Why the observed look enters the likelihood
When a trial stops early, the observed data include more than the final estimate: the stopping rule also tells us which paths could have produced that stopping time. Conditional maximum likelihood uses that information by treating the observed look as given. Across several possible looks, the calculation has the same structure at each one, but the probability of stopping at the observed look changes with the look and the stopping rule.
We use a canonical normal model with known variance scale. Let the prespecified information times be \(0<t_1<\cdots<t_K\), and let \(X_j\) denote the cumulative score at look \(j\). Assume independent increments \(X_j-X_{j-1}\sim N(\theta(t_j-t_{j-1}),t_j-t_{j-1})\), where \(X_0=t_0=0\) and \(\theta\) is the parameter of interest. Equivalently, \(Z_j=X_j/\sqrt{t_j}\) has mean \(\theta\sqrt{t_j}\), variance 1, and the canonical correlations across looks. The stopping rule is fixed in advance and is applied to the observed path.
This definition conditions on the stopping look, not just on the terminal statistic. In general, the observed path includes the interim statistics that establish that earlier stopping did not occur. If only a summary of that path is available, its conditional density must be obtained by integrating out the unobserved components; one cannot automatically use the full-path formula.
The score equation across looks
For the canonical model, the joint density factors into normal densities for the increments. Differentiating its log with respect to \(\theta\) gives a particularly simple numerator score. For an increment \(d_j=x_j-x_{j-1}\) with information increment \(\Delta t_j=t_j-t_{j-1}\), its log density, up to terms not involving \(\theta\), is \(\theta d_j-\theta^2\Delta t_j/2\). Its derivative is \(d_j-\theta\Delta t_j\). Summing over the observed increments gives \(x_k-\theta t_k\).
The conditional log likelihood subtracts \(\log\pi_k(\theta)\). Therefore an interior conditional maximum likelihood estimate solves the following equation. The stopping probability is determined by the entire prespecified stopping rule, including all earlier looks.
This derivation also clarifies what must be evaluated numerically. The observed-path score is explicit, but \(\pi_k(\theta)\) may require a multivariate normal probability or a sequential integration over the paths that reach look \(k\). When the stopping rule has several boundaries, the region defining \(T=k\) includes the no-stop conditions at all earlier looks and the condition that causes stopping at \(k\), unless \(k\) is the forced final look.
Computing and checking a root
A reliable calculation separates the statistical definition from the numerical method. First specify the stopping region for the observed look. Then evaluate \(\pi_k(\theta)\) and, preferably, its derivative using the same region and model. Solve \(U_c(\theta)=0\), and check the resulting likelihood as well as the score. Numerical integration error in a small stopping probability can otherwise produce a misleading root.
Record the observed values through look \(k\) and state which inequalities define stopping at \(k\).
Use \(x_k-\theta t_k-\pi_k'(\theta)/\pi_k(\theta)\), with the probability calculated for that exact stopping event.
Evaluate the score over a reasonable parameter range, bracket a sign change when one exists, and use a bracketed method or a safeguarded Newton method.
Check that the score is near zero, that the conditional log likelihood is locally maximized, and that probability and derivative calculations are numerically stable.
A sign-changing bracket is useful but does not by itself prove that the root is the global maximum. A score can have more than one root, and a constrained parameter space can place the maximum at its edge. Compare the conditional log likelihood at all candidate roots and relevant parameter-space boundaries. If the stopping probability is zero at a parameter value, the conditional distribution given that stopping event is undefined there.
Worked Example: Efficacy Stop at the First Look
Suppose \(t_1=1\), \(X_1\sim N(\theta,1)\), and the trial stops for efficacy when \(X_1\geq2\). A hypothetical trial stops with \(x_1=2.5\). The model is known-variance normal, and the observed value satisfies the stopping condition.
Here \(\pi_1(\theta)=P_\theta(X_1\geq2)=1-\Phi(2-\theta)\), where \(\Phi\) and \(\phi\) are the standard normal distribution and density functions. The conditional score is
At \(\theta=0.870\), \(2-\theta=1.130\), \(\phi(1.130)\approx0.2107\), and \(1-\Phi(1.130)\approx0.1292\). Substitution gives \(U_c(0.870)\approx2.5-0.870-(0.2107/0.1292)\approx-0.0002\). Solving the equation numerically gives \(\hat{\theta}_c\approx0.8689\), or about \(0.869\) to three decimal places. The value is on the same scale as \(X_1\), here the standard-deviation units implied by \(t_1=1\). The observed estimate \(2.5\) is not the conditional estimate: conditioning on an efficacy crossing accounts for the selection of paths with large values.
Final-look stopping after an earlier noncrossing
At a final look, the probability in the denominator is the probability of reaching that look under the stopping rule. In a multi-boundary design it can involve the joint distribution of every interim statistic. The following two-look example has a simple reach probability, making the correction visible without a multivariate integration.
Worked Example: Reaching a Forced Final Look
Consider \(t_1=1,t_2=2\). At look 1, stop if \(X_1\geq0\); otherwise continue. At look 2, the trial always stops. In a hypothetical trial, \(X_1=-0.2\) and \(X_2=1.0\), so the observed path reaches the final look. The increments are \(-0.2\) and \(1.2\), consistent with the observed cumulative values.
The probability of stopping at look 2 is the probability of not crossing at look 1: \(\pi_2(\theta)=P_\theta(X_1<0)=\Phi(-\theta)\). The full-path numerator score is \(X_2-2\theta=1-2\theta\), so
At \(\theta=1.450\), \(\phi(-1.450)\approx0.1394\) and \(\Phi(-1.450)\approx0.0735\). Thus \(U_c(1.450)\approx1-2(1.450)+0.1394/0.0735\approx-0.00374\). Numerically solving gives \(\hat{\theta}_c\approx1.4468\). The ordinary, unconditioned normal likelihood based on the terminal cumulative score alone would give \(X_2/t_2=0.500\). That is a different likelihood calculation: the conditional likelihood here uses the observed interim path and divides by the probability of reaching look 2.
Worked Example: A Different Final-Look Path
Keep the same design, but suppose a second hypothetical trial has \(X_1=-0.2\) and \(X_2=2.0\). It also reaches look 2 because \(X_1<0\). The stage increments are now \(-0.2\) and \(2.2\). The conditional score uses the same stopping probability, but the terminal path value changes:
For a numerical bracket, at \(\theta=2.300\), \(\phi(-2.300)/\Phi(-2.300)\approx2.64143\), giving \(U_c(2.300)\approx0.04143\). At \(\theta=2.400\), the ratio is approximately \(2.732\), giving \(U_c(2.400)\approx-0.068\). A bracketed root solver therefore has a sign change between these values; solving gives \(\hat{\theta}_c\approx2.338\). The positive-to-negative change is consistent with a local maximum in this example, and direct evaluation of the conditional log likelihood confirms it. The root differs from the previous example because the observed terminal path differs, even though the stopping rule is identical.
Numerical diagnostics and common mistakes
For more than two looks, \(\pi_k(\theta)\) is often calculated by integrating the joint normal density over the stopping region. The same information times, correlation structure, and boundary conventions used to define the trial must be used in this integration. Useful checks include comparing probabilities across all stopping outcomes (their sum should be 1), increasing integration accuracy, and checking the score with a small finite difference of the conditional log likelihood.
- Omitting the stopping probability. Solving \(x_k-\theta t_k=0\) is the ordinary full-path likelihood equation, not the conditional likelihood equation.
- Using the wrong event in \(\pi_k\). For a later look, include every earlier condition required to reach it. A crossing probability at look \(k\) alone is not generally the probability of stopping there.
- Confusing the observed density with a stopping probability. The numerator is a density for the observed path; the denominator is a probability of the stopping event. They are not interchangeable.
- Trusting an unbracketed Newton iteration. A Newton step can diverge or find a nonmaximizing root. Use bracketing where possible and compare conditional log likelihoods.
- Ignoring numerical instability. If \(\pi_k(\theta)\) is very small, direct probability and derivative calculations can underflow or lose precision. Use stable tail probabilities or log-probability calculations and verify convergence.
Conditional maximum likelihood is not a universal correction for stopping-induced bias. It answers a conditional likelihood question given the observed stopping look; it does not, by itself, guarantee unbiasedness or a particular unconditional bias property. The ordinary MLE instead uses the unconditioned likelihood for the observed data under its model. Other stopping-aware estimators can target different properties, so the estimation target should be stated rather than inferred from the word “adjusted.”
Check Your Understanding
Use the conditional likelihood definition and score equation to answer each question.
- In a canonical model, what quantity must be subtracted from the ordinary path score to obtain the conditional score at look \(k\)?
- A two-look design stops at look 1 if \(X_1\geq c\), and otherwise stops at the forced final look 2. Write \(\pi_2(\theta)\) when \(X_1\sim N(\theta t_1,t_1)\).
- For the design in Worked Example 2, why is the event probability at look 2 \(\Phi(-\theta)\), rather than \(P_\theta(X_2\geq0)\)?
- A numerical solver finds two roots of the conditional score. What additional check is needed before reporting an estimate?
Solutions. (1) Subtract \(\pi_k'(\theta)/\pi_k(\theta)\), the derivative of the log probability of stopping at the observed look. (2) Reaching look 2 means \(X_1<c\), so \(\pi_2(\theta)=\Phi((c-\theta t_1)/\sqrt{t_1})\). (3) Stopping at look 2 is determined by failure to cross the look-1 boundary; the final look is forced, so its statistic does not define the event probability. (4) Evaluate the conditional log likelihood at both roots and at any relevant parameter-space boundaries, and report a maximizing candidate rather than an arbitrary score root.