Two-Parameter Logistic CRM

Why Allow Both Location and Slope to Vary?

A logistic CRM can represent dose toxicity with a curve whose position and steepness are both uncertain. The position determines which dose range has substantial toxicity probability; the slope determines how quickly that probability changes across doses. Estimating both from accumulating trial data is flexible, but it also means that posterior dose recommendations must account for uncertainty in both parameters together.

Let the trial have ordered dose levels $j=1,\ldots,J$, with fixed, increasing numerical scores $x_1<\cdots<x_J$. The scores are the model's dose scale; they need not equal the physical doses. For a participant treated at dose $j$, let $Y=1$ indicate a dose-limiting toxicity and $Y=0$ its absence.

Definition: In the two-parameter logistic CRM, the toxicity probability at dose score $x_j$, conditional on parameters $\alpha$ and $\beta$, is $$ p_j(\alpha,\beta)=P(Y=1\mid x_j,\alpha,\beta) =\frac{1}{1+\exp[-(\alpha+\beta x_j)]}. $$ Here $\alpha$ is the intercept and $\beta$ is the slope. To impose an increasing dose-toxicity curve, the model restricts $\beta>0$. The score where the conditional toxicity probability is $0.5$ is $x=-\alpha/\beta$, so uncertainty in both parameters entails uncertainty in the curve's location as well as its steepness.

This specification differs from the logistic CRM with a fixed intercept: the data can now update the intercept and slope jointly. The dose scores, logistic link, and positive-slope restriction are modeling choices that should be stated before the likelihood is formed. The parameters are not themselves dose-toxicity probabilities; those probabilities are obtained by applying the logistic curve at a score.

Likelihood and Joint Posterior

Suppose $n$ participants have observed dose scores $x_i$ and toxicity outcomes $y_i\in\{0,1\}$, for $i=1,\ldots,n$. Conditional on $(\alpha,\beta)$, assume participant outcomes are independent Bernoulli variables with probabilities given by the logistic model. This likelihood assumes that the recorded outcome and assigned dose are correctly represented, and that there is no additional dependence between participants after conditioning on the model parameters.

L(\alpha,\beta\mid \mathbf{y},\mathbf{x}) =\prod_{i=1}^n p_i(\alpha,\beta)^{y_i} \{1-p_i(\alpha,\beta)\}^{1-y_i}, \qquad p_i(\alpha,\beta)=\frac{1}{1+\exp[-(\alpha+\beta x_i)]}.

A prior density $\pi(\alpha,\beta)$ represents uncertainty before the observed outcomes. It must assign probability only to the permitted parameter space, including $\beta>0$ for a monotone model. The prior may associate $\alpha$ and $\beta$; independence is not required. Bayes' rule gives the joint posterior density as the likelihood times the prior, divided by the integral of that product over the parameter space.

\pi(\alpha,\beta\mid\mathbf{y},\mathbf{x}) = \frac{L(\alpha,\beta\mid\mathbf{y},\mathbf{x})\pi(\alpha,\beta)} {\displaystyle \int_{\beta>0}\int_{-\infty}^{\infty} L(a,b\mid\mathbf{y},\mathbf{x})\pi(a,b)\,da\,db}.

The posterior toxicity probability at dose $j$ is the posterior expectation of the conditional probability:

P_j^{\mathrm{post}} =P(Y_{\mathrm{new}}=1\mid x_j,\mathbf{y},\mathbf{x}) =\int_{\beta>0}\int_{-\infty}^{\infty} p_j(\alpha,\beta)\, \pi(\alpha,\beta\mid\mathbf{y},\mathbf{x})\,d\alpha\,d\beta.

This integral is the posterior predictive probability of toxicity for a new participant at dose $j$, under the model and prior. It averages the conditional probability over all parameter values according to their posterior support. In general, it is not equal to the logistic probability evaluated at the posterior means of the parameters.

Deriving the Posterior Dose Probability

The averaging formula follows directly from conditioning on the unknown parameters. Given $(\alpha,\beta)$, the new participant's toxicity probability is $p_j(\alpha,\beta)$. Integrating this conditional probability over the posterior distribution gives the probability conditional only on the observed trial data:

Condition on the parameters.
At dose score $x_j$, the probability of a new toxicity is $p_j(\alpha,\beta)$ by the Bernoulli-logistic model.

Weight by posterior support.
Parameter values with greater posterior density contribute more to the predictive probability than values with little posterior support.

Integrate over both parameters.
The law of total probability gives $P_j^{\mathrm{post}}=\int p_j(\alpha,\beta)\pi(\alpha,\beta\mid\mathbf{y},\mathbf{x})\,d\alpha\,d\beta$. Both location and slope uncertainty are included in this integral.

For continuous priors, the integral is usually evaluated by numerical quadrature or posterior simulation. For Monte Carlo draws $(\alpha^{(m)},\beta^{(m)})$, $m=1,\ldots,M$, from the joint posterior, the corresponding estimate is the average of the conditional probabilities:

\widehat P_j^{\mathrm{post}} =\frac{1}{M}\sum_{m=1}^{M} \frac{1}{1+\exp[-(\alpha^{(m)}+\beta^{(m)}x_j)]}.

This calculation averages probabilities, not parameter values. Substituting the posterior means into the logistic formula is a different calculation and generally does not produce the posterior predictive probability because the logistic function is nonlinear.

Worked Example: Updating a Joint Finite Prior

Consider a hypothetical trial with one participant treated at score $x=0$ who experiences a toxicity and one treated at score $x=1$ who does not. Let the prior put equal probability on four parameter pairs: $(-\ln 3,\ln 3)$, $(-\ln 3,2\ln 3)$, $(0,\ln 3)$, and $(0,2\ln 3)$. All four slopes are positive, and all prior weights are $0.25$.

For the observed data, the likelihood at each pair is $p_0(1-p_1)$: the first participant contributes a toxicity probability at $x=0$, and the second contributes a non-toxicity probability at $x=1$. The model gives $p_0=0.25$ when $\alpha=-\ln 3$, and $p_0=0.5$ when $\alpha=0$. At $x=1$, the four probabilities in the listed order are $0.5, 0.75, 0.75,$ and $0.9$. Therefore the likelihoods are:

\begin{aligned} 0.25(1-0.5)&=0.125,\\ 0.25(1-0.75)&=0.0625,\\ 0.5(1-0.75)&=0.125,\\ 0.5(1-0.9)&=0.05. \end{aligned}

With equal prior weights, the normalized posterior weights are proportional to these likelihoods. Their sum is $0.3625$, so the weights are $0.125/0.3625=10/29$, $0.0625/0.3625=5/29$, $0.125/0.3625=10/29$, and $0.05/0.3625=4/29$. They sum to one.

At score $x=0$, the posterior predictive probability is $ (10/29)(0.25)+(5/29)(0.25)+(10/29)(0.5)+(4/29)(0.5)=10.75/29\approx0.3707$. At $x=1$, it is $ (10/29)(0.5)+(5/29)(0.75)+(10/29)(0.75)+(4/29)(0.9)=19.85/29\approx0.6845$. Both predictions average across all four joint parameter possibilities, rather than selecting one parameter pair.

Interpreting Posterior Predictions on a Dose Grid

A trial often has only a discrete set of candidate doses. Once $P_j^{\mathrm{post}}$ is calculated at each score, a target-based rule can compare those probabilities with the prespecified toxicity target. For example, the dose with posterior probability closest to a target is selected if that is the protocol's rule and no safety restriction overrides it. The next example uses the posterior from the finite-grid calculation; it illustrates the comparison, not a universal selection rule.

Worked Example: Comparing Dose Scores With a Target

Suppose the available hypothetical dose scores are $x=-1,0,1$, and the target toxicity probability is $0.30$. Use the four posterior weights $10/29,5/29,10/29,4/29$ from the preceding calculation. At score $-1$, the four conditional probabilities in the same parameter order are $0.1,1/28,0.25,$ and $0.1$, since the corresponding linear predictors are $-2\ln 3,-3\ln 3,-\ln 3,$ and $-2\ln 3$.

The posterior prediction at $x=-1$ is $ (10/29)(0.1)+(5/29)(1/28)+(10/29)(0.25)+(4/29)(0.1)\approx0.1406$. The already calculated predictions at $x=0$ and $x=1$ are $0.3707$ and $0.6845$. Their absolute distances from the target are approximately $0.1594$, $0.07069$, and $0.3845$, respectively. Thus, under a rule that selects the available dose with posterior predictive probability closest to $0.30$, score $x=0$ is selected. This conclusion is conditional on that rule and does not replace protocol-specific safety constraints.

The target comparison is based on posterior predictive probabilities, not on a claim that the selected dose has exactly the target toxicity probability. Here, for example, the selected score has an estimated posterior predictive probability of about $0.3707$, which is near but above the target.

Sequential Updating and Practical Computation

When a new outcome is observed, the posterior is updated by multiplying its current density by the new participant's Bernoulli likelihood contribution and renormalizing. The same procedure works whether the current posterior is represented by a grid, a numerical approximation, or simulation draws with appropriate updating.

Worked Example: Updating After Another Toxicity

Continue the hypothetical finite-grid calculation and suppose a second participant treated at $x=1$ experiences toxicity. Multiply each current posterior weight by its conditional probability at $x=1$, namely $0.5,0.75,0.75,$ and $0.9$. The unnormalized values are $5/29,3.75/29,7.5/29,$ and $3.6/29$; their sum is $19.85/29=397/580$. Normalizing gives updated weights $100/397,75/397,150/397,$ and $72/397$.

At $x=0$, the updated predictive probability is $ (100/397)(0.25)+(75/397)(0.25)+(150/397)(0.5)+(72/397)(0.5)=154.75/397\approx0.3897$. At $x=1$, it is $ (100/397)(0.5)+(75/397)(0.75)+(150/397)(0.75)+(72/397)(0.9)=283.55/397\approx0.7142$. The additional toxicity at $x=1$ raises the posterior predictive probability at that score from approximately $0.6845$ to $0.7142$. It also changes the distribution over both intercept and slope; it is not an update to a single scalar parameter.

With continuous priors, direct grid calculations can be replaced by quadrature or posterior simulation. Numerical checks should confirm that posterior weights or density integrate to one, that probabilities stay between zero and one, and that $P_j^{\mathrm{post}}$ does not decrease with increasing dose scores when the model restricts $\beta>0$. Monte Carlo estimates should also be checked for stability across additional draws or independent runs.

Assumptions, Alternatives, and Limitations

The two-parameter logistic CRM is useful when both the dose range where toxicity rises and the sharpness of that rise should be learned from trial data. Its assumptions are consequential: the dose scores are fixed and ordered; the logistic curve is an adequate representation of dose toxicity; outcomes are conditionally independent Bernoulli observations; and the positive-slope restriction is appropriate. The prior must also be specified on the joint parameter space, with its scale and dependence structure understood.

A practical alternative is the one-parameter logistic CRM with a fixed intercept, which leaves less curve flexibility and has fewer uncertain parameters. A skeleton-based one-parameter CRM similarly concentrates uncertainty in a single parameter that maps the dose skeleton to probabilities. These simpler approaches can be easier to calibrate and may be more stable with limited data, but they cannot update location and slope independently in the way the two-parameter model can. The two-parameter version should not be treated as automatically superior: additional flexibility is useful only if supported by reasonable priors and adequate trial information.

A posterior may be highly sensitive to the prior, particularly when only a few outcomes have accrued or observations cover a narrow range of doses. Posterior dependence between $\alpha$ and $\beta$, extreme parameter values, or markedly different predictions under plausible priors are warnings to examine the model and its operating characteristics. The logistic form also imposes a smooth monotone curve; it may not represent abrupt changes or non-monotone toxicity patterns. Finally, posterior toxicity probabilities are model-based estimates, not guarantees of the true risk at a dose.

Common mistake: Evaluating the logistic curve at the posterior mean parameters and calling that the posterior toxicity probability. The correct quantity is the posterior average of $p_j(\alpha,\beta)$. Since the logistic function is nonlinear, averaging its values generally differs from applying it to averaged parameters. Integrate or average the conditional probabilities over the joint posterior.

Check Your Understanding

Answer each question, then compare with the solutions below.

In the model $p_j(\alpha,\beta)=\{1+\exp[-(\alpha+\beta x_j)]\}^{-1}$, what restriction on $\beta$ makes toxicity probability increase with dose score?
Write the likelihood contribution from one participant with no toxicity at score $x_j$.
Why is the posterior predictive toxicity probability not generally the logistic curve evaluated at the posterior mean parameters?
In the finite-grid example, what is the posterior predictive probability at $x=1$ after the additional toxicity, to four significant figures?

Solutions. (1) Require $\beta>0$, so the linear predictor and logistic probability increase with $x_j$. (2) The contribution is $1-p_j(\alpha,\beta)$, because $y=0$. (3) The logistic function is nonlinear, so the posterior average of its values need not equal its value at the posterior means; the predictive probability averages over the full joint posterior. (4) The updated calculation is $283.55/397\approx0.7142$, conditional on the stated finite prior, model, and observed outcomes.

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