Estimation When the Trial Reaches Its Maximum

Why Reaching Maximum Information Is a Selection Event

When a group sequential trial continues to its maximum information, it has not crossed an efficacy or futility boundary at any earlier look. That history matters for estimation: the final estimate is computed from the data available at maximum information, but the trials contributing such estimates are a selected subset of all trials. Even if the ordinary maximum likelihood estimator (MLE) is unbiased at a fixed sample size, its mean among trials that reach the maximum need not equal the true effect.

The focus here is the estimate among trials that reach maximum information, whether or not the final test statistic crosses a boundary at the maximum look. Conditioning additionally on the final test result defines a different selected group and can give a different bias.

Definition: Let $K$ be the maximum look, with information $I_1<\cdots<I_K$. Let $Y_k$ be the efficacy statistic at look $k$, and let $l_k$ and $u_k$ be the lower and upper stopping boundaries. The event of reaching maximum information is $$ A_K=\bigcap_{j=1}^{K-1}\{l_j<Y_j<u_j\}. $$ If no boundary is crossed before the final look, the ordinary estimate at maximum information is $\hat\theta_K=Y_K/\sqrt{I_K}$. Its conditional bias is $b_K(\theta)=E_\theta(\hat\theta_K-\theta\mid A_K)$.

Canonical Model and Bias Formula

Assume the canonical normal model with a fixed parameter $\theta$, known information values, and a prespecified group sequential design. In this model, $Y_k$ has mean $\sqrt{I_k}\theta$, variance 1, and the correlation between two looks $j$ and $k$ is $\sqrt{I_j/I_k}$ when $j<k$. Equivalently, write $Y_k=\sqrt{I_k}\theta+W_k$, where the $W_k$ are jointly normal with mean zero and the stated correlations. The estimate $Y_K/\sqrt{I_K}$ is unbiased at a fixed, unselected look because $E_\theta(Y_K/\sqrt{I_K})=\theta$.

Selection enters through $A_K$, which restricts the earlier statistics. The terminal error $W_K$ is correlated with those earlier statistics. Thus, restricting their values can shift the conditional mean of $W_K$, and the shift translates directly into bias in the terminal estimate.

Result: Under the canonical normal model, $$ b_K(\theta)=\frac{E_\theta(W_K\mid A_K)}{\sqrt{I_K}}. $$ The probability of reaching maximum information is $P_\theta(A_K)$. The contribution of the maximum-information stage to the estimator’s overall bias is therefore $$ E_\theta\big[(\hat\theta_K-\theta)\mathbf{1}_{A_K}\big] =P_\theta(A_K)b_K(\theta). $$

This result is a direct consequence of writing $\hat\theta_K-\theta=W_K/\sqrt{I_K}$ and conditioning on the event that the trial reaches the final look. Its key implication is that the ordinary fixed-look unbiasedness statement, $E_\theta(W_K)=0$, does not imply $E_\theta(W_K\mid A_K)=0$.

A Two-Look Derivation

For two looks, suppose the trial stops early if $Y_1$ crosses a boundary and otherwise reaches look 2. Let $I_1/I_2=r$, so $0<r<1$. The canonical model gives $W_2=\sqrt{r}\,W_1+\sqrt{1-r}\,\varepsilon$, where $\varepsilon\sim N(0,1)$ is independent of $W_1$. If reaching look 2 requires $l_1<Y_1<u_1$, then, under the true parameter, this is equivalent to $a<W_1<d$, where $a=l_1-\sqrt{I_1}\theta$ and $d=u_1-\sqrt{I_1}\theta$. Taking conditional expectations, $$ E_\theta(W_2\mid A_2)=\sqrt{r}\,E(W_1\mid a<W_1<d), $$ because the independent $\varepsilon$ has conditional mean zero. For a standard normal variable, integrating $w\phi(w)$ over $(a,d)$ gives $\phi(a)-\phi(d)$, where $\phi$ is the standard normal density. Since $P(a<W_1<d)=\Phi(d)-\Phi(a)$, where $\Phi$ is the standard normal distribution function, $$ b_2(\theta)=\frac{\sqrt{r}}{\sqrt{I_2}}\, \frac{\phi(a)-\phi(d)}{\Phi(d)-\Phi(a)}. $$ This formula makes the selection effect explicit: survival is centered at zero only when the truncation is symmetric around zero. For a one-sided upper boundary, take $d$ to be the boundary relative to the true mean and let $a$ tend to $-\infty$. Then the conditional residual mean is $-\phi(d)/\Phi(d)$, which is negative. Surviving trials have, on average, less positive early noise, and the correlated terminal noise is also pulled downward.

Worked Example: Symmetric Boundaries at the Null Effect

Consider a hypothetical two-look trial with $I_1=50$, $I_2=100$, and $\theta=0$. At look 1, the trial stops if $Y_1\leq-2$ or $Y_1\geq2$; otherwise it continues. The model assumptions are those stated above, and at the true value $\theta=0$, $Y_1=W_1\sim N(0,1)$. Thus, reaching the maximum means $-2<W_1<2$, a region symmetric around zero.

The conditional residual mean is $$ E(W_1\mid -2<W_1<2) =\frac{\phi(-2)-\phi(2)}{\Phi(2)-\Phi(-2)} =\frac{0.0540-0.0540}{0.9772-0.0228}=0. $$ Here $r=50/100=0.5$, so the two-look formula gives $$ b_2(0)=\frac{\sqrt{0.5}}{10}(0)=0. $$ The terminal estimate is conditionally unbiased among trials that reach the maximum in this particular case. The reason is the symmetry of both the true null model and the survival interval, not the fact that the estimate is an ordinary MLE by itself.

Worked Example: Upper Efficacy Boundary and Downward Terminal Bias

Now consider a hypothetical two-look trial with $I_1=50$, $I_2=100$, true effect $\theta=0.2$ effect units, and an upper efficacy boundary $Y_1\geq2$ at look 1. There is no lower stopping boundary in this illustration. The mean of $Y_1$ is $\sqrt{50}(0.2)=1.414$, so reaching the maximum is the event $Y_1<2$, or $W_1<2-1.414=0.5858$. The normal model implies that the probability of reaching the maximum is $\Phi(0.5858)$, approximately $0.721$.

For a standard normal variable truncated above at $d$, $E(W_1\mid W_1<d)=-\phi(d)/\Phi(d)$. With $d=0.5858$, $\phi(d)\approx0.336$ and $\Phi(d)\approx0.721$, so the conditional residual mean is approximately $-0.466$. Therefore, $$ b_2(0.2)=\frac{\sqrt{50/100}}{\sqrt{100}}(-0.466) =\frac{0.7071}{10}(-0.466) \approx-0.0330 $$ effect units. The mean estimate among trials reaching maximum information is approximately $0.2-0.0330=0.1670$ effect units. Its contribution to overall bias is approximately $0.721(-0.0330)=-0.0238$ effect units. This is only the maximum-stage contribution; the early-stopped trials also contribute to overall bias.

Worked Example: Asymmetric Boundaries and Positive Terminal Bias

Consider another hypothetical two-look design with $I_1=50$, $I_2=100$, and true effect $\theta=0$. At look 1, stop for futility if $Y_1\leq-1.5$, or for efficacy if $Y_1\geq2$; otherwise continue. Under the canonical model at $\theta=0$, reaching the maximum is the event $-1.5<W_1<2$. This interval is not symmetric, so the zero-mean errors are not retained symmetrically.

The probability of reaching the maximum is $$ P(-1.5<W_1<2)=\Phi(2)-\Phi(-1.5) =0.9772-0.0668=0.9104. $$ The conditional early residual mean is $$ E(W_1\mid -1.5<W_1<2) =\frac{\phi(-1.5)-\phi(2)}{0.9104} =\frac{0.1295-0.0540}{0.9104} \approx0.0829. $$ The positive sign occurs because the lower tail removed by the futility boundary is more extreme, in density-weighted terms, than the upper tail removed by the efficacy boundary. With $r=0.5$, the maximum-stage bias is $$ b_2(0)=\frac{\sqrt{0.5}}{10}(0.0829)\approx0.00586 $$ effect units. Its contribution to overall bias is $0.9104(0.00586)\approx0.00533$ effect units. The direction of bias is therefore not determined merely by the presence of an efficacy boundary: both boundaries and the true parameter determine the selection among survivors.

How Terminal Bias Fits Into Overall Bias

Let $A_k$ denote the event that the trial stops at look $k$, for $k=1,\ldots,K$; $A_K$ is the event of reaching the maximum. These stopping events partition the possible trial histories. If the reported estimate at look $k$ is $\hat\theta_k$, then the overall bias of that stage-specific estimator is $$ E_\theta(\hat\theta-\theta) =\sum_{k=1}^{K}P_\theta(A_k)\, E_\theta(\hat\theta_k-\theta\mid A_k). $$ Consequently, the maximum-stage term is $P_\theta(A_K)b_K(\theta)$. A sizable conditional bias may contribute little to overall bias if reaching the maximum is rare; conversely, even a modest conditional bias may matter when most trials continue to the maximum. The weighted contributions can also offset one another, so a small overall bias does not establish that each stage-specific conditional bias is small.

This decomposition clarifies the relationship to the earlier material on unconditional bias correction: an overall correction targets the weighted expectation across stopping outcomes, whereas $b_K(\theta)$ describes the selected terminal group alone. Conditional likelihood methods and median-unbiased estimation after stopping use different estimation principles. They should not be confused with the ordinary MLE merely evaluated among trials that survived to the maximum.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to examine the distribution of the earlier statistics among simulated trials that reach maximum information. Compare their conditional mean with the unconditional mean, and compare the simulated mean terminal estimate with the true parameter. The canonical model predicts the same selection mechanism through the dependence between earlier and terminal errors. A marked discrepancy between simulated and calculated conditional bias can indicate an implementation error, incorrect boundaries, or failure of the assumed model.

Check the event being conditioned on. Reaching the maximum means no earlier crossing. Conditioning also on crossing a final efficacy boundary defines a different event.
Check the boundary scale and direction. Boundaries apply to $Y_k$, whose mean depends on $\theta$; the survivor region for the residual $W_k$ must therefore be shifted by $\sqrt{I_k}\theta$.
Check symmetry rather than assuming zero bias. A symmetric boundary region gives zero conditional residual mean in the symmetric null model, but this conclusion need not hold away from the null or with asymmetric boundaries.

The calculation relies on a known-information canonical normal model and fixed, prespecified boundaries. Estimated standard errors, nuisance-parameter estimation, model misspecification, or changes to the design can alter the joint distribution and selection mechanism. Those situations require methods beyond the formulas derived here. The ordinary fixed-sample estimate remains a useful reference, but it does not by itself adjust for the selection induced by reaching the maximum.

Common mistake: “The estimate at the maximum is an unbiased MLE, so among trials that reach the maximum its mean is the true effect.” The fixed-look statement is $E_\theta(W_K)=0$, not $E_\theta(W_K\mid A_K)=0$. Since the earlier statistics are correlated with $W_K$, selection on earlier boundaries can shift its conditional mean. Check the survivor event and calculate or estimate the conditional expectation before claiming zero terminal-stage bias.

Check Your Understanding

Use the canonical normal model and distinguish conditional bias at the maximum from overall bias.

In the two-look formula, what feature of the truncation region makes the conditional bias zero at $\theta=0$?
With an upper-only interim boundary, why is the conditional terminal bias typically negative among trials that reach the maximum?
If the probability of reaching the maximum is $0.4$ and its conditional bias is $0.05$ effect units, what is its contribution to overall bias?
Why can the terminal estimate’s conditional bias be zero under symmetric null boundaries without being zero under an asymmetric design?

Solutions. (1) The survival interval for the centered residual is symmetric about zero, so its conditional mean is zero; correlation then gives zero terminal residual mean. (2) Survival restricts the earlier statistic to values below the efficacy boundary, producing a negative conditional early residual mean; the correlated terminal residual is pulled downward as well. (3) The contribution is $0.4(0.05)=0.020$ effect units. (4) Symmetric truncation at the null preserves a zero conditional residual mean, whereas asymmetric truncation generally does not; the removed tails need not balance.