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 | 0.00701 (0.00226) | −0.8 | 95.4 | 84.8 | 0.0245 (0.0348) | 7.9 | 93.8 | 13.4 |
0.5 | 2.0 | 0.00247 (0.00160) | −65.1 | 19.0 | 32.0 | 0.0060 (0.0253) | −73.6 | 91.4 | 8.4 |
1.25 | 0.00314 (0.00202) | −55.6 | 51.2 | 37.2 | 0.0108 (0.0314) | −52.4 | 91.6 | 7.0 | |
1 | 0.00360 (0.00226) | −49.1 | 64.4 | 38.0 | 0.0114 (0.0345) | −49.8 | 92.6 | 6.6 | |
0.75 | 0.00410 (0.00261) | −42.0 | 77.8 | 34.4 | 0.0119 (0.0395) | −47.6 | 93.4 | 8.6 | |
0.5 | 0.00510 (0.00319) | −27.9 | 89.4 | 36.4 | 0.0134 (0.0464) | −41.0 | 92.4 | 8.6 | |
0.6 | 2.0 | 0.00299 (0.00160) | −57.7 | 27.2 | 45.0 | 0.0107 (0.0252) | −52.9 | 89.8 | 9.0 |
1.25 | 0.00365 (0.00202) | −48.4 | 59.4 | 43.6 | 0.0108 (0.0314) | −52.4 | 90.2 | 7.4 | |
1 | 0.00424 (0.00226) | −40.0 | 74.8 | 48.8 | 0.0166 (0.0347) | −26.9 | 91.8 | 9.8 | |
0.75 | 0.00519 (0.00261) | −26.6 | 90.2 | 51.2 | 0.0142 (0.0392) | −37.4 | 92.6 | 9.4 | |
0.5 | 0.00601 (0.00319) | −15.0 | 92.2 | 48.8 | 0.0219 (0.0466) | −3.5 | 91.0 | 11.6 | |
0.7 | 2.0 | 0.00358 (0.00160) | −49.4 | 41.0 | 61.0 | 0.0130 (0.0253) | −42.7 | 93.2 | 7.6 |
1.25 | 0.00441 (0.00202) | −37.6 | 74.4 | 59.4 | 0.0138 (0.0314) | −39.2 | 92.2 | 8.4 | |
1 | 0.00478 (0.00226) | −32.4 | 82.0 | 57.0 | 0.0168 (0.0346) | −26.0 | 92.6 | 9.4 | |
0.75 | 0.00546 (0.00261) | −22.8 | 88.0 | 53.0 | 0.0184 (0.0393) | −18.9 | 94.8 | 10.0 | |
0.5 | 0.00701 (0.00320) | −0.8 | 94.6 | 60.6 | 0.0217 (0.0467) | −4.4 | 91.6 | 10.6 | |
0.8 | 2.0 | 0.00406 (0.00160) | −42.6 | 50.6 | 71.0 | 0.0131 (0.0253) | −42.3 | 93.8 | 7.8 |
1.25 | 0.00510 (0.00202) | −27.9 | 83.2 | 72.0 | 0.0149 (0.0313) | −34.4 | 92.4 | 9.8 | |
1 | 0.00558 (0.00226) | −21.1 | 87.8 | 70.6 | 0.0155 (0.0347) | −31.7 | 92.4 | 10.6 | |
0.75 | 0.00660 (0.00261) | −6.6 | 94.2 | 73.6 | 0.0232 (0.0391) | 2.2 | 94.6 | 10.6 | |
0.5 | 0.00791 (0.00319) | 11.9 | 95.8 | 69.4 | 0.0260 (0.0465) | 14.5 | 93.2 | 11.0 | |
0.9 | 2.0 | 0.00447 (0.00160) | −36.8 | 64.0 | 80.6 | 0.0147 (0.0252) | −35.2 | 92.6 | 9.4 |
1.25 | 0.00574 (0.00202) | −18.8 | 89.6 | 80.0 | 0.0209 (0.0315) | −7.9 | 93.4 | 11.6 | |
1 | 0.00637 (0.00226) | −9.9 | 92.0 | 80.2 | 0.0225 (0.0346) | −0.9 | 90.6 | 12.8 | |
0.75 | 0.00722 (0.00261) | 2.1 | 94.4 | 79.4 | 0.0237 (0.0394) | 4.4 | 91.2 | 13.2 | |
0.5 | 0.00899 (0.00320) | 27.2 | 92.6 | 82.4 | 0.0317 (0.0465) | 39.6 | 91.6 | 13.6 |