Why Condition on the Observed Look?
When a group sequential trial stops, the observed look is part of the data. A trial that stops at the first look has crossed its stopping boundary then; a trial that stops at a later look did not stop at any earlier look. Consequently, an analysis that conditions on the observed look must account for the event that led to that stopping time. The resulting likelihood is not just the ordinary likelihood for the observed statistic.
This distinction matters because stopping depends on the data, and the probability of the stopping event generally depends on the treatment effect. The likelihood conditional on the observed look therefore contains a parameter-dependent normalizing probability. The introductory material on likelihood versus conditional likelihood after stopping distinguishes these targets; here the focus is constructing the conditional likelihood under the canonical normal model.
Model and Stopping Event
Assume a one-sided efficacy design with \(K\) planned looks, cumulative information \(0<I_1<\cdots<I_K\), and upper efficacy boundaries \(a_1,\ldots,a_K\). Let \(\theta\) be the treatment effect on the scale for which the canonical model is
where \(W(I)\) is a standard Brownian motion. Thus \(Z_i\sim N(\theta\sqrt{I_i},1)\), and for \(i<j\), \(\operatorname{Corr}(Z_i,Z_j)=\sqrt{I_i/I_j}\). The model assumes the planned information times and boundaries are specified, and that the standardized statistics have this joint normal distribution. In applications, departures from the model or use of estimated standard errors may require additional assessment.
Write \(f_{\theta,k}(z)\) for the marginal density of \(Z_k\). The event \(D_k\) includes information about all earlier looks, so the conditional density at \(Z_k=z\) is
The numerator is the joint density contribution for observing \(Z_k=z\) and satisfying the earlier non-crossing conditions. The denominator is the probability, under the same \(\theta\), of stopping at look \(k\). It makes the conditional density integrate to one over the stopping region. In particular, \(P_\theta(D_k)\) is not a constant that can be omitted when comparing treatment effects.
Deriving the Conditional Density
For a continuous statistic, the joint probability of observing \(Z_k\) in a small interval around \(z\) and stopping at look \(k\) is approximately the numerator density times the interval width. Dividing that joint probability by \(P_\theta(D_k)\) gives the conditional density. More formally, for a small interval \(B\) lying in the stopping region,
Taking the interval width to zero yields the displayed conditional density. This also explains why the density is zero outside the stopping region: values there could not have produced the specified stopping look under the stated rule. For the first look, there are no earlier looks to consider, so the numerator reduces to the ordinary density \(f_{\theta,1}(z)\) on \(z\ge a_1\).
Worked Example: Conditioning at a Single Efficacy Look
Suppose there is one look, the statistic satisfies \(Z\sim N(\theta,1)\), the trial stops for efficacy when \(Z\ge1\), and the observed value is \(z=1.4\). The event is \(D_1=\{Z\ge1\}\). The conditional likelihood is
where \(\phi\) and \(\Phi\) are the standard normal density and distribution functions. At \(\theta=0\), the numerator is \(\phi(1.4)\approx0.1497\), and the crossing probability is \(1-\Phi(1)\approx0.1587\). Therefore
The conditional density is approximately \(0.944\) per \(Z\)-unit. It is a density, not a probability that the effect equals zero. For a different candidate value, such as \(\theta=0.2\), both the numerator and the crossing probability must be evaluated at that value: \(\phi(1.2)/[1-\Phi(0.8)]\approx0.9166\). The ratio of these conditional likelihood values is approximately \(0.971\), not a p-value.
Two Looks: The Earlier Non-Crossing Condition
At a later look, conditioning only on the current crossing is insufficient. The trial is known not to have crossed at any earlier look. The canonical bivariate normal distribution gives the conditional distribution of the earlier statistic given the current one, which supplies the numerator’s non-crossing factor. The denominator is the probability of crossing now while having stayed below the earlier boundary.
Worked Example: Conditional Likelihood at the Second Look
Consider two looks with \(I_1=1\), \(I_2=4\), and boundary 1 at each look. Then \(\operatorname{Corr}(Z_1,Z_2)=0.5\), \(E_\theta(Z_1)=\theta\), and \(E_\theta(Z_2)=2\theta\). The event of stopping at look 2 is \(D_2=\{Z_1<1,Z_2\ge1\}\). Suppose \(Z_2=1.4\) is observed.
For these information times, the stopping probability can be written as a difference between a marginal crossing probability and a joint crossing probability:
Numerical bivariate normal integration gives \(P_0(D_2)\approx0.09614\). At \(\theta=0.25\), \(Z_2\sim N(0.5,1)\), so \(P_{0.25}(Z_2\ge1)=1-\Phi(0.5)\approx0.30854\); the joint crossing probability is approximately \(0.12944\). Hence \(P_{0.25}(D_2)\approx0.30854-0.12944=0.17910\).
Given \(Z_2=1.4\), the conditional distribution of \(Z_1\) is normal with standard deviation \(\sqrt{1-0.5^2}=\sqrt{0.75}\). Its mean is \(0.5Z_2=0.7\) at \(\theta=0\), and is \(0.25+0.5(1.4-0.5)=0.7\) at \(\theta=0.25\). Thus, for either value of \(\theta\),
The marginal densities are \(f_{0,2}(1.4)=\phi(1.4)\approx0.1497\) and \(f_{0.25,2}(1.4)=\phi(0.9)\approx0.2661\). The conditional likelihoods are therefore approximately
Their ratio, for \(\theta=0.25\) relative to \(\theta=0\), is equivalently
The ratio compares the relative conditional likelihood support for these two specified effect values, given this stopping look and statistic. It does not state the probability that either effect value is true.
Boundary Definitions Change the Conditioning Event
The event in the denominator must match the actual stopping rule. If a one-look design stops for efficacy at either tail, the conditioning event is \(|Z|\ge c\), not just \(Z\ge c\). The same principle applies with multiple looks: include all required earlier non-crossings and the current crossing condition, using the actual boundaries and stopping rules. A futility boundary, for example, changes the event if it determines whether the trial stops.
Worked Example: A Two-Sided One-Look Stop
Suppose \(Z\sim N(0,1)\), the trial stops when \(|Z|\ge2\), and the observed value is \(z=2.2\). The stopping event has two tails, so its probability under \(\theta=0\) is \(P(|Z|\ge2)=2[1-\Phi(2)]\approx0.04550\). The conditional density at the observed positive value is
Using only the upper-tail probability in the denominator would give the wrong conditional likelihood for this two-sided rule. Although the observation lies in the upper tail, the conditioning event includes either way of stopping.
Calculation Checks and Limitations
Conditional likelihoods can require multivariate normal probabilities, especially when the observed look follows several earlier looks. Numerical integration, deterministic methods, or Monte Carlo checks can be used to evaluate those probabilities, but the event must first be specified correctly. Useful checks include verifying that the denominator is between zero and one, that the conditional density integrates to one over the current stopping region, and that the numerator uses the joint model implied by the information times.
The main alternative is the ordinary likelihood based on the observed statistic without conditioning on the stopping look. These are different inferential targets: the ordinary likelihood does not divide by the probability of the observed stopping event, whereas the conditional likelihood does. Neither construction makes the stopping mechanism irrelevant in every inferential task; the appropriate target depends on the question. Conditional likelihoods also rely on the assumed model and known stopping rule, and can be sensitive to inaccurate boundary, timing, or covariance specifications.
- Wrong: “The observed statistic crossed at look \(k\), so I can use only \(P_\theta(Z_k\ge a_k)\) in the denominator.” Correction: use \(P_\theta(D_k)\), which also incorporates all earlier non-crossings.
- Wrong: “The denominator is just a normalizing constant, so it can be dropped when comparing \(\theta\).” Correction: it generally depends on \(\theta\), and dropping it changes the conditional likelihood.
- Wrong: “A conditional likelihood value is the probability of the observed statistic.” Correction: it is a density; probabilities require integrating it over an interval.
Check Your Understanding
Answer each question by identifying the event or calculation needed; concise solutions follow.
- In a two-look upper-boundary design, what event describes stopping at look 2?
- In the two-look example, why does the conditional likelihood denominator depend on \(\theta\)?
- For a one-look design stopping when \(|Z|\ge c\), what probability belongs in the denominator?
- What does a conditional likelihood ratio of 0.5 for \(\theta_1\) relative to \(\theta_0\) mean, and what does it not mean?
Solutions. (1) The event is \(\{Z_1<a_1,Z_2\ge a_2\}\): no earlier crossing and a crossing at look 2. (2) The distribution of the statistics, and therefore the chance of this crossing path, changes with the treatment effect. (3) Use \(P_\theta(|Z|\ge c)=P_\theta(Z\ge c)+P_\theta(Z\le-c)\), including both tails. (4) Given the specified stopping event and observed statistic, the conditional likelihood at \(\theta_1\) is half its value at \(\theta_0\). This ratio is not a probability that either parameter is true.