The Interpretation of Interaction in Contingency Tables.¶
Simpson, E. H. (1951). The Interpretation of Interaction in Contingency Tables. Journal of the Royal Statistical Society, Series B, 13(2), 238-241.
Cited by¶
27 citations across 27 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Aggregate-Marginal Divergence
- The heterogeneous population (role). A mix of units — customers, cases, students, regions, years — whose characteristics shift under the average over time. The aggregate (role). A widely tracked total or average: an integral over the mix, dominated by an inertial stock of past contributions, and a lagging summary. The margin (role). The contribution of the next unit — the latest cohort, the leading edge — a leading indicator of the system's trajectory. The opposite-direction trends (relation). The aggregate trends one way while the marginal contribution trends the other — sustainably, not as a transient or accounting artefact. The masking duration (relation). The aggregate keeps moving favourably for as long as the past stock outweighs the new flow, which can be years — during which the decision-relevant quantity is invisible on the headline metric. The integration invariant. The aggregate is integration over a shifting mix, and the trend in that integral can diverge from the trend at the leading edge — distinct from Simpson's paradox (cross-tabular reversal at one time-point) and from outlier leverage (no outliers required).
This sourceOrigin of Simpson's paradox — cross-tabular reversal of association under stratification — cited for contrast with the temporal aggregate-marginal divergence.
Supported in partVerified against the publisher's abstract
Simpson 1951 is the origin paper for the stratified-interaction phenomenon the claim names "Simpson's paradox", but its abstract states no time-point or outlier-leverage contrast.
“The definition of second order interaction in a (2 × 2 × 2) table given by Bartlett is accepted, but it is shown by an example that the vanishing of this second order interaction does not necessarily justify the mechanical procedure of forming the three component 2 × 2 tables and testing each of these for significance by standard methods.”
- The heterogeneous population (role). A mix of units — customers, cases, students, regions, years — whose characteristics shift under the average over time. The aggregate (role). A widely tracked total or average: an integral over the mix, dominated by an inertial stock of past contributions, and a lagging summary. The margin (role). The contribution of the next unit — the latest cohort, the leading edge — a leading indicator of the system's trajectory. The opposite-direction trends (relation). The aggregate trends one way while the marginal contribution trends the other — sustainably, not as a transient or accounting artefact. The masking duration (relation). The aggregate keeps moving favourably for as long as the past stock outweighs the new flow, which can be years — during which the decision-relevant quantity is invisible on the headline metric. The integration invariant. The aggregate is integration over a shifting mix, and the trend in that integral can diverge from the trend at the leading edge — distinct from Simpson's paradox (cross-tabular reversal at one time-point) and from outlier leverage (no outliers required).
- Aggregation
- The second tension—the silent imposition of homogeneity—is exemplified by the contingency-table reversal Simpson (1951) formalized, in which an aggregated association can vanish or invert relative to its within-stratum counterparts.
This sourceCanonical exposition of the contingency-table reversal in which an aggregated association can vanish or invert relative to within-stratum counterparts.
Supported in partVerified against the publisher's abstract
The 1951 abstract establishes that a vanishing second-order interaction does not license mechanically pooling the component 2×2×2 tables, but does not itself state the reversal of association.
“it is shown by an example that the vanishing of this second order interaction does not necessarily justify the mechanical procedure of forming the three component 2 × 2 tables and testing each of these for significance by standard methods”
- The second tension—the silent imposition of homogeneity—is exemplified by the contingency-table reversal Simpson (1951) formalized, in which an aggregated association can vanish or invert relative to its within-stratum counterparts.
- Partition Dependence of Aggregates
- The result is a sign reversal: the aggregated correlation shows the treatment associated with a worse outcome, the exact opposite of the within-group relationship.
Supported in partVerified against the publisher's abstract
Simpson's own abstract supports only the frame that aggregate and component 2 × 2 tables can give divergent answers, not the specific directional sign reversal asserted.
“it is shown by an example that the vanishing of this second order interaction does not necessarily justify the mechanical procedure of forming the three component 2 × 2 tables and testing each of these for significance by standard methods.”
- The result is a sign reversal: the aggregated correlation shows the treatment associated with a worse outcome, the exact opposite of the within-group relationship.
- Simpson's Paradox
- Simpson's paradox transfers because its arithmetic is identical across substrates — a weighted average can flip sign relative to its components whenever the weights vary across groups — even though it is named after a person, its structure is medium-neutral.
This sourceThe eponymous paper formalizing the reversal of an association on aggregation across a third variable.
Supported in partVerified against the publisher's abstract
Read only as an abstract, Simpson 1951 concerns second-order interaction in 2x2x2 tables and the danger of mechanically splitting them; it supplies the aggregation frame, not the weighted-average sign-flip or medium-neutrality claim.
“it is shown by an example that the vanishing of this second order interaction does not necessarily justify the mechanical procedure of forming the three component 2 × 2 tables and testing each of these for significance by standard methods.”
- Simpson's paradox transfers because its arithmetic is identical across substrates — a weighted average can flip sign relative to its components whenever the weights vary across groups — even though it is named after a person, its structure is medium-neutral.
Mechanisms¶
- Aggregate Trend Overlay
- The guard is to hold Simpson's paradox
This sourceDemonstrates that associations in stratified contingency tables can differ materially from the association after the tables are pooled.
- The guard is to hold Simpson's paradox
- Aggregation Sensitivity Test
- This is a live instance of Simpson's paradox territory
This sourceShows that an association can reverse direction when the same component data are aggregated rather than stratified.
- This is a live instance of Simpson's paradox territory
- Baseline Delta Table
- Worse, summing deltas across heterogeneous units can produce a comforting grand total while nearly every unit worsens — a Simpson's paradox in the subtotal row.
This sourceDemonstrates that a pooled association can reverse the direction present within every subgroup.
- Worse, summing deltas across heterogeneous units can produce a comforting grand total while nearly every unit worsens — a Simpson's paradox in the subtotal row.
- Benefit-Barrier Split Matrix
- The average was dominated by the frequent filers and hid the group that only loses; the same reversal a Simpson's paradox produces when subgroups run against the pooled trend.
This sourceShows that a pooled association can differ from or reverse the associations observed within its component subgroups.
- The average was dominated by the frequent filers and hid the group that only loses; the same reversal a Simpson's paradox produces when subgroups run against the pooled trend.
- Decay Segment Comparison
- Its failure mode is the aggregation trap in reverse — read only the blended curve and you can conclude the opposite of what holds for key subgroups, a Simpson's-paradox reversal where the aggregate hides or inverts the segment truth.
This sourceShows Simpson’s paradox: an aggregate association can hide or reverse the direction present within component groups.
- Its failure mode is the aggregation trap in reverse — read only the blended curve and you can conclude the opposite of what holds for key subgroups, a Simpson's-paradox reversal where the aggregate hides or inverts the segment truth.
- Demand Segmentation Dashboard
- Its signature hazard is Simpson's paradox — the direction of an effect in the pooled data can reverse once the population is split, so a dashboard that segments carelessly (or fails to) can point the decision the wrong way.
This sourceShows Simpson’s paradox: an association in pooled data can reverse after the population is divided into relevant subgroups.
- Its signature hazard is Simpson's paradox — the direction of an effect in the pooled data can reverse once the population is split, so a dashboard that segments carelessly (or fails to) can point the decision the wrong way.
- Direction-Sensitive Metric Dashboard
- Its strength is catching direction and drift that a blended metric hides — the aggregate that stays healthy while one side erodes is a textbook Simpson's-paradox trap, and side-paired metrics are the standard antidote.
This sourceShows how aggregating strata can hide or alter relationships visible in the disaggregated data.
- Its strength is catching direction and drift that a blended metric hides — the aggregate that stays healthy while one side erodes is a textbook Simpson's-paradox trap, and side-paired metrics are the standard antidote.
- Equivalence Class Crosswalk Table
- When an old class genuinely splits, forcing a single new target (or silently apportioning by a guessed share) manufactures precision the data can't support — and aggregates built on that guess can even reverse direction relative to the truth, a Simpson's-paradox trap where the roll-up says the opposite of the parts.
This sourceShows that an aggregate association can reverse direction relative to the associations within its component groups.
- When an old class genuinely splits, forcing a single new target (or silently apportioning by a guessed share) manufactures precision the data can't support — and aggregates built on that guess can even reverse direction relative to the truth, a Simpson's-paradox trap where the roll-up says the opposite of the parts.
- Frame Compatibility Review
- Its failure mode is that a review can manufacture comparability by aggressively repairing frames that should have stayed apart, producing a number that looks reconciled but averages away the very heterogeneity that mattered — the trap behind Simpson's paradox, where subgroup-honest rates reverse when force-pooled
This sourceShows how associations within subgroups can reverse when the groups are combined.
- Its failure mode is that a review can manufacture comparability by aggressively repairing frames that should have stayed apart, producing a number that looks reconciled but averages away the very heterogeneity that mattered — the trap behind Simpson's paradox, where subgroup-honest rates reverse when force-pooled
- Frame-of-Reference Shift
- Simpson's paradox is the standing warning here: an association can reverse sign depending purely on how the data are grouped
This sourceShows that an association can reverse when contingency-table data are partitioned rather than aggregated.
- Simpson's paradox is the standing warning here: an association can reverse sign depending purely on how the data are grouped
- Grouped Reporting Table
- Its dangerous failure is Simpson's paradox — a trend visible at one grouping can reverse when the records are pooled or grouped another way, so a table can tell the literal truth and still point the reader the wrong direction.
This sourceShows with identical contingency-table probabilities that stratum-specific associations can disappear in the pooled table, creating paradox and interpretive error.
- Its dangerous failure is Simpson's paradox — a trend visible at one grouping can reverse when the records are pooled or grouped another way, so a table can tell the literal truth and still point the reader the wrong direction.
- Mix-Shift and Base-Effect Audit
- Its failure mode is over-adjustment: standardizing until a real signal is "explained away," or falling into the Simpson's-paradox trap where a mix reversal is dismissed as mere composition when the composition change is the story worth acting on
This sourceDemonstrates the contingency-table association reversal later known as Simpson’s paradox, in which aggregate and stratified results can point in different directions.
- Its failure mode is over-adjustment: standardizing until a real signal is "explained away," or falling into the Simpson's-paradox trap where a mix reversal is dismissed as mere composition when the composition change is the story worth acting on
- Multi-Resolution Dashboard
- The same data can tell opposite stories at different aggregations — a metric rising hour over hour can be falling day over day — an aggregation cousin of Simpson's paradox that a poorly-labeled board invites viewers to walk straight into.
This sourceDemonstrates that associations in contingency-table data can reverse when observations are combined rather than examined by subgroup.
- The same data can tell opposite stories at different aggregations — a metric rising hour over hour can be falling day over day — an aggregation cousin of Simpson's paradox that a poorly-labeled board invites viewers to walk straight into.
- Normalized Metric Check
- Segment Funnel Comparison
- Its strength is defeating the average: it exposes Simpson's paradox cases, where an aggregate trend reverses or vanishes once the population is split, and it turns a diffuse "everything is a bit worse" into a located "this segment, this stage."
This sourceDemonstrates that an association present within strata can disappear when the strata are combined.
- Its strength is defeating the average: it exposes Simpson's paradox cases, where an aggregate trend reverses or vanishes once the population is split, and it turns a diffuse "everything is a bit worse" into a located "this segment, this stage."
- Segment Stratification Table
- Its strength is catching Simpson's-paradox reversals and subgroup effects that a single coefficient averages away, and making the group mix — and its influence on the headline number — explicit and governable.
This sourceShows how an aggregate contingency table can reverse or obscure the associations in its constituent strata, making stratum composition material to the headline result.
- Its strength is catching Simpson's-paradox reversals and subgroup effects that a single coefficient averages away, and making the group mix — and its influence on the headline number — explicit and governable.
- Stratified Analysis
- Conditioning on the confounder reverses the headline — the hallmark of Simpson's paradox.
This sourceShows that an aggregate association can reverse when a contingency table is subdivided, the pattern now called Simpson’s paradox.
- Conditioning on the confounder reverses the headline — the hallmark of Simpson's paradox.
- Stratified Benchmark Suite
- Its strength is exposing the subgroup and tail failures that a pooled metric structurally conceals — the territory of Simpson's paradox, where an aggregate can flatter or even reverse the pattern inside its strata
This sourceShows that aggregation can conceal subgroup patterns and even reverse an association visible within the strata.
- Its strength is exposing the subgroup and tail failures that a pooled metric structurally conceals — the territory of Simpson's paradox, where an aggregate can flatter or even reverse the pattern inside its strata
- Stratified Rate Table
- The signature danger is Simpson's paradox — the direction of an association can flip between the stratified table and the pooled marginal, so a table and its own total row can tell opposite stories, and which is "right" depends on what's being asked.
This sourceDemonstrates reversal between pooled and stratified associations and shows that the appropriate interpretation depends on the question being asked.
- The signature danger is Simpson's paradox — the direction of an association can flip between the stratified table and the pooled marginal, so a table and its own total row can tell opposite stories, and which is "right" depends on what's being asked.
- System-Wide Net-Risk Dashboard
- A well-meaning average can reverse the subgroup story entirely, and a dashboard can be quietly gamed by choosing a flattering boundary.
This sourceShows that aggregate and subgroup associations can tell different—even reversed—stories.
- A well-meaning average can reverse the subgroup story entirely, and a dashboard can be quietly gamed by choosing a flattering boundary.
- Top-Driver Analysis
- Its failure modes are the ones the causal screen exists to catch: a top driver can be large because it is a broad catch-all, over-reported, or downstream of another cause, and aggregate rankings can reverse within segments — Simpson's paradox.
This sourceDemonstrates that aggregating stratified contingency tables can produce a materially different association from the relationships observed within strata.
- Its failure modes are the ones the causal screen exists to catch: a top driver can be large because it is a broad catch-all, over-reported, or downstream of another cause, and aggregate rankings can reverse within segments — Simpson's paradox.
- Unit-Time Dashboard
- Its strength is refusing to average: by keeping every unit-period cell, it protects against the aggregation trap where a pooled mean reverses
This sourceShows that pooling strata can conceal or reverse associations present within the separate groups.
- Its strength is refusing to average: by keeping every unit-period cell, it protects against the aggregation trap where a pooled mean reverses
- Variance Analysis
- A factor can dominate the pooled decomposition only because the case mix behind it is unbalanced, and the ranking can reverse the moment the data are properly split — Simpson's paradox in operational clothes.
This sourceFormalizes Simpson’s paradox as a reversal between pooled and conditionally stratified comparisons.
- A factor can dominate the pooled decomposition only because the case mix behind it is unbalanced, and the ranking can reverse the moment the data are properly split — Simpson's paradox in operational clothes.
Verification¶
Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.
Does it back the claim? Read against the text for 4 of 27 citations: 4 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.
See how references were verified.
Registry ID ref:586bb7e70915 · see in the full table