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 | ||||||
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\( \widehat{\beta_1}\times 10 \) 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 |