Recovering Conditional Error From an Original Design

Why Recover the Original Conditional Error?

When a trial reaches an interim analysis, its original design assigns a specific probability to eventual rejection under the null hypothesis, conditional on the data observed so far. That probability is the design’s conditional error. Recovering it matters when evaluating how much Type I error the original design leaves for the future: the quantity depends on the interim result, not just on the nominal significance level.

Here we derive the function for a two-look, one-sided group sequential test with an upper efficacy boundary at each look and no futility stopping. The earlier material on conditional error gives the general principle; the focus here is to obtain the function explicitly from the original stopping rule and the canonical joint normal distribution.

Definition: For interim data summarized by \(Z_1=z\), the conditional error of an original design is the probability, under \(H_0\), that the original design will reject \(H_0\), conditional on \(Z_1=z\). It includes rejection at the interim look if that boundary has already been crossed.

Model and Original Stopping Rule

Let \(I_1\) and \(I_2\) be the information at the interim and final analyses, where \(0<I_1<I_2\). Write \(Z_1\) and \(Z_2\) for the standardized test statistics at those looks. The canonical model under \(H_0\) is

\begin{pmatrix}Z_1\\ Z_2\end{pmatrix} \sim N\left( \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}1&\rho\\ \rho&1\end{pmatrix} \right), \qquad \rho=\sqrt{\frac{I_1}{I_2}}.

Thus each statistic is standard normal under \(H_0\), and their correlation is determined by the information fraction. This model follows from the independent-increments structure of the accumulated score: information added after the interim analysis is independent of the score accumulated before it. It is exact in the corresponding normal-score model and is commonly used as a large-sample model for standardized statistics.

Let \(a_1\) and \(a_2\) be the original upper critical values. The stopping rule rejects \(H_0\) at the interim look if \(Z_1\geq a_1\). If \(Z_1<a_1\), the trial continues and rejects at the final look if \(Z_2\geq a_2\). The assumptions for the calculation are that these boundaries and the planned information values are fixed, the stated canonical joint distribution applies under \(H_0\), and there are no other stopping or rejection rules.

Key result: Under this model, the conditional error after observing \(Z_1=z\) is

A(z)= \begin{cases} 1, & z\geq a_1,\\[4pt] 1-\Phi\!\left(\dfrac{a_2-\rho z}{\sqrt{1-\rho^2}}\right), & z<a_1, \end{cases} \qquad \rho=\sqrt{\frac{I_1}{I_2}}.

Here \(\Phi\) is the standard normal cumulative distribution function. The second line is the probability of crossing the original final boundary, conditional on not having crossed the interim boundary.

Deriving the Conditional Error Function

The canonical model can be represented as

Z_2=\rho Z_1+\sqrt{1-\rho^2}\,\varepsilon, \qquad \varepsilon\sim N(0,1),

where \(\varepsilon\) is independent of \(Z_1\). Given \(Z_1=z\), the only random part of \(Z_2\) is the independent standard normal term. Therefore,

Z_2\mid Z_1=z \sim N\!\left(\rho z,\,1-\rho^2\right).

If \(z\geq a_1\), rejection has already occurred, so the probability that the original design rejects is 1. If \(z<a_1\), rejection can occur only at the final look. Standardizing the conditional final statistic gives

\begin{aligned} A(z) &=P(Z_2\geq a_2\mid Z_1=z,H_0)\\ &=P\left( \frac{Z_2-\rho z}{\sqrt{1-\rho^2}} \geq \frac{a_2-\rho z}{\sqrt{1-\rho^2}} \ \middle|\ Z_1=z,H_0 \right)\\ &=1-\Phi\!\left(\frac{a_2-\rho z}{\sqrt{1-\rho^2}}\right), \qquad z<a_1. \end{aligned}

This derivation is specific to the original boundaries and the null joint distribution. In particular, the probability is not the marginal probability \(P(Z_2\geq a_2)\): conditioning on \(Z_1=z\) changes the distribution of \(Z_2\).

Worked Example: Conditional Error at an Interim Result

Suppose a hypothetical two-look design has \(I_1/I_2=0.50\), interim boundary \(a_1=2.40\), and final boundary \(a_2=2.00\). The observed interim statistic is \(z=1.20\). Calculate the original conditional error.

Since \(1.20<2.40\), the interim boundary has not been crossed. The correlation and conditional standard deviation are

\rho=\sqrt{0.50}=0.7071, \qquad \sqrt{1-\rho^2}=\sqrt{0.50}=0.7071.

The conditional mean of \(Z_2\) is \(\rho z=(0.7071)(1.20)=0.8485\). Hence

\begin{aligned} A(1.20) &=1-\Phi\!\left(\frac{2.00-(0.7071)(1.20)}{0.7071}\right)\\ &=1-\Phi(1.628)\\ &\approx 0.0517. \end{aligned}

Under \(H_0\), conditional on this interim statistic and on the original design’s model, the probability of eventual rejection is about 5.17%. This is a conditional Type I error probability for this interim result; it is not the probability that \(H_0\) is true.

How the Function Changes With Interim Evidence

For \(z<a_1\), the conditional final mean \(\rho z\) increases with \(z\). The final boundary is therefore less extreme relative to the conditional distribution as the interim statistic increases, so the conditional error increases. The conditional error is not constant across interim results.

Worked Example: Comparing Interim Results

Use the same hypothetical design as above: information fraction \(0.50\), \(a_1=2.40\), and \(a_2=2.00\). Compare the conditional errors at \(z=0\), \(z=0.80\), and \(z=1.80\). All three values are below the interim efficacy boundary.

At \(z=0\),

A(0)=1-\Phi\!\left(\frac{2.00}{0.7071}\right) =1-\Phi(2.828) \approx 0.0023.

At \(z=0.80\), the conditional mean is \((0.7071)(0.80)=0.5657\), giving

A(0.80) =1-\Phi\!\left(\frac{2.00-0.5657}{0.7071}\right) =1-\Phi(2.028) \approx 0.0213.

At \(z=1.80\), the conditional mean is \((0.7071)(1.80)=1.273\), so

A(1.80) =1-\Phi\!\left(\frac{2.00-1.273}{0.7071}\right) =1-\Phi(1.028) \approx 0.1519.

The increasing values reflect the stronger interim evidence: under \(H_0\), a larger observed \(Z_1\) shifts the conditional distribution of \(Z_2\) upward. These calculations use the same boundaries and information fraction; only the observed interim statistic changes.

At \(z\geq a_1\), the conditional error is 1 because the original design has already rejected. Thus the full function has a jump at the interim boundary if the limiting continuation probability just below \(a_1\) is less than 1. This is a consequence of the stopping rule, not a numerical defect. In a trial that has already stopped for efficacy, there is no remaining decision to make.

Worked Example: A Different Interim Information Fraction

Consider another hypothetical design with \(I_1/I_2=0.36\), \(a_1=2.50\), and \(a_2=2.10\). Suppose the interim statistic is \(z=1.00\). The interim boundary has not been crossed, and

\rho=\sqrt{0.36}=0.60, \qquad \sqrt{1-\rho^2}=\sqrt{0.64}=0.80.

The conditional mean of the final statistic is \(0.60\), and the conditional standard deviation is \(0.80\). Therefore,

\begin{aligned} A(1.00) &=1-\Phi\!\left(\frac{2.10-(0.60)(1.00)}{0.80}\right)\\ &=1-\Phi(1.875)\\ &\approx 0.0304. \end{aligned}

The original design assigns about a 3.04% conditional rejection probability to this interim result under \(H_0\). The information fraction matters because it determines both the correlation with the final statistic and the conditional variance.

Connection to the Original Type I Error

The conditional error function must agree with the original design’s unconditional rejection probability when averaged over the interim statistic under \(H_0\). For the rule above, the original Type I error is

\begin{aligned} \alpha_{\mathrm{orig}} &=P(Z_1\geq a_1) +P(Z_1<a_1,\ Z_2\geq a_2)\\ &=P(Z_1\geq a_1) +\int_{-\infty}^{a_1} A(z)\phi(z)\,dz\\ &=\int_{-\infty}^{\infty} A(z)\phi(z)\,dz, \end{aligned}

where \(\phi\) is the standard normal density, and \(A(z)=1\) for \(z\geq a_1\). The second equality follows by conditioning on \(Z_1\); the final equality separates the early-rejection region from the continuation region. This identity is a useful check on a numerical implementation: integrating the conditional error over the null distribution of \(Z_1\) must recover the original design’s Type I error.

The conditional error is not generally “the nominal alpha minus the alpha already spent.” That subtraction gives a single residual amount, whereas \(A(z)\) varies with the observed interim result. Nor is conditional error the same as conditional power: conditional error is calculated under \(H_0\) and concerns a rejection probability, while conditional power concerns rejection under a specified alternative.

Diagnostics, Alternatives, and Limitations

A direct calculation from the conditional normal distribution is usually the clearest way to recover the function. Bivariate normal integration gives the same probabilities, but may be less transparent about the dependence on \(z\). Simulation can check an implementation or assess departures from the assumed model, but a finite simulation does not replace the analytic conditional probability and carries Monte Carlo error.

Useful checks include confirming that \(0<I_1/I_2<1\), that the resulting \(\rho\) lies between 0 and 1, and that \(A(z)\) increases with \(z\) throughout the continuation region. Check also that the implemented rule assigns \(A(z)=1\) after crossing the interim efficacy boundary and uses the correct final boundary. A decreasing continuation-region function, a probability outside \([0,1]\), or disagreement between integration and the original Type I error indicates a likely coding or model-specification error.

The result relies on the canonical joint normal model and the stated stopping rule. If the statistic is not adequately normal, the information values are misrepresented, the interim timing changes the joint distribution, or an omitted futility rule changes which paths can reject, this formula need not describe the original design’s conditional error. Under a futility rule, for example, the continuation region and rejection event must include that rule; the formula above cannot simply be reused unchanged.

The practical distinction is between calculating the original conditional error and choosing a future procedure. This tutorial supplies the probability budget implied by the original design at each possible interim result. It does not assert that every subsequent design has this same conditional error; any proposed continuation must be assessed against the conditional error available at the observed result.

Common mistake: “The original test has significance level 0.025, so after the interim analysis the remaining conditional error is 0.025 minus the interim alpha spent.” This is generally wrong because the remaining rejection probability depends on the observed \(z\), through the conditional distribution of \(Z_2\mid Z_1=z\). Calculate \(A(z)\) from the original stopping rule and canonical joint distribution instead.

Check Your Understanding

Use the definition and derivation above. The solutions follow the questions.

In the continuation region, what are the conditional mean and variance of \(Z_2\) given \(Z_1=z\)?
A design has information fraction \(0.25\), final boundary \(a_2=1.96\), and observed \(z=0\), with no interim crossing. Write the conditional-error calculation and give its approximate value.
Why is the conditional error equal to 1 when the interim statistic has crossed the original efficacy boundary?
State an integral identity that connects the conditional error function to the original unconditional Type I error.

Solutions. (1) The correlation is \(\rho=\sqrt{I_1/I_2}\), so \(Z_2\mid Z_1=z\sim N(\rho z,1-\rho^2)\). (2) Here \(\rho=\sqrt{0.25}=0.50\) and \(\sqrt{1-\rho^2}=\sqrt{0.75}=0.8660\). Thus \(A(0)=1-\Phi(1.96/0.8660)=1-\Phi(2.263)\approx 0.0118\). (3) The original rule has already rejected at the interim look, so eventual rejection under that rule is certain for that observed path. (4) With \(\phi\) the standard normal density, \(\alpha_{\mathrm{orig}}=\int_{-\infty}^{\infty}A(z)\phi(z)\,dz\), because averaging the conditional rejection probability over the null distribution of the interim statistic recovers the unconditional rejection probability.