Why a Changed Correlation Matters
In a group sequential trial, the interim and final test statistics are usually dependent: both contain information from participants observed before the interim analysis. If the design changes, the final statistic may have a different correlation with the statistic already observed. Even if the final statistic still has a standard normal null distribution and the numerical rejection boundary is unchanged, its distribution conditional on the interim result may have changed. That change can alter the probability of eventual rejection for the observed interim data.
Conditional error makes this issue precise. It is the probability, under the null hypothesis, that the original design rejects after the interim point, conditional on the interim data observed so far. To preserve the original design’s Type I error after an adaptation, a modified design can be required to have conditional rejection probability no greater than this original conditional error for each relevant interim result. When the interim statistic is unchanged and the conditional distributions are continuous, the new boundary can be chosen to make the two conditional probabilities equal.
The calculation in this tutorial concerns a one-sided normal test with one interim statistic and one final statistic. The same principle applies when the original rejection rule is more complicated, but then its conditional error function must first be calculated for that rule. The key step here is to obtain the correct conditional distribution when the correlation changes.
Conditional Distribution Under the Null
Assume that under \(H_0\), the interim and final statistics have a bivariate standard normal distribution:
Here, \(\rho\) is their correlation. The assumptions are that both statistics are standardized to have mean 0 and variance 1 under the null, their joint null distribution is bivariate normal, and the correlation is known for the design being analyzed. Under these conditions, the conditional distribution of the final statistic given \(Z_I=z\) is
Thus the conditional mean is \(\rho z\), and the conditional standard deviation is \(\sqrt{1-\rho^2}\). The correlation affects both: changing \(\rho\) changes the conditional center and the conditional spread. It is not generally sufficient to adjust only one of them.
Deriving the Conditional Error and the Adjusted Boundary
Suppose the original design rejects for \(Z_F\ge c\), where \(c\) is a fixed one-sided boundary. Given \(Z_I=z\), the original conditional error is the upper-tail probability under the conditional normal distribution:
where \(\Phi\) is the standard normal cumulative distribution function. Now suppose the modified design uses a final statistic \(Z_F^*\) that is still standard normal under \(H_0\), remains jointly normal with the same interim statistic, but has a new correlation \(\rho^*\). Its conditional distribution is
To preserve the original conditional error, choose a boundary \(c^*(z)\) satisfying \(P_{H_0}(Z_F^*\ge c^*(z)\mid Z_I=z)=A(z)\). Equating the upper-tail probabilities gives the boundary directly:
For the original fixed-boundary rule, the expression simplifies because \(\Phi^{-1}(1-A(z))=(c-\rho z)/\sqrt{1-\rho^2}\). Therefore,
This is a result for a one-sided upper-tail test under the stated normal model. It is conditional on the observed interim statistic; the adjusted boundary will generally vary with \(z\). If \(A(z)=0\), the matching upper-tail boundary is \(+\infty\); if \(A(z)=1\), it is \(-\infty\). In applications, practical rejection rules and allowable boundaries may require additional constraints.
Worked Example: Calculating the Original Conditional Error
Suppose the original design has \(\rho=0.50\), rejects when \(Z_F\ge 2.00\), and observes \(Z_I=0.80\). Assume the bivariate standard normal null model above. Find the conditional distribution of \(Z_F\) and the original conditional error.
The conditional mean is \(\rho z=(0.50)(0.80)=0.400\). The conditional variance is \(1-\rho^2=1-0.25=0.75\), so the conditional standard deviation is \(\sqrt{0.75}=0.8660\). Therefore,
The conditional error is
Under the original design and the null model, a trial with interim statistic \(0.80\) has about a 3.234% chance of crossing the final boundary. This is a conditional probability, not the unconditional Type I error of the whole design.
What Happens If the Correlation Changes?
Keeping the same numerical boundary does not, in general, keep the same conditional error. For the new correlation \(\rho^*\), the conditional error from retaining boundary \(c\) is
The difference between \(A^*(z)\) and \(A(z)\) depends on both the observed interim value and the correlation change. It need not have the same direction for every interim value. The following examples show how to calculate both the changed conditional error and a boundary that restores the original value.
Worked Example: A Lower Correlation With the Same Boundary
Use the original design with \(\rho=0.50\), \(c=2.00\), and observed \(Z_I=1.00\). Now suppose the modified final statistic has correlation \(\rho^*=0.30\) with the unchanged interim statistic. First find the conditional error if the boundary remains 2.00, then find a boundary that preserves the original conditional error.
Under the original design, the conditional mean is \((0.50)(1.00)=0.500\) and the standard deviation is \(\sqrt{0.75}=0.8660\). Hence
Under the modified design, the conditional mean is \((0.30)(1.00)=0.300\) and the standard deviation is \(\sqrt{1-0.30^2}=\sqrt{0.91}\). If the boundary is left at 2.00, then
Thus the unchanged boundary gives a conditional error of about 3.74%, below the original value of about 4.163% at this interim result. To match the original conditional error exactly, use the original standardized boundary distance \(1.7321\) in the new conditional distribution:
Indeed, \((1.952-0.300)/\sqrt{0.91}\approx1.732\), so the new upper-tail probability is approximately \(0.04163\), matching the original conditional error up to rounding.
Worked Example: A Higher Correlation and an Interim-Dependent Boundary
Return to the first example, with original \(\rho=0.50\), \(c=2.00\), and \(Z_I=0.80\). Suppose the modified final statistic instead has correlation \(\rho^*=0.60\) with the same interim statistic. Calculate the conditional error if the boundary stays at 2.00, and then calculate the preserving boundary.
The modified conditional mean is \((0.60)(0.80)=0.480\), and its standard deviation is \(\sqrt{1-0.60^2}=0.800\). Retaining the boundary gives
The original conditional error was approximately \(0.03234\), so the old boundary does not preserve it. To match it, the new boundary is
Substitution verifies the result: \((1.958-0.480)/0.800=1.8475\), giving an upper-tail probability of approximately \(0.03234\). The boundary has decreased from 2.00 in this case; preserving conditional error does not mean retaining the old boundary or always moving it in the same direction.
Why the Calibration Preserves Error
The justification has two parts. First, the bivariate normal model gives the conditional distribution of the final statistic at each observed interim value. Second, the calibrated boundary equates the new conditional rejection probability to the original conditional error. Across all possible interim results, the law of total probability then connects these conditional probabilities to the overall null rejection probability.
For each \(z\), calculate the original \(A(z)\) and the modified conditional distribution using \(\rho^*\).
Choose \(c^*(z)\) so that the modified probability of rejection given \(Z_I=z\) equals \(A(z)\), or is no greater than \(A(z)\) for conservative preservation.
Under \(H_0\), \(P(\text{modified rejection})=E_{H_0}[P(\text{modified rejection}\mid Z_I)]\). If the conditional probability is no greater than \(A(Z_I)\) for every relevant \(Z_I\), the overall rejection probability is no greater than that of the original design.
The last step is a direct consequence of taking expectations of the pointwise inequality. If the conditional probabilities are equal for every interim result, their expectations are equal as well. This argument concerns Type I error under the null model used to calculate the conditional distributions; it does not establish equality of power under alternatives.
Assumptions, Diagnostics, and Alternatives
The calculation is reliable only if the conditional model represents the design actually being proposed. In particular, verify that the interim statistic is the same statistic in both designs, that both statistics have the stated null standardization, and that the new interim–final correlation is appropriate for the modified data structure. If the new statistic is not jointly normal with the interim statistic, the normal conditional formula may not apply. If nuisance parameters affect the null distribution or correlation, a single plug-in correlation may not provide valid error control across the null; nuisance-parameter handling requires separate justification.
Useful checks include computing the conditional mean and variance independently from the covariance matrix, checking that \(1-\rho^2\) and \(1-(\rho^*)^2\) are positive, and substituting the proposed boundary back into the conditional tail formula. Numerical calculations should also be checked at more than one interim value: a boundary calibrated at one \(z\) does not establish preservation at other possible \(z\) values. If the adaptation rule randomizes, conditions on a decision, or uses several interim observations, the conditional event and conditioning information must reflect that design.
| Approach | What it does | Key limitation |
|---|---|---|
| Retain the old numerical boundary | Uses the same final cutoff despite a changed conditional distribution | Does not generally preserve conditional error |
| Recalibrate the conditional boundary | Matches the modified conditional tail to the original conditional error | Depends on a valid conditional model and may require a boundary that varies with interim data |
| Calibrate only the unconditional null rejection rate | Targets an overall Type I error under a specified design distribution | May not preserve error conditional on the observed adaptation data |
The distinction between conditional and unconditional calibration matters when the design change is selected after observing interim data. An unconditional calculation can average over interim values in a way that does not control the rejection probability for the particular observed value. Conditional calibration addresses that value directly. It can be exact under the specified model; when exact matching is not feasible, requiring the modified conditional rejection probability to be no greater than the original one gives conservative preservation.
Common Mistakes
- Using the old correlation after the design changes. The conditional distribution must use the correlation for the design whose final statistic is being evaluated.
- Changing the conditional mean but not the variance. The conditional standard deviation is \(\sqrt{1-\rho^2}\), so it also changes with the correlation.
- Assuming the old boundary preserves error. A fixed marginal null distribution for the final statistic does not imply an unchanged conditional distribution given the interim statistic.
- Claiming overall error control from one interim value. Conditional error must be preserved or bounded over the relevant interim values, not just at one observed value.
- Interpreting conditional error as power. The calculations here are under \(H_0\); they concern false rejection probability, not the probability of rejection under a treatment effect.
Check Your Understanding
For each question, use the one-sided bivariate normal model and show how the conditional distribution informs the answer.
- If \(\rho=0.40\) and \(Z_I=1.50\), what are the conditional mean and variance of \(Z_F\)?
- An original design has \(\rho=0.50\), \(c=2.00\), and \(Z_I=0\). What is its conditional error? If the new correlation is \(0.20\), what boundary preserves that conditional error at this interim value?
- Why is it insufficient to verify the adjusted boundary at only the observed interim statistic when the design must control error for all possible interim results?
- State two model checks needed before using the correlation-based conditional distribution.
Solutions. (1) The mean is \((0.40)(1.50)=0.600\), and the variance is \(1-0.40^2=0.840\); thus \(Z_F\mid Z_I=1.50\sim N(0.600,0.840)\). (2) At \(Z_I=0\), the original conditional mean is zero and standard deviation is \(\sqrt{0.75}=0.8660\). Thus \(A(0)=1-\Phi(2.00/0.8660)=1-\Phi(2.309)\approx0.0105\). The original standardized boundary distance is \(2.00/\sqrt{0.75}\). The new standard deviation is \(\sqrt{1-0.20^2}=\sqrt{0.96}\), so \(c^*(0)=\sqrt{0.96}(2.00/\sqrt{0.75})\approx2.263\), which preserves the same conditional tail. (3) The conditional error is a function of the interim value. Error control for the design requires the relevant conditional probability inequality across possible interim results, so one pointwise check cannot establish the general condition. (4) For example, verify that both statistics are standardized under \(H_0\), that their joint null distribution is adequately modeled as bivariate normal, and that the correlation used corresponds to the proposed design.