Logistic CRM With a Fixed Intercept

Why Fix the Intercept?

A CRM represents toxicity risk at each dose as a function of a shared model parameter. In the one-parameter logistic version considered here, the intercept is fixed and the parameter controls how rapidly toxicity changes across doses. Fixing the intercept gives the model a specified reference risk; the dose scores then determine how each dose lies relative to that reference. This is a different parameterization from the power CRM, even though both can use one parameter to update a dose-toxicity curve.

The distinction matters in practice. A dose score is not itself a toxicity probability, and the logistic parameter is not interpreted as a power applied to a probability. Instead, the score is a covariate on the log-odds scale. A sensible choice of score ordering and parameter support is therefore needed before the model can be used for dose finding.

Definition: A one-parameter logistic CRM with a fixed intercept specifies the probability of dose-limiting toxicity at dose $j$ as $$ p_j(\beta)=\frac{\exp(a+\beta x_j)}{1+\exp(a+\beta x_j)}. $$ Here $a$ is a fixed intercept, $x_j$ is the fixed dose score for dose $j$, and $\beta$ is the unknown model parameter. Equivalently, $\operatorname{logit}\{p_j(\beta)\}=a+\beta x_j$, where $\operatorname{logit}(p)=\log\{p/(1-p)\}$.

The outcome is assumed to be binary for each participant: toxicity occurs or does not occur within the protocol-defined assessment window. Conditional on $\beta$ and the assigned doses, participant outcomes are modeled as independent Bernoulli observations with the corresponding probabilities $p_j(\beta)$. The model also treats the specified scores and intercept as fixed, rather than estimating them from the trial. These assumptions describe the likelihood; they do not replace protocol rules for delayed outcomes, safety exclusions, or dose assignment.

Choosing the Scores and Reading the Parameter

A common construction starts with a skeleton $q_1,\ldots,q_J$, where $q_j$ is the prior reference toxicity probability for dose $j$. Suppose the target dose has skeleton probability $q_*$. Set the intercept to $a=\operatorname{logit}(q_*)$, and set each score to the difference between the skeleton log-odds and the intercept:

a=\operatorname{logit}(q_*), \qquad x_j=\operatorname{logit}(q_j)-a.

This makes $x_*=0$ at the target dose, and at $\beta=1$ the model reproduces the skeleton exactly: $p_j(1)=q_j$. In this construction, $\beta$ changes the slope of log-odds across scores while the target-dose probability remains fixed at $q_*$, since $p_*(\beta)=\operatorname{logit}^{-1}(a)=q_*$. Thus a fixed intercept has a concrete consequence: evidence updates the curve through its slope, not through a freely moving reference risk.

A useful mathematical check is the derivative with respect to the score:

\frac{\partial p_j(\beta)}{\partial x_j} =\beta\,p_j(\beta)\{1-p_j(\beta)\}.

For $0<p_j(\beta)<1$, this derivative is positive when $\beta>0$. Therefore, if dose scores increase with dose and the prior restricts $\beta$ to positive values, the model orders toxicity probabilities in the same way as the scores. If scores decrease with dose, the direction is reversed. If $\beta$ can be negative, the model can reverse the ordering; that is usually inconsistent with an intended monotone dose-toxicity relationship. A zero value makes all dose risks equal.

Worked Example: Building a Skeleton-Based Logistic Curve

Consider four hypothetical doses with skeleton probabilities $0.05,\ 0.12,\ 0.25,\ 0.40$, and use the second dose as the target reference. Set $a=\operatorname{logit}(0.12)$. The logit is $\log\{p/(1-p)\}$, so $a=\log(0.12/0.88)=-1.992430$. The scores are each skeleton logit minus this intercept:

(x_1,x_2,x_3,x_4) =(-0.952009,\ 0,\ 0.893818,\ 1.586965).

For instance, $x_1=\log(0.05/0.95)-a=-2.944439-(-1.992430)=-0.952009$, while $x_4=\log(0.40/0.60)-a=-0.405465-(-1.992430)=1.586965$. The scores increase across doses, and the target score is zero.

At $\beta=1$, the probabilities are the skeleton by construction. To see how the curve changes, take $\beta=1.5$. The linear predictors $a+\beta x_j$, in dose order, are approximately $-3.420444,\ -1.992430,\ -0.651703,\ 0.388018$. Applying the inverse-logit to each gives toxicity probabilities approximately $0.0317,\ 0.1200,\ 0.3426,\ 0.5958$. For example, the fourth probability is

p_4(1.5)=\frac{\exp(0.388018)}{1+\exp(0.388018)} \approx 0.5958.

The reference dose remains at $0.12$, while probabilities below and above it move away from that reference as the slope increases. This shape is a consequence of the centered scores, not a universal feature of every possible fixed-intercept score construction.

Updating the Parameter With Trial Outcomes

Once participants have been treated and their binary outcomes are observed, the likelihood is obtained by multiplying their Bernoulli probabilities. If participant $i$ receives dose $d_i$ and has outcome $y_i$, where $y_i=1$ denotes toxicity and $y_i=0$ denotes no toxicity, then

L(\beta)=\prod_{i=1}^{n} p_{d_i}(\beta)^{y_i} \{1-p_{d_i}(\beta)\}^{1-y_i}.

A prior distribution for $\beta$ is combined with this likelihood using Bayes' rule. For a discrete prior with candidate values $\beta_k$ and prior probabilities $\pi_k$, the posterior probability at candidate $k$ is proportional to $\pi_kL(\beta_k)$. This is the same basic posterior updating logic used in other CRM parameterizations; the important model-specific input here is the logistic probability at each dose and candidate parameter value.

Worked Example: A Discrete Posterior Update

Suppose a hypothetical trial uses a fixed intercept $a=\operatorname{logit}(0.25)=-1.098612$. The reference dose has score $0$, and a higher dose has score $x=\log(3)=1.098612$. Assign equal prior probabilities to $\beta=0,\ 1,\ 2$. At the higher dose, the three toxicity probabilities are $0.25,\ 0.50,\ 0.75$, respectively, because the linear predictors are $-1.098612,\ 0,\ 1.098612$.

Assume three participants have fully assessed outcomes at that higher dose, with two toxicities and one non-toxicity. The binomial coefficient is common to all candidate values, so use the proportional likelihood $p^2(1-p)$. The likelihoods are

\begin{aligned} L(0)&\propto (0.25)^2(0.75)=0.046875,\\ L(1)&\propto (0.50)^2(0.50)=0.125000,\\ L(2)&\propto (0.75)^2(0.25)=0.140625. \end{aligned}

With equal prior weights, their sum is $0.312500$, so the normalized posterior probabilities are

P(\beta=0\mid\text{data})=0.150,\qquad P(\beta=1\mid\text{data})=0.400,\qquad P(\beta=2\mid\text{data})=0.450.

The outcomes make larger values of $\beta$ more plausible in this example because the observed toxicity frequency is high at a dose whose risk increases with $\beta$. The posterior is about the model parameter, not a probability that any particular dose is safe. Dose selection still requires applying the trial's prespecified decision rule to the posterior dose-toxicity curve.

How It Differs From the Power CRM

The power CRM starts with skeleton probabilities $q_j$ and models dose $j$'s toxicity probability as $q_j^\alpha$, typically with $\alpha>0$. Its parameter acts directly on probabilities: $\alpha=1$ reproduces the skeleton. In the fixed-intercept logistic model, $\beta$ acts linearly on dose scores on the log-odds scale, and the chosen score construction makes $\beta=1$ reproduce the skeleton. Equal numerical parameter values therefore do not have equal interpretations across the two models.

When $q_j$ increases with dose and $\alpha>0$, the power model preserves that ordering. In the logistic model, ordering is determined by the dose scores and the sign of $\beta$. With increasing scores and positive $\beta$, it also preserves the ordering. But the curves can differ substantially in shape, even when both reproduce the same skeleton at their reference parameter values.

Worked Example: Comparing Predictions at the Same Skeleton

Use the four-dose skeleton from the first example. At $\beta=1.5$, the fixed-intercept logistic predictions were approximately $0.0317,\ 0.1200,\ 0.3426,\ 0.5958$. Under the power model at $\alpha=1.5$, the predictions are the skeleton probabilities raised to $1.5$:

\begin{aligned} 0.05^{1.5}&=0.01118, & 0.12^{1.5}&=0.04157,\\ 0.25^{1.5}&=0.1250, & 0.40^{1.5}&=0.2530. \end{aligned}

Both sets are increasing across dose, but they differ at every dose except none in this particular calculation: the power model changes even the target dose probability, whereas the centered fixed-intercept logistic model keeps it at $0.12$. At $\beta=1$ or $\alpha=1$, both recover the common skeleton. Thus the comparison concerns the parameter-dependent curve away from the skeleton, not which model is inherently correct. Model choice should reflect how the trial team wants deviations from the elicited skeleton to behave.

One way to understand the contrast is to consider limiting values. In the power model, as $\alpha$ approaches zero, every $q_j$ strictly between zero and one approaches a predicted probability of one. In the centered logistic construction, as $\beta$ approaches zero, every dose probability approaches the common value $\operatorname{logit}^{-1}(a)=q_*$. These limits reveal distinct structural assumptions. Neither model can freely move the reference risk and independently change the slope; that would require another parameter and is outside this one-parameter specification.

Diagnostics, Limitations, and a Common Pitfall

Before interpreting a fitted curve, check that the numerical score ordering matches the intended clinical dose ordering and that the prior support for $\beta$ preserves monotonicity. Evaluate predicted probabilities across the full plausible range of $\beta$, not just at the prior's central value. Probabilities extremely close to zero or one indicate near-saturation: additional changes in the linear predictor then have little effect on the probability scale. Also check prior predictive curves against clinical expectations. If credible prior values imply implausible toxicity risks or a sharply distorted curve, reconsider the score scale or prior rather than assuming the outcomes will repair the problem.

A score of zero deserves particular attention. At any dose with $x_j=0$, the probability is fixed at $\operatorname{logit}^{-1}(a)$ for every $\beta$. Outcomes at that dose contribute to the likelihood, but by themselves cannot distinguish candidate values of $\beta$. More generally, observations concentrated at scores close to zero may provide limited information about the slope. A posterior that remains close to its prior can reflect weak information in the observed dose locations, not a computational failure.

The fixed intercept is also a substantive restriction. If the true curve differs from the chosen reference risk or has a shape not expressible by changing one slope parameter, the model may fit poorly. The model imposes a particular relationship among risks at all doses; a good fit at one dose does not validate that relationship elsewhere. The skeleton sensitivity analysis and dose-selection safeguards remain relevant, and the logistic specification does not eliminate uncertainty about model form.

Pitfall: confusing the score with risk. A score such as $x_j=1.59$ is not a 159% toxicity probability. It enters the linear predictor, and its effect depends on $a$ and $\beta$.
Pitfall: assuming any score sequence preserves dose ordering. The correction is to verify the score order and parameter support. Increasing scores imply increasing risks only when $\beta>0$.
Pitfall: treating $\beta$ as interchangeable with the power parameter. The logistic parameter changes log-odds per score unit; the power parameter exponentiates probabilities. Their scales and effects differ.
Pitfall: believing the fixed reference dose is learned from the data. If its score is zero, its risk is fixed by $a$; updating $\beta$ cannot move it.

Key takeaway: A fixed-intercept logistic CRM models toxicity log-odds as $a+\beta x_j$. With logit-centered skeleton scores, $\beta=1$ reproduces the skeleton, and positive $\beta$ preserves increasing-dose ordering when scores increase. Its slope interpretation and fixed reference risk distinguish it from the power CRM.

Check Your Understanding

Answer each question, then compare with the solutions.

For the model $p_j(\beta)=\operatorname{logit}^{-1}(a+\beta x_j)$, what does the condition $x_j=0$ imply about the dose probability?
If dose scores increase with dose, what restriction on $\beta$ ensures the model's toxicity probabilities also increase with dose?
A skeleton probability is $0.20$, and the fixed intercept is its logit. What score makes this dose the reference dose?
At a dose with probability $0.60$, four participants have three toxicities and one non-toxicity. What is the likelihood contribution, omitting the common binomial coefficient?
State one structural difference between the fixed-intercept logistic CRM and the power CRM.

Solutions. (1) Its probability is $\operatorname{logit}^{-1}(a)$, independent of $\beta$. (2) Require $\beta>0$; then increasing scores give increasing probabilities. (3) Set $x_j=0$, consistent with $x_j=\operatorname{logit}(0.20)-a=0$. (4) The contribution is $0.60^3(1-0.60)=0.0864$. (5) The logistic parameter changes log-odds linearly with score and a zero-score probability stays fixed; the power parameter raises each skeleton probability to a power and generally changes the reference probability too.