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Table 3 Summary of the regression coefficients, their standard errors (SE)(×10− 4) and relative bias of 144,000 simulated datasets on the impact of mixture error model on 2-pollutant Poisson regression by the error variability of PM2.5 and NO2. Results presented for all scenarios (N = 3000 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

CRFsa: PM2.5: β1 = 5.4a NO2: β2 = 6a
Correlation between exposures PM2.5 error variabilityb NO2 error variabilityb \( {\hat{\boldsymbol{\beta}}}_{\mathbf{1}} \) (SEW)/(SEB)c Bias (%)d \( {\hat{\boldsymbol{\beta}}}_{\mathbf{2}} \) (SEW)/(SEB)c Bias (%)d
Low Very low Very low 5.44 (1.76)/(4.15) + 0.7 5.90 (1.54)/(3.62) −1.7
Low 5.49 (1.76)/(4.00) + 1.6 5.74 (1.58)/(3.66) −4.3
Moderate 6.10 (1.74)/(4.12) + 13.0 4.45 (1.71)/(4.02) −25.8
High 6.57 (1.73)/(4.08) + 21.7 2.85 (1.88)/(4.56) −52.5
Low Very low 5.20 (1.77)/(4.23) −3.7 6.04 (1.54)/(3.62) + 0.7
Low 5.22 (1.76)/(4.16) −3.4 5.69 (1.57)/(3.72) −5.1
Moderate 5.88 (1.75)/(4.11) + 8.9 4.56 (1.70)/(4.17) −23.9
High 6.36 (1.74)/(4.05) + 17.8 3.00 (1.88)/(4.56) −50.0
Moderate Very low 4.31 (1.81)/(4.30) −20.2 6.35 (1.53)/(3.58) + 5.8
Low 4.58 (1.80)/(4.29) −15.3 6.10 (1.56)/(3.78) + 1.7
Moderate 4.94 (1.79)/(4.31) −8.5 4.88 (1.70)/(4.02) −18.7
High 5.42 (1.78)/(4.21) + 0.3 3.16 (1.88)/(4.59) −47.4
High Very low 3.46 (1.85)/(4.31) −36.0 6.64 (1.52)/(3.57) + 10.6
Low 3.72 (1.85)/(4.40) −31.1 6.37 (1.56)/(3.67) + 6.1
Moderate 4.07 (1.84)/(4.48) −24.6 5.16 (1.69)/(4.05) −14.0
High 4.32 (1.84)/(4.37) −20.1 3.21 (1.88)/(4.64) −46.5
Moderate Very low Very low 5.36 (1.77)/(4.19) −0.8 5.99 (1.56)/(3.71) −0.2
Low 5.52 (1.77)/(4.34) + 2.2 5.59 (1.60)/(3.69) −6.9
Moderate 6.12 (1.76)/(4.19) + 13.4 4.39 (1.72)/(4.07) −26.9
High 6.78 (1.74)/(4.15) + 25.5 2.82 (1.89)/(4.56) −53.0
Low Very low 5.06 (1.79)/(4.17) −6.4 6.06 (1.56)/(3.62) + 0.9
Low 5.15 (1.78)/(4.23) −4.6 5.73 (1.59)/(3.79) −4.5
Moderate 5.87 (1.76)/(4.07) + 8.6 4.53 (1.71)/(4.14) −24.6
High 6.46 (1.74)/(4.16) + 19.6 3.07 (1.89)/(4.54) −48.9
Moderate Very low 4.20 (1.83)/(4.32) −22.2 6.30 (1.55)/(3.74) + 5.1
Low 4.31 (1.81)/(4.37) −20.2 6.17 (1.57)/(3.75) + 2.9
Moderate 4.90 (1.80)/(4.26) −9.3 5.03 (1.71)/(4.13) −16.2
High 5.64 (1.79)/(4.30) + 4.4 3.11 (1.88)/(4.66) −48.2
High Very low 3.11 (1.86)/(4.42) −42.3 6.72 (1.54)/(3.60) + 12.1
Low 3.57 (1.85)/(4.40) −33.9 6.39 (1.56)/(3.67) + 6.5
Moderate 4.17 (1.84)/(4.37) −22.8 5.26 (1.70)/(4.10) −12.3
High 4.55 (1.83)/(4.39) −15.8 3.17 (1.88)/(4.64) −47.2
High Very low Very low 5.43 (1.78)/(4.15) + 0.5 5.89 (1.57)/(3.74) −1.9
Low 5.51 (1.78)/(4.21) −2.1 5.57 (1.60)/(3.86) −7.3
Moderate 6.21 (1.76)/(4.09) + 15.0 4.30 (1.73)/(4.09) −28.3
High 6.81 (1.74)/(4.08) + 26.1 2.49 (1.90)/(4.73) −58.4
Low Very low 4.90 (1.80)/(4.24) −9.3 6.16 (1.57)/(3.70) + 2.7
Low 4.99 (1.82)/(4.27) −7.5 5.46 (1.62)/(3.91) −9.0
Moderate 5.77 (1.78)/(4.19) + 6.8 4.36 (1.73)/(4.11) −27.3
High 6.35 (1.76)/(4.12) + 17.6 2.60 (1.90)/(4.67) −56.7
Moderate Very low 3.97 (1.83)/(4.32) −26.5 6.27 (1.56)/(3.79) + 4.5
Low 4.17 (1.83)/(4.46) −22.7 6.07 (1.59)/(3.70) + 1.2
Moderate 4.87 (1.81)/(4.28) −9.8 4.87 (1.71)/(4.05) −18.9
High 5.55 (1.79)/(4.23) + 2.8 2.87 (1.89)/(4.73) −52.1
High Very low 3.03 (1.87)/(4.45) −44.0 6.61 (1.54)/(3.66) + 10.1
Low 3.30 (1.86)/(4.51) −38.9 6.38 (1.57)/(3.63) + 6.4
Moderate 4.07 (1.85)/(4.42) −24.7 5.07 (1.70)/(4.12) −15.5
High 4.54 (1.84)/(4.41) −15.8 3.39 (1.88)/(4.60) −43.4
  1. a Concentration-response functions for the generation of the health outcome
  2. b Moderate error variability as defined in Table 1. Very low = 0.1 x Moderate, Low = 0.5 x Moderate, High = 1.3 x Moderate
  3. c SEW: Within-simulations (or model-based) standard error, SEB: Between-simulations (or empirical) standard error
  4. d Relative bias = \( \frac{\left({\hat{\boldsymbol{\beta}}}_{\boldsymbol{\iota}}-{\boldsymbol{\beta}}_{\boldsymbol{\iota}}\right)}{{\boldsymbol{\beta}}_{\boldsymbol{\iota}}} \)
  5. (Classical, Berkson) percentages: (43,57%) for PM2.5, (33,67%) for NO2