The One-Parameter Power CRM

From a Dose Skeleton to a Toxicity Model

In a dose-finding study, the toxicity probability at each candidate dose is unknown. The continual reassessment method (CRM) represents those probabilities using a model rather than estimating each dose’s rate independently. In the one-parameter power CRM, a prespecified dose-toxicity skeleton supplies the relative pattern across doses, and one unknown parameter transforms that pattern. The transformation is simple, but its consequences matter: it determines how the model can respond to observed outcomes and which toxicity curves it can represent.

Let there be $J$ ordered dose levels, from lowest to highest. Write $p_{0j}$ for the skeleton probability at dose $j$, and let $\alpha$ be the unknown model parameter. The skeleton is fixed for a given model specification; it is not a set of observed toxicity rates. Each $p_{0j}$ is a starting probability that encodes the anticipated toxicity pattern before the study data are used.

Definition: The one-parameter power CRM specifies the probability of dose-limiting toxicity at dose $j$, conditional on the model parameter $\alpha$, as $$ p_j(\alpha)=p_{0j}^{\exp(\alpha)}, \qquad j=1,\ldots,J. $$ Here $p_{0j}$ is the fixed skeleton probability and $\alpha$ is a single unknown real-valued parameter shared across all dose levels. The quantity $\exp(\alpha)$ is positive, so the power applied to every skeleton probability is positive.

For a standard specification, the skeleton values satisfy $0<p_{01}<p_{02}<\cdots<p_{0J}<1$. Each is a probability strictly between zero and one, and higher doses have higher skeleton probabilities. The outcome being modeled must also be defined consistently: for example, whether toxicity means a dose-limiting toxicity observed within a prespecified assessment window. A participant’s outcome is coded as toxicity or no toxicity for that window.

The model assumes that participants treated at dose $j$ have the same toxicity probability $p_j(\alpha)$, conditional on $\alpha$, and that their toxicity outcomes are conditionally independent given their assigned doses and $\alpha$. It uses one common parameter to relate all doses to the skeleton. In particular, it does not include separate parameters for each dose, participant-level risk factors, or a different time-to-toxicity distribution. These are substantive modeling restrictions, not consequences of the formula.

What the Parameter Does

The value $\alpha=0$ gives $\exp(\alpha)=1$, so the model probabilities equal the skeleton exactly. Positive and negative values move the probabilities in opposite directions. Because skeleton values lie between zero and one, raising them to a power greater than one makes them smaller; raising them to a positive power less than one makes them larger.

Interpretation: If $\alpha=0$, the toxicity curve is the skeleton. If $\alpha>0$, then $p_j(\alpha)<p_{0j}$ at every dose. If $\alpha<0$, then $p_j(\alpha)>p_{0j}$ at every dose. The same parameter acts on every dose; it shifts the whole skeleton rather than independently adjusting its entries.

This shared movement is a defining feature of the one-parameter model. It does not mean that all doses change by the same number of percentage points. Rather, each skeleton probability is transformed by the same exponent, and the resulting absolute change depends on its starting value.

Worked Example: A Parameter Above Zero

Suppose a hypothetical four-dose study uses the skeleton $0.05, 0.12, 0.25, 0.40$, and consider the candidate value $\alpha=\ln(2)$. Then $\exp(\alpha)=2$, so each skeleton probability is squared.

At dose 1, $p_1(\alpha)=0.05^2=0.0025$. At dose 2, $p_2(\alpha)=0.12^2=0.0144$. At dose 3, $p_3(\alpha)=0.25^2=0.0625$. At dose 4, $p_4(\alpha)=0.40^2=0.1600$.

The resulting probabilities are $0.0025, 0.0144, 0.0625,$ and $0.1600$, all below their corresponding skeleton values. These are probabilities, not observed rates from a real study; the example illustrates what the model implies at this specified parameter value.

Worked Example: A Parameter Below Zero

Consider a different hypothetical skeleton, $0.04, 0.16, 0.36, 0.64$, and set $\alpha=-\ln(2)$. Now $\exp(\alpha)=0.5$, so the transformation takes the square root of each skeleton probability.

The probabilities are $p_1(\alpha)=\sqrt{0.04}=0.20$, $p_2(\alpha)=\sqrt{0.16}=0.40$, $p_3(\alpha)=\sqrt{0.36}=0.60$, and $p_4(\alpha)=\sqrt{0.64}=0.80$.

Each transformed probability exceeds its skeleton value, as expected for $\alpha<0$. The example also shows why the parameter’s effect should not be described as adding a fixed amount to every dose: the changes are $0.16, 0.24, 0.24,$ and $0.16$, respectively.

Why Dose Ordering Is Preserved

The model assumes an increasing skeleton, and the power transformation preserves that ordering for every finite value of $\alpha$. To see why, set $c=\exp(\alpha)$, so $c>0$. For a fixed $c$, the transformation of a skeleton probability $u$ is $f(u)=u^c$. Its derivative is positive for $u>0$:

f'(u)=c u^{c-1}>0.

Thus $f$ is strictly increasing on the interval of possible skeleton probabilities. If $p_{0j}<p_{0k}$, applying $f$ to both sides gives $p_{0j}^{c}<p_{0k}^{c}$, or $p_j(\alpha)<p_k(\alpha)$. This justifies the implication: an increasing skeleton produces an increasing modeled toxicity curve at every parameter value. If two skeleton entries are equal, their modeled probabilities remain equal; the model cannot separate them.

There is also a useful way to see the direction of the parameter effect. Differentiating with respect to $\alpha$ gives

\frac{\partial p_j(\alpha)}{\partial \alpha} = p_{0j}^{\exp(\alpha)}\exp(\alpha)\ln(p_{0j})<0,

because $p_{0j}$ is between zero and one, so $\ln(p_{0j})<0$. Increasing $\alpha$ therefore decreases every modeled toxicity probability. This is a local rate-of-change result; it does not say that the decreases are equal across doses.

Connecting the Model to Cohort Outcomes

Suppose $n_j$ participants are treated at dose $j$, and $y_j$ experience the defined toxicity outcome. Under the model’s conditional independence assumption, the probability contribution from those outcomes, as a function of $\alpha$, is proportional to

L_j(\alpha) = p_j(\alpha)^{y_j} \left[1-p_j(\alpha)\right]^{n_j-y_j}.

Across dose levels, the likelihood is the product of these contributions. If outcomes are recorded as counts, binomial coefficients can be included in the binomial likelihood; they do not depend on $\alpha$, so they do not change comparisons of likelihood across parameter values. The likelihood connects the specified probability model to observed outcomes. It does not, by itself, specify a prior distribution or a dose-selection rule.

Worked Example: Comparing Two Parameter Values Against Cohort Data

In a hypothetical study, use the skeleton $0.10, 0.20, 0.40$. Suppose three participants are observed at each dose, with $0, 1,$ and $2$ toxicities, respectively. Compare the likelihood at $\alpha=0$ with the likelihood at $\alpha=\ln(2)$. The conditional independence and common-probability assumptions are being used here; the calculation does not establish that either assumption is true.

At $\alpha=0$, the model probabilities are $0.10, 0.20, 0.40$. Omitting the parameter-independent binomial coefficients, the likelihood is

$$ L(0) =(0.10)^0(0.90)^3 (0.20)^1(0.80)^2 (0.40)^2(0.60)^1 =0.729(0.128)(0.096) =0.008958. $$

At $\alpha=\ln(2)$, the skeleton values are squared, giving model probabilities $0.01, 0.04, 0.16$. Therefore

L(\ln 2) =(0.01)^0(0.99)^3 (0.04)^1(0.96)^2 (0.16)^2(0.84)^1 =0.970299(0.036864)(0.021504) \approx 0.0007692.

For these data, the likelihood at $\alpha=0$ is about $0.008958/0.0007692=11.65$ times the likelihood at $\alpha=\ln(2)$. The observed toxicity counts are relatively high for the smaller probabilities implied by the latter value. This comparison concerns only these two specified parameter values; it is not a dose recommendation or a probability that either value is correct.

Assumptions, Diagnostics, and Alternatives

A power CRM is most interpretable when the skeleton has been specified for the actual dose grid and the toxicity outcome is consistently defined. Its main structural assumption is that one exponent can adequately adjust all doses together. That is a strong form of borrowing information: outcomes at different doses inform a common parameter because the model ties their probabilities to one shared transformation.

The model can be a poor description if the true pattern departs substantially from the skeleton’s shape, even when the dose ordering is correct. For example, a skeleton that rises too slowly at low doses and too quickly at high doses may not be repaired by a single exponent: the parameter changes every dose through the same power transformation. Other warning signs include a proposed skeleton that is not increasing despite an assumed increasing dose-toxicity relationship, probabilities outside the interval from zero to one, or toxicity outcomes defined inconsistently across participants. Such problems call for checking the specification and data, not simply interpreting a fitted parameter more confidently.

Compared with estimating a separate toxicity probability at every dose, the one-parameter model is more constrained and can share information across dose levels. That constraint can stabilize estimates when data are sparse, but it can also force the fitted curve to follow an unsuitable shape. Rule-based escalation methods avoid specifying a full parametric toxicity curve, but they use different decision structures and do not provide this same one-parameter probability model. Neither approach is universally preferable: the choice depends on the study’s aims, available information, and the credibility of its assumptions.

A common mistake is to say, “A positive $\alpha$ raises toxicity because it raises the exponent.” That reasoning treats a larger exponent as if it always made a number larger. Here the base is a probability below one: for $0<u<1$, increasing the exponent decreases $u$ raised to that exponent. The correction is to check the base and exponent together: $\alpha>0$ means $\exp(\alpha)>1$, so $p_{0j}^{\exp(\alpha)}<p_{0j}$.

Key takeaway: The one-parameter power CRM is $p_j(\alpha)=p_{0j}^{\exp(\alpha)}$. The skeleton sets the ordered pattern, while $\alpha$ moves all probabilities in a linked way: zero reproduces the skeleton, positive values lower its probabilities, and negative values raise them.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

For a skeleton value $p_{0j}=0.25$ and $\alpha=\ln(2)$, calculate $p_j(\alpha)$ and state whether it is above or below the skeleton value.
For $p_{0j}=0.09$ and $\alpha=-\ln(3)$, calculate $p_j(\alpha)$.
Why does an increasing skeleton remain increasing for every finite $\alpha$?
State two assumptions used when writing the cohort likelihood from toxicity counts.
Give one advantage and one limitation of using a shared parameter instead of a separate probability at every dose.

Solutions. (1) Since $\exp(\ln 2)=2$, $p_j=0.25^2=0.0625$, which is below $0.25$. (2) Since $\exp(-\ln 3)=1/3$, $p_j=0.09^{1/3}\approx0.4481$, which is above $0.09$. (3) For $c=\exp(\alpha)>0$, the transformation $f(u)=u^c$ has derivative $c u^{c-1}>0$ for $u>0$, so it preserves strict ordering. (4) For example, the model assumes a common toxicity probability for participants at the same dose, conditional independence given dose and $\alpha$, and a consistently defined toxicity outcome; any two suffice. (5) A shared parameter can borrow information across doses and reduce the number of free parameters, but it restricts the curve to shapes obtainable by transforming the specified skeleton.

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