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Table 4 Summary of the regression coefficients, their standard errors (SE)(x10-4) and the percentage decrease from single- to multi-pollutant model estimates for 144,000 simulated datasets on the impact of three error models (classical, Berkson and mixture) on 2-pollutant Poisson regression. Results presented for all scenarios (N = 144,000 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

Exposure Model \( {\hat{\boldsymbol{\beta}}}_{\mathbf{1}} \) (SEW)/(SEB)a Bias (%)b Change (%)c \( {\hat{\boldsymbol{\beta}}}_{\mathbf{2}} \) (SEW)/(SEB)a Bias (%)b Change (%)c
True:
 Multi-Pollutant 5.40 (1.77)/(4.18) + 32.6 5.99 (1.55)/(3.66) + 20.5
 Single-Pollutant 7.16 (1.71)/(4.06) 7.22 (1.50)/(3.55)
Classical:
 Multi-Pollutant 4.65 (1.57)/(3.84) −13.8 + 24.7 4.77 (1.35)/(3.40) −20.5 + 17.2
 Single-Pollutant 5.80 (1.54)/(3.80) + 7.3 5.59 (1.32)/(3.38) −6.9
Berkson:
 Multi-Pollutant 5.75 (2.41)/(6.32) + 6.4 + 24.7 5.93 (2.17)/(5.49) −1.2 + 18.0
 Single-Pollutant 7.17 (2.36)/(6.10) + 32.8 7.00 (2.13)/(5.37) + 16.7
Mixture:
 Multi-Pollutant 5.03 (1.80)/(4.37) −6.9 + 22.5 4.97 (1.68)/(4.25) −17.1 + 17.7
 Single-Pollutant 6.16 (1.76)/(4.28) + 14.1 5.85 (1.62)/(4.21) −2.4
  1. a SEW: Within-simulations (or model-based) standard error, SEB: Between-simulations (or empirical) standard error
  2. b Relative bias = \( \frac{\left({\hat{\boldsymbol{\beta}}}_{\boldsymbol{\iota}}-{\boldsymbol{\beta}}_{\boldsymbol{\iota}}\right)}{{\boldsymbol{\beta}}_{\boldsymbol{\iota}}} \)
  3. c Percentage change from multi- to single-pollutant estimate = \( \frac{\left({\hat{\boldsymbol{\beta}}}_{\boldsymbol{S}}-{\hat{\boldsymbol{\beta}}}_{\boldsymbol{M}}\right)}{{\hat{\boldsymbol{\beta}}}_{\boldsymbol{M}}} \)
  4. (Classical, Berkson) percentages: (43,57%) for PM2.5, (33,67%) for NO2