Information Across Strata

Why Information Must Be Combined Across Strata

A stratified trial may divide participants by site, region, disease severity, or another baseline characteristic. Each stratum can provide its own estimate of the treatment contrast, with its own precision. To describe the precision of a common treatment effect, the stratum-specific information must be combined in a way that reflects how precisely each contrast was estimated.

The central result is simple: under independence and a common treatment contrast, information adds across strata. A stratum with more information receives more weight in the combined estimate and contributes a larger share of the total precision. Its share is not generally the same as its share of enrollment.

Information is on the scale of inverse variance. If the contrast is measured in outcome points, for example, information is measured in inverse squared points. A larger information value means a smaller variance and therefore greater precision. Stratum shares are dimensionless proportions that add to one.

Assumptions and the Combining Formula

The basic calculation uses a common-effect model for the contrast. Specifically, each \(\hat{\theta}_s\) is an unbiased estimate of the same \(\theta\), and the estimates from different strata are independent. For the exact normal calculation below, assume \(\hat{\theta}_s\sim N(\theta,1/I_s)\), with each \(I_s\) known and positive. In large-sample applications, the same formulas are often used with estimated variances as an approximation, provided the estimates are sufficiently well behaved.

The stratum outcomes themselves need not have the same baseline means or distributions. What must be common for this calculation is the treatment contrast being estimated, on the stated effect scale. For example, a common difference in mean outcome is different from a common ratio of means; one must define the contrast before combining information.

Thus the combined estimate is an information-weighted average, not necessarily an average weighted by stratum sample size. Its standard error is \(SE(\hat{\theta}_{\text{comb}})=1/\sqrt{I_{\text{total}}}\). This formula does not say that stratum estimates are identical; it gives the precision-weighted estimate of their stipulated common contrast.

Why the Information Adds

A short likelihood argument shows why the weights and total information take this form. Under the normal model, the log likelihood for \(\theta\), apart from a constant, is the sum of the independent stratum log likelihoods:

Differentiate with respect to \(\theta\). The score is \(\sum_s I_s(\hat{\theta}_s-\theta)\). Setting it equal to zero gives \(\hat{\theta}_{\text{comb}}=\sum_s I_s\hat{\theta}_s/\sum_s I_s\). The second derivative of the log likelihood is \(-\sum_s I_s\), so the information in the joint likelihood is \(I_{\text{total}}=\sum_s I_s\). Independently, the variance of the weighted estimate is

The variance calculation uses independence: covariance terms between distinct stratum estimates are zero. If participants or measurements connect strata, or if the estimates are otherwise correlated, simply adding the stratum information may not be justified.

Calculating Information From Group Sizes

For a difference in means between two independent treatment groups within stratum \(s\), suppose outcomes have a common variance \(\sigma_s^2\) in that stratum, and the treatment and control sample sizes are \(n_{Ts}\) and \(n_{Cs}\). The variance of the estimated difference is \(\sigma_s^2(1/n_{Ts}+1/n_{Cs})\). Therefore its information is

This expression shows why enrollment alone does not determine information. The allocation between treatment and control matters, and a stratum with more variable outcomes can contribute less precision even when it has many participants. The formula assumes independent observations within each treatment group, the stated variance model, and a well-defined mean difference in every included stratum.

Information Shares at an Interim Analysis

At an interim analysis, it is useful to distinguish each stratum’s information from its share of the total information. If information in stratum \(s\) has reached \(I_s(t)\) at a particular analysis and its planned final value is \(I_s(F)\), its within-stratum progress is \(I_s(t)/I_s(F)\). The overall information fraction is instead the ratio of the summed current information to the summed planned final information:

This is a weighted combination of the strata’s progress, with planned information \(I_s(F)\) providing the weights. It is not generally the simple average of the within-stratum progress values, nor is it the fraction of all participants enrolled. The distinction matters when strata accrue at different rates or have different planned contributions.

Interpretation, Diagnostics, and Alternatives

A useful diagnostic is to calculate both the absolute information values and their shares. A very small share may arise from small enrollment, an unfavorable treatment-control allocation, high variance, or a combination of these. Check whether the planned allocation and variance assumptions match the design. At an interim, compare observed information by stratum with planned values; substantial departures can reveal that the overall information target is being reached through a different mix of strata than expected.

Also check that every stratum estimate targets the same contrast on the same scale. For example, differences in means cannot be combined by this formula with log-odds ratios merely because both are treatment effects. A common numerical value can have a different interpretation on different effect scales. If treatment effects genuinely differ across strata, the information-weighted average is not automatically a universal common effect: it is a weighted summary whose weights depend on the information distribution.

The main alternative is to report separate stratum-specific estimates rather than impose a common contrast. That preserves evidence about differences across strata but does not yield the same single precision-weighted summary. An unstratified analysis is another alternative, but pooling raw outcomes without accounting for strata can mix treatment effects with differences in stratum composition or baseline outcome levels. Which analysis is appropriate depends on the estimand and design, not merely on which calculation is easiest.

The additive formula is limited to information calculations that respect the model and independence assumptions. If stratum estimates are correlated, the variance of their weighted combination includes covariance terms; the simple reciprocal of summed information need not hold. Likewise, if the effect is not common, a single weighted estimate may conceal clinically important variation. Information shares describe precision contribution, not the scientific importance of a stratum or the treatment effect’s size there.

Common Mistake

• Wrong reasoning: “Stratum A has twice as many participants as stratum B, so it must contribute twice as much information.”
• Correction: Information depends on the variance of the contrast, not enrollment alone. For a two-group mean difference, \(I_s=1/\{\sigma_s^2(1/n_{Ts}+1/n_{Cs})\}\); allocation and outcome variability both affect the result.
• Wrong reasoning: “The overall interim information fraction is the average of the strata’s interim fractions.”
• Correction: Weight each within-stratum progress value by its planned final information, or directly divide summed interim information by summed final information.