Interpreting Fisher Contributions by Stage

What a Stage Contribution Shows

A combined test gives one final statistic, but that number can conceal how the stagewise results shaped it. Fisher’s method makes this structure visible: each stage contributes an amount determined by its p-value, and the total is the sum of those amounts. A small stagewise p-value contributes more than a larger one. This breakdown helps describe which stage numerically dominates the combined statistic, while not establishing that the stage is more reliable, more important clinically, or independently decisive.

The setting here is a trial with two stages, each producing a p-value for the same null hypothesis \(H_0\). Assume that, under \(H_0\), the p-values are independent and each has a continuous uniform distribution on \((0,1)\). Under these conditions, the usual Fisher statistic has a chi-square reference distribution with four degrees of freedom. The independence and uniformity conditions matter: merely having two reported p-values does not ensure that reference distribution is appropriate.

A contribution is a component of the test statistic, not a p-value, an effect estimate, or a measure of clinical benefit. It has no physical unit. The ratio \(C_i/X\) can be reported as that stage’s arithmetic share of the observed statistic, but it is not a percentage of the trial’s “true evidence” attributable to that stage. The p-values themselves depend on the stagewise tests, their assumptions, and the data.

Why the Contributions Add

The decomposition follows directly from Fisher’s product form. Taking the logarithm of the product turns multiplication of p-values into addition of their log values; multiplying by \(-2\) gives the conventional Fisher statistic. This algebra also explains why a stage with a very small p-value can have a large contribution.

On the log scale, reducing a p-value by a fixed factor adds the same amount to its contribution, regardless of the starting value. If a p-value is multiplied by \(r\), where \(0<r<1\), then its contribution increases by \(-2\ln(r)\). Equivalently, the local rate of change is \(dC_i/dp_i=-2/p_i\): the contribution changes more sharply for a fixed absolute p-value change when the p-value is already small. These facts describe the statistic’s arithmetic, not the evidential quality of either stage.

Under the assumptions stated above, the two-stage reference result is \(X\sim\chi^2_4\) under \(H_0\). For an observed value \(x\), the upper-tail p-value is \(P(\chi^2_4\ge x)=e^{-x/2}(1+x/2)\). This gives a way to assess the combined result; the individual contributions help explain how the observed statistic was formed. The combined p-value is not obtained by adding the stagewise p-values.

Dominance and the Combined Decision

A dominant contribution means that one term is much larger than the other in the observed sum. Since \(C_i=-2\ln(p_i)\), dominance is driven by the relative sizes of the p-values: the smaller one has the larger contribution. It does not mean that Fisher’s method discards the other stage. Both contributions remain in \(X\), and the chi-square comparison uses the total.

A large contribution can arise from a small stagewise p-value, but that p-value may reflect a large estimated effect, high precision, random variation, or features of the stagewise analysis. The contribution does not separate these possibilities. For clinical interpretation, effect estimates, uncertainty intervals, design details, and the intended estimand remain important; Fisher’s contribution alone is not an effect-size summary.

Different Patterns Can Give the Same Fisher Result

Because the two-stage statistic depends on the product \(p_1p_2\), different pairs of p-values can yield the same total and hence the same combined p-value under the same reference distribution. Their contribution profiles can nevertheless look quite different. The following comparison isolates that point.

Assumptions, Diagnostics, and Alternatives

Before interpreting a contribution breakdown, check the conditions that support the combined reference distribution. The stagewise p-values must test the intended null hypothesis in a compatible way, and under that null they must satisfy the distributional assumptions used for calibration. The usual chi-square-four result requires independent, continuous uniform p-values under \(H_0\). Independent stage data can support independence of stagewise results when the analyses are correctly specified, but independence should not be assumed simply because the trial labels observations as different stages.

In an adaptive trial, stage 2 design choices may depend on stage 1 results. The ordinary reference distribution is justified only if the resulting stagewise p-values retain the required null distribution and independence properties, or if an appropriate calibration is used. For discrete tests, p-values may not be exactly uniform, and the usual continuous-uniform derivation may not provide exact calibration. If these conditions are doubtful, the reported contributions can still describe the arithmetic of the observed statistic, but a chi-square-based combined p-value may not be valid.

A practical review should check that each p-value is for the intended hypothesis, that the stagewise procedures are valid for their designs, and that the relevant dependence and calibration assumptions are defensible. Also compare the unrounded p-values and contributions when assessing relative sizes: rounding can distort small differences or produce a mismatch between the displayed components and total. A contribution table is a diagnostic summary, not a substitute for checking the analysis that generated the p-values.

Fisher’s method combines p-values through their product, so a very small p-value can strongly influence the result. The weighted inverse-normal method instead transforms p-values to normal scores and combines them using chosen weights. These procedures therefore summarize stagewise evidence differently. Neither contribution dominance nor a particular combination rule automatically identifies the clinically best interpretation; the method must match the trial’s hypotheses and design. A separate stagewise analysis can describe each stage’s result, but it answers a different inferential question from the prespecified combined test.

Common Mistake

Wrong reasoning: “Stage 2 contributed 87% of the Fisher statistic, so 87% of the evidence came from stage 2, and stage 1 can be ignored.” This treats an arithmetic ratio as a scientific attribution and turns a combined test into a stage-specific conclusion.

Correction: The ratio \(C_2/X\) says only that stage 2’s transformed p-value accounts for 87% of the observed sum. The combined test still uses both stages, and its calibration depends on the joint behavior of both p-values under \(H_0\). A separate conclusion about a stage requires an appropriate stage-specific inferential analysis and context.