proportionality-considerations

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 from “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 \(\rightarrow\) ZIP \(\neq\) ZIP \(\rightarrow\) 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 \(\Delta(\text{VAR})\), defined as the change in the value of a variable induced solely by geographic transformation and allocation choices, holding the underlying data source constant:

\[\Delta(\text{VAR}) = \text{VAR}_1 - \text{VAR}_0\]

where \(\text{VAR}_0\) is the baseline value at geography A and \(\text{VAR}_1\) the post-transformation value at the same geography. The pipeline is a composition of hops:

\[T = T_2 \circ T_1 \circ \dots \quad \text{(e.g., ZCTA } \rightarrow \text{ ZIP } \rightarrow \text{ County)}\]

Transformation composition is not commutative: in general, \(T_2 \circ T_1 \neq T_1 \circ T_2\). \(\Delta\) therefore depends on the pathway of transformation, not only its endpoints (pathway dependence); on the choices made at each decision point—membership definition, crosswalk selection, allocation ratio (sensitivity); and on the data itself.

In this example, \(\Delta_x(\text{population})\) for Hennepin County under a ZCTA \(\rightarrow\) ZIP \(\rightarrow\) County transformation using HUD TOT_RATIO allocation is \(-12.6\%\) relative to the relationship-based baseline.

\(\Delta(\text{VAR})\) does not imply:

• Which representation is “correct,”
• Which direction of allocation recovers the “true” value, or
• That a zero delta is desirable.

\(\Delta(\text{VAR})\) states only that the variable is not invariant under this transformation.