Appendix 2—table 3. Estimated Average Bias of ${\alpha }_x$ Under Various Scenarios.

Where $\tau ,{\beta }_x,{\beta }_z$ are selected to Induce a Zero Correlation Between $X$ and $Z$ After Selecting on $Y_1$ . Results are based on sample size of n=50,000 and 1000 samples obtained from the data-generating model described above.

$a$	$b$	${\lambda }_z\left(max\right)$	${\beta }_x$	${\beta }_z$	${\lambda }_z$	$\tau$	All data		Select on ${\displaystyle Y_1>\tau }$
							${\alpha }_{x.z}$	${\alpha }_x$	${\alpha }_{x.z}$	${\alpha }_x$
0.2	0.2	0.0422	0.2000	0.1952	0.0200	–0.5774	–0.0000	0.0118	–0.0002	–0.0002
0.2	0.2	0.0422	0.1999	0.1946	0.0350	1.3809	–0.0001	0.0208	–0.0004	–0.0008
0.2	0.9	0.5519	0.1931	0.8361	0.2700	0.4647	–0.0001	0.1620	–0.0002	–0.0001
0.2	0.9	0.5519	0.1857	0.8175	0.4000	1.8276	–0.0001	0.2398	–0.0002	–0.0003
0.9	0.2	0.2277	0.8936	0.0683	0.1200	0.6717	0.0001	0.0721	0.0005	0.0009
0.9	0.2	0.2277	0.8842	0.0598	0.1900	3.0584	0.0001	0.1140	–0.0080	–0.0118
0.9	0.9	0.5156	0.8710	0.2383	0.2600	–0.1627	–0.0002	0.1556	–0.0001	–0.0003
0.9	0.9	0.5156	0.8207	0.1818	0.4500	2.3590	–0.0002	0.2699	0.0011	–0.0005