Why the Likelihood Changes When We Condition
A group sequential trial may stop because an interim statistic crosses a boundary. Once that happens, the observed estimate and the fact that the trial stopped are linked: the stopping event selects which data paths are observed. This makes it important to distinguish two questions. What parameter value makes the observed data most likely under the sampling model? And what parameter value makes those data most likely among paths satisfying the observed stopping event?
The first question uses the ordinary likelihood for the observed data. The second uses a conditional likelihood, which divides the data density by the probability of the event on which we condition. That probability depends on the parameter in many designs, so the two likelihoods need not have the same maximizer.
Here \(f_\theta(d)\) is a density or probability mass under the sampling model. If the data are continuous, the conditional density is understood over the region defining \(E\). The event must be specified precisely: for example, “cross the upper boundary at the first look” is not the same event as “stop at the first look,” which might include crossing either boundary.
Deriving the Conditioning Term
For data values \(d\) in the event region, the conditional density follows from the definition of conditional probability. In a continuous model, it is the original density restricted to that region and divided by the probability mass of the region. Thus
Because the observed \(d\) satisfies \(E\), the indicator is one and does not affect maximization. Taking logarithms gives a direct comparison:
where \(\ell\) is the log-likelihood and \(U\) is its score, the derivative with respect to \(\theta\). The ordinary and conditional estimates coincide if the conditioning probability is constant in \(\theta\), or if the extra score happens to be zero at the ordinary maximum. Otherwise, the conditional estimate solves a different score equation.
This comparison assumes a correctly specified sampling model, a stopping rule and event defined in advance, and a known event probability under each candidate \(\theta\). In a group sequential model, \(P_\theta(E)\) is calculated from the design and the joint distribution of the interim statistics. It is not generally the probability of stopping at any look; it is the probability of the particular event being conditioned on.
Worked Example: Stopping at an Upper Boundary
Suppose a hypothetical one-look design observes \(X\sim N(\theta,1)\) and stops for efficacy when \(X>1\). The observed value is \(x=1.5\). Compare the ordinary estimate with the estimate from the likelihood conditional on stopping for efficacy.
Model and event. The variance is known to be one, and the event is \(E=\{X>1\}\). Its probability is \(P_\theta(E)=1-\Phi(1-\theta)=\Phi(\theta-1)\), where \(\phi\) and \(\Phi\) are the standard normal density and distribution functions. The observed value \(1.5\) satisfies the event.
Ordinary likelihood. Apart from a constant, \(\ell_o(\theta)=-\tfrac12(1.5-\theta)^2\). Its score is \(1.5-\theta\), so the ordinary maximum likelihood estimate is \(\hat{\theta}_o=1.5\).
Conditional likelihood. The conditional log-likelihood and score are
Setting the score to zero gives \(\theta+\phi(\theta-1)/\Phi(\theta-1)=1.5\). Numerical solution gives \(\hat{\theta}_c\approx -0.131\). For example, at \(\theta=-0.127\), the left side is approximately \(-0.127+1.62776=1.50076\), so \(-0.127\) does not solve the equation; the root is about \(-0.131\). The conditional estimate is much lower than the ordinary estimate because, for large \(\theta\), crossing the boundary is very likely. Conditional on crossing, \(x=1.5\) is then not especially large relative to the parameter.
The Event Probability Can Move the Estimate Substantially
The next example uses two independent normal measurements to make the numerator and conditioning term easy to see. One measurement, \(T\), also determines whether the event occurs. In this setup the data retained are both observed values, while the conditional likelihood is explicitly conditional on the event involving \(T\).
Worked Example: Conditioning on a Non-Efficacy Event
Suppose a hypothetical model has independent \(Y,T\sim N(\theta,1)\), with observed values \(y=t=1.1\). The stopping event under consideration is \(E=\{T\leq1.5\}\). Compare the ordinary likelihood for both measurements with their likelihood conditional on \(E\).
Ordinary estimate. The log-likelihood, up to a constant, is \(-\tfrac12[(1.1-\theta)^2+(1.1-\theta)^2]\). Its score is \(2.2-2\theta\), giving \(\hat{\theta}_o=1.1\).
Conditional estimate. Since \(P_\theta(E)=\Phi(1.5-\theta)\), the conditional score is
The root is approximately \(\hat{\theta}_c=1.49845\). As a numerical check, at \(\theta=1.499\), the score is \(2.2-2(1.499)+0.39894208/0.50039894\approx-0.0007519\), which is close to zero but not zero. The root is therefore slightly below 1.499. In this constructed example, conditioning on \(T\leq1.5\) changes the estimate substantially; it does not establish that conditional estimates are always closer to the true parameter.
These examples illustrate why one must name the conditioning event rather than speak vaguely of “adjusting for stopping.” Changing the event changes its probability term and therefore changes the conditional score. Even with the same observed data, conditioning on crossing an upper boundary, not crossing it, or stopping at a particular look generally produces different likelihoods.
A Group Sequential Illustration
Worked Example: Continuing After the First Look
Consider a hypothetical two-look model with independent increments \(Z_1,Z_2\sim N(\theta,1)\). The design stops for efficacy at the first look if \(Z_1\geq1\); otherwise it continues. Suppose \(z_1=0.2\) and \(z_2=0.9\) are observed. Compare the ordinary estimate with the estimate conditional on the event \(E=\{Z_1<1\}\), which allowed the trial to continue.
Ordinary estimate. The ordinary log-likelihood is, up to a constant, \(-\tfrac12[(0.2-\theta)^2+(0.9-\theta)^2]\). Its score is \(1.1-2\theta\), so \(\hat{\theta}_o=0.55\).
Conditional estimate. Under the model, \(P_\theta(E)=\Phi(1-\theta)\). Dividing the joint density by that probability gives the conditional score
Numerically solving \(U_c(\theta)=0\) gives \(\hat{\theta}_c\approx0.926\). At the ordinary estimate, the conditional score is \(U_c(0.55)=\phi(0.45)/\Phi(0.45)\), approximately \(0.54\), rather than zero. The continuing event therefore changes the score equation. The event is defined here as not crossing the first-look boundary; conditioning on a different stopping event would require a different probability term.
How to Choose and Check the Likelihood
The ordinary likelihood and the conditional likelihood answer different inferential questions. The ordinary likelihood describes the observed data under the sampling model and the stopping rule. The conditional likelihood describes their distribution within the subset of possible data paths satisfying \(E\). The latter can be useful when the analysis specifically targets inference conditional on that event, but it is not automatically a bias correction or a universally preferable estimate.
| Feature | Ordinary likelihood | Likelihood conditional on \(E\) |
|---|---|---|
| Expression | \(f_\theta(d)\) | \(f_\theta(d)/P_\theta(E)\) |
| Question | Which \(\theta\) best fits the observed data? | Which \(\theta\) best fits the data among paths satisfying \(E\)? |
| Stopping-event term | No extra factor when the full path is observed and the rule is deterministic | Subtracts \(\log P_\theta(E)\) from the log-likelihood |
| When maximizers agree | When the conditioning term does not change the maximizer, for example if \(P_\theta(E)\) is constant in \(\theta\) | |
A practical diagnostic is to write down the event before calculating anything. Then check that its probability matches the design, including the relevant boundary, look, and direction. The probability must be computed under the parameter value being evaluated; treating it as a fixed number when it varies with \(\theta\) drops the very term that distinguishes the conditional likelihood.
A further limitation is that conditional likelihood does not guarantee unbiasedness, smaller mean squared error, or improved coverage. Its properties depend on the model and the chosen event. If the event is rare for some parameter values, \(P_\theta(E)\) may be very small and the log-conditioning term can dominate numerically. A near-zero denominator, a computed event probability inconsistent with the design, or a conditional score that fails to include the derivative of that probability are warning signs requiring investigation.
Common Mistake
- Wrong reasoning: “The trial stopped at this boundary, so multiply the ordinary likelihood by the probability of stopping there.”
- Correction: When the full observed path is available and the stopping rule is deterministic, the event is already determined by that path. The ordinary likelihood is the sampling density of the observed path, not that density multiplied by a second copy of the event probability. A conditional likelihood instead divides by \(P_\theta(E)\), because it renormalizes the sampling distribution over paths in \(E\). If only the event label is observed, with no underlying path, its probability may itself be the likelihood contribution.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- If \(P_\theta(E)\) is constant in \(\theta\), do the ordinary and conditional likelihoods have different maximizers? Why?
- For \(X\sim N(\theta,1)\) and \(E=\{X\leq c\}\), write the conditional likelihood for an observed \(x\leq c\).
- In the upper-boundary example, why is the conditional score not simply the ordinary score?
- Suppose only the indicator \(I(E)\) is recorded, not the data that generated it. What likelihood contribution describes the observation \(I(E)=1\)?
- Name one diagnostic that can reveal an incorrectly constructed conditional likelihood.
Solutions. (1) No: dividing by a positive constant changes the likelihood’s scale but not its maximizer. (2) It is \(\phi(x-\theta)/\Phi(c-\theta)\), because \(P_\theta(X\leq c)=\Phi(c-\theta)\). (3) Conditioning subtracts the derivative of \(\log P_\theta(E)\) from the ordinary score, and that probability depends on \(\theta\). (4) The contribution is \(P_\theta(E)\), since the event indicator is the observed data. (5) Examples include a missing derivative of \(\log P_\theta(E)\), a denominator based on the wrong boundary or look, or an event probability that is nearly zero but treated as numerically stable.