Reusing a Shared Control Group Across Stages

Why Reusing Controls Changes the Analysis

In a seamless phase 2/3 trial, investigators may use the same control participants to form both the phase 2 and phase 3 treatment contrasts. This can conserve participants, but it changes the sampling relationship between the stagewise estimates. Each contrast subtracts the same control mean, so a high or low realization of that mean moves both contrasts in the same direction. Treating the contrasts as independent can therefore miscalibrate a combined test.

The focus here is the dependence created by literal reuse of control outcomes: the same control participants, measured for the same endpoint, contribute to both stagewise comparisons. This is distinct from having separate control cohorts in the two stages. It is also distinct from dependence caused by selecting a treatment using interim results. The contrasts and their covariance must be specified before choosing how to combine evidence.

Model and Covariance From Shared Controls

For a direct derivation, suppose participant outcomes are independent and normally distributed, with a common variance \(\sigma^2\). Within each stage, treatment and control participants are distinct; across stages, the treatment participants are also distinct. Only the control participants overlap. Write their mean as \(\bar{Y}_C\). The treatment means and shared control mean are then mutually independent, except that the same \(\bar{Y}_C\) appears in both contrasts.

The stagewise contrasts have variances \(\sigma^2(1/n_j+1/m)\). Their covariance follows by expanding the definition and using independence of the two treatment groups from each other and from the controls:

The covariance is positive: a control mean that happens to be unusually high reduces both treatment-minus-control contrasts, while an unusually low control mean increases both. Standardizing each contrast gives \(Z_j=(D_j-\Delta_j)/SE_j\), where \(\Delta_j\) is its true mean contrast and \(SE_j=\sigma\sqrt{1/n_j+1/m}\). Thus the correlation of the standardized contrasts is

Under the global null hypothesis \(H_0:\Delta_1=\Delta_2=0\), the vector \((Z_1,Z_2)\) is bivariate standard normal with correlation \(\rho\). This exact normal result uses the stated normal model and known \(\sigma\). The covariance formula itself also follows from the independent-participant variance calculation without normality, though normality or a suitable large-sample approximation is needed for the normal-reference combined test described below.

Adjusting a Prespecified Combined Test

A common stagewise combination uses fixed weights \(w_1,w_2\) on standardized stage statistics, with \(w_1^2+w_2^2=1\). Under independent stages, \(w_1Z_1+w_2Z_2\) has variance 1. With shared controls, its variance is instead \(1+2w_1w_2\rho\). Therefore, under the null, a correctly standardized statistic is

For a prespecified upper-tailed test at level \(\alpha\), reject \(H_0\) when \(Z_{\mathrm{comb}}>z_{1-\alpha}\), where \(z_{1-\alpha}\) is the \(1-\alpha\) quantile of the standard normal distribution. The one-sided p-value is \(1-\Phi(Z_{\mathrm{comb}})\), with \(\Phi\) the standard normal cumulative distribution function. This calibration assumes the null joint distribution and weights have been correctly specified. If the test uses a different combination function, its reference distribution must likewise account for the joint dependence; an independence-based reference distribution is not justified merely because each stage has a valid marginal p-value.

A Covariance-Matrix View

The two-stage calculation can be written compactly using the covariance matrix. Under the common-variance model, the covariance matrix of the unstandardized contrasts is

Its diagonal entries are the stagewise variances, and its off-diagonal entries are the shared-control covariance. Standardizing the rows and columns gives a correlation matrix with diagonal entries 1 and off-diagonal entries \(\rho\). This representation is useful because it makes the source of the dependence explicit and extends directly to more than two contrasts: every pair that reuses the same control observations receives a covariance contribution from those observations.

A weighted sum of the stagewise statistics has variance \(w^\mathsf{T}Rw\), where \(R\) is their correlation matrix. In the two-stage case this is \(w_1^2+w_2^2+2w_1w_2\rho\); for normalized weights it reduces to the denominator variance used above. The adjustment is not an arbitrary penalty. It is the standardization required by the actual null variance of the chosen linear combination.

Assumptions, Diagnostics, and Alternatives

Before using the formulas, verify the data structure rather than infer it from stage labels. The derivation assumes that the same \(m\) control outcomes enter both control means, that treatment participants are not reused across stages, and that outcomes from different participants are independent. The simple displayed variance formulas also assume a common variance. If treatment and control variances differ, replace the common-variance terms with the relevant group variances; for example,

For binary outcomes, the same shared-observation logic applies to risk-difference contrasts: the covariance includes the variance of the reused control proportion. The exact standard errors depend on the endpoint model and estimand, so the continuous normal formulas should not be substituted automatically.

Useful checks include confirming participant identifiers across stage datasets, verifying that each reused control outcome is counted once in the shared control mean, and comparing the implemented covariance matrix with the design’s overlap pattern. A zero off-diagonal entry is appropriate for disjoint control cohorts under participant independence; it is not appropriate when the same controls contribute to both contrasts. A materially non-positive-semidefinite covariance matrix is a warning that entries, weights, or overlap counts have been assembled inconsistently.

The independence alternative is to use separate controls in each stage. Then the control contributions are independent and the stagewise covariance is zero, at the cost of enrolling separate control cohorts. A different analysis can fit a joint model to all participants and estimate the contrasts using the model’s covariance matrix. That can be useful when the design or outcome model is more complex, but it does not remove the need to represent the shared observations correctly. For combination procedures based on stagewise p-values, the reference distribution must account for their dependence; the earlier material on Fisher tests does not license using the independent-p-value calibration when control data overlap.

These calculations also presume the treatment contrast and combination rule are fixed for the test being calibrated. If phase 2 results determine which treatment is carried forward, selection can alter the null distribution of the evidence used at the transition. Shared-control covariance is then one dependence source, not a substitute for the selection-valid procedures covered in the earlier material on conditional error and confirmatory validity after treatment selection.

Common Pitfall and Practical Interpretation

A common mistake is to calculate a valid standard error for each stage, observe that both stagewise statistics are individually standard normal under the null, and then combine them as if they were independent. The error is in the joint distribution: correct marginal distributions do not imply independence. In this setting, both contrasts contain \(-\bar{Y}_C\), which produces covariance \(\sigma^2/m\). Ignoring that positive covariance usually understates the variance of a positive weighted sum and can make the combined test anti-conservative.

The correction is to use the covariance induced by the actual participant overlap, standardize the combination using its joint null variance, and report the resulting test in context. A p-value is the probability, under the specified null model and design, of a statistic at least as extreme in the prespecified direction as the one observed. It is not the probability that the null hypothesis is true. A finding of statistical significance also does not by itself establish clinical importance; the stagewise contrasts and their units remain important for interpretation.