Identifiability Problems in Two-Parameter CRM

Why Two Parameters Can Be Hard to Learn

A two-parameter logistic CRM can flexibly represent dose-toxicity curves, but that flexibility creates a practical challenge: early in a trial, there may be too little information to determine both the curve’s level and its steepness. Several combinations of parameters can then fit the observed data similarly while implying different toxicity probabilities at untried doses. This is an identifiability problem, not merely a computational inconvenience. It can make a posterior dose recommendation depend substantially on the prior, even when the posterior calculation is performed correctly.

The issue is especially clear when all participants have been treated at one dose. Their outcomes can inform the toxicity probability at that dose, but the data alone do not generally determine how the curve changes on either side of it. A dose recommendation may nevertheless be produced by the prespecified CRM rule; interpreting that recommendation requires attending to the uncertainty behind it.

Model and Meaning of Identifiability

Let \(x_j\) be the fixed numerical score assigned to dose level \(j\), with larger scores representing higher doses. For participant \(i\), let \(Y_i=1\) indicate a dose-limiting toxicity (DLT) and \(Y_i=0\) otherwise. The two-parameter logistic CRM specifies

Here \(a\) is the intercept, \(b\) is the slope, and the usual monotone dose-toxicity specification constrains \(b>0\). Conditional on the assigned doses and model parameters, the DLT outcomes are assumed independent Bernoulli observations with the probabilities specified above. The dose scores are fixed and known, DLT status is completely observed for each participant, and no unmodelled time trend or other covariate is included in this model. Bayesian inference additionally requires a specified prior on \((a,b)\), supported on the allowed parameter space.

For this model, each observed dose informs the linear predictor \(a+bx\) at that dose. Identifying both \(a\) and \(b\) requires information about the curve at different dose scores. A prior can regularize the posterior when the likelihood is weak, but it cannot make the observed data contain information they do not contain. In that situation, it is important to separate what the data support from what follows from the prior.

Why One Observed Dose Creates a Ridge

Suppose \(n\) participants are all treated at the same score \(x_*\), and \(y\) experience a DLT. Apart from the binomial coefficient, their likelihood is

The likelihood depends on \(a\) and \(b\) only through \(a+bx_*\). Thus, any two parameter pairs satisfying \(a+bx_*=\eta_*\) have identical likelihood for these data. They form a line in the parameter plane, restricted only by the allowed parameter space. This line is often called a likelihood ridge.

The same point appears in the Fisher information. If the probability at \(x_*\) is \(p_*\), the information from these \(n\) observations is proportional to

The determinant of this matrix is zero: its second row is \(x_*\) times its first row. There is therefore no information in this one-dose design to estimate both parameters separately. With observations at two distinct dose scores, the information matrix can have full rank. But full rank is not a guarantee of precision: few participants, few DLTs, or dose scores that are close together can still leave a broad, strongly correlated posterior for \(a\) and \(b\).

How the Posterior Carries Uncertainty Into a Dose Rule

Let \(D\) denote all observed dose assignments and DLT outcomes. The posterior distribution is proportional to the likelihood times the prior. For each candidate dose \(j\), define its posterior mean toxicity probability as \(q_j=E[p(x_j;a,b)\mid D]\). Under the target-dose rule considered here, the recommended dose is the eligible candidate whose \(q_j\) is closest to \(\phi\). The rule selects using a posterior summary; it does not imply that the toxicity probability at that dose is known precisely.

Averaging probabilities over the posterior is not the same as averaging the parameters and then applying the logistic function. The dose rule here uses \(E[p(x_j;a,b)\mid D]\), the posterior mean of the toxicity probability. In general, because the logistic function is nonlinear, \(E[\operatorname{expit}(a+bx_j)\mid D]\) need not equal \(\operatorname{expit}(E[a\mid D]+x_jE[b\mid D])\).

What Additional Dose Information Can—and Cannot—Do

With observations at scores \(x_1\) and \(x_2\), the information matrix is a sum of contributions in the directions \((1,x_1)\) and \((1,x_2)\). If the contributions have positive weights \(w_1\) and \(w_2\), its determinant is \(w_1w_2(x_1-x_2)^2\). Distinct scores make that determinant positive, but a small separation or small weights can make it small. In a trial, the weights depend on the number of observations and the probabilities: information is limited when there are few participants or when observed outcomes are nearly all the same at an informative dose.

Diagnosing Instability and Choosing a Comparison

A useful review of a two-parameter CRM asks not only whether the computation converged, but also whether the available dose-outcome pattern supports the curve used for the recommendation. Examine the posterior distributions jointly and marginally. A pronounced intercept-slope trade-off, broad marginal distributions, or a posterior concentrated near a parameter constraint can indicate weak information. Also compare posterior predictions at each candidate dose and assess whether the recommended dose changes under reasonable prior choices or modest changes in the observed data. Sensitivity is especially informative when the early observations are concentrated at one dose.

These checks are not automatic proof of failure. For example, a correlation between \(a\) and \(b\) is expected when a likelihood ridge runs through the parameter space. It is the resulting uncertainty in clinically relevant predictions and decisions that matters. Report uncertainty in toxicity probabilities, and avoid describing a posterior mean as if it were the risk for every plausible curve. If one dose is clearly closest to target under all plausible posterior curves, a recommendation may be stable despite parameter uncertainty; if plausible curves point to different doses, the rule’s single selected dose can conceal decision uncertainty.

The fixed-intercept model is a meaningful alternative when its intercept is justified by prior clinical information or a prespecified calibration. It is not a general cure for sparse data: fixing a parameter transfers uncertainty and assumptions into that constraint. Conversely, retaining two parameters does not by itself make the model unsuitable; it makes careful prior specification and uncertainty assessment particularly important.

Common Pitfall

Wrong reasoning: “The posterior dose rule selected a dose, so the data must have identified both logistic parameters.” A rule can return a dose even when the likelihood has a ridge, because the prior and the decision rule still produce a posterior summary and an ordering of candidates.

Correction: Check whether the observed dose levels provide information that distinguishes the parameters, then examine the posterior predictions and recommendation sensitivity. A computed recommendation is a decision under the model and prior; it is not evidence that the curve is precisely identified. When the recommendation is prior-sensitive, state that limitation rather than presenting the selected dose as a data-determined conclusion.