Persistent underprediction of severe grades, for example, warns that a model may give misleading escalation risks.
• Look for evidence against a shared slope. If the dose trends in different cumulative grade probabilities appear incompatible with a common log-odds shift, the proportional-odds structure may be inadequate. Sparse counts at high grades can make this difficult to assess, so uncertainty should be taken seriously.
• Check the outcome definition and assessment window. Inconsistent grading, missing assessments, or different opportunities to observe toxicity can undermine comparisons. The model as written does not account for those complications.
• Inspect posterior behavior. Implausible prior-driven risks, poorly explored parameter distributions, or unstable posterior risk estimates can undermine a recommendation. Report uncertainty at the doses relevant to the decision, not just a fitted central curve.

A binary DLT model is simpler and can be appropriate when the decision truly concerns only one threshold, but it loses information about other grades. A cumulative-logit model is useful when ordering is meaningful and the shared-slope assumption is reasonable. Other ordinal approaches, such as adjacent-category or continuation-ratio models, define different comparisons among grades and may fit different data-generating structures. A nominal multinomial model avoids assuming an ordering, but also gives up the structure supplied by ordered grades and may require more parameters. None of these alternatives removes the need for clinically justified grade definitions and decision rules.

Common Mistakes and Limitations

Ordinal grades provide more information than a single binary indicator only if the grades are recorded consistently and their ordering is meaningful. The proportional-odds model can be too restrictive, and the assumption that dose raises each cumulative risk monotonically may not be suitable in every setting. Moreover, an escalation rule based on toxicity alone does not identify a dose with adequate efficacy or an optimal benefit-risk balance. These are limitations of the decision problem, not issues solved by choosing a more detailed toxicity outcome.