Deriving Weights From Stage Information

Why Derive Weights From Information?

In a planned multi-stage trial, the inverse normal method combines evidence from each stage using weights. The weights determine how strongly each stage contributes to the final statistic. If the stages are expected to provide different amounts of information about the same treatment effect, equal weights need not be an efficient choice. The information-proportional rule gives greater weight to stages expected to estimate that effect more precisely.

The derivation depends on more than knowing the sample sizes. It assumes a common treatment-effect scale, independent stage contributions, and a suitable normal approximation for the stagewise evidence under alternatives. Separately, exact null calibration of the inverse normal statistic requires independent uniform stagewise p-values. Validity alone does not imply uniformity, so these conditions must not be conflated.

Assumptions Behind the Information Rule

Consider \(K\) planned stages. Let \(Z_i\) denote the oriented, one-sided stagewise normal score, with larger values supporting the prespecified beneficial direction. The information-based derivation uses the following working model under a local alternative:

where \(\delta\) is the common treatment effect on the model's standardized scale and the stage contributions are independent. The dot indicates an approximation when the stage statistics are only approximately normal. For a normally distributed stage estimate with variance \(1/I_i\), standardizing its estimate gives this mean structure exactly under the corresponding normal model. In other settings it is a large-sample or local-alternative approximation.

For an inverse normal test, a one-sided p-value \(p_i\) is transformed as \(Z_i=\Phi^{-1}(1-p_i)\), where \(\Phi\) is the standard normal cumulative distribution function. If the p-values are independent and each is exactly uniform under a continuous exact null, the transformed scores are independent standard normal variables under that null. For discrete or conservative p-values, or at interior points of a composite null, this exact standard normal statement need not hold. A design must use appropriate null calibration for its actual stagewise tests.

The information rule is appropriate when stagewise evidence concerns the same estimand and direction, the planned \(I_i\) values represent the stages' relative precision, and independence is justified by independent increments or another suitable design argument. It is not a general rule for choosing weights when stages measure different effects, have dependent evidence, or have substantially different alternative mean structures.

Deriving the Information-Proportional Weights

Write the weighted combination statistic as \(C=\sum_{i=1}^K w_i Z_i\). Under independent standard normal stage scores at the exact null, its variance is \(\sum_i w_i^2\). The usual normalization \(\sum_i w_i^2=1\) therefore makes \(C\) standard normal under that null. Under the working alternative model, its expected value is \(\delta\sum_i w_i\sqrt{I_i}\). For a beneficial effect \(\delta>0\), maximizing this mean maximizes the normal-approximation noncentrality for a fixed one-sided critical value.

The restriction to nonnegative weights aligns each stage's contribution with the same beneficial direction. The normalization fixes the null variance; without it, multiplying all weights by a positive constant would change the statistic's scale without changing its ordering.

Why the Square-Root Rule Is Optimal

The justification is the Cauchy–Schwarz inequality. For any weights with \(\sum_i w_i^2=1\),

Equality holds precisely when the weight vector is proportional to the vector with entries \(\sqrt{I_i}\). Normalizing that vector to have squared length one gives the stated weights. Thus, under the working model, the maximum alternative mean of \(C\) is \(\delta\sqrt{\sum_i I_i}\). This result is a justification for a planned weighting choice, not a claim that this choice is optimal for every possible alternative or every combination function.

Obtaining Information From Standard Errors

In many designs, information is available through planned standard errors rather than a direct information table. If a stage-specific treatment-effect estimate has variance \(SE_i^2\), then \(I_i=1/SE_i^2\) under the information convention used here. These quantities carry the reciprocal squared units of the effect estimate, while the weights themselves are dimensionless.

Information Is Not Always Sample Size

For equal per-participant information and otherwise comparable observations, information is proportional to sample size. In that special case, square-root information weights are also proportional to the square roots of planned sample sizes. If per-participant information differs across stages, using sample size alone can misrepresent the stages' relative precision.

Diagnostics, Alternatives, and Limitations

Before adopting information-proportional weights, check whether the design assumptions match the planned trial. In particular, verify that the stagewise statistics use the same effect scale and direction; that information estimates represent the intended precision; and that stagewise evidence is independent under the planned design. If stages share observations or otherwise have correlated statistics, the variance of the weighted sum includes covariance terms, so the unit-squared-weight normalization and derivation above no longer apply unchanged.

Also distinguish planned weights from weights chosen after seeing stage results. The derivation uses information values fixed by the design and does not by itself establish validity for outcome-dependent weight selection. If information changes, its source matters: a change known from design features is different from a change selected using observed treatment effects. Any adaptation requires a method whose calibration accounts for that adaptation.

The information rule is a power-oriented choice under a specified alternative structure, not a universal guarantee of greatest power. If treatment effects vary by stage, endpoints differ, or information estimates are poorly specified, an alternative set of prespecified weights may better reflect the scientific objective. The earlier material on selecting weights distinguishes such design goals from the algebraic normalization of a combination statistic.

Common Mistake

A frequent error is to reason, “Stage 2 has twice as much information, so it should get twice the weight.” That confuses information fractions with inverse normal weights. The information fractions are proportional to \(w_i^2\), not \(w_i\), because the statistic's variance is determined by squared weights. The correction is to calculate \(w_i=\sqrt{I_i/\sum_j I_j}\) and then check that \(\sum_i w_i^2=1\).