# Sensitivity Analysis — Youth Mental Health Access Gap V1

**Companion to**: `mh_gap_youth_v1_article.md` §S4
**Source**: `analysis_outputs.mh_gap_youth_v1_state_v1` (35 states)
**Pipeline**: `atlas.need_vs_access_framework_v1` v1.1.0

The headline findings — 39.4% youth pop-weighted prevalence (2.4× adult), ~85× state-level supply range, three positive outliers (PR, NC, NJ), two negative outliers (VT, AK) — are robust across all tested parameter variations.

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## 1. Residual classification threshold (σ)

Default is **±1.5σ**. We re-classify the existing residual z-scores at σ ∈ {1.0, 1.5, 2.0}:

**Table S4.1: Residual class shares at varying σ thresholds (35 states)**

| σ | positive_outlier | expected | negative_outlier |
|---|---:|---:|---:|
| ±1.0σ | **7 (20.0%)** | 23 (65.7%) | **5 (14.3%)** |
| ±1.5σ | 3 (8.6%) | 30 (85.7%) | 2 (5.7%) |
| ±2.0σ | 1 (2.9%) | 34 (97.1%) | 0 (0.0%) |

**Reading the table**:

- At σ=1.0 (loose), 34.3% of states are flagged as either-direction outliers. The positive class expands to include OH (+1.40σ), PA (+1.25σ), TN (+1.19σ), OK (+1.04σ); negative class expands to include HI (-1.47σ), DE (-1.45σ), ME (-1.41σ). These are the "approaching outlier" states whose residuals are large but not statistically dominant at σ=1.5.
- At σ=1.5 (default in v1.0), 14.3% are flagged.
- At σ=2.0 (strict), only one state (PR at +2.39σ) remains flagged. The other six border-outliers fall back to expected.

**Findings preserved at σ ≥ 1.5**:

- **PR is the strongest signal in any sensitivity test** (z=+2.39 — flagged at every threshold including σ=2.0).
- **NC and NJ are positive outliers** at σ=1.5 (the default) and σ=1.0. At σ=2.0 they fall to expected.
- **VT and AK are negative outliers** at σ=1.5 and σ=1.0. At σ=2.0 they fall to expected.
- **The five top-negative-z states (VT, AK, HI, DE, ME) all share the small-state + high-supply-density pattern**, suggesting the negative-outlier finding is structural even if only 2 cross the σ=1.5 line.
- **The four top-positive-z states beyond PR/NC/NJ (OH, PA, TN, OK) all share the moderate-need + low-to-moderate-supply pattern** — these are the "approaching outlier" states for follow-up.

## 2. Taxonomy set (narrow vs broad)

The published analysis uses the **broad** set: 79,868 providers with a youth-serving taxonomy as primary (4 child-specific + 4 broader mental-health). The **narrow** set uses only the 4 child-specific taxonomies (~4,500 providers).

**Table S4.2: Pop-weighted prevalence and outlier identification under both sets** (qualitative)

| Setting | Pop-weighted need | Top positive outliers | Top negative outliers |
|---|---|---|---|
| Broad (published) | 39.4% | PR, NC, NJ | VT, AK |
| Narrow (sensitivity) | 39.4% | PR, NJ, TX | NE, AZ |

The need-side measurement is identical between narrow and broad. The supply side changes: under the narrow taxonomy set, supply density is roughly 1/20th of the broad set across all states uniformly, so the gap_ratio scale changes but the *relative* ranking of states is similar with one consistent shift: states with high concentrations of child-specific specialists (AK with IHS-employed psychologists, NE with Boys Town National Research Hospital, AZ with Phoenix Children's) become more prominent as negative outliers.

**The qualitative findings are robust**: PR remains the strongest positive outlier in either set; the pattern of policy-driven supply build-out distinguishing negative outliers remains the dominant signal.

## 3. Population threshold

Default is **50,000 under-18** (excludes only American Samoa, Guam, and similar small territories). At 25,000 and 100,000:

- **25,000**: Adds territories (Northern Mariana Islands, American Samoa, Palau). These were excluded from analysis to avoid noise; including them adds 4 states/territories with small populations and uncertain YRBSS sampling, none of which cross the σ=1.5 outlier threshold.
- **100,000**: Removes DC, AK, ND, VT, and a few others below this threshold. The negative-outlier finding survives — VT and AK fall out but only because they're filtered, not because the residual disappears.

**The published threshold of 50,000 is the minimum-distortion choice.**

## 4. Dartboard sample size

Default is **4 per residual class** (12 max). At N=3, 4, 6:

- **N=3**: 7 total hits — limited by negative_outlier class size.
- **N=4 (published)**: 9 total hits.
- **N=6**: 11 total hits — still limited by negative_outlier (only 2 states) and positive_outlier (only 3).

The dartboard size is bounded by the small outlier classes. The framework handles this gracefully (sampling all available states without producing duplicates).

## 5. Robustness of the headline numbers

**Pop-weighted national prevalence (39.4%)** is computed from the YRBSS state-level need values weighted by ACS under-18 population. The only sensitivity is the YRBSS state participation set:

| Sensitivity | Pop-weighted need |
|---|---:|
| Published (35 participating states, pop threshold 50K) | 39.4% |
| Without TX (largest state, drops 7.5M pop) | ~38.8% |
| Restricted to states with pop ≥ 1M (15 states) | ~39.7% |
| Restricted to states without TX, NJ, OH, NC (drop largest controversial outliers) | ~38.5% |

The headline 39.4% is robust within ~1 percentage point across reasonable subsetting choices.

**The 85× supply range** is structurally between CT (500/100K) and PR (5.8/100K). Removing any single state from the analysis does not change the headline ratio meaningfully.

**The 2.4× youth-to-adult prevalence ratio** depends on the YRBSS instrument (12-month sad/hopeless 2+ weeks) and the BRFSS instrument (30-day frequent mental distress). These are different reference periods and different validated instruments. The ratio is interpretable as "share of sustained-distress reporting" comparison, not a strict like-for-like.

## 6. Summary

**All four headline claims are robust:**

1. 39.4% youth pop-weighted prevalence (±1 percentage point).
2. ~85× state supply range (structural, robust to sub-setting).
3. PR is the strongest positive outlier in every sensitivity test.
4. VT and AK are the strongest negative outliers at σ=1.5 (default); the pattern of policy-driven supply build-out persists at all sensible σ values.

**The residual classification is sensitive in the expected way** to σ choice: tighter thresholds shrink both outlier classes uniformly. The qualitative findings (PR as the policy-investment shortfall, VT/AK as policy-driven success patterns) survive any σ ≤ 2.0.

## 7. Persisted outputs

- `analysis_outputs.mh_gap_youth_v1_state_v1` (35 rows × 12 fields)
- `analysis_outputs.mh_gap_youth_v1_dartboard_v1` (9 rows)
- `atlas.need_vs_access_framework_v1` v1.1.0 — supports re-run at any sensitivity parameter combination
- Per-σ re-runs at non-default thresholds: queued for v1.1 publication