Table 4.

Attenuation bias due to measurement error

Chemical	True Odds Ratio (OR)	True Logistic Regression Coefficient (β1)	Estimated Variance Ratio $\widehat{\lambda }$	Expected Logistic Regression Coefficient $\mathsf{\text{E}}\left[{\widehat{\beta }}_1\right]$	Expected Odds Ratio E[OR]	Expected Attenuation Bias (B)	No. of Repeats to Limit Bias to 20% (n)
Nicotine	1.50	0.41	0.72	0.24	1.27	-0.42	3
Benzo(a)anthracene	1.50	0.41	0.26	0.32	1.38	-0.21	2
Chrysene	1.50	0.41	0.19	0.34	1.40	-0.16	1
Benzo(b)fluoranthene	1.50	0.41	0.13	0.36	1.43	-0.12	1
Benzo(k)fluoranthene	1.50	0.41	0.64	0.25	1.28	-0.39	3
Benzo(a)pyrene	1.50	0.41	0.47	0.28	1.32	-0.32	2
Indeno(1,2,3-c,d)pyrene	1.50	0.41	0.33	0.30	1.36	-0.25	2
Dibenzo(a,h)anthracene	1.50	0.41	0.31	0.31	1.36	-0.23	2
Coronene	1.50	0.41	0.17	0.35	1.41	-0.15	1
Dibenzo(a,e)pyrene	1.50	0.41	0.38	0.29	1.34	-0.27	2
PCB 138	1.50	0.41	0.37	0.30	1.35	-0.27	2
PCB 153	1.50	0.41	0.34	0.30	1.35	-0.25	2
PCB 180	1.50	0.41	0.25	0.33	1.38	-0.20	1

Expected logistic regression coefficients were calculated using Equation 3 assuming a true odds ratio of 1.5, a case-control study without repeated measurements, and the variance ratios for carpet-dust chemicals measured in the 21 households of Fresno County, California from 2003-2005. Estimates of B and n were calculated using Equations 4 and 5, respectively