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 PM_2.5 and NO₂. Results presented for all scenarios (N = 3000 in each row)

Correlation between exposures

PM_2.5 error variability^b

NO₂ error variability^b

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

(SE_W)/(SE_B)^c

Bias (%)^d

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

(SE_W)/(SE_B)^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