Truncated-Normal Bias at an Efficacy Boundary

Why Crossing Selects Large Estimates

At a fixed analysis, an efficacy decision is often made when an estimate is sufficiently far above the null value. Among repetitions that cross the boundary, estimates therefore tend to be higher than the true parameter. The size of this selection effect is not simply the distance from the null to the boundary: it depends on the true effect, the standard error, and how rarely crossing occurs.

This tutorial derives that effect for a one-sided boundary at one fixed look. The result is a truncated-normal calculation, expressed through the inverse Mills ratio. It quantifies the conditional mean and bias when crossing has occurred; it does not by itself describe the average bias across every trial or a design with multiple possible stopping looks.

Model and Boundary

Assume that at the look under study the estimate $\hat{\theta}$ has a normal sampling distribution with known, fixed standard error $\tau>0$:

\hat{\theta}\sim N(\theta,\tau^2).

Here $\theta$ is the true parameter and $\theta_0$ is the null value. Define the one-sided test statistic and its standardized true effect by

Z=\frac{\hat{\theta}-\theta_0}{\tau}, \qquad \delta=\frac{\theta-\theta_0}{\tau}.

It follows that $Z\sim N(\delta,1)$. Suppose efficacy is declared at this look when $Z>c$, for a specified boundary $c$. Equivalently, the estimate must satisfy $\hat{\theta}>\theta_0+c\tau$. The calculation below conditions only on crossing this boundary at this look.

Definition: For a standard normal random variable, let $\phi(x)$ denote its density and $\Phi(x)$ its cumulative distribution function. The inverse Mills ratio for an upper tail is $$ \lambda(x)=\frac{\phi(x)}{1-\Phi(x)}. $$ Under the fixed-look normal model, the conditional mean and conditional bias given efficacy crossing are $$ E(\hat{\theta}\mid Z>c)=\theta+\tau\lambda(c-\delta), \qquad E(\hat{\theta}\mid Z>c)-\theta=\tau\lambda(c-\delta). $$

The bias here is defined relative to the true parameter, conditional on the event $Z>c$. Since $\lambda(x)>0$, this conditional bias is upward. Its size is in the units of $\theta$, because $\tau$ has those same units.

Derivation of the Conditional Mean

Standardize the estimation error by writing $X=(\hat{\theta}-\theta)/\tau$, so $X\sim N(0,1)$. Since $\hat{\theta}=\theta+\tau X$, the crossing event can be rewritten as

Z>c \quad\Longleftrightarrow\quad \frac{\hat{\theta}-\theta_0}{\tau}>c \quad\Longleftrightarrow\quad X>c-\delta.

Thus the selected estimation error is a standard normal variable truncated below at $a=c-\delta$. To find its conditional mean, use the standard normal density and the identity $\phi'(x)=-x\phi(x)$:

E(X\mid X>a) =\frac{\int_a^\infty x\phi(x)\,dx}{1-\Phi(a)} =\frac{\phi(a)}{1-\Phi(a)} =\lambda(a).

For the middle equality, integrating $-\phi'(x)$ from $a$ to infinity gives $\phi(a)$, since $\phi(x)$ tends to zero as $x$ tends to infinity. Substituting $a=c-\delta$ into $\hat{\theta}=\theta+\tau X$ gives both the conditional mean and the conditional bias in the definition.

Worked Example: A Rare Crossing Under the Null

Suppose a hypothetical estimate is measured in points, with $\theta_0=\theta=5.0$ points, known standard error $\tau=0.4$ points, and efficacy boundary $c=2.0$. Find the crossing probability and the conditional mean estimate.

The standardized true effect is $\delta=(5.0-5.0)/0.4=0$. Therefore $Z\sim N(0,1)$, and the crossing probability is

P(Z>2.0)=1-\Phi(2.0)=0.02275.

For the conditional mean, $c-\delta=2.0$. Using $\phi(2.0)=0.05399$ and $1-\Phi(2.0)=0.02275$,

\lambda(2.0)=\frac{0.05399}{0.02275}=2.373, \qquad E(\hat{\theta}\mid Z>2.0)=5.0+0.4(2.373)=5.949\text{ points}.

The conditional bias is $0.4(2.373)=0.949$ points. Crossing requires an estimate above $5.0+2(0.4)=5.8$ points, and the mean among crossing estimates is 5.949 points. The result is a conditional average over hypothetical repetitions, not a prediction that any particular estimate equals 5.949.

How the True Effect Changes the Bias

The boundary $c$ is fixed on the test-statistic scale, but the truncation point for the standardized estimation error is $c-\delta$. If the true effect is farther above the null, crossing is less selective: a larger share of estimates cross, and their average excess above $\theta$ is smaller. If crossing is rare under the true effect, the selected estimates tend to be farther above $\theta$.

Worked Example: A Positive True Effect

Consider a hypothetical setting with $\theta_0=10.0$ units, true value $\theta=10.5$ units, $\tau=0.5$ units, and boundary $c=2.0$. The model assumes the estimate is normally distributed with this known standard error.

Here $\delta=(10.5-10.0)/0.5=1.0$, so $Z\sim N(1,1)$. The crossing probability is

P(Z>2.0)=1-\Phi(2.0-1.0)=1-\Phi(1.0)=0.1587.

The inverse Mills ratio is $\lambda(1.0)=\phi(1.0)/[1-\Phi(1.0)]=0.2420/0.1587=1.525$. Hence

E(\hat{\theta}\mid Z>2.0)=10.5+0.5(1.525)=11.263\text{ units}.

The conditional bias is $0.763$ units. The efficacy threshold for the estimate is $10.0+2(0.5)=11.0$ units, so the conditional mean is above that threshold, as it must be for estimates restricted to exceed it.

Worked Example: Crossing Is Almost Certain

Now suppose a hypothetical study has $\theta_0=0$ units, $\theta=1.2$ units, $\tau=0.3$ units, and $c=2.0$. The normal, fixed-standard-error assumptions are retained.

The standardized true effect is $\delta=1.2/0.3=4.0$, and the truncation point is $c-\delta=-2.0$. Thus

P(Z>2.0)=1-\Phi(-2.0)=0.9772.

Using $\phi(-2.0)=0.05399$ and $1-\Phi(-2.0)=0.9772$,

\lambda(-2.0)=\frac{0.05399}{0.9772}=0.05525, \qquad E(\hat{\theta}\mid Z>2.0)=1.2+0.3(0.05525)=1.217\text{ units}.

The conditional bias is about $0.0166$ units. Because crossing is already very likely at the true effect, conditioning on crossing changes the mean only slightly.

Interpretation, Diagnostics, and Limits

The formula depends on the model as well as the boundary. In particular, it assumes a normal estimate, a known fixed standard error, a fixed one-sided threshold, and conditioning on crossing at this single look. The standard error may be fixed by the design; it is not being re-estimated after observing the data. The calculation is exact under these assumptions.

A practical diagnostic is to check whether the assumed sampling distribution and standard error are credible at the analysis look, and whether the efficacy decision really is the event $Z>c$. A threshold on a different statistic, a data-dependent standard error, or a discrete test statistic may not yield this exact truncated-normal calculation. In those settings the selected sampling distribution must be derived for the actual statistic and rule, or the normal formula must be justified as an approximation.

The result also concerns one conditional target only: $E(\hat{\theta}\mid Z>c)-\theta$. It is not the unconditional bias $E(\hat{\theta})-\theta$ over all repetitions. Nor does it account for the possibility of crossing at one of several looks, stopping for another reason, or continuing to a later analysis. Those events change what is being conditioned on and require a calculation that reflects the full stopping rule.

Quantity or approach	What it answers	Key distinction
Unselected estimate	Its mean under the model is $\theta$.	Does not condition on an efficacy decision.
One-look truncated-normal calculation	The mean and bias among repetitions crossing $Z>c$.	Uses $\tau\lambda(c-\delta)$ under the stated normal model.
Full sequential analysis	Estimation behavior under a rule with multiple looks or stopping outcomes.	Must incorporate the complete set of possible analysis paths; the one-look formula alone is insufficient.

A frequent mistake is to use $\lambda(c)$ for every true effect. That argument is correct only when $\delta=0$, so that the null value is the true value. In general, the estimate's error is truncated at $c-\delta$, and the correct bias is $\tau\lambda(c-\delta)$. Another mistake is to call this conditional bias the bias of the estimator across all trials: it describes only repetitions satisfying the specified crossing event.

Key takeaway: At a fixed one-sided efficacy look, a normal estimate has conditional upward bias $\tau\lambda(c-\delta)$ among boundary-crossing repetitions. The inverse Mills ratio uses the boundary relative to the true mean, $c-\delta$, not the boundary alone.

Check Your Understanding

Use the fixed-look normal model and the definitions above. Each problem concerns conditioning only on crossing at the stated look.

For $\delta=0$ and $c=1.0$, write the crossing probability and conditional bias in terms of $\Phi$, $\phi$, and $\tau$.
Suppose $\theta_0=20$ units, $\theta=21$ units, $\tau=0.5$ units, and $c=2$. Find $\delta$, $c-\delta$, and the conditional mean using $\lambda(0)=0.7979$.
Why is $\lambda(c)$ not generally the right multiplier for $\tau$?
Does the one-look conditional-bias formula give the unconditional bias across trials in a design that can stop at several looks? Explain.

Solutions. (1) The crossing probability is $1-\Phi(1)$, and the conditional bias is $\tau\phi(1)/[1-\Phi(1)]=1.525\tau$. (2) $\delta=(21-20)/0.5=2$, so $c-\delta=0$. The conditional mean is $21+0.5(0.7979)=21.399$ units. (3) The error is truncated at $c-\delta$; only under the null, when $\delta=0$, does this equal $c$. (4) No. The formula conditions on crossing at one specified look; an unconditional calculation for a multiple-look design must account for all stopping and continuation paths.

Quantity or approach	What it answers	Key distinction
Unselected estimate	Its mean under the model is \(\theta\).	Does not condition on an efficacy decision.
One-look truncated-normal calculation	The mean and bias among repetitions crossing \(Z>c\).	Uses \(\tau\lambda(c-\delta)\) under the stated normal model.
Full sequential analysis	Estimation behavior under a rule with multiple looks or stopping outcomes.	Must incorporate the complete set of possible analysis paths; the one-look formula alone is insufficient.