# Stress tests of negative control calibration

1 October 2026. These post hoc sensitivity analyses use the full fifteen-context synthesis. They add no independent observations and do not replace its assumed-bias primary result. The original calibrated models and their reported numbers are preserved.

## Autism-specific residual bias

A new shared bias b ~ N(0, s²) allows autism-specific ascertainment or counterfactual error that the control outcomes do not measure. The existing calibrated likelihood is evaluated at a = theta + b and integrated with the original N(0, (log 2)²) effect prior. This is a likelihood-level extension, not an ad hoc widening of a confidence interval. The SD values are sensitivity assumptions, not estimates from these data. V1 retains the control-derived mean shifts; V2 omits them.

| Calibration | Residual shared log SD | Common RR (95% CrI) | P(RR > 1.05) | P(RR > 1.25) |
|---|---:|---:|---:|---:|
| V1 | 0.00 | 0.871 (0.741–1.020) | 0.010 | <0.001 |
| V1 | 0.05 | 0.871 (0.722–1.049) | 0.024 | <0.001 |
| V1 | 0.10 | 0.873 (0.680–1.119) | 0.073 | 0.002 |
| V1 | 0.15 | 0.876 (0.632–1.212) | 0.137 | 0.016 |
| V1 | 0.20 | 0.880 (0.587–1.318) | 0.196 | 0.044 |
| V1 | 0.30 | 0.890 (0.510–1.551) | 0.279 | 0.115 |
| V2 | 0.00 | 0.946 (0.806–1.111) | 0.102 | <0.001 |
| V2 | 0.05 | 0.947 (0.785–1.142) | 0.139 | 0.002 |
| V2 | 0.10 | 0.947 (0.738–1.216) | 0.210 | 0.015 |
| V2 | 0.15 | 0.949 (0.685–1.314) | 0.271 | 0.049 |
| V2 | 0.20 | 0.950 (0.634–1.425) | 0.315 | 0.092 |
| V2 | 0.30 | 0.955 (0.547–1.665) | 0.369 | 0.171 |

These rows inherit the calibrated fits’ low local effective sample sizes and control-model assumptions. The JSON reports each independent scramble separately. Agreement between two scrambles is a diagnostic, not an error bound. They do not establish that vaccines prevent autism.

## Dependence and control type

The control error model is refitted with covariance C = diag((1−rho)s_j² + omega²) + rho ssᵀ within each context. Positive rho values are assumed sensitivity scenarios, not estimated correlations. The de-nested set follows the existing rule-literal sensitivity. Injury-only and non-injury rows are also separated. This checks calibration inputs; the autism synthesis is not refitted under these covariance scenarios, so these rows cannot be read as corrected vaccine-risk estimates.

| Control set | Rows | Contexts | rho | Between-context SD | Within-context SD | Mean bias (posterior SD) |
|---|---:|---:|---:|---:|---:|---:|
| all_controls_rho0.0 | 45 | 9 | 0.00 | 0.204 | 0.025 | +0.038 (0.091) |
| all_controls_rho0.5 | 45 | 9 | 0.50 | 0.066 | 0.073 | +0.029 (0.060) |
| all_controls_rho0.9 | 45 | 9 | 0.90 | 0.092 | 0.377 | +0.061 (0.105) |
| de_nested_rho0.0 | 38 | 9 | 0.00 | 0.158 | 0.062 | +0.050 (0.081) |
| de_nested_rho0.5 | 38 | 9 | 0.50 | 0.078 | 0.128 | +0.035 (0.073) |
| de_nested_rho0.9 | 38 | 9 | 0.90 | 0.100 | 0.377 | +0.071 (0.115) |
| injury_only_rho0.0 | 25 | 5 | 0.00 | 0.206 | 0.054 | +0.069 (0.139) |
| injury_only_rho0.5 | 25 | 5 | 0.50 | 0.122 | 0.087 | +0.071 (0.121) |
| injury_only_rho0.9 | 25 | 5 | 0.90 | 0.162 | 0.427 | +0.117 (0.178) |
| non_injury_only_rho0.0 | 13 | 6 | 0.00 | 0.222 | 0.121 | +0.073 (0.139) |
| non_injury_only_rho0.5 | 13 | 6 | 0.50 | 0.189 | 0.149 | +0.053 (0.141) |
| non_injury_only_rho0.9 | 13 | 6 | 0.90 | 0.172 | 0.212 | +0.056 (0.150) |

Correlations between the two U.S. exposure-path controls are still unknown; these covariance scenarios do not solve that cross-context dependence. Injury categories are not guaranteed unaffected outcomes: the control exclusion restriction and bias comparability remain substantive assumptions.

## Held-out context checks

Every row from a context is removed before predicting its controls. Both U.S. exposure paths are removed together. Unlike within-context row deletion, this prevents a held-out nested category from being predicted using its own context’s other categories. Coverage is descriptive across dependent outcomes; neither the row nor group averages are independent-binomial validation rates. No result selects a model.

| Set | Row coverage | Equal-group coverage | Held-out groups |
|---|---:|---:|---:|
| de_nested | 0.974 | 0.984 | 8 |
| injury_only | 1.000 | 1.000 | 5 |

## Verification and limits

Both zero-residual-bias baselines reproduce the published calibrated common-effect summaries to an absolute tolerance of 1e-9. At rho = 0 the covariance likelihood reproduces the original error model. Unit tests compare its sufficient statistics with a direct multivariate-normal likelihood and verify that correlated copies do not contribute independent precision. Input and cache hashes are recorded in the JSON.

Negative controls require an unaffected outcome and a defensible relation between its bias and the target outcome’s bias; empirical calibration does not verify those assumptions by itself. [Lipsitch et al. (2010)](https://pmc.ncbi.nlm.nih.gov/articles/PMC3053408/); [Schuemie et al. (2018)](https://pmc.ncbi.nlm.nih.gov/articles/PMC5856503/).

This audit does not complete the proposed 3374–3376 extensions, resolve the Monte Carlo limitation, or supply a universal causal risk bound. Those are distinct tasks, not additional evidence obtained here.