Computing Conditional Error in a Canonical Design

Why Conditional Error Depends on the Interim Statistic

In a two-look design, the interim data provide information about the final statistic, but they do not determine it. Conditional error quantifies the remaining chance of rejection under the null hypothesis, given the interim result and the original design’s rejection rule. It is therefore a probability conditional on observed interim information, not the overall Type I error rate.

This tutorial considers a one-sided test in a canonical normal model. The goal is to calculate the probability that the final statistic crosses its original boundary, conditional on the interim statistic. The calculation assumes the original final boundary is fixed; how to choose a replacement stage-two boundary is a separate question.

Definition: For a design with rejection event \(R\) and interim data \(D_1\), the conditional error at the interim data is \(P_{H_0}(R\mid D_1)\). In a two-look design that continues to the final look whenever it has not rejected early, this is the conditional probability of final-look rejection on a continuing path.

Canonical Two-Look Normal Model

Let \(Z_1\) be the standardized statistic at the interim look and \(Z_2\) the cumulative standardized statistic at the final look. Write \(t=I_1/I_2\) for the interim information as a fraction of the final information, where \(0<t<1\). Under the null hypothesis, assume the canonical independent-increments representation

Z_1\sim N(0,1),\qquad Z_2=\sqrt{t}\,Z_1+\sqrt{1-t}\,Z_{\mathrm{inc}}, \qquad Z_{\mathrm{inc}}\sim N(0,1),

where \(Z_{\mathrm{inc}}\) is independent of \(Z_1\). Thus \(Z_2\) also has a standard normal distribution under \(H_0\), and the correlation between \(Z_1\) and \(Z_2\) is \(\sqrt{t}\). The model applies when the standardized score has independent increments and information is measured on the scale that gives this canonical representation. A normal approximation may support the same calculation for large-sample tests, but its adequacy then needs assessment.

Suppose the original final-look rule rejects \(H_0\) when \(Z_2\ge c\), with fixed boundary \(c\). If there is also an early efficacy boundary \(b_1\), the design rejects early when \(Z_1\ge b_1\); the conditional error formula below applies on paths that have not already rejected and that continue to the final look. Any futility rule must also be respected: if it stops the trial without rejection, the conditional probability of a later rejection is zero.

Deriving the Conditional-Error Formula

Once the interim value \(Z_1=z_1\) is observed, the only randomness remaining in the final statistic is the independent increment. Substituting the observed value into the canonical representation gives a normal conditional distribution with mean \(\sqrt{t}\,z_1\) and variance \(1-t\).

Z_2\mid Z_1=z_1 \sim N\!\left(\sqrt{t}\,z_1,\;1-t\right).

The conditional error is the probability that a draw from this conditional distribution is at least the original final boundary. Standardizing by its conditional standard deviation yields

A(z_1) =P_{H_0}(Z_2\ge c\mid Z_1=z_1) =1-\Phi\!\left(\frac{c-\sqrt{t}\,z_1}{\sqrt{1-t}}\right),

where \(\Phi\) is the standard normal cumulative distribution function. This is a pointwise conditional probability for a continuing path. It is not the unconditional probability of rejection, and it is not the probability that \(H_0\) is true.

Derivation: From \(Z_2=\sqrt{t}\,z_1+\sqrt{1-t}\,Z_{\mathrm{inc}}\), final rejection is equivalent to \(Z_{\mathrm{inc}}\ge (c-\sqrt{t}\,z_1)/\sqrt{1-t}\). Since \(Z_{\mathrm{inc}}\sim N(0,1)\) independently of the interim data, the probability of this event is the upper standard-normal tail in the formula for \(A(z_1)\).

The formula also shows how the interim statistic affects the remaining rejection probability. Differentiating with respect to \(z_1\) gives

A'(z_1)= \sqrt{\frac{t}{1-t}}\, \phi\!\left(\frac{c-\sqrt{t}\,z_1}{\sqrt{1-t}}\right)>0,

where \(\phi\) is the standard normal density. Thus higher interim statistics imply higher conditional error for this one-sided upper-boundary test. This monotonicity follows from the assumed positive association between the interim and final statistics; it should not be carried over automatically to a different test statistic or rejection region.

Worked Example: Conditional Error at a Zero Interim Statistic

Suppose a hypothetical one-sided design has information fraction \(t=0.75\), final boundary \(c=1.645\), and observed interim statistic \(z_1=0\). Assume the canonical model holds and the path continues to the final look. Calculate the conditional error.

The conditional mean is \(\sqrt{0.75}(0)=0\), and the conditional standard deviation is \(\sqrt{1-0.75}=0.5\). Therefore,

A(0)=1-\Phi\!\left(\frac{1.645-\sqrt{0.75}(0)}{\sqrt{0.25}}\right) =1-\Phi(3.290) \approx 0.00050.

The interim statistic is exactly zero, but the final statistic has only \(0.5\) units of conditional standard deviation remaining. The original boundary is therefore 3.29 conditional standard deviations above its conditional mean, making the remaining rejection probability about \(0.00050\). This is not the design’s overall Type I error; it is the conditional probability at this particular interim value.

Worked Calculations at Other Interim Values

Worked Example: A Less Advanced Interim Look

Consider a hypothetical design with \(t=0.75\), the same one-sided final boundary \(c=1.645\), and an interim statistic \(z_1=-1\). Assume the canonical model and continuation to the final look.

Here \(\sqrt{t}=0.866025\) and \(\sqrt{1-t}=0.5\). Substitution gives

\frac{c-\sqrt{t}\,z_1}{\sqrt{1-t}} = \frac{1.645-(0.866025)(-1)}{0.5} = \frac{2.511025}{0.5} =5.02205.

Consequently,

A(-1)=1-\Phi(5.02205)\approx 2.6\times 10^{-7}.

Under the model, the conditional probability of crossing the original final boundary is extremely small at this interim result. It is still a probability, not a conclusion that rejection is impossible.

Worked Example: Conditional Error at a Positive Interim Statistic

For a hypothetical design with \(t=0.50\) and \(c=1.96\), suppose the interim statistic is \(z_1=1.00\). Assume independent normal increments under \(H_0\) and that the trial continues.

The conditional mean is \(\sqrt{0.50}(1.00)=0.707107\), and the conditional standard deviation is \(\sqrt{0.50}=0.707107\). The standardized distance from the conditional mean to the boundary is

\frac{1.96-(0.707107)(1.00)}{0.707107} \approx 1.7719.

Therefore,

A(1.00)=1-\Phi(1.7719)\approx 0.0382.

Conditional on this interim result, the original final-look rule has about a 3.82% probability of rejecting under \(H_0\). This can exceed or fall below the overall nominal level at a particular interim value: conditional error varies with the data, while the overall error averages over possible interim results and also accounts for any early rejection rule.

Checking the Calculation by Averaging

A useful check is the law of total probability. If the interim statistic is observed but does not itself cause stopping or rejection, averaging the conditional probability over its null distribution must recover the marginal final rejection probability. In particular, if there is no early stopping and the only rejection rule is \(Z_2\ge c\), then

\int_{-\infty}^{\infty} A(z_1)\phi(z_1)\,dz_1 =P_{H_0}(Z_2\ge c) =1-\Phi(c).

This identity is exact under the canonical model. It follows because \(A(z_1)\) is the conditional probability of the event \(Z_2\ge c\), and integrating a conditional probability over the distribution of the conditioning variable gives the unconditional probability of that event.

Worked Example: Verify the Unconditional Error by Integration

Suppose a hypothetical design uses \(t=0.75\), \(c=1.645\), and has no early stopping: the interim statistic is observed, but every path proceeds to the final look. Its conditional error function is

A(z_1)=1-\Phi\!\left(\frac{1.645-\sqrt{0.75}\,z_1}{0.5}\right).

The overall null rejection probability is the average of this function over \(Z_1\sim N(0,1)\):

\int_{-\infty}^{\infty} \left[1-\Phi\!\left(\frac{1.645-\sqrt{0.75}\,z_1}{0.5}\right)\right] \phi(z_1)\,dz_1 =P_{H_0}(Z_2\ge 1.645).

Because the canonical representation gives \(Z_2\sim N(0,1)\), the last probability is \(1-\Phi(1.645)\approx 0.0500\). Thus integrating the conditional errors recovers the intended 5% one-sided error for this final-only rejection rule. If an early efficacy boundary is added, the overall rejection probability must instead include early rejections and integrate final rejection only over paths that continue.

Assumptions, Diagnostics, and Alternatives

The calculation is reliable only to the extent that its model and conditioning event match the design. Before using it, verify that the final boundary \(c\) is the original boundary for the statistic \(Z_2\), that \(t\) is the information fraction at the interim look, and that the independent-increments representation is appropriate under \(H_0\). The information fraction is not automatically the fraction of participants enrolled if information per participant differs across stages.

Check the scale. The formula uses standardized cumulative statistics and an information fraction, not raw stage means or an unstandardized treatment difference.
Check the event. A path that already crossed an early efficacy boundary is already a rejection; a path stopped for futility has no later final rejection under that rule. Do not apply the continuation formula as if those paths still faced the final decision.
Check the direction. The upper-tail formula assumes rejection for large positive \(Z_2\). A lower-tail or two-sided rejection region requires its own conditional probability calculation.
Check numerical behavior. For fixed \(t\) and \(c\), \(A(z_1)\) should increase with \(z_1\). Values should remain between zero and one. At \(z_1=0\), the standardized boundary is \(c/\sqrt{1-t}\), which is a convenient arithmetic check.

A common mistake is to use \(1-\Phi(c)\) as the conditional error at every interim result. That is the marginal final rejection probability under the null, not the probability conditional on \(Z_1=z_1\). The conditional calculation must use the shifted mean \(\sqrt{t}\,z_1\) and the remaining standard deviation \(\sqrt{1-t}\). Another error is to use \(\sqrt{t}\) as the conditional standard deviation; it scales the conditional mean, whereas the remaining increment has standard deviation \(\sqrt{1-t}\).

This direct calculation is one way to express the original design’s remaining rejection probability. Combination tests provide an alternative framework for constructing tests from stagewise evidence, while conditional error tracks the conditional probability of the original rejection event. These approaches are related through error control but are not interchangeable calculations: each requires its own test statistic, assumptions, and calibration. The present formula also has limits: it does not by itself address nonnormal or studentized statistics, a changed second-stage test, or a new final boundary.

Key takeaway: In a canonical two-look normal design, conditional error on a continuing path is the upper-tail probability beyond the original final boundary under \(Z_2\mid Z_1=z_1\sim N(\sqrt{t}\,z_1,1-t)\). Preserve the original boundary, information fraction, conditioning event, and tail direction in every calculation.

Check Your Understanding

Answer each question using the canonical one-sided model, assuming the path continues unless the question says otherwise.

For \(t=0.50\), \(c=1.645\), and \(z_1=0\), write the conditional distribution and calculate the conditional error.
For fixed \(t\) and \(c\), does conditional error increase or decrease as \(z_1\) increases? Give the reason.
Why is the conditional error not generally equal to \(1-\Phi(c)\)?
Suppose the trial stops for futility after the interim look. What is the probability of a later final-look rejection under that stopping rule?

Solutions. (1) The conditional distribution is \(N(0,0.50)\), so the conditional standard deviation is \(\sqrt{0.50}=0.707107\). Hence \(A(0)=1-\Phi(1.645/0.707107)=1-\Phi(2.3264)\approx 0.0100\). (2) It increases, since \(A'(z_1)=\sqrt{t/(1-t)}\,\phi((c-\sqrt{t}z_1)/\sqrt{1-t})>0\). (3) \(1-\Phi(c)\) is the marginal final rejection probability; conditioning shifts the mean of \(Z_2\) to \(\sqrt{t}z_1\) and leaves variance \(1-t\). (4) It is zero, because the design does not proceed to a final rejection on a futility-stopped path.