The Ordinary MLE Under a Stopping Rule

Why an unchanged estimate can still be biased

A group-sequential trial may stop early when interim results cross a prespecified boundary. The estimate reported at that stop is often the ordinary maximum likelihood estimate (MLE) computed from the data observed so far. That can be the correct maximizer of the likelihood for the observed path and still have a biased repeated-sampling distribution. There is no contradiction: maximization describes how an estimate is obtained from one dataset, whereas bias describes its average over hypothetical repetitions of the whole design.

The key qualification is that the stopping rule must be accounted for correctly. When a prespecified rule determines whether to stop from the data already observed, and the full stopped data path is recorded, the event of following that rule generally contributes no parameter-dependent factor to the likelihood. The observed path can therefore have the same likelihood maximizer as it would under a fixed sample size equal to the number of observations actually collected. That fact alone says nothing about whether the estimator is unbiased.

Definition: For a parameter \(\theta\) and an estimator \(\hat{\theta}\), the bias under the full sampling design is \(E_\theta(\hat{\theta})-\theta\). Conditional bias given a stopping event \(A\) is \(E_\theta(\hat{\theta}\mid A)-\theta\), when \(P_\theta(A)>0\). The ordinary MLE is a value of \(\theta\) maximizing the likelihood of the observed data; being an MLE does not imply either bias quantity is zero.

When the stopped-path likelihood has the ordinary form

Suppose observations arrive in stages, and let \(\tau\) be the stage at which the design stops. At stage \(j\), write the newly observed data as \(Y_j\), with model density or probability mass function \(f_\theta(y_j\mid y_1,\ldots,y_{j-1})\). The observed data are the complete path \((y_1,\ldots,y_\tau)\), including the fact that the trial stopped at \(\tau\). Assume the sampling model is correctly specified, the stopping rule was prespecified and is a function of observed data and known design constants, and the rule itself does not depend on the unknown parameter \(\theta\). Also assume the stopping time is finite, as it is in a design with a fixed maximum number of looks.

For an observed path that follows the rule, its likelihood is proportional to the product of the sampling contributions along that path:

L(\theta;y_1,\ldots,y_\tau) = \prod_{j=1}^{\tau} f_\theta(y_j\mid y_1,\ldots,y_{j-1}).

The path must satisfy the stopping rule: it did not stop at an earlier stage and did stop at \(\tau\), or reached the maximum stage. For the observed path, this condition is already met. Its indicator is either one or zero, and it does not vary with \(\theta\). Thus it does not change which value of \(\theta\) maximizes the likelihood. In independent-increment models, the product above is simply the product of the stage-specific sampling contributions.

This argument is about the likelihood of the full observed stopped path. If only a summary is retained, such as the stopping stage and a rounded test statistic, its likelihood may require summing or integrating over paths that produce that summary. The resulting likelihood need not have the same maximizer. The argument also needs reconsideration if the rule depends on an unknown parameter or if the sampling model is misspecified.

Worked Example: Bernoulli Sampling Until a Success or Three Trials

Suppose independent observations \(X_i\) satisfy \(P_p(X_i=1)=p\) and \(P_p(X_i=0)=1-p\). The prespecified rule is to stop at the first success, or after three trials if no success occurs. The complete observed sequence is recorded. For a path with \(s\) successes in \(n\) observed trials, the likelihood is \(p^s(1-p)^{n-s}\), so the ordinary MLE is \(\hat{p}=s/n\), including the boundary estimates 0 or 1 when all observations are alike.

For example, the path \((0,0,1)\) has likelihood \((1-p)^2p\). Differentiating its log likelihood gives \(1/p-2/(1-p)=0\), hence \(\hat{p}=1/3\). The stop rule restricts which paths can be observed, but it adds no parameter-dependent multiplier to this path's likelihood.

Now consider repeated trials of this design when the true success probability is \(p=0.2\). The estimate is 1 if the first trial succeeds, \(1/2\) if the first success is on trial two, \(1/3\) if it is on trial three, and 0 if all three trials fail. Therefore

E_p(\hat{p}) =p+(1-p)p\left(\frac12\right) +(1-p)^2p\left(\frac13\right).

Substituting \(p=0.2\) gives \(0.2+0.8(0.2)(0.5)+0.8^2(0.2)(1/3)=0.2+0.08+0.0426667\approx0.3227\). The unconditional bias is approximately \(0.3227-0.2=0.1227\). Thus the sample proportion is still the MLE for each observed path, but its average under this stopping design exceeds the true success probability.

Stopping can shift conditional and unconditional averages

The effect of stopping depends on which repetitions are averaged. Conditional bias restricts attention to a specified event, such as crossing an efficacy boundary at an interim look. Unconditional bias averages over every possible stopping outcome, including early efficacy stops, later stops, and reaching the maximum sample size. As the introductory material on conditional and unconditional bias targets emphasizes, these are different summaries and need not have the same value or even the same sign.

A particularly clear mechanism arises when the trial stops for efficacy after an unusually large estimate. Among trials that cross a positive boundary, the estimates are selected from the upper tail of their sampling distribution. Their conditional mean can exceed the true effect, even though the estimate at each observed look is an ordinary MLE. Whether the overall unconditional bias is also positive depends on the complete design and the probabilities and estimates associated with all stopping outcomes.

Worked Example: Conditional Mean at a One-Look Efficacy Boundary

For illustration, suppose a hypothetical interim estimate \(\bar{X}\) has distribution \(N(\theta,0.5^2)\), with known standard error \(0.5\) mmHg. The rule stops for efficacy if \(\bar{X}\geq1.0\) mmHg. At this look, the normal likelihood is maximized at \(\hat{\theta}=\bar{X}\). Consider the repeated-sampling behavior under \(\theta=0\) mmHg, conditional on crossing the boundary.

Standardizing gives \(Z=\bar{X}/0.5\sim N(0,1)\), and crossing means \(Z\geq2\). For a standard normal variable, \(E(Z\mid Z\geq c)=\phi(c)/[1-\Phi(c)]\), where \(\phi\) and \(\Phi\) are the standard normal density and distribution functions. Hence

E_0(\bar{X}\mid\bar{X}\geq1.0) =0.5\frac{\phi(2)}{1-\Phi(2)} =0.5\frac{0.05399097}{0.02275013} \approx1.1866\text{ mmHg}.

The conditional bias among boundary-crossing trials is about \(1.1866-0=1.1866\) mmHg. This does not mean the MLE at a particular crossing is wrong: for an observed value of \(\bar{X}\), it still maximizes that look's normal likelihood. The calculation instead shows how selecting repetitions by a high-estimate event changes their average.

A two-look calculation of unconditional bias

The same distinction applies when all trial outcomes are included. The following small model gives an explicit unconditional calculation. Assume two independent observations \(X_1,X_2\), each distributed as \(N(\theta,1)\), with known variance \(1\). Stop after the first observation if \(X_1\geq1\); otherwise collect \(X_2\) and stop. This is a prespecified, finite stopping rule based only on the observed first-stage data.

Worked Example: MLE and Unconditional Bias in a Two-Look Design

If the trial stops after one observation, its likelihood is proportional to \(\exp[-(X_1-\theta)^2/2]\), maximized at \(\hat{\theta}=X_1\). If it continues, the likelihood is proportional to \(\exp[-\{(X_1-\theta)^2+(X_2-\theta)^2\}/2]\), maximized at \(\hat{\theta}=(X_1+X_2)/2\). These are the usual normal MLEs for the data observed at each stopping time.

Let \(\hat{\theta}\) denote this estimate under the rule, and evaluate its expectation at \(\theta=0\). Since \(X_1,X_2\) are independent standard normal variables under this value, and \(E_0(X_2)=0\),

E_0(\hat{\theta}) =E_0[X_1 I(X_1\geq1)] +\frac12 E_0[X_1 I(X_1<1)] +\frac12 E_0[X_2 I(X_1<1)].

The last term is zero by independence and \(E_0(X_2)=0\). Also \(E_0(X_1)=0\), so \(E_0[X_1 I(X_1<1)]=-E_0[X_1 I(X_1\geq1)]\). For a standard normal variable, \(\int_1^\infty x\phi(x)\,dx=\phi(1)\), because \(\phi'(x)=-x\phi(x)\). Consequently,

E_0(\hat{\theta}) =\frac12\phi(1) =\frac12(0.2419707) \approx0.120985.

The unconditional bias is therefore approximately \(0.1210\) on the measurement scale of \(X_1\) and \(X_2\). The estimate is the MLE for either observed path type, yet the repeated-sampling mean is not the true value 0. The positive bias arises because high first observations trigger stopping, while lower first observations are averaged with a second observation.

Why the MLE property is not a bias guarantee

Likelihood maximization is a within-dataset criterion: it selects the parameter value that makes the observed data comparatively most compatible with the model. Bias is an expectation across datasets generated repeatedly under a fixed parameter and the same stopping rule. Stopping changes which paths are observed and how much information is available on each path. It can therefore alter the sampling distribution of an estimator without changing the maximizer for any given complete path.

It is useful to compare the ordinary MLE with alternatives by asking what each is designed to do. The ordinary MLE preserves the usual likelihood maximization for the observed path under the stated assumptions. A bias-adjusted estimate instead targets a specified repeated-sampling bias property, and may no longer maximize that ordinary path likelihood. Likewise, a fixed-sample analysis that ignores the actual stopping design may not describe the estimator's repeated-sampling behavior under the design that was run. These approaches answer distinct questions; none should be treated as automatically interchangeable.

The result also has practical limitations. It does not establish that standard errors, confidence intervals, or p-values computed as if the sample size were fixed have their nominal properties after stopping. It does not guarantee that the same estimate is obtained from a reduced summary rather than the full path. And it does not imply that bias must always be upward: futility stopping, asymmetric boundaries, parameter values, and maximum-sample outcomes can alter the overall direction and magnitude. A bias assessment must use the actual stopping rule and the intended target, conditional or unconditional.

Diagnostics and common pitfalls

Confusing “is the MLE” with “is unbiased.” The MLE statement concerns maximizing a likelihood for observed data. Unbiasedness requires \(E_\theta(\hat{\theta})=\theta\) under the full repeated-sampling design.
Reporting only the boundary-crossing average as overall bias. That is conditional bias for the crossing event. Unconditional bias must average over all stopping outcomes with their design probabilities.
Using an inappropriate observed-data likelihood. Check whether the complete stopped path is available and whether the stopping rule is parameter-free. A compressed record or a parameter-dependent rule may require a different likelihood calculation.
Assuming the direction of bias from one outcome. Large estimates among efficacy stops suggest upward conditional bias there, but do not by themselves determine the unconditional bias across the design.
Checking only the estimate and not the design assumptions. Verify that the sampling model, dependence structure, interim timing, and actual stopping rule match the assumptions used in the likelihood argument.

Key takeaway: Under a correctly specified model and a prespecified, parameter-free stopping rule, the likelihood of a complete observed stopped path can have the same maximizer as the ordinary fixed-sample likelihood for the data actually observed. That MLE property does not remove stopping-induced conditional or unconditional bias.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In one sentence, distinguish the MLE property from unbiasedness under a stopping design.
For the Bernoulli design that stops at the first success or after three trials, what estimate follows the path \((0,1)\), and why?
Why is \(E_0(\bar{X}\mid\bar{X}\geq1.0)\) not the unconditional bias in the one-look normal example?
Name two conditions used in the stopped-path likelihood argument and explain what can fail if one is absent.

Solutions. (1) The MLE maximizes the likelihood for the observed data, whereas unbiasedness requires its expectation over repetitions of the full design to equal the true parameter. (2) The estimate is \(1/2\), because the observed sequence has one success in two trials; its likelihood is \(p(1-p)\), maximized at \(p=1/2\). (3) It averages only trials crossing the efficacy boundary, so it is a conditional mean; unconditional bias averages estimates over every stopping outcome and subtracts the true parameter. (4) For example, the rule must be prespecified and parameter-free, and the full stopped path must be represented under the correct sampling model. A parameter-dependent rule can introduce parameter dependence through selection, while retaining only a summary can require integrating over unobserved paths.