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Table 2 Results of simulations for PM10 setting: β1 × 10 = 0.00509, [24] and β2 × 10 = 0.0677, [23]

From: Measurement error in a multi-level analysis of air pollution and health: a simulation study

Correlation between “true” and modelled data (τ) Ratio of variances: modelled vs “true” (λ) Estimating the health effect of short-term exposure Estimating the health effect of long-term exposure
\( \widehat{\beta_1}\times 10 \) a
\( \left( se\left(\widehat{\beta_1}\right)\times 10\right) \) a
Biasb (%) Coverage probability (%) Powerc (%) \( \widehat{\beta_2}\times 10 \) a
\( \left( se\left(\widehat{\beta_2}\right)\times 10\right) \) a
Biasb (%) Coverage probability
(%)
Powerc (%)
1 1 0.00531 (0.00351) 4.3 95.8 33.8 0.0621 (0.0975) −8.3 93.0 13.0
0.5 2.0 0.00179 (0.00248) −64.8 72.8 12.0 0.0229 (0.0725) −66.2 87.8 6.4
1.25 0.00233 (0.00314) −54.2 85.4 10.4 0.0268 (0.0892) −60.4 89.0 8.4
1 0.00266 (0.00351) −47.7 91.2 12.4 0.0424 (0.0977) −37.4 90.0 11.0
0.75 0.00285 (0.00405) −44.0 90.8 10.2 0.0383 (0.1092) −43.4 91.4 11.8
0.5 0.00339 (0.00496) −33.4 95.8 11.0 0.0592 (0.1271) −12.6 84.6 15.2
0.6 2.0 0.00232 (0.00248) −54.4 79.8 17.4 0.0305 (0.0725) −54.9 90.4 8.6
1.25 0.00301 (0.00314) −40.9 90.6 16.4 0.0378(0.0884) −44.2 89.2 11.6
1 0.00289 (0.00352) −43.2 90.0 13.8 0.0412 (0.0971) −39.1 89.4 8.8
0.75 0.00358 (0.00406) −29.7 93.4 15.4 0.0531 (0.1097) −21.6 90.4 14.2
0.5 0.00422 (0.00496) −17.1 95.6 14.8 0.0594 (0.1264) −12.3 89.4 13.0
0.7 2.0 0.00262 (0.00248) −48.5 82.6 17.4 0.0338 (0.0726) −50.1 90.8 8.2
1.25 0.00336 (0.00314) −34.0 91.4 19.2 0.0415 (0.0884) −38.7 90.8 9.4
1 0.00369 (0.00351) −27.5 92.0 19.6 0.0464 (0.0968) −31.5 91.8 10.4
0.75 0.00407 (0.00405) −20.0 94.6 17.6 0.0591 (0.1083) −12.7 91.0 12.4
0.5 0.00511 (0.00496) 0.4 95.2 16.2 0.0578 (0.1261) −14.6 89.4 14.0
0.8 2.0 0.00282 (0.00248) −44.6 85.0 19.4 0.0363 (0.0727) −46.4 91.8 10.2
1.25 0.00378 (0.00314) −25.7 94.0 23.4 0.0532 (0.0890) −21.4 93.2 11.4
1 0.00395 (0.00351) −22.4 93.6 20.8 0.0633 (0.0973) −6.5 90.4 13.6
0.75 0.00459 (0.00406) −9.8 94.0 19.2 0.0504 (0.1100) −25.6 88.2 13.2
0.5 0.00602 (0.00496) 18.3 93.8 22.6 0.0707 (0.1269) 4.4 89.0 15.0
0.9 2.0 0.00329 (0.00248) −35.4 87.8 25.4 0.0458 (0.0726) −32.3 91.4 10.2
1.25 0.00418 (0.00314) −17.9 93.0 29.0 0.0545 (0.0891) −19.5 90.8 13.4
1 0.00466 (0.00351) −8.4 95.8 24.8 0.0557 (0.0970) −17.7 92.2 13.4
0.75 0.00516 (0.00405) 1.4 94.8 24.8 0.0759 (0.1096) 12.1 91.2 15.6
0.5 0.00667 (0.00496) 31.0 95.4 27.6 0.0909 (0.1265) 34.3 89.8 16.4
  1. aCoefficients and standard errors are averages of their respective within-simulation estimates. bPercent bias is highlighted in bold when positive (i.e. away from the null). cThe percentage of effect estimates that were statistically significant (p < 0.05)