Table 3.

Results of a quantitative analysis of prebaseline selection bias according to the causal structure in Figure 4 across 2,000 simulated cohorts modeled after participants in our sample of Chicago Health and Aging Project participants ($n=\mathrm{10,000}$ in each simulated cohort).

$\text{exp\hspace{0.17em}}\left({\mathrm{\gamma }}_1\right)$	$\text{exp\hspace{0.17em}}\left({\mathrm{\gamma }}_2\right)$	$\text{exp\hspace{0.17em}}\left({\mathrm{\gamma }}_3\right)$	$\text{exp\hspace{0.17em}}\left({\mathrm{\gamma }}_4\right)$	$\text{exp\hspace{0.17em}}\left({\mathrm{\gamma }}_5\right)$	$\overline{\widehat{\mathrm{\beta }}}$ (bias %) age 65 y	$\overline{\widehat{\mathrm{\beta }}}$ (bias %) age 75 y
1.05	1.25	0.9	1.0	1.0	$-0.002$ ($-32\%$)	$-0.003$ ($-14\%$)
1.25	1.75	0.7	1.0	1.0	0.000 ($-100\%$)	$-0.001$ ($-52\%$)
1.25	1.75	0.8	1.0	1.0	$-0.001$ ($-65\%$)	$-0.002$ ($-33\%$)
1.50	2.25	0.7	1.0	1.0	$-0.000$ ($-94\%$)	$-0.002$ ($-49\%$)
1.50	2.25	0.8	1.0	1.0	$-0.001$ ($-59\%$)	$-0.002$ ($-30\%$)
1.25	1.75	0.7	0.9	0.7	0.003 ($-212\%$)	0.002 ($-177\%$)
1.25	1.75	0.8	0.9	0.7	0.003 ($-186\%$)	0.002 ($-166\%$)
1.50	2.25	0.7	0.8	0.5	0.005 ($-276\%$)	0.005 ($-254\%$)
1.50	2.25	0.8	0.8	0.5	0.005 ($-260\%$)	0.005 ($-253\%$)

^Note:The parameters ${\mathrm{\gamma }}_1$ to ${\mathrm{\gamma }}_5$ correspond to log OR, and therefore, $\text{exp}\left({\mathrm{\gamma }}_1\right)$ to $\text{exp}\left({\mathrm{\gamma }}_5\right)$ correspond to ORs. An OR of 1.0 is the null value, indicating no association (or in the case of interactions, no interaction). We assumed that the probability of death was given by the following equation in agreement with Figure 2 in the main text:

$\mathrm{Pr}\left(S_i=0\right)=\frac{\text{\hspace{0.17em}exp\hspace{0.17em}}\left\{{\mathrm{\gamma }}_{0,c}+{\mathrm{\gamma }}_{1}I\left(Q_i=2\right)+{\mathrm{\gamma }}_{2}I\left(Q_i=3\right)+{\mathrm{\gamma }}_3{U}_i+{\mathrm{\gamma }}_{4}I\left(Q_i=2\right)U_i+{\mathrm{\gamma }}_{5}I\left(Q_i=3\right)U_i\right\}}{1+\text{exp\hspace{0.17em}}\left\{{\mathrm{\gamma }}_{0,c}+{\mathrm{\gamma }}_{1}I\left(Q_i=2\right)+{\mathrm{\gamma }}_{2}I\left(Q_i=3\right)+{\mathrm{\gamma }}_3{U}_i+{\mathrm{\gamma }}_{4}I\left(Q_i=2\right)U_i+{\mathrm{\gamma }}_{5}I\left(Q_i=3\right)U_i\right\}},$

where I is an indicator function and $Q_i$ denotes the pseudo-individual’s TRAP exposure, with 1 corresponding to the first/lowest tertile of exposure (i.e., “low”), 2 corresponding to the second tertile of exposure (i.e., “medium”), and 3 corresponding to the third/highest tertile of exposure (i.e., “high”). The values of ${\mathrm{\gamma }}_1$ through ${\mathrm{\gamma }}_5$ are chosen as part of the simulation design (i.e., they are user-specified), and a root finding procedure is applied to estimate the ${\mathrm{\gamma }}_{0,c}$ that gives approximately the correct marginal probability of death, as given by life tables in Arias.⁵⁶ OR, odds ratio; TRAP, traffic-related air pollution.