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Table 5 Summary of the regression coefficients, their standard errors (SE)(x10-4) and relative bias of 48,000 simulated datasets on the impact of mixture error model on 2-pollutant Poisson regression. Results presented for the core scenario (Area: Europe, Error type: Additive-Mixture) and sensitivity analyses (N = 48,000 in each row)

From: Quantifying the short-term effects of air pollution on health in the presence of exposure measurement error: a simulation study of multi-pollutant model results

Sensitivity Analysis CRFsa: PM2.5: β1 = 5.4a NO2: β2 = 6a
Scenario \( {\hat{\boldsymbol{\beta}}}_{\mathbf{1}} \) (SEW)/(SEB)b Bias (%)c \( {\hat{\boldsymbol{\beta}}}_{\mathbf{2}} \) (SEW)/(SEB)b Bias (%)c
Main Analysis (Europe-Mixture) 5.33 (1.49)/(3.57) −1.3 4.36 (2.00)/(5.10) −27.4
Different “true” CRFs Low effect CRFd 2.66 (1.50)/(3.53) −1.5 2.19 (2.00)/(4.79) −27.0
High effect CRFe 10.66 (1.47)/(3.73) −1.3 8.78 (1.97)/(6.05) −26.9
Only PM2.5 effect 4.85 (1.50)/(3.57) −10.2 0.13 (2.02)/(4.76)
Only NO2 effect 0.44 (1.50)/(3.55) 4.35 (2.02)/(5.05) −27.5
Mixture error percentages (Classical,Berkson)
PM2.5: (55,45%), NO2: (45,55%)
5.20 (1.46)/(3.54) −3.6 4.47 (1.83)/(4.59) −25.6
(Classical,Berkson)
PM2.5: (70,30%), NO2: (60,40%)
5.08 (1.42)/(3.45) −6.0 4.48 (1.70)/(4.26) −25.4
Error type Multiplicative 0.83 (0.27)/(1.92) −84.5 0.61 (0.27)/(1.68) −90.0
  1. aConcentration-response functions for the generation of the health outcome
  2. b SEW: Within-simulations (or model-based) standard error, SEB: Between-simulations (or empirical) standard error
  3. c Relative bias = \( \frac{\left({\hat{\boldsymbol{\beta}}}_{\boldsymbol{\iota}}-{\boldsymbol{\beta}}_{\boldsymbol{\iota}}\right)}{{\boldsymbol{\beta}}_{\boldsymbol{\iota}}} \)
  4. d Half the CRF from Mills et al. 2006
  5. e Twice the CRF from Mills et al. 2006