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Table 2 Summary of the true and error-prone regression coefficients, their standard errors (SE) x 10-4 and relative bias from 144,000 simulated datasets on the impact of three error models (classical, Berkson and mixture) on 2-pollutant Poisson regression by area of study. Results presented for all scenarios (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

Exposure

CRFsa:

PM2.5: β1 = 5.4a

NO2: β2 = 6a

Area

\( {\hat{\boldsymbol{\beta}}}_{\mathbf{1}} \)

(SEW)/(SEB)b

Bias (%)c

Coverage Probability (%)

Power (%)

\( {\hat{\boldsymbol{\beta}}}_{\mathbf{2}} \)

(SEW)/(SEB)b

Bias (%)c

Coverage Probability (%)

Power (%)

True

Europe

5.37

(1.48)/(3.46)

6.00

(1.79)/(4.20)

East NA

5.42

(1.88)/(4.41

5.98

(1.39)/(3.25)

West NA

5.40

(1.95)/(4.58)

5.99

(1.48)/(3.45)

Classical

Europe

4.86

(1.36)/(3.27)

−10.0

55.3

76.0

4.49

(1.52)/(3.82)

−25.1

51.1

67.6

East NA

4.58

(1.67)/(4.02)

−15.1

54.6

65.2

4.97

(1.25)/(3.09)

−17.2

53.9

80.4

West NA

4.52

(1.72)/(4.17)

−16.3

54.3

63.7

4.85

(1.30)/(3.24)

−19.2

52.8

77.2

Berkson

Europe

5.84

(1.70)/(4.02)

+ 8.2

59.3

75.4

5.53

(2.46)/(5.89)

−7.9

58.8

62.2

East NA

5.78

(2.73)/(6.43)

+ 7.1

59.3

62.2

6.17

(1.95)/(4.58)

+ 2.8

59.7

75.6

West NA

5.62

(3.29)/(7.88)

+ 4.1

59.3

60.2

6.09

(2.50)/(5.87)

+ 1.5

59.2

71.1

Mixtured

Europe

5.33

(1.49)/(3.57)

−1.3

57.9

76.1

4.36

(2.00)/(5.10)

−27.4

55.0

60.8

East NA

4.91

(1.91)/(4.61)

−9.1

57.1

63.2

5.37

(1.48)/(3.58)

−10.6

58.0

76.9

West NA

4.84

(1.99)/(4.81)

−10.3

57.2

61.5

5.20

(1.58)/(3.85)

−13.3

57.1

72.8

  1. aConcentration-response functions for 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 (Classical,Berkson) percentages: (43,57%) for PM2.5, (33,67%) for NO2