Why the Adaptation Rule Belongs in the Likelihood
In an adaptive trial, later assignments, sample size, or continuation decisions can depend on outcomes already recorded. The likelihood must describe the data that could have been observed under the design actually used. A useful starting point is to separate the outcome model from the rule that determines what happens next. This prevents two opposite errors: treating the observed sample as if it came from a fixed design when it did not, and multiplying by a design probability that is constant in the parameter when forming an ordinary likelihood.
Let \(\theta\) denote the parameter of interest, and let \(H_t\) denote the recorded history through stage \(t\), including observed outcomes and assignments. Write \(A_{t+1}\) for the next assignment or design action. The outcome model specifies the distribution of the next outcomes given the history and action; the adaptation rule specifies how \(A_{t+1}\) is chosen from \(H_t\). The likelihood is the joint probability or density of the observed history, regarded as a function of \(\theta\).
The chain rule gives the factorization of a sequential likelihood. If the design chooses action \(A_{t+1}\) with known probability \(q(A_{t+1}\mid H_t)\), and the next outcomes have density \(f_\theta(Y_{t+1}\mid H_t,A_{t+1})\), then each stage contributes both factors to the joint model. For a deterministic rule \(A_{t+1}=g(H_t)\), the action factor is an indicator: it is one for histories consistent with the rule and zero otherwise. These expressions assume the outcome model correctly describes outcomes conditional on the recorded history and assigned treatment, and that the design rule is correctly specified. If the rule uses additional randomization, its probabilities must also be specified.
For a deterministic rule, replace \(q(a_{t+1}\mid h_t)\) by \(\mathbf{1}\{a_{t+1}=g(h_t)\}\). The factorization does not require outcomes from different stages to be unconditionally independent; it uses their conditional distributions given the history. When the design probabilities are known and do not involve \(\theta\), they are constant as functions of \(\theta\) for the observed history. They can therefore be omitted when maximizing the ordinary full-data likelihood for \(\theta\), but they still describe the joint sampling process and can matter for a likelihood conditioned on selection or adaptation.
Full Data, Conditioning, and the Normalizing Factor
There are two legitimate ways to use a likelihood after adaptation, and they answer different questions. The full-data likelihood uses all observed outcomes and the design rule that generated the actions. A conditional likelihood instead asks about the distribution of some data after conditioning on a stated event or on an observed part of the history. That conditioning changes the probability model.
For an event \(E\) determined by the recorded data, the conditional density of data \(Z=z\) given \(E\) is the joint density restricted to \(E\), divided by the probability of \(E\). The denominator generally depends on \(\theta\). Thus, when the analysis conditions on the fact that an adaptation event occurred, omitting this denominator does not produce the conditional likelihood.
In contrast, if the interim data \(X=x\) are treated as fixed and inference uses only later outcomes \(Y\), the relevant conditional likelihood is \(f_\theta(y\mid X=x,A=a)\). It includes no factor for the probability of the observed interim data because that is not the data being used in this conditional analysis. It is still essential to state what is conditioned on. The introductory material on conditioning on adaptation events establishes why conditioning choices change the sampling distribution; here the practical task is to construct the corresponding likelihood correctly.
Worked Example: A Normal Outcome With Adaptive Continuation
Suppose a hypothetical trial records \(X_1\sim N(\theta,0.25)\). It continues to one additional observation if \(X_1\geq 0.2\). Conditional on continuation, \(X_2\sim N(\theta,0.25)\), independently of \(X_1\). The observed values are \(x_1=0.3\) and \(x_2=0.6\). The observed rule is consistent with continuation because \(0.3\geq0.2\).
The normal density with variance \(0.25\) has standard deviation \(0.5\). Under the stated independence model, the full likelihood for these two observed outcomes is
At \(\theta=0\), the squared deviations sum to \(0.3^2+0.6^2=0.09+0.36=0.45\). Therefore
This is the joint density of the observed outcomes, in inverse squared outcome units. The deterministic continuation rule contributes an indicator equal to one for this observed history, so it does not change the displayed likelihood. It would be incorrect to pretend that two observations were fixed in advance if the analysis is meant to represent the complete sampling process; the rule explains why the second observation exists.
If the analysis instead conditions on continuation, then \(P_\theta(X_1\geq0.2)=1-\Phi((0.2-\theta)/0.5)\), where \(\Phi\) is the standard normal cumulative distribution function. The conditional density of both values given continuation is the full density divided by this probability. If one conditions on the exact interim value \(X_1=0.3\) and uses only the new observation, the conditional likelihood is instead the density of \(X_2=0.6\), namely \(\{1/\sqrt{2\pi(0.25)}\}\exp[-(0.6-\theta)^2/0.5]\). These are distinct analyses with distinct data bases.
Worked Example: Binomial Outcomes With Randomized Continuation
In a hypothetical two-stage trial, 20 patients in stage one yield 10 responses. Given that history, the design continues with probability \(0.7\), using a parameter-free randomization. If it continues, 12 more patients yield 7 responses. Assume all responses are independent Bernoulli outcomes with common response probability \(p\), conditional on the design and treatment. The observed continuation decision is therefore possible under the rule.
The two observed stage counts have binomial coefficients \(\binom{20}{10}=184756\) and \(\binom{12}{7}=792\). The continuation probability is also part of the joint probability of this observed history. Thus
The coefficient product is \(184756\times792=146326752\). At \(p=0.6\), the likelihood is
This is the joint probability of the two stage counts and the observed continuation decision. Because \(0.7\) is constant in \(p\), it can be dropped for ordinary likelihood-based estimation of \(p\) from the full observed history. If the analysis conditions on continuation, however, the normalizing probability must be included. In this example the continuation probability is \(0.7\) at every possible stage-one history, so conditioning on continuation alone divides the joint probability by \(0.7\). If continuation probability varied with the stage-one response count, the conditional distribution of that count among continued trials would also be altered.
Conditioning Correctly: A Truncated Normal Illustration
A small example isolates the normalization issue. Suppose a hypothetical interim statistic \(X\) has distribution \(N(\theta,1)\), and the trial continues only when \(X>0\). The observed value is \(x=0.5\). The unconditional likelihood contribution is the normal density at \(0.5\). For an analysis explicitly conditional on continuation, the likelihood contribution is that density divided by the probability of continuation.
Worked Example: Likelihood Given a Continuation Threshold
The unconditional density is \(f_\theta(0.5)=\phi(0.5-\theta)\), where \(\phi\) is the standard normal density. Since \(X>0\) has probability \(\Phi(\theta)\) under \(X\sim N(\theta,1)\), the conditional likelihood is
At \(\theta=0\), \(\phi(0.5)\approx0.3521\) and \(\Phi(0)=0.5\), giving \(L_c(0;0.5)\approx0.7041\) per unit of \(X\). At \(\theta=0.5\), \(\phi(0)=0.3989\) and \(\Phi(0.5)\approx0.6915\), giving \(L_c(0.5;0.5)\approx0.5769\) per unit. The normalizing probability varies with \(\theta\), so it changes the relative likelihood across parameter values. Using only \(\phi(0.5-\theta)\) would be the unconditional likelihood, not the likelihood conditional on having crossed the threshold.
Practical Checks and Common Failure Modes
Before using a likelihood from an adaptive trial, make the observation process explicit. List which outcomes were observed, which actions were taken, and what history determined each action. Then write the outcome densities in their chronological conditional form. This often exposes missing factors, impossible histories, or a mismatch between the assumed model and the actual rule.
- Dropping the interim outcomes: If the full-data analysis uses interim and later outcomes, their outcome-model contributions remain in the likelihood. Dropping them is appropriate only when the analysis deliberately conditions on them or otherwise excludes them from the data being used.
- Forgetting the conditional denominator: A likelihood conditional on a selection, continuation, or stopping event requires division by that event's probability under the model. The denominator can depend on \(\theta\).
- Confusing a parameter-free factor with no design rule: A known randomization probability can be omitted from an ordinary likelihood for \(\theta\) when it is constant in \(\theta\). It must not be mistaken for evidence that the design was fixed or irrelevant to a conditional analysis.
- Including outcomes that were never observed: Do not multiply by densities for hypothetical future patients as if their outcomes had been recorded. If unobserved outcomes are part of a different target likelihood, they must be integrated out under an explicit model.
- Ignoring design-dependent observation or missingness: The factorization assumes the recorded history and outcome model adequately represent what was observed. If missing outcomes or unrecorded decision inputs depend on outcomes, the analysis may require modeling those mechanisms or conditioning on additional recorded information.
The main alternatives are the full-data likelihood and a specified conditional likelihood. The full-data approach retains all recorded outcomes and models the design path; it is usually the natural likelihood for inference based on the complete observed history. The conditional approach can be appropriate when the inferential target is deliberately conditional on a selection or adaptation event, but it must use the corresponding conditional distribution. A further option is to condition on the exact interim data and analyze only later outcomes; this is a conditional analysis that deliberately does not use the interim outcomes as evidence about \(\theta\). These approaches need not give the same likelihood shape or estimate.
Likelihood construction is not a guarantee that the resulting model is adequate. Check that conditional outcome distributions match the trial's treatment assignments, that the adaptation rule is implemented as stated, and that the observed path has positive probability under the model. Sparse outcomes, misspecified outcome distributions, or an incompletely recorded rule can undermine the likelihood even if its algebra is correct. In particular, a procedure that multiplies stage-wise likelihoods must justify each stage's conditional model rather than assume independence without checking the design and data structure.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A trial continues when an interim response count is at least 4. The observed count is 5 and later outcomes are recorded. What must a full-data likelihood include?
- A randomized continuation decision has probability 0.4 at every interim history and does not involve \(\theta\). Can its factor be dropped from the ordinary full-data likelihood for estimating \(\theta\)? Does that make the rule irrelevant to a conditional analysis?
- If inference is explicitly conditional on \(X>0\), what factor must be applied to the density \(f_\theta(x)\) for an observed \(x>0\)?
- A sample-size rule leads to 30 observed outcomes, although the planned maximum was 50. Should the likelihood include density factors for the 20 unobserved outcomes?
- Explain the difference between conditioning on the exact interim value \(X=x\) and conditioning only on the event \(X>c\).
Solutions. (1) Include the outcome-model contributions for the observed interim and later outcomes, in their conditional sequence, and represent the continuation rule; for a deterministic rule its indicator is one for the observed history. (2) Yes, the factor \(0.4\) is constant in \(\theta\) and can be dropped for ordinary full-data likelihood estimation. No: a conditional analysis still has to use the correct distribution given its conditioning event. (3) Divide by \(P_\theta(X>0)\), so the conditional density is \(f_\theta(x)/P_\theta(X>0)\). (4) No. Those outcomes were not observed; include them only if a distinct model requires integrating over their possible values. (5) Conditioning on \(X=x\) fixes the interim value and uses the distribution of later data given that value; conditioning on \(X>c\) retains variation in \(X\) above the threshold and requires normalization by \(P_\theta(X>c)\).