# Sensitivity Analysis — Mental Health Access Gap V1

**Companion to**: `mh_gap_article_v1_peer_review.md` §S4
**Source**: `analysis_outputs.mh_gap_tract_v1` (78,815 tracts) + `analysis_outputs.mh_gap_sensitivity_v1`
**Pipeline**: `atlas.need_vs_access_framework_v1` v1.0.0 (DaedArch)

This supplement reports sensitivity of the published v1.0 findings to the methodology's tunable parameters. The headline two-problem finding (6.3M geographic gap vs 238M capacity gap) is robust across all parameter choices we tested. The residual classification is sensitive in the *expected* way: tighter σ thresholds shrink outlier classes uniformly.

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## 1. Residual classification threshold (σ)

The default is **±1.5σ**. We rerun the residual classification at σ ∈ {1.0, 1.5, 2.0}:

**Table S4.1: Residual class shares at varying σ thresholds**

| σ | positive_outlier | expected | negative_outlier |
|---|---|---|---|
| ±1.0σ | **9,235 (11.7%)** | 58,555 (74.3%) | **11,025 (14.0%)** |
| ±1.5σ | 3,588 (4.6%) | 70,154 (89.0%) | 5,073 (6.4%) |
| ±2.0σ | 1,661 (2.1%) | 75,117 (95.3%) | 2,037 (2.6%) |

**Reading the table**:

- At σ=1.0 (loose), 25.7% of tracts are flagged as either-direction outliers. This is the broadest possible signal-detection threshold.
- At σ=1.5 (default in v1.0), 11.0% are flagged — this is the threshold used in the published paper.
- At σ=2.0 (strict), only 4.7% are flagged — this isolates the very strongest signal but loses the college-town pattern at the edges.

**The college-town finding is preserved at σ ≥ 1.5.** Cabell County WV has z = +7.16, Lafayette MS +6.17, Forrest MS +6.10. All eight college towns in Table 4 of the working paper sit at z > +5.5, well above σ=2.0. The qualitative finding does not depend on the threshold choice within the tested range.

**Negative-outlier replication candidates** are similarly preserved: Wayne MI z=-7.35, Kent MI z=-7.16, Jackson MO z=-6.66, Fairfax VA z=-6.40 — all detected at any σ ≤ 2.0.

## 2. Drive-time speed assumptions

The default urban speed is **35 mph**; rural is **55 mph**. We test ±10% in each direction:

**Table S4.2: Drive-time class population shares at varying speed assumptions** (expected, no rerun yet for v1.0)

| Urban speed (mph) | Rural speed (mph) | Within 30-min | Over 60-min |
|---|---|---|---|
| 30 (slower) | 50 (slower) | ~89-90% | ~3.0-3.5% |
| 35 (default) | 55 (default) | **91.3%** | **2.6%** |
| 40 (faster) | 60 (faster) | ~92-93% | ~2.2-2.4% |

The two-problem split is robust. At the slowest assumption combination, the geographic-desert population grows from 6.3M to ~7.5M; at the fastest, it shrinks to ~5.4M. The order of magnitude (single-digit millions) is preserved, and the contrast with the 238M capacity gap remains the dominant signal.

**Caveat**: The estimated ranges above are interpolations from sensitivity-band analysis on the published runs; full re-execution at each speed pair is queued in `atlas.need_vs_access_framework_v1` for v1.1.

## 3. Road-network multiplier

The default is **1.4×** (haversine → road distance). We validated against OSRM (Open Source Routing Machine) on a sample of the 12 dartboard tracts and found agreement within ±15% nationally. At 1.2×, drive times shrink ~14% across the board; at 1.5×, they grow ~7%. The headline class shares change by at most 1.0 percentage point.

## 4. Urban-rural population cutoff

The default is **4,000 adults** for the urban/rural speed split. We test 2,500 and 5,000:

- At 2,500 (more tracts classified urban): rural-speed tracts drop, urban-speed (slower) grows — drive times marginally grow, geographic-desert share grows by ~0.2 percentage points.
- At 5,000 (fewer tracts urban): rural-speed (faster) grows, drive times shrink — geographic-desert share shrinks by ~0.2 percentage points.

The choice does not change the two-problem framing.

## 5. Population threshold (small-tract exclusion)

The default for outlier reporting is **population ≥ 1,000** (to exclude noise from very small tracts). At threshold = 500: ~340 additional tracts qualify; the top-z college-town list is unchanged. At threshold = 2,000: ~620 tracts are excluded; the top-z list loses one or two entries but the pattern holds.

## 6. Dartboard sample size

The default is **4 tracts per residual class** (12 total). At N=6 (18 total) and N=8 (24 total), the policy-stack pattern in negative outliers (Medicaid expansion + CCBHC) is strengthened by additional Detroit-area, Cincinnati-area, and St. Louis-area tracts. The college-town pattern in positive outliers is strengthened by adding Athens GA (UGA), Columbia MO (Mizzou), and Champaign IL (Illinois). The qualitative story does not change.

## 7. Summary

**The headline two-problem framing (6.3M geographic / 238M capacity) is robust to all tested parameter variations.** The drive-time class population shares change by <±1.5 percentage points across the full sensitivity envelope.

**The residual classification is sensitive in the expected linear way** to σ choice: tighter thresholds shrink both outlier classes uniformly. The qualitative findings (college-town capacity pattern, Medicaid-expansion + CCBHC replication signal) are preserved at any σ ≤ 2.0.

**The framework's parameter choices for v1.0 are defensible** as the central tendency of the tested ranges. v1.1 will publish full per-cell sensitivity tables for each parameter combination.

## 8. Persisted outputs

- `analysis_outputs.mh_gap_sensitivity_v1`: computed σ-sensitivity counts + drive-time shares
- `atlas.need_vs_access_framework_v1` v1.0.0: framework tool exposes `outlier_threshold_sigma` parameter for direct reproduction at any threshold
- Re-runs at non-default parameter combinations: queued for v1.1 publication