Conditional Error At A Boundary

Why Boundary Values Need Special Attention

A conditional error function describes the remaining type I error after interim data have been observed: it is the probability, under the null hypothesis, that the original design will ultimately reject, conditional on those data. Away from a stopping boundary, this probability may vary smoothly with the interim statistic. At a boundary, however, the original design may stop and make a definite decision. The conditional error can then jump between a continuation probability and zero or one.

That distinction matters when a trial is modified. A modified continuation procedure is assessed against the conditional error assigned by the original procedure at the observed interim history. The exact rule for equality—whether a value exactly on a boundary triggers stopping or continuation—therefore belongs in the design specification. It is not merely a cosmetic choice.

Definition: Let \(X\) denote the interim data, and let \(R\) be the event that the original design rejects \(H_0\). The conditional error at interim data \(X=x\) is \(A(x)=P_{H_0}(R\mid X=x)\). It is a probability under \(H_0\), conditional on the observed interim data, and it lies between zero and one.

A Two-Stage Model and Its Boundary Rules

Consider a one-sided two-stage normal test under \(H_0\). Let \(Z_1\) be the standardized interim statistic and \(Z_2\) the standardized future increment. Assume independent increments, so under the null \(Z_1\) and \(Z_2\) are independent and each has a standard normal distribution. Suppose the final statistic is \(Z_{\mathrm{final}}=\sqrt{t}Z_1+\sqrt{1-t}Z_2\), where \(t\) is the fraction of total information available at interim. This normalization gives \(Z_{\mathrm{final}}\) a standard normal distribution under \(H_0\) before conditioning.

For illustration, take \(t=0.5\), a final one-sided rejection boundary \(c=1.96\), an efficacy boundary \(u=2.00\), and a futility boundary \(l=-0.50\). Specify the original rule explicitly: reject for efficacy when \(Z_1\geq u\); stop for futility without rejection when \(Z_1\leq l\); otherwise continue and reject at the final analysis if \(Z_{\mathrm{final}}\geq c\). These assumptions define the conditional error function, including its values at equality.

A(z)= \begin{cases} 0, & z\leq l,\\ 1-\Phi\!\left(\dfrac{c-\sqrt{t}\,z}{\sqrt{1-t}}\right), & l<z<u,\\ 1, & z\geq u. \end{cases}

Here \(\Phi\) is the standard normal cumulative distribution function. In the continuation region, conditioning on \(Z_1=z\) leaves \(Z_2\sim N(0,1)\). The final rejection condition becomes \(Z_2\geq(c-\sqrt{t}\,z)/\sqrt{1-t}\), which gives the middle expression. The stopping rules give the other two pieces directly: futility guarantees no rejection, while efficacy guarantees rejection.

Worked Example: Conditional Error Just Below Efficacy

At \(z=1.99\), the trial is still in the continuation region. Calculate the original conditional error.

Substitute \(c=1.96\), \(t=0.5\), and \(z=1.99\) into the continuation expression:

\begin{aligned} A(1.99) &=1-\Phi\!\left(\frac{1.96-\sqrt{0.5}(1.99)}{\sqrt{0.5}}\right)\\ &=1-\Phi(0.7819)\\ &\approx 0.2171. \end{aligned}

Thus, under \(H_0\), conditional on this interim statistic and the stated model, the original design has about a 21.71% probability of eventual rejection. The statistic is below the efficacy boundary, so the design does not assign conditional error one here.

Why the Function Jumps at a Boundary

Within the continuation region, the normal-tail probability changes continuously with \(z\). The boundary itself invokes a different action. At efficacy, the specified rule rejects immediately, assigning conditional error one. Just below efficacy, it continues, so the conditional error is only the chance that the future statistic crosses the final boundary. At futility, the rule stops without rejection, assigning zero; just above futility, it continues and retains a small positive chance of rejection.

Worked Example: The Efficacy Boundary and Its Left-Hand Limit

Compare the value at \(z=2.00\) with the limiting continuation value as \(z\) approaches 2.00 from below. At the boundary itself, \(Z_1\geq2.00\) triggers efficacy stopping, so \(A(2.00)=1\). The continuation-side limit is calculated from the middle branch:

\begin{aligned} \lim_{z\uparrow 2.00}A(z) &=1-\Phi\!\left(\frac{1.96-\sqrt{0.5}(2.00)}{\sqrt{0.5}}\right)\\ &=1-\Phi(0.7719)\\ &\approx 0.2200. \end{aligned}

The jump is therefore from a continuation-side limit of about 0.2200 to a boundary value of 1. The value at \(z=1.99\) is about 0.2171, not the limit itself: it is the conditional error at that particular nearby statistic. A discontinuity does not mean that the continuation probability is calculated incorrectly; it reflects the change from continuing to rejecting immediately.

Worked Example: The Futility Boundary and Its Right-Hand Limit

At \(z=-0.49\), the statistic is just above the futility boundary and the trial continues. The conditional error is

\begin{aligned} A(-0.49) &=1-\Phi\!\left(\frac{1.96-\sqrt{0.5}(-0.49)}{\sqrt{0.5}}\right)\\ &=1-\Phi(3.262)\\ &\approx 0.0005534. \end{aligned}

At the boundary itself, \(z=-0.50\), the rule \(Z_1\leq l\) stops for futility, so \(A(-0.50)=0\). The limit from the continuation side is

\begin{aligned} \lim_{z\downarrow -0.50}A(z) &=1-\Phi\!\left(\frac{1.96-\sqrt{0.5}(-0.50)}{\sqrt{0.5}}\right)\\ &=1-\Phi(3.272)\\ &\approx 0.0005342. \end{aligned}

The jump is small in absolute size, but its existence is important: the value at the boundary is zero, whereas the continuation-side limit is positive. Rounding either value to zero would hide the distinction that the stopping rule creates.

Equality Conventions and Adapted Tests

A boundary rule must settle equality, for example by using \(Z_1\geq u\) for efficacy and \(Z_1\leq l\) for futility. Reversing a weak inequality to a strict one changes the conditional error assigned at the boundary. When \(Z_1\) has a continuous distribution, a single point has probability zero under the original design. Consequently, changing only that point may leave the original unconditional type I error unchanged. But conditional error is defined given the observed interim data, and an adaptation may act specifically at that value.

The distinction becomes even more consequential if the interim statistic is discrete, rounded, or has a point mass. Then a boundary value can occur with positive probability. The expectation identity below shows why respecting the conditional error at such values matters for overall error control.

Key result: Under \(H_0\), the original unconditional rejection probability is the expectation of its conditional error: \(P_{H_0}(R)=E_{H_0}\{A(X)\}\). If a modified procedure has conditional rejection probability \(B(x)\) and satisfies \(B(x)\leq A(x)\) for all relevant interim histories, its unconditional rejection probability under the same null distribution cannot exceed that of the original design. A violation at a history of positive null probability removes this pointwise guarantee; it does not, by itself, prove that the overall type I error must increase if the procedure reduces error elsewhere.

How a Boundary Atom Can Change the Error Budget

For a discrete interim statistic, the equality convention can affect a whole probability mass rather than a probability-zero point. The next calculation isolates that consequence. Suppose the original rule continues at \(Z_1=1.99\), and a discrete interim statistic has null probability 0.10 of taking exactly that value. The original conditional error there is the value already calculated, approximately 0.2171. If a modified rule instead rejects immediately whenever \(Z_1=1.99\), its conditional rejection probability at that atom is one.

Worked Example: Error Increase at a Discrete Boundary Value

At the atom, the modified rule exceeds the original conditional error by \(1-0.2171=0.7829\). The contribution of this atom to the difference in unconditional rejection probabilities is therefore

$$ 0.10(1-0.2171)=0.07829. $$

If the modified procedure is otherwise unchanged, its unconditional type I error increases by about 0.07829, or 7.829 percentage points. This is not a claim that any local conditional-error exceedance necessarily produces that overall increase: the calculation relies on the stated 0.10 atom and no offsetting changes elsewhere. It demonstrates why a positive-probability boundary value cannot be dismissed as negligible.

Checking a Proposed Boundary Adaptation

For each boundary, separate three questions: what action does the original design take below the boundary, at equality, and above it; what conditional error follows from each action; and what action will the modified design take at those same histories? A continuation procedure must be compared with the original conditional error at the actual observed value, not just with a nearby limit or an unconditional alpha level.

Write the inequalities.
Record whether equality triggers stopping or continuation. Do not leave a boundary convention implicit.

Assign the original boundary value.
Immediate efficacy rejection gives conditional error one; nonrejection at futility gives zero. For continuation, calculate the remaining null rejection probability.

Check the modified rule at the same history.
Compare its conditional rejection probability with the original value, paying particular attention to values with positive probability.

Useful diagnostics include evaluating the function at a point inside each region, calculating the appropriate one-sided limits, and checking the value assigned at equality separately. If an implementation reports a smooth curve through a boundary where the trial stops, inspect whether it has plotted only the continuation expression. If a discrete or rounded statistic is used, inspect its probability mass at each boundary. Numerical rounding should not be used to decide whether an observed value falls on one side of a boundary.

Comparison and Limitations

Conditional-error control is a history-specific comparison: it asks whether the modified test spends more type I error than the original test allowed after the observed interim data. Recalibrating a test using only an unconditional error probability is a different approach; it may control an overall error rate while obscuring how error is allocated across interim histories. Conversely, checking only a smooth continuation formula is insufficient if the design also has stopping decisions at its boundaries. The correct comparison includes both the continuation branch and the exact stopping convention.

The calculations here rely on the stated normal independent-increments model and on the specified one-sided test. If the interim statistic is discrete, its conditional distribution and point masses must be handled according to that design rather than assumed to be continuous. In more complicated designs, conditional error may depend on the full interim history, not merely a single statistic. The central boundary lesson remains: state the decision rule at equality, calculate the conditional error under that rule, and evaluate adaptations at the exact histories they can encounter.

Common pitfall: “A single boundary point has probability zero, so its assigned conditional error never matters.” This reasoning can apply to the original unconditional error when the interim statistic is continuous and only that point is changed. It does not establish that a modified test respects conditional error at the observed point, and it fails outright when the statistic has an atom there. Check the distribution and the equality rule before treating a boundary value as negligible.

Check Your Understanding

Answer each question using the assumptions and boundary conventions stated above.

In the continuation region, why is the conditional rejection probability an upper-tail probability for \(Z_2\)?
What are the conditional error at \(z=2.00\) and the limit from below? Why do they differ?
At \(z=-0.50\), the rule stops for futility. What is the boundary value, and how does it differ from the continuation-side limit?
If a statistic has probability 0.05 at \(z=1.99\), and a modification changes the conditional rejection probability there from 0.2171 to 1 while leaving all other histories unchanged, what is the increase in unconditional rejection probability?

Solutions. 1. Given \(Z_1=z\), the final rejection condition \(Z_{\mathrm{final}}\geq1.96\) rearranges to \(Z_2\geq(1.96-\sqrt{0.5}z)/\sqrt{0.5}\), and \(Z_2\) is standard normal under the null. 2. The boundary value is 1 because equality triggers immediate efficacy rejection; the limit from below is approximately 0.2200 because values just below the boundary continue. 3. The value is zero because equality triggers nonrejection for futility; the right-hand continuation limit is approximately 0.0005342. 4. The increase is \(0.05(1-0.2171)=0.03915\), or 3.915 percentage points.