# proportionality-considerations

## R Markdown

## Why Crosswalks Create Delta

Much like antigen or diagnostic formulation, where sensitivity and specificity must be balanced, geographic transformations operate within a similar trade-off space. A method that maximizes coverage may reduce geographic precision, while a method that preserves tight geographic alignment may shed or distort counts.

Crosswalks, allocation rules, and lookup tables implicitly select a point along this trade-off. This audit makes that selection visible. ## Proportional Allocation Is an Assumption This is verbatim for “How to Apply Proportional Allocation” 1) Identify the Split: If a Census tract is split by two different ZIP codes (e.g., 30% in ZIP A, 70% in ZIP B), the RES_RATIO will be 0.30 and 0.70. 2) Calculate Allocation: If a Census tract has 100 housing units, 30 units (100 * 0.30) are allocated to ZIP A, and 70 units (100 * 0.70) are allocated to ZIP B. 3) Handle Multiple Records: A single tract or ZIP code may appear multiple times if it overlaps with multiple, opposing boundaries.

## Direction Matters (ZCTA → ZIP ≠ ZIP → ZCTA)

This construction (crosswalks in general) does not imply bidirectionality (it is not a valid inverse crosswalk) and does not encode proportional allocation. It is used solely to quantify how a typical boundary-translation workflow can alter aggregate estimates under an explicit allocation rule.

## What geoDeltaAudit Measures

We quantify Δx(VAR), defined as the change in the value of a variable induced solely by geographic transformation and allocation choices, holding the underlying data source constant. Δx(VAR) Δ sensitivity o perturbation o pathway dependence In this example, Δx(population) for Hennepin County under a ZCTA → ZIP → County transformation using HUD proportional allocation is −12.6% relative to the relationship-based baseline. • Input: VAR₀ at geography A • Transformation: T₁ ∘ T₂ ∘ … • Output: VAR₁ at geography A • Result: Δx(VAR) = VAR₁ − VAR₀ Δx(VAR) does not imply:directionality of truth • which representation is “correct” • that zero delta is desirable Δx(VAR) is saying that the variable is not invariant under this transformation.