Multivariate Allocation Proportions in K-Arm Trials

Why K-Arm Allocation Fluctuations Are Joint

In a trial with \(K\) treatment arms, the allocation proportions are not \(K\) unrelated estimates. At every sample size, they sum to one: allocating a larger fraction to one arm necessarily leaves a smaller total fraction for the others. Consequently, even when a Gaussian limit describes the fluctuations around an interior allocation target, its covariance matrix must reflect this exact constraint.

The goal here is to express that joint structure precisely. The key result is that the limiting covariance is singular in the full \(K\)-dimensional space: fluctuations lie in the \((K-1)\)-dimensional subspace whose coordinates sum to zero. This does not prevent useful inference. It means that one should work with \(K-1\) free coordinates or with contrasts between arms, rather than treating all \(K\) proportions as independent.

Allocation Vectors and the Simplex Constraint

Let \(N_{n,i}\) be the number assigned to arm \(i\) among the first \(n\) trial participants, for \(i=1,\ldots,K\). Define the allocation-proportion vector by \(\hat{\boldsymbol v}_n=(N_{n,1}/n,\ldots,N_{n,K}/n)'\), and let \(\boldsymbol v=(v_1,\ldots,v_K)'\) be its target. Each vector of proportions belongs to the probability simplex.

For the Gaussian limit considered here, assume the target is fixed and interior and that the design satisfies conditions yielding a centered, root-\(n\) Gaussian limit:

The assumptions include that the allocation proportions converge to \(\boldsymbol v\) and that the relevant fluctuations have a Gaussian limit under the design's regularity conditions. In an adaptive design, this limit and its covariance must account for the allocation mechanism and its feedback; convergence to the target alone does not establish a Gaussian limit. The earlier material on Gaussian fluctuations and estimator-allocation feedback provides the relevant asymptotic setting. This tutorial focuses on the additional joint structure imposed by \(K\)-arm proportions.

Why the Limiting Covariance Must Be Singular

At every \(n\), \(\boldsymbol 1'\hat{\boldsymbol v}_n=1\), and \(\boldsymbol 1'\boldsymbol v=1\). Subtracting these equalities gives an exact identity, not an approximation:

Therefore any distributional limit \(Z\) must also satisfy \(\boldsymbol 1'Z=0\) with probability one. In particular, \(\Sigma=\operatorname{Cov}(Z)\) obeys \(\Sigma\boldsymbol 1=\boldsymbol 0\). The covariance is positive semidefinite and has rank at most \(K-1\); it cannot be positive definite as a \(K\times K\) matrix. For a nondegenerate Gaussian limit in every tangent direction, its rank is exactly \(K-1\).

This conclusion follows directly from the exact constraint. If \(Z\) is centered and \(\boldsymbol 1'Z=0\) almost surely, then \(\Sigma\boldsymbol 1=E[Z(Z'\boldsymbol 1)]=\boldsymbol 0\). Because covariance matrices are symmetric, the column sums and row sums are both zero. The result does not require independent allocations or a particular adaptive rule.

A Free-Coordinate Representation

One way to work with the constrained limit is to retain the first \(K-1\) components. The last component is determined by their sum. Define the \((K-1)\times(K-1)\) matrix \(\Gamma\) as the covariance matrix of \((Z_1,\ldots,Z_{K-1})'\), and set \(B=\begin{pmatrix}I_{K-1}\\-\boldsymbol 1_{K-1}'\end{pmatrix}\). Since \(Z_K=-\sum_{i=1}^{K-1}Z_i\), \(Z=B(Z_1,\ldots,Z_{K-1})'\), and therefore:

This is a parameterization of every covariance matrix supported on \(\mathcal T\): any positive semidefinite \(\Gamma\) yields a positive semidefinite \(\Sigma\) with zero row and column sums, and a tangent-supported \(\Sigma\) is recovered by taking its upper-left \((K-1)\times(K-1)\) block as \(\Gamma\). If the limit is nondegenerate on the tangent subspace, \(\Gamma\) is positive definite. Other choices of \(K-1\) coordinates are possible; the last arm is not statistically special.

Contrasts and the Meaning of Covariance

A contrast is a linear combination \(c'Z\) with \(\boldsymbol 1'c=0\). Its variance is \(c'\Sigma c\), and it measures uncertainty in a comparison of allocation fluctuations that does not depend on an arbitrary overall level. Differences between two arms use \(c=e_i-e_j\), where \(e_i\) is the \(i\)th coordinate vector. The corresponding variance is:

The negative covariance often present between arms matters: ignoring it can substantially misstate the uncertainty of a difference. More generally, even if a vector \(c\) does not sum to zero, the constraint implies that \(c'Z\) equals the contrast formed by \(c-(\boldsymbol 1'c/K)\boldsymbol 1\), because \(\boldsymbol 1'Z=0\). Only the component of a coefficient vector along \(\mathcal T\) affects the random limit.

What Changes Under Adaptive Allocation

For an adaptive design, let \(\boldsymbol\pi_t\) denote the conditional allocation-probability vector immediately before participant \(t\) is assigned. Conditional on the observed history, the one-hot allocation vector \(X_t\) has mean \(\boldsymbol\pi_t\). Its allocation innovation, \(X_t-\boldsymbol\pi_t\), has conditional covariance \(\operatorname{diag}(\boldsymbol\pi_t)-\boldsymbol\pi_t\boldsymbol\pi_t'\). If \(\boldsymbol\pi_t\) approaches \(\boldsymbol v\) suitably, this describes the limiting covariance of the allocation innovations under the corresponding martingale conditions.

But the final allocation fluctuation also reflects how the probabilities \(\boldsymbol\pi_t\) themselves move in response to past observations. That feedback can add variation, reduce it in some directions, and create covariance terms. Thus the multinomial formula is a useful benchmark, not a universal answer for an adaptive design. The full asymptotic covariance must come from the applicable adaptive-limit result, including its feedback and estimation contributions. Whatever that full covariance is, the exact sum-to-one constraint still requires \(\Sigma\boldsymbol 1=\boldsymbol 0\).

Diagnostics, Alternatives, and Limitations

A covariance calculation should be checked against the geometry of the allocation vector before it is used. The following checks catch common implementation and interpretation errors:

• Check the constraint: verify that each row and column of the reported full covariance sums to zero, up to rounding. Equivalently, check that \(\Sigma\boldsymbol 1\) is numerically close to zero.
• Check positive semidefiniteness: all eigenvalues should be nonnegative in exact arithmetic. Tiny negative eigenvalues can result from rounding; a substantial negative eigenvalue indicates an invalid covariance calculation.
• Check the rank: a nondegenerate limit on the tangent subspace has \(K-1\) positive eigenvalues and one zero eigenvalue in the full \(K\)-dimensional representation.
• Check the scale: \(\Sigma\) describes the limit of \(\sqrt n(\hat{\boldsymbol v}_n-\boldsymbol v)\). The approximate covariance of \(\hat{\boldsymbol v}_n\) is \(\Sigma/n\), not \(\Sigma\).

There are several valid representations, each useful for a different purpose. The full singular covariance displays all arm-to-arm covariances; a \(K-1\) coordinate covariance \(\Gamma\) avoids a redundant coordinate; and contrast variances directly answer questions about differences among arms. The multinomial covariance is appropriate for independent categorical assignments with fixed probabilities. It is not a substitute for an adaptive covariance that includes response-driven changes in allocation.

The interior Gaussian result also has limits. At a boundary target, some proportions approach zero and root-\(n\) Gaussian behavior can fail or require different scaling; the tangent-space covariance result alone does not establish a suitable limit there. Likewise, a covariance matrix does not establish asymptotic normality: the limit may be non-Gaussian or degenerate in additional directions. Finally, the covariance depends on the design and its response feedback, not only on the target proportions.