Checking the Canonical Covariance Matrix

Why Check the Covariance Matrix?

At sequential analyses, statistics from different looks usually reuse some of the same accumulated information. Their covariance therefore reflects how much information the looks share. A proposed covariance matrix must do more than have plausible entries: it must agree with the look schedule and be a mathematically valid covariance matrix. These checks matter when specifying a joint distribution for the statistics or using it in a group sequential calculation.

The calculations below use the canonical cumulative-information model. Its covariance structure follows directly from independent increments. In an application, that model may be exact under a specified probability model or an approximation justified by the score-statistic theory. The checks here establish whether a proposed matrix has the canonical form; they do not, by themselves, establish that the model assumptions hold for a particular study.

Model and Canonical Covariance

Suppose there are \(m\) ordered looks with cumulative information \(0<I_1\le I_2\le\cdots\le I_m\). Write \(\Delta I_k=I_k-I_{k-1}\), taking \(I_0=0\). Let \(S_j\) be the cumulative statistic at look \(j\). Under the null hypothesis, assume its increments \(S_k-S_{k-1}\) are independent, normally distributed with mean zero and variance \(\Delta I_k\). An increment with \(\Delta I_k=0\) is identically zero. Under a canonical fixed alternative, the increments may have means proportional to their information increments; the covariance calculations below are unchanged.

For each positive-information look, define the standardized cumulative statistic \(Z_j=S_j/\sqrt{I_j}\). Under the null its mean is zero and its variance is one. Under an alternative, its mean need not be zero, but the covariance matrix is still computed by centering the statistics. Since \(S_i\) and \(S_j\) share all increments through the earlier look, their covariance is the information at that earlier look.

The formula is also a useful direct check: a candidate correlation between looks \(i\) and \(j\) is determined by the earlier information level and the two marginal standard deviations. For the unstandardized cumulative statistics, the corresponding covariance is \(\operatorname{Cov}(S_i,S_j)=I_{\min(i,j)}\).

Why the Entries Have This Form

For \(i\le j\), write \(S_i\) as the sum of increments through look \(i\), and \(S_j\) as the sum through look \(j\). The increments through \(i\) occur in both sums. Every later increment in \(S_j\) is independent of \(S_i\) and has zero covariance with it. Thus \(\operatorname{Cov}(S_i,S_j)=\sum_{k=1}^{i}\Delta I_k=I_i\). Dividing by the standard deviations gives the canonical correlation.

This argument relies on the independent-increment model, not merely on the fact that the looks are ordered. If increments are dependent or the statistic does not have variance equal to information, the same formula need not apply. That is why checking a matrix against the schedule and checking the assumptions behind the schedule are separate tasks.

Positive Semidefiniteness From the Increments

A covariance matrix must be positive semidefinite. The canonical matrix satisfies this because each standardized cumulative statistic is a linear combination of independent, unit-variance increment variables. To handle tied information levels correctly, define \(U_k=(S_k-S_{k-1})/\sqrt{\Delta I_k}\) only for indices \(k\) with \(\Delta I_k>0\). These \(U_k\) are independent standard normal variables under the null. A zero-information increment is centered and equal to zero almost surely, so it contributes no term and no \(U_k\) is defined for it.

For any real vector \(a=(a_1,\ldots,a_m)^\mathsf{T}\), substitute this representation into the quadratic form and collect the coefficient of each \(U_k\). Independence and unit variances give the following expression, which is a sum of squares:

The inequality proves positive semidefiniteness, including for nondecreasing schedules with zero increments. It also explains why a tied look can make the matrix singular: if two looks have the same information, their cumulative statistics are identical under this model, so a nonzero linear combination of them can have variance zero. Positive semidefinite therefore does not always mean positive definite.

Practical Checks and Failure Modes

For a proposed canonical covariance or correlation matrix, use the schedule as well as general matrix checks. First confirm that every \(I_j\) is positive and the levels are in nondecreasing order. Then check symmetry, unit diagonal for the standardized statistics, and each off-diagonal entry against \(\sqrt{I_{\min(i,j)}/I_{\max(i,j)}}\). Finally, check positive semidefiniteness. The increment factorization proves this property for a correctly constructed canonical matrix, while eigenvalues or principal minors can help diagnose a supplied matrix.

Small negative eigenvalues from floating-point rounding should be interpreted relative to the numerical tolerance and scale of the calculation; a materially negative eigenvalue indicates that the matrix is not positive semidefinite. A matrix with a correlation outside \([-1,1]\), a non-unit diagonal, or an entry inconsistent with the information ratio is already suspect, even before an eigenvalue check. When information levels repeat, exact duplicate rows and columns are expected, not evidence of a computational error.

A direct alternative is to build the matrix from standardized increments and multiply the resulting loading matrix by its transpose. This makes positive semidefiniteness transparent and is especially helpful with tied looks. A generic eigenvalue or Cholesky-based validity check answers a different question: whether the supplied matrix is positive semidefinite (or positive definite), not whether it has the canonical schedule-specific entries. Cholesky factorization without modification requires positive definiteness and may fail for a valid singular canonical matrix.

The canonical form has limitations. It depends on the assumed cumulative-information structure and independent increments; departures from those assumptions can change the covariance entries. It also presumes positive information at each standardized look, because \(Z_j=S_j/\sqrt{I_j}\) is undefined when \(I_j=0\). The matrix check is therefore a consistency and algebra check, not a substitute for assessing whether the statistical model supports the canonical representation.