Reusing Shared-Control Data In Multiplicity Calculations

Why Shared-Control Contrasts Are Dependent

When several experimental treatments are compared with one control group, each treatment-versus-control estimate contains the same control estimate. The treatment groups may be independently randomized and their sample means independent, but the resulting contrasts are not independent: a high or low observed control mean moves every contrast in the same direction. This dependence matters when calculating simultaneous error rates and intervals. Treating the contrasts as independent can give an incorrect critical value.

The calculations below begin with independent treatment arms and a common control arm. They focus on the covariance structure of the treatment-control contrasts and how that structure enters joint inference. The introductory material on Dunnett and maxT procedures gives the broader multiplicity context; here the key task is to build the dependence correctly from the trial design.

Deriving the Covariance

Assume first that observations are independent across arms, independent within each arm, and have finite variances. Let the variance in arm \(i\) be \(\sigma_i^2\), and let the control-arm variance be \(\sigma_0^2\). Then the sample mean variances are \(\operatorname{Var}(\bar{X}_i)=\sigma_i^2/n_i\) and \(\operatorname{Var}(\bar{X}_0)=\sigma_0^2/n_0\). For two distinct treatment arms \(i\) and \(j\), independence across arms means \(\operatorname{Cov}(\bar{X}_i,\bar{X}_j)=\operatorname{Cov}(\bar{X}_i,\bar{X}_0)=\operatorname{Cov}(\bar{X}_j,\bar{X}_0)=0\).

Expanding the covariance of the two differences leaves the shared control variance as the only nonzero term:

The variance of a single contrast includes both arms. Consequently, the correlation between distinct contrasts is positive under this model:

If all arms have a common variance \(\sigma^2\), the factor \(\sigma^2\) cancels in the correlation. With equal treatment-arm sizes \(n_T\), this simplifies to \(\rho=n_T/(n_T+n_0)\). The more of each contrast’s uncertainty comes from the common control mean, the more strongly the contrasts move together.

Unequal Sample Sizes and the Covariance Matrix

Unequal allocation changes each contrast’s variance and usually changes the correlation. It does not change the basic covariance result when the same control sample mean is used: under the independent-arm model, every off-diagonal covariance among distinct treatment-control estimates is \(\sigma_0^2/n_0\). In matrix form, the diagonal entries are the individual contrast variances, and all off-diagonal entries equal this common-control contribution when arm variances are common. This matrix, rather than a collection of marginal standard errors alone, describes the joint sampling distribution needed for correlated simultaneous procedures.

Using the Joint Distribution for Simultaneous Inference

Suppose the vector of contrast estimates is jointly normal, with mean vector \((\Delta_1,\ldots,\Delta_m)\) and covariance matrix \(\Sigma_D\) given by the arm variances and allocation. If variances are known, or treated as known for a planning calculation, standardizing each contrast gives correlated standard normal statistics under its null value. If variances are estimated, the appropriate joint \(t\) or model-based procedure must reflect that estimation; using a normal critical value without justification can understate uncertainty.

For one-sided claims that each treatment is better than control, a single-step maxT rule uses the maximum of the standardized treatment-control statistics. The critical value is chosen so that the probability that this maximum exceeds the critical value is \(\alpha\) under the relevant joint null distribution. The correlation matrix is essential: it determines the distribution of the maximum. Simultaneous lower confidence bounds use the same critical value and have the form \(D_i-c\,SE(D_i)\).

Selection, Diagnostics, and Limits

If a treatment is selected from the same candidate family using the observed trial data, a simultaneous statement that covers all candidate treatment-control effects also covers the selected effect on the event that all those statements hold. This yields unconditional selection-safe coverage under the model and the familywise calibration. It does not, by itself, guarantee the same coverage conditional on the particular selection event. Nor does computing the shared-control covariance alone correct every consequence of adaptive selection: if interim data influence which arms continue, the joint distribution and testing rule must also account for that selection and any reused observations.

A maxT critical value calibrated under the global null gives weak familywise error control (control when all tested nulls are true). Strong control across configurations with some false nulls requires additional justification. One common sufficient condition is subset pivotality: the joint distribution of the statistics for any subset of true null hypotheses does not depend on the parameter values for the false null hypotheses. If that condition fails, calibration must be valid for the relevant true-null subsets or use a method that otherwise establishes strong control.

Before relying on the simple formula, check that the arms are independent, that the same control observations enter each contrast, and that the analysis uses the planned allocation and outcome model. In stratified or covariate-adjusted analyses, the covariance is determined by the fitted model and contrast vectors; it need not equal the unadjusted expression \(\sigma_0^2/n_0\). Repeated measurements, clustering, shared patients, unequal variances, missing outcomes, or changes to the analysis population can also alter the covariance. Estimate or derive the full covariance matrix from the actual analysis model rather than forcing the simple formula onto a design it does not describe.

The main alternatives differ in how they handle multiplicity. Bonferroni uses a critical value valid without knowing the correlations, but can be conservative because it ignores the positive dependence. A correlated maxT calculation uses the covariance matrix and can be less conservative when the model is correct. Holm’s step-down procedure also controls familywise error without requiring the shared-control correlation for its basic validity, though it does not use that correlation to obtain the same joint maxT calibration. The choice depends on the inferential target, assumptions, and whether reliable joint calibration is available.