Worked Example With Unequal Group Variances

Why Unequal Group Variances Matter

In a multi-arm study, several treatment-control comparisons may use the same control participants. That shared group contributes uncertainty to every contrast, so the contrasts are correlated. When the outcome variances differ by group, the covariance still comes from the shared control mean, but the variance of each contrast also includes its own treatment group's variance. The calculation must therefore keep the group-specific variances and sample sizes separate.

This tutorial focuses on two treatment arms compared with one control, with each contrast written as treatment minus control. It builds on the earlier material on shared-control correlation and heteroscedastic shared-control correlation by working through the calculations with specified, potentially unequal group variances.

Assume observations are independent within each group, each group has a finite outcome variance, and the samples from different groups are independent. The variance of \(\bar{X}_g\) is then \(\sigma_g^2/n_g\). The arm-specific variances need not be equal. Normal outcome distributions are not required for these covariance and correlation formulas; if outcomes are normal, the group means and the contrasts are jointly normal as well.

Deriving the Unequal-Variance Formula

The two contrasts contain the same random quantity, \(\bar{X}_0\), with the same negative sign. The treatment means themselves come from independent groups, so they do not contribute cross-covariance. Expanding the covariance makes this structure explicit.

Substituting these quantities in the definition of correlation gives the result for two treatment-minus-control contrasts:

The covariance has units of outcome squared. Each variance in the denominator has the same units, so the correlation is dimensionless. For these same-direction contrasts, the shared-control covariance is nonnegative and is positive when the control variance is positive. Unequal treatment variances alter the denominator, and therefore can alter the correlation, without changing the shared-control covariance.

How Allocation Changes the Correlation

The formula separates two roles. Increasing control sample size \(n_0\) reduces the shared-control variance contribution \(\sigma_0^2/n_0\), which is the covariance. It also reduces that contribution to both contrast variances. Changing a treatment arm's sample size or variance changes that contrast's variance, but not the covariance with the other treatment contrast, provided control allocation and variance stay fixed.

Correlation is not determined by covariance alone: its denominator changes too. For example, a large treatment-arm variance makes that contrast less precise and can reduce its correlation with the other contrast. Nor does a larger sample in a treatment arm necessarily change the covariance; under this independent-groups model, it changes only the variance of that arm's contrast. These statements concern contrast dependence, not the association between individual outcomes across groups.

Using Estimated Variances

In applications, population variances may be unknown. If separate sample variances \(s_0^2,s_1^2,s_2^2\) are available, replacing each \(\sigma_g^2\) by its group-specific estimate gives a plug-in estimate of the contrast covariance matrix and correlation. This preserves the unequal-variance structure. It does not make the unknown population correlation known exactly: the estimated correlation also has sampling uncertainty.

Do not confuse this plug-in calculation with an analysis that assumes a common variance and pools information across groups. Pooling is appropriate only under a common-variance model; if variances differ, it changes the variance inputs and can give an unsuitable estimate of contrast dependence. The introductory material on estimating contrast covariance from a common variance treats that distinct assumption.

Assumptions, Diagnostics, and Alternatives

The covariance identity relies on independent samples across groups and on the stated definition of each contrast. If participants contribute to more than one group, or if outcomes across groups are otherwise dependent, the cross-covariance terms in the derivation need not be zero. The group means must also have finite variances for the formulas to apply. Normality is useful when a joint normal model for the contrasts is needed, but is not required for the covariance calculation itself.

A quick diagnostic is to check the components before calculating the correlation: the covariance should be \(\sigma_0^2/n_0\); each variance should add the control term to that arm's own term; and the final correlation must lie between \(-1\) and \(1\). For the specified same-direction contrasts, it should be nonnegative. A negative value or a value exceeding 1 signals a sign, arithmetic, or model-input problem. Check especially that variances, not standard deviations, were divided by sample sizes.

The main alternatives answer different modeling or design questions. A pooled-variance calculation imposes a common-variance assumption rather than retaining separate arm variances. Separate control groups remove this particular shared-control covariance under independent sampling, but they change the design and the amount of information per group. A randomization-based or simulation-based assessment can examine dependence under a specified design and analysis; it does not remove the need to define the contrast and its variance model. The formula here is limited to unadjusted sample-mean contrasts under independent groups; other estimators or dependence structures require their own covariance calculation.