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¶
13 citations across 13 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.
- 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.
- 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
- 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.
- 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¶
- 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.
- Normalized Metric Check
- 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
- 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
Verification¶
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