---
title: "proportionality-considerations"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Proportionality considerations}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

## 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.

