Price Equation Decomposition Table¶
Decomposition ledger — instantiates Selection–Transmission Change Attribution
Lays out every unit's weight and value in both states as a ledger and recomposes the weighted-mean change into an exact selection term plus a transmission term.
When a weighted average moves, this mechanism refuses to guess why. Price Equation Decomposition Table is the exact unit-level ledger: one row per unit, columns for its weight and value in the starting state and the comparison state, and two derived totals that must add back to the observed change in the weighted mean. Its defining move is that it produces both channels at once and closes them — a selection term built from how units' relative weights shifted crossed against their values, and a transmission term built from how much each unit's own value changed, weighted by its representation. Neither term is a guess; together with a reported residual they recompose the observed Δmean by construction. Where a covariance calculation isolates one channel and a dashboard displays the result, this table is the primary accounting artifact the others read from.
Example¶
A dairy herd's average butterfat percentage rose from 3.90% to 4.05% over a breeding season, and the manager wants to know whether the cows improved or the herd's makeup changed. The table gives each cow a row: her milk-volume share (weight) and butterfat percent (value) at the start and at the end. Two columns are then derived. The selection column captures cows whose share of total milk grew being, on average, higher-butterfat cows — high-fat animals came to dominate the tank. The transmission column captures the same cows' butterfat drifting up or down within the season. Summed, the selection term contributes +0.11 points and the transmission term +0.05 points; they recompose to the +0.15-point rise with a −0.01 residual flagged for review. The verdict is legible and exact: most of the gain was composition — the herd reweighted toward fatter milk — not the cows individually improving.
How it works¶
Each unit gets a weight and a value in both states. From those columns the ledger derives two totals directly: the selection term as the sum, over units, of the change in a unit's relative weight multiplied by its value, and the transmission term as the sum of each unit's weighted within-unit value change. The construction follows the Price equation, an algebraic identity in which the two terms plus a residual equal the change in the weighted mean.[n1] Because the identity is exact, the table's job is bookkeeping discipline, not estimation: enter every unit, handle descendants and splits by an explicit apportionment rule, and let any failure to recompose surface as a residual rather than be smoothed away. Run once, it yields the split; the residual line is the built-in check that nothing was dropped.
Tuning parameters¶
- Weight basis — whether weights are frequency, revenue share, survival, or head count. This fixes what "selection" even means; changing it re-scales both terms.
- Descendant apportionment — how a split or merged unit's value and weight are inherited across states. Cleaner rules shrink the residual but demand a defensible lineage claim.
- Residual tolerance — how large a non-closure is accepted before the table is declared unreliable. Tighter tolerance catches bookkeeping errors but rejects more messy real data.
- Nesting depth — a flat two-term split versus a multi-level Price expansion for hierarchical units (herds within farms). Deeper nesting attributes change to more levels at the cost of far more rows.
- Row granularity — individual units versus grouped strata. Grouping is legible but can hide within-group selection as spurious transmission.
When it helps, and when it misleads¶
Its strength is exactness and reviewability: it delivers both channels simultaneously, forces every unit onto the page, and its residual is an honest self-audit that another analyst can re-run. It is the artifact that makes a claim like "the improvement was composition, not the units" defensible rather than rhetorical.
Its central failure mode is mistaking an algebraic split for a causal proof. The identity always closes; that a term is large says the arithmetic attributes change there, not that intervening on that channel would move the aggregate. The classic misuse is reporting "selection dominates" while the weights and unit matches the table consumed were themselves shaky — a tidy ledger built on a bad correspondence map launders error into confidence. The guarding discipline is to treat the table as accounting rather than mechanism: recompute the residual, defend the upstream weights and matches, and hand the causal question to a routing and robustness step rather than answering it from the ledger alone.
How it implements the components¶
unit_value_measure— the value column: each unit's trait or score recorded comparably in both states.unit_weight_measure— the weight column: each unit's share or frequency in both states.selection_term_definition— the differential-weighting total, computed from how relative weights shifted against values.transmission_term_definition— the within-unit total, computed from each unit's own value change weighted by representation.decomposition_identity_and_residual_check— the recomposition line where the two terms plus residual equal the observed Δmean, the table's built-in audit.
It does not build the identity links it consumes (unit_correspondence_map — that is the Lineage or Panel Correspondence Matrix) and it does not route the split to an action (intervention_attribution_rule — that is the Composition-vs-Transformation Dashboard). Its nearest twin is the Covariance Selection-Term Calculation: the table produces both terms and recomposes them exactly, while the covariance calculation yields only the selection term as a single statistic.
Related¶
- Instantiates: Selection–Transmission Change Attribution — it is the canonical exact-accounting mechanism, the ledger the others read.
- Consumes: Lineage or Panel Correspondence Matrix for unit identity across states; Entry/Exit Normalization Protocol for comparable weights.
- Sibling mechanisms: Covariance Selection-Term Calculation · Within-Unit Change Assay · Lineage or Panel Correspondence Matrix · Composition-vs-Transformation Dashboard · Entry/Exit Normalization Protocol · Decomposition Residual Reconciliation Workflow · Selection–Transmission Sensitivity Analysis
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: The mechanism computes exact selection and transmission terms from unit weights and values to explain the change in a weighted mean.
Nearest alternative: Representation, Specification & Plan — A ledger presents the terms, but the defining contribution is the algebraic decomposition.
Review outcome: Adjudicated after independent review; high confidence.
Origin Attribution¶
Primary origin: Biology & Ecology
Origin pattern: Cross-disciplinary synthesis
Present-day reach: Multi-domain
Rationale: The Price equation originated in evolutionary biology as an exact decomposition of selection and transmission change.
Related originating lineages:
- Mathematics — The mathematics tradition materially shaped Price Equation Decomposition Table through its own practice of formal definition, proof, mapping, and quantitative structure.
- Statistics & Experimental Design — The statistics_experimental_design tradition materially shaped Price Equation Decomposition Table through its own practice of probability, calibrated inference, experimental design, and uncertainty analysis.
Review resolution: Both blind reviewers agree that biology ecology is the primary origin. Explicit reconciliation resolves reported ambiguity, origin mode disagreement, domain reach disagreement, encyclopedia synthesis disagreement. Formative alternate lineages are retained as mathematics, statistics_experimental_design; later breadth of use is recorded separately as domain_reach=multi_domain, while origin_mode=cross_disciplinary_synthesis describes the relationship among origin lineages.
Attribution caveat: The exact encyclopedia label appears to synthesize established practices; the primary domain identifies the strongest formative lineage, while the alternates record material ingredients rather than downstream uses.
Encyclopedia synthesis: The exact catalogued form synthesizes established practice rather than reproducing a single standard historical label.
Review outcome: Reconciled after independent review; medium confidence.
Notes¶
[n1] The Price equation, formulated by George R. Price, expresses the change in a population's mean trait as the sum of a covariance term (differential representation of units) and an expectation term (within-unit change), plus a residual. It is an exact algebraic identity, which is precisely why it disciplines bookkeeping without itself proving causation. ↩