Canonical Joint Distribution Of Sequential Logrank Z Statistics

Why Sequential Logrank Statistics Are Correlated

At successive analyses of a time-to-event trial, the logrank statistic at a later look uses events and risk-set information that have already contributed to earlier statistics. The statistics therefore share accumulated evidence. Treating them as unrelated standard normal variables discards this overlap and gives the wrong joint model for evaluating a sequence of analyses.

The canonical joint distribution describes this dependence through the information available at each look. Its central result is simple: when the cumulative score has the usual large-sample covariance structure, the correlation between two standardized cumulative scores is the square root of the ratio of their information. The result applies to cumulative logrank scores, including scores combined across strata, provided their information and covariance are correctly represented.

The score's sign depends on the chosen treatment-group convention, which must be kept fixed across looks. For the covariance derivation, that choice does not matter: reversing the sign of every score reverses all the \(Z_k\)'s but leaves their correlations unchanged. Earlier material on combining stratum scores and information supplies the corresponding cumulative score and information when a trial is stratified.

Assumptions Behind the Canonical Model

The model is a large-sample approximation, not an exact finite-sample identity. Its use requires more than dividing each observed score by the square root of its information. At a minimum:

• The score is computed consistently at each look from the same trial, treatment contrast, and score convention. Later cumulative scores include the earlier event contributions.
• The information \(I_k\) is positive and increases with accumulated event information. The information used in the standardization represents the variance of the corresponding score on the scale used in the analysis.
• Under the null hypothesis \(H_0\), the cumulative score is approximately centered at zero, with variance \(I_k\).
• For looks \(j\leq k\), the covariance of cumulative scores is approximately \(I_j\). This is the shared-score structure: the earlier score is included in the later score, while the additional score accumulated afterward contributes no covariance with the earlier score under the working model.
• The joint score distribution is adequately approximated by a multivariate normal distribution. This generally relies on sufficient event information and regularity of the risk-set score contributions.

These assumptions concern the joint distribution across looks, not just the marginal distribution at each look. Marginally, each \(Z_k\) may be approximately standard normal under \(H_0\), yet the marginal statements alone do not determine how the statistics relate to one another.

Deriving the Covariance and Correlation

For two looks \(j\leq k\), standardizing their cumulative scores means dividing each score by its own information-based standard deviation. Applying this to the shared-score covariance gives:

Under \(H_0\), each standardized statistic has approximate mean zero and variance one, so this covariance is also the correlation. In particular, later looks are more correlated with an earlier look when they add relatively little information, and less correlated when they add much more information.

Thus every diagonal entry is one. For example, with three looks, the matrix has the form:

The formula also has a useful information-fraction form. If the maximum information is \(I_K\), define \(t_k=I_k/I_K\). Then \(R_{jk}=\sqrt{t_j/t_k}\) for \(j\leq k\). The correlations depend on relative information, not on the unit used to measure it.

A Short Justification Using Accumulated Scores

The covariance formula can be understood by separating the score already accumulated from the score added afterward. For \(j<k\), write \(U_k=U_j+(U_k-U_j)\). Under the shared-score model, \(\operatorname{Var}(U_j)\approx I_j\) and \(\operatorname{Cov}(U_j,U_k-U_j)\approx 0\). Hence:

Substitution into the standardization gives \(R_{jk}=\sqrt{I_j/I_k}\), as above. This argument establishes the covariance structure needed here. It does not by itself establish that the finite-sample score increments have every property required for a particular sequential procedure; those properties depend on the trial and analysis conditions.

Mean Structure Under a Constant Local Drift

A common extension describes a sustained effect by a constant score drift per unit information. Let \(\theta\) denote that signed drift. Under the corresponding local alternative, suppose \(E(U_k)\approx \theta I_k\), while the covariance structure remains approximately the same. Then:

The canonical model under this alternative has mean vector \((\theta\sqrt{I_1},\ldots,\theta\sqrt{I_K})\) and the same covariance matrix \(R\). The sign of \(\theta\) follows the chosen score orientation. This is a model for a constant drift on the information scale; it should not be assumed automatically for a time-varying treatment effect.

Unequal Information Increments and Model Checks

The canonical formula uses cumulative information, so it accommodates unequal increases between analyses. If the information values are \(I_1=30\), \(I_2=70\), and \(I_3=120\), the increments are \(30\), \(40\), and \(50\). The score covariance between looks two and three is based on \(I_2=70\), not on the latest increment of \(50\).

In practice, check that information values are strictly increasing and that the reported score at a later look genuinely includes the earlier accumulation under a consistent analysis definition. Compare the actual information estimates with the planned information scale, and verify that standard errors are compatible with the score variances being used. Large discrepancies, very few events, unstable risk-set contributions, or changes in the analysis population can make the normal and covariance approximations questionable.

Stratification does not change the algebra if the combined score and information correctly represent all strata. However, substantial stratum-specific information imbalance, changes in stratum contributions over time, or a mismatch between the score covariance and the reported total information may undermine the simple shared-information representation. Likewise, under non-proportional hazards, the null covariance model may remain a useful approximation while the constant-drift mean model fails: effects that vary over event time need not produce \(E(U_k)\) proportional to \(I_k\).

How the Canonical Model Compares With Alternatives

The canonical model is a compact approximation for the joint law of cumulative standardized scores. It is often more appropriate than treating each look as a separate test, because it explicitly accounts for their overlap. Two other representations can be useful, but answer different questions:

When the canonical assumptions are inadequate, a more detailed model of the score contributions or trial-specific simulation can be preferable. Such alternatives require defensible assumptions about event times, censoring, risk sets, and treatment effects; they are not automatically more accurate simply because they use more detail.

Common Mistake and Limitations

The canonical covariance is not a guarantee that every realized trial follows the model exactly. The approximation can be weakened by sparse events, substantial departures from the assumed score-variance relationship, incorrectly handled ties or censoring, unplanned changes to the analysis, or information values that do not reflect actual score variability. Random or data-dependent analysis times also require care: the covariance formula should not be taken as proof that any stopping rule preserves the intended joint distribution.

The model also does not determine a monitoring boundary, a stopping decision, or the overall error rate of a sequential design. It supplies the joint distributional structure needed for such calculations, but those conclusions require their own design and calibration assumptions.