Winner's Curse After Treatment Selection

Why Selecting the Largest Interim Response Inflates an Estimate

In an adaptive trial, an interim analysis may be used to choose which treatment arm to continue. If the arm with the largest estimated benefit is selected, its estimate is likely to include not only a genuine treatment effect but also a favourable random error. Selection makes that error part of the reason the arm won. The selected estimate can therefore be too high, even when every arm estimate was unbiased before the comparison.

This tutorial quantifies that upward bias under a deliberately clear model: each arm has an unbiased, independent, normally distributed interim effect estimate, and the arm with the largest estimate is selected. The result describes the estimate used for selection. It does not automatically describe an independent estimate collected after selection, or estimates subject to other adaptations.

Definition: Suppose arm \(i\) has true effect \(\theta_i\) and interim estimate \(Y_i\). Let \(J=\arg\max_i Y_i\) be the selected arm. The winner’s-curse bias is \(E[Y_J-\theta_J]\): the expected excess of the selected estimate over the true effect of the arm selected. The reference is the selected arm’s own true effect, which can vary with the random selection.

Assume there are \(K\) candidate arms and that, before selection, \(Y_i\sim N(\theta_i,\sigma_i^2)\) independently for \(i=1,\ldots,K\). Thus each estimate is unbiased, its variance is known, the normal model is adequate, and the estimates have no shared random component. The response scale is oriented so that a larger value represents a more favourable response. For an outcome where lower values are better, reverse the direction before applying the maximum-selection formulas.

With continuous normal distributions, ties have probability zero. The selected arm’s estimate has a different distribution from an unselected arm’s estimate: it is the maximum of the candidates. In general, its bias relative to its own true effect is

E[Y_J-\theta_J] = \sum_{i=1}^{K} \int_{-\infty}^{\infty} (y-\theta_i) f_i(y) \prod_{j\ne i} F_j(y)\,dy,

where \(f_i\) and \(F_i\) are the density and cumulative distribution function of \(Y_i\). For arm \(i\) to be selected with estimate \(y\), its estimate must equal \(y\), and every other estimate must be at most \(y\). Independence gives the product of cumulative probabilities. This expression quantifies bias even when the true effects or variances differ, though it may require numerical integration.

Equal Effects: A Direct Measure of the Winner’s Curse

A particularly informative case has a common true effect \(\theta\) and common standard error \(\sigma\): \(Y_i=\theta+\sigma Z_i\), where the \(Z_i\) are independent standard normal variables. Since all arms have the same true effect, the selected arm’s true effect is still \(\theta\), regardless of which arm wins. The bias is therefore the expected maximum estimation error:

E[Y_J-\theta] = \sigma E\left[\max_{1\le i\le K} Z_i\right] = \sigma\int_{-\infty}^{\infty} z K\phi(z)\Phi(z)^{K-1}\,dz,

Here \(\phi\) and \(\Phi\) are the standard normal density and cumulative distribution function. The density of the maximum at \(z\) is \(K\phi(z)\Phi(z)^{K-1}\): any one of the \(K\) variables can be the maximum at \(z\), while the other \(K-1\) must be no greater than \(z\). The integral increases with \(K\). More candidate arms create more opportunities for a large positive random error to determine the winner.

Worked Example: Two Equally Effective Arms

Suppose two hypothetical treatment arms each have true effect \(2.0\) points, and each independent interim estimate has standard error \(1.5\) points. The arm with the larger estimate is selected. Calculate the expected upward bias.

For two standard normal variables, the maximum can be written as \((Z_1+Z_2+|Z_1-Z_2|)/2\). The expected value of \(Z_1+Z_2\) is zero. Also, \(Z_1-Z_2\sim N(0,2)\), so \(E|Z_1-Z_2|=2/\sqrt{\pi}\). Consequently, \(E[\max(Z_1,Z_2)]=1/\sqrt{\pi}\), and the bias is

1.5\left(\frac{1}{\sqrt{\pi}}\right) = \frac{1.5}{1.77245} \approx 0.8463\text{ points}.

The expected selected estimate is therefore \(2.0+0.8463=2.8463\) points, although either arm’s estimate considered before selection has expectation \(2.0\) points. The difference is caused by choosing the larger of two noisy estimates; it does not imply that either arm’s underlying effect has changed.

Worked Example: Four Arms With a Common Effect

Consider four hypothetical arms, each with true effect \(1.5\) mmHg and interim-estimate standard error \(0.8\) mmHg. Assume independent normal estimates. In one possible interim data set, the estimates are \(1.2\), \(2.1\), \(2.4\), and \(1.8\) mmHg. The third arm wins, with an observed estimate of \(2.4\) mmHg.

For four independent standard normal variables, numerical evaluation of the maximum’s density gives \(E[\max(Z_1,Z_2,Z_3,Z_4)]\approx1.0294\). Thus the expected bias due to selecting the winner is

0.8(1.0294)=0.8235\text{ mmHg}.

Across repetitions of this selection process under the stated model, the selected estimate has expectation approximately \(1.5+0.8235=2.3235\) mmHg. In the particular data set shown, the winning estimate exceeds the common true effect by \(2.4-1.5=0.9\) mmHg. That realized excess is not the expected bias; it is one outcome of the random process. The numerical calculation also shows why simply counting arms is insufficient: the expected bias is expressed in the same units as the estimates and scales with their standard error.

Two Arms With Different True Effects

When the true effects differ, selection can favour the genuinely better arm as well as an arm with a favourable error. The bias definition remains \(E[Y_J-\theta_J]\), not \(E[Y_J]-\theta_1\) for some fixed arm. For two arms with the same standard error \(\sigma\), let \(\delta=\theta_1-\theta_2\). The exact bias has a simple form:

E[Y_J-\theta_J] = \frac{\sigma}{\sqrt{\pi}} \exp\left(-\frac{\delta^2}{4\sigma^2}\right).

This result is positive for every finite \(\delta\). It is largest when the true effects are equal, and it decreases as the absolute difference between them grows. If one arm is very clearly better relative to the estimation noise, it usually wins, so the selection rule has less opportunity to choose an arm because of an unusually favourable error.

Worked Example: Unequal True Effects and Equal Standard Errors

Suppose two hypothetical arms have true effects \(5.0\) and \(4.0\) units, with independent interim estimates that each have standard error \(2.0\) units. Selection still goes to the arm with the larger estimated effect. Here \(\delta=5.0-4.0=1.0\) unit. The bias is

\frac{2.0}{\sqrt{\pi}} \exp\left(-\frac{1.0^2}{4(2.0)^2}\right) = 1.1284\exp(-0.0625) \approx 1.060\text{ units}.

For a check on the scale of this result, arm 1 is selected when \(Y_1-Y_2\ge0\). Since \(Y_1-Y_2\sim N(1.0,8.0)\), its selection probability is \(\Phi(1/\sqrt{8})\approx0.6382\). The expected true effect of the selected arm is therefore \(0.6382(5.0)+0.3618(4.0)=4.638\) units. Adding the \(1.060\)-unit bias gives an expected selected estimate of about \(5.698\) units. This is an expectation over repeated trials, not a prediction that a particular selected estimate will equal that value.

Why the Two-Arm Formula Holds

Write the estimation errors as \(\varepsilon_i=Y_i-\theta_i\), so each has mean zero and variance \(\sigma^2\). Arm 1 wins when \(\delta+\varepsilon_1-\varepsilon_2\ge0\). Set \(U=\varepsilon_1-\varepsilon_2\) and \(S=\varepsilon_1+\varepsilon_2\). Under independent normal errors with equal variances, \(S\) and \(U\) are independent, each has mean zero, and \(U\sim N(0,2\sigma^2)\). The error of the selected arm is

\varepsilon_J = \frac{S}{2} + \frac{U}{2}\operatorname{sign}(U+\delta).

The first term has expectation zero. For the second, \(\operatorname{sign}(U+\delta)\) is \(1\) when \(U\ge-\delta\) and \(-1\) otherwise. Using \(E[U]=0\) and the normal tail identity \(E[U\,1(U<-\delta)]=-\sqrt{2}\sigma\phi(\delta/(\sqrt{2}\sigma))\), we obtain

E[\varepsilon_J] = \sqrt{2}\sigma\phi\left(\frac{\delta}{\sqrt{2}\sigma}\right) = \frac{\sigma}{\sqrt{\pi}} \exp\left(-\frac{\delta^2}{4\sigma^2}\right).

The calculation makes the mechanism explicit: the winner’s error is not an ordinary zero-mean error after selection. The selection event depends on that error, so its conditional contribution to the selected estimate is positive on average.

Assumptions, Diagnostics, and Alternatives

The formulas are useful when interim arm estimates are approximately unbiased and normal, have known or suitably estimated standard errors, and are independent across arms. Independence is a substantive condition, not a mathematical convenience that can be ignored. For example, arm comparisons that share a control group can have correlated estimates. In that case, the product of cumulative probabilities in the general formula is invalid; the joint distribution must account for the correlation. Likewise, unequal standard errors, non-normal estimates, changing estimands, or selection based on several outcomes require a model that represents those features.

A practical diagnostic is to write down exactly what was compared and what was selected. Confirm that the selection rule really chooses the largest estimate on a common, directionally consistent scale. Then check whether the assumed variances and dependence structure match the trial’s allocation and analysis. If there are several candidate arms, compare the calculated bias with the standard error and with the observed differences among arms. A predicted bias that is negligible relative to the decision’s precision may have little practical impact; a bias similar to the estimated effect is consequential.

The calculation concerns reusing the interim estimates that determined the winner. If a new, independent sample estimates the selected arm’s effect after selection, and the new estimate is unbiased with sampling independent of the selection data, then its expected error conditional on the selected arm is zero. The estimate used to choose the arm and an independent estimate of the chosen arm are therefore not interchangeable.

Several design or analysis choices address different parts of the problem. Pre-specifying a single arm avoids selecting the largest among several noisy estimates, but removes the opportunity to choose among candidates. An independent validation stage separates selection data from estimation data, at the cost of additional observations. Selection-aware estimation or inference instead models the selection event when using the data that produced the winner. None of these choices makes the simple independent-normal formula applicable when its assumptions fail.

Common Mistakes and Limitations

Assuming an unbiased estimate stays unbiased after selection. Each \(Y_i\) has expectation \(\theta_i\) before selection, but \(Y_J\) is chosen precisely because it is large. Its error is consequently positive on average under the stated model.
Comparing the selected estimate with the wrong true effect. When arm effects differ, define bias relative to \(\theta_J\), the true effect of the arm actually selected, not automatically relative to the best arm or a fixed arm.
Treating the standard error as the selection bias. The standard error describes the spread of an individual estimate before selection. The bias depends on the number of candidates, their effects, their variances, and their dependence.
Applying the independence formula to correlated comparisons. Shared controls or other common data can induce correlation. Ignoring it can misstate how often an extreme estimate wins and how large the resulting bias is.

The normal model gives transparent quantitative results, but its answer is only as credible as the assumptions and selection rule. It does not by itself provide a corrected estimate, a confidence interval, or a testing procedure. Those require an analysis that represents the complete adaptive design rather than only the act of choosing the largest interim estimate.

Key takeaway: Selecting the largest among unbiased, independent normal estimates creates positive expected error in the winner. With equal true effects and a common standard error, the bias is the standard error multiplied by the expected maximum of \(K\) standard normal variables.

Check Your Understanding

Answer each question using the assumptions and definitions in this tutorial. Solutions follow the questions.

Two equally effective arms have independent normal estimates with standard error \(0.6\) units. What is the expected winner’s-curse bias?
Why is the bias not generally zero just because each arm estimate is unbiased before selection?
Two arms have equal standard errors \(\sigma=1.0\) and true effects differing by \(2.0\) units. Write the two-arm bias formula with these values. Is the bias larger or smaller than when the true effects are equal?
A trial selects an arm using interim data, then estimates that arm using a new independent sample. Under what conditions is the new estimate’s expected error zero conditional on the selected arm?

Solutions. (1) For two equally effective arms, the bias is \(0.6/\sqrt{\pi}=0.3385\) units. (2) Selection depends on the estimates: a positive random error makes an arm more likely to win, so the error of the winner has positive expectation even though each unselected error has mean zero. (3) Substituting \(\delta=2.0\) and \(\sigma=1.0\) gives \((1/\sqrt{\pi})\exp(-2^2/4)\approx0.2076\) units. It is smaller than the equal-effect bias \(1/\sqrt{\pi}\approx0.5642\) units because the true effects are farther apart relative to the standard error. (4) The new estimate must be unbiased for the selected arm’s effect, and its sampling data must be independent of the data and rule that selected that arm. Then its conditional expected estimation error is zero.