Table 4.

Validity of WQS regression and quantile g-computation under nonnull estimates when directional homogeneity holds, individual exposure effects are nonadditive, and the overall exposure effect includes terms for linear (${\mathrm{\psi }}_1$) and squared (${\mathrm{\psi }}_2$) exposure (e.g., quadratic polynomial) for 1,000 simulated samples of $n=500$. Corresponding estimates for $n=100$ are provided in Table S2.

Scenario	Method	da	Biasb		MCSEc		RMVARd
			${\mathrm{\psi }}_1$	${\mathrm{\psi }}_2$	${\mathrm{\psi }}_1$	${\mathrm{\psi }}_2$	${\mathrm{\psi }}_1$	${\mathrm{\psi }}_2$
7. Validity when the true exposure effect is nonadditive/nonlinear	WQSe	4	0.21	$–0.07$	0.34	0.11	0.31	0.10
		9	0.21	$–0.07$	0.73	0.24	0.64	0.21
		14	0.13	$–0.04$	1.12	0.37	1.02	0.34
	Q-gcompf	4	$-0.01$	0.00	0.13	0.04	0.13	0.04
		9	0.00	0.00	0.16	0.03	0.16	0.04
		14	0.00	0.00	0.19	0.04	0.18	0.04
8. Validity when the overall exposure effect is nonlinear due to underlying nonlinear effects	WQSe	4	$-0.20$	0.07	0.31	0.12	0.31	0.10
		9	$-0.15$	0.07	0.61	0.22	0.61	0.20
		14	$-0.11$	0.05	0.97	0.34	0.98	0.32
	Q-gcompf	4	$-0.01$	0.00	0.15	0.04	0.15	0.04
		9	0.00	0.00	0.18	0.05	0.18	0.05
		14	0.00	0.00	0.20	0.05	0.20	0.05

Note: MCSE, Monte Carlo standard error; RMVAR, root mean variance: .

^aTotal number of exposures in the model.

^bEstimate of ${\mathrm{\psi }}_1$ or ${\mathrm{\psi }}_2$ minus the true value.

^cStandard deviation of the bias across 1,000 iterations.

^dSquare root of the mean of the variance estimates from the 1,000 simulations, which should equal MCSE if the variance estimator is unbiased.

^eWeighted quantile sum regression (R package gWQS defaults, allowing for quadratic term for total exposure effect).

^fQuantile g-computation (R package qgcomp defaults, including an interaction term between $X_1$ and $X_2$ (scenario 7) or a term for $X_1{X}_1$ (scenario 8) as well as quadratic term for total exposure effect).