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How statement and proof provenance work

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Reproducibility and transparent reporting — Examples

1 · Prerequisites

Reproducibility and transparent reporting

2 · Summary

The example uses small synthetic datasets to distinguish re-running a fixed calculation on the same data, changing a documented grouping rule, and conducting a separate study with its own new data. It makes only descriptive arithmetic claims and does not certify whether the two study estimates are consistent.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-6-luna)Open item page →

A documented grouping change and a new-data study

Example

All observations below are synthetic. The descriptive question is how the mean outcome differs between the older and younger age groups. In Study 1, the pre-specified rule assigns ages at most 20 to the younger group and ages above 20 to the older group. The reported contrast is older-group mean minus younger-group mean.

Study 1 record	Age	Outcome
A	18	10
B	19	11
C	21	13
D	22	14

An exploratory reanalysis of the same four rows instead uses a cut point of 18: ages at most 18 are younger, and ages above 18 are older. This changes the grouping rule and therefore the descriptive contrast being computed.

A separate Study 2 collects new observations while addressing the same pre-specified age-group question and using the original cut point of 20.

Study 2 record	Age	Outcome
E	17	9
F	20	10
G	21	12
H	24	14

A transparent account states both cut points, that the second Study 1 analysis was exploratory and chosen after inspecting the first table, and that no rows were excluded. It gives the data and arithmetic below, marks the values as synthetic, and makes no population, causal, or inferential claim. No software is needed for these calculations; a computational report using software would also identify relevant software and environment details. The Study 2 estimate alone does not decide whether the studies' results are consistent: no uncertainty assessment or consistency criterion is supplied here.

Facts & Assumptions

Given: For a nonempty finite group $G$ of records with real outcomes $y_{i}$ , let ${\overset{ˉ}{y}}_{G} = ∣ G ∣^{- 1} \sum_{i \in G} y_{i}$ , where $∣ G ∣$ is the number of records; the mean of an empty group is undefined. Let $Δ$ be the older- group mean minus the younger-group mean, with the age cut points and the two tables as specified in the Example.

[F1]

Computational reproducibility, replicability, and transparent reporting: computational reproducibility keeps the input data, computational steps, methods and code, and analysis conditions the same.

[F2]

Computational reproducibility, replicability, and transparent reporting: replicability compares studies answering the same scientific question using each study's own newly obtained data.

[A1]

The two tables are constructed solely for this example; all four rows in each study are used, group means are unweighted arithmetic means, and no uncertainty model or consistency threshold is assumed.

Verification

1.1givenA1algebra

Under Study 1's pre-specified cut point of 20, the younger outcomes are $10, 11$ and the older outcomes are $13, 14$ . Their means are $21 / 2$ and $27 / 2$ , respectively, so $Δ = 27 / 2 - 21 / 2 = 3$ .

2.1givenA1step 1.1algebra

Under the alternate cut point of 18, the younger mean is $10$ and the older mean is $(11 + 13 + 14) / 3 = 38 / 3$ , so the contrast is $38 / 3 - 10 = 8 / 3$ . The raw rows are unchanged from step 1.1, but the grouping rule and resulting contrast differ.

2.2givenF1step 1.1algebra

Reapplying the Study 1 pre-specified grouping and the same mean calculation to the same rows returns $Δ = 3$ . The table, cut point, formula, row set, and manual arithmetic are unchanged, so this illustrates the same-input computation comparison in [F1]; it concerns reproduction of the calculation, not whether the grouping or conclusion is correct.

3.1givenF2A1algebra∎

Study 2 has its own new rows. Under the same cut point of 20, the younger mean is $(9 + 10) / 2 = 19 / 2$ and the older mean is $(12 + 14) / 2 = 13$ , giving $Δ = 13 - 19 / 2 = 7 / 2$ . It is a replication attempt under [F2], but these point estimates alone cannot determine consistency without uncertainty and a stated comparison criterion.