Why Each Cohort Outcome Matters
In dose finding, a cohort may contribute several toxicity outcomes at one dose, while later cohorts receive different doses. A likelihood gathers those observations into a function of the dose-toxicity probabilities implied by a model. It makes explicit that a toxicity supports larger toxicity probabilities at the dose received, whereas a non-toxicity supports smaller ones. Neither outcome is discarded merely because a cohort is summarized by its number of toxicities.
A dose-toxicity skeleton supplies reference probabilities for ordered dose levels. To form a likelihood for a model, we need the model’s probability of toxicity at each dose. Write \(p_j(\theta)\) for that probability at dose level \(j\), under model parameter \(\theta\). The skeleton may help specify how these probabilities relate across doses, but it is not itself a set of observed outcomes. Unless a model explicitly equates its probabilities to the skeleton values, the likelihood uses \(p_j(\theta)\), not the skeleton values in place of data.
From Individual Outcomes to Cohort Likelihood
Suppose cohort \(c\) contains \(m_c\) participants, all treated at dose level \(d_c\). Let \(Y_{ci}\) be the binary outcome for participant \(i\) in cohort \(c\), and let \(r_c=\sum_{i=1}^{m_c}Y_{ci}\) be the cohort’s toxicity count. Assume, for now, that outcomes are conditionally independent given the assigned doses and \(\theta\), and that every participant at dose \(j\) has the same toxicity probability \(p_j(\theta)\). Then the likelihood of the observed, individually labelled outcomes is
The second expression follows because each toxicity at the cohort’s dose contributes one factor of \(p_{d_c}(\theta)\), and each non-toxicity contributes one factor of \(1-p_{d_c}(\theta)\). If several cohorts were treated at the same dose, their exponents add. Let \(n_j\) be the total number treated at dose \(j\), and \(y_j\) the total number with toxicity at that dose. The likelihood can then be written
These expressions give the likelihood kernel for the observed individual outcomes: a function of \(\theta\) proportional to the probability of those data. If the data are instead recorded as counts, the probability of each count includes a binomial coefficient. For a cohort count \(r_c\), that probability is
The coefficient \(\binom{m_c}{r_c}\) counts how many different arrangements of \(r_c\) toxicities among \(m_c\) participants produce the same count. It does not depend on \(\theta\), so including or omitting it does not change comparisons of likelihood values across \(\theta\). It does matter when calculating the probability of the count event itself.
Worked Likelihood Calculations
Worked Example: Two Cohorts at Different Doses
Consider hypothetical data from two cohorts of three participants each. At dose 1, one participant has toxicity and two do not. At dose 2, two have toxicity and one does not. Compare two candidate model settings: setting A gives \(p_1=1/3\) and \(p_2=2/3\); setting B gives \(p_1=0.20\) and \(p_2=0.50\). Assume independent binary outcomes conditional on dose and model setting, with a common probability within each dose.
The observed likelihood kernel is \(p_1(1-p_1)^2p_2^2(1-p_2)\): the dose 1 cohort contributes one toxicity factor and two non-toxicity factors, and the dose 2 cohort contributes two toxicity factors and one non-toxicity factor.
Under setting A, substitution gives
Under setting B,
For these observed outcomes, setting A has the larger likelihood kernel; the ratio is \(L(A)/L(B)\approx1.372\). This is a relative comparison of how well the two settings assign probability to these data. It is not a probability that setting A is true, nor a posterior probability for either setting.
Worked Example: Counts Across Repeated Cohorts
Suppose two hypothetical cohorts of three participants each receive dose 1, with counts 0/3 and 1/3 toxicities. A third cohort of three receives dose 2 and has 2/3 toxicities. Evaluate the likelihood kernel at \(p_1=0.20\) and \(p_2=0.50\). Assume all nine outcomes are independent conditional on dose and have the stated dose-specific probabilities.
At dose 1, the total is one toxicity and five non-toxicities. At dose 2, there are two toxicities and one non-toxicity. Therefore the labelled-outcome likelihood kernel is
If the observation is represented as the exact three cohort counts, the probability of that count pattern is the kernel multiplied by \(\binom{3}{0}\binom{3}{1}\binom{3}{2}=1\cdot3\cdot3=9\), giving \(0.073728\). If instead the event is only that there was one toxicity among all six participants at dose 1 and two among the three at dose 2, its probability uses \(\binom{6}{1}\binom{3}{2}=18\) and is \(0.147456\). These are probabilities of different count events. Their parameter-dependent likelihood kernels are the same because the coefficients do not involve \(p_1\) or \(p_2\).
Adaptive Dose Assignment and the Likelihood
Dose levels in an adaptive trial are chosen using accumulating information. That does not by itself invalidate the product likelihood. If the assignment rule is known and its choice at each step depends only on observed history, then for a realized sequence that follows a deterministic rule, the assignment adds no parameter-dependent factor: once the history is given, the rule determines the next dose. The likelihood for the observed outcomes, conditional on the assigned doses, is still the product of their outcome contributions.
More generally, suppose a known randomization policy assigns a dose with probability \(g_c\), conditional on the observed history before cohort \(c\), and \(g_c\) has no unknown parameter \(\theta\). The joint probability of assignments and outcomes contains the factor \(\prod_c g_c\) as well as the outcome likelihood. Since that assignment factor is constant as a function of \(\theta\), it does not change the likelihood comparison across values of \(\theta\). This reasoning does not license ignoring an assignment mechanism that depends on unknown model parameters or on unrecorded prognostic information.
Worked Example: Outcomes Along an Adaptive Path
In a hypothetical trial, two participants are treated at dose 1 and neither has toxicity. The protocol then assigns two participants to dose 2; one has toxicity and one does not. Suppose \(p_1=0.25\) and \(p_2=0.60\). Assume independent outcomes conditional on dose, complete assessment for all four participants, and a deterministic rule that assigns dose 2 after the observed dose 1 outcomes.
The outcome likelihood is
The realized assignment of the second cohort adds no further factor under this deterministic rule: the observed first-cohort outcomes already determine that assignment. If a known randomization probability of 0.40 had instead been used to assign dose 2 after that history, the joint likelihood of assignments and outcomes would have the extra factor 0.40, or \(0.05400\). That factor is constant across candidate values of \(\theta\), so the outcome likelihood kernel remains \(0.1350\) for comparing those values.
Why the Product Form Holds—and When It Does Not
The key result is a direct consequence of conditional independence. For a single participant, the Bernoulli probability of the observed outcome is \(p_j(\theta)\) if \(Y=1\), and \(1-p_j(\theta)\) if \(Y=0\). The exponent form expresses both cases at once. If all participant outcomes are conditionally independent, their joint probability is the product of these individual probabilities. Regrouping the factors by dose yields the count-based likelihood.
This derivation relies on substantive assumptions, not just algebra. The endpoint must be defined consistently, such as toxicity within a specified window; outcomes must be observed or handled by a model appropriate to their observation status; participants at a given dose must share the modelled toxicity probability, conditional on variables included in the model; and the outcomes must be conditionally independent. The dose labels and outcome records must also be correct.
If outcomes within a cohort are dependent, the product of Bernoulli terms is generally not the correct joint probability. If risk differs systematically between participants, a single \(p_j(\theta)\) may be inadequate unless relevant differences are modelled. If some participants have incomplete follow-up, treating an unobserved outcome as a non-toxicity falsely contributes \(1-p_j(\theta)\); censoring or delayed assessment requires a suitable likelihood contribution rather than that substitution.
A useful diagnostic is to compare observed cohort patterns with what the assumed model could plausibly produce, while remembering that small cohorts make such checks noisy. Look for inconsistent endpoint windows, missing or delayed assessments, patient characteristics associated with toxicity, and evidence of clustering or shared exposure. Strong departures from the assumed structure are reasons to revisit the likelihood model, not reasons to reinterpret the same product as automatically valid.
Common Pitfalls and Alternatives
| Approach or claim | What it means | Important caution |
|---|---|---|
| Individual Bernoulli likelihood | Multiplies one toxicity or non-toxicity factor for each participant. | Requires the stated conditional outcome model and independence assumption. |
| Binomial count probability | Uses the cohort toxicity count and includes a binomial coefficient. | The coefficient matters for the count-event probability, but is constant in \(\theta\). |
| Pooling across dose levels | Treats outcomes at several doses as if they shared one toxicity probability. | Usually discards the dose-specific information the model is meant to represent. |
| Likelihood versus posterior | The likelihood compares how well parameter values account for observed data. | A likelihood is not a probability distribution over \(\theta\); a prior is needed to form a posterior. |
A common mistake is to say, “There were three toxicities in nine participants, so the likelihood is one binomial likelihood with \(n=9\).” That calculation ignores dose level when the participants received different doses. The correction is to group outcomes by dose and use the appropriate \(p_j(\theta)\) for each participant. Pooling is justified only if the model genuinely assigns the same toxicity probability to the pooled observations.
The count representation is often compact and, under the shared-probability and independence assumptions, preserves the parameter-dependent part of the likelihood when outcomes are grouped by dose. Keeping cohort identities can still be useful for checking patterns, accounting for cohort-level differences, or applying a model that does not give all participants at a dose the same risk. The likelihood’s compact form is therefore a consequence of the assumptions, not a reason to overlook the trial’s structure.
Check Your Understanding
Try each question first, then check the solutions below.
- At one dose, four hypothetical participants have outcomes \(1,0,0,1\). Write their likelihood contribution using the dose toxicity probability \(p\).
- Two doses have probabilities \(p_1(\theta)\) and \(p_2(\theta)\). At dose 1, five participants have one toxicity; at dose 2, three participants have two toxicities. Write the likelihood kernel.
- Explain why a binomial coefficient can be omitted when comparing likelihood values across \(\theta\), but not when reporting the probability of a specific count event.
- A participant’s toxicity status is unobserved at the end of follow-up. Why is it generally wrong to count that participant as a non-toxicity?
Solutions. (1) The two toxicities contribute \(p^2\), and the two non-toxicities contribute \((1-p)^2\), so the contribution is \(p^2(1-p)^2\). (2) There are one toxicity and four non-toxicities at dose 1, and two toxicities and one non-toxicity at dose 2, so the kernel is \(p_1(\theta)[1-p_1(\theta)]^4p_2(\theta)^2[1-p_2(\theta)]\). (3) The coefficient counts arrangements of outcomes and is constant in \(\theta\), so it cannot change the relative likelihood across parameter values. It is part of the probability of the count event and must be included for that probability. (4) A non-toxicity means the endpoint was assessed and did not occur within its defined window. An unobserved status does not establish that outcome, so it needs an appropriate missing-data or follow-up likelihood treatment.