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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