. 2020 Apr 7;128(4):047004. doi: 10.1289/EHP5838

Table 3.

Validity of WQS regression and quantile g-computation under the null (no exposures affect the outcome or exposures counteract) and nonnull estimates when directional homogeneity holds; 1,000 simulated samples of $n = 500$ . Corresponding estimates for $n = 100$ are provided in Table S1.

Scenario	Method	d^a	Truth^b	Bias^c	MCSE^d	RMVAR^e	Coverage^f	Power/type 1 error^g
1. Validity under the null, no exposures are causal	WQS^h	4	0	0.00	0.09	0.09	0.95	0.05
		9	0	$- 0.01$	0.13	0.12	0.94	0.06
		14	0	$- 0.01$	0.15	0.15	0.95	0.05
	Q-gcompⁱ	4	0	0.00	0.08	0.08	0.94	0.06
		9	0	0.00	0.12	0.12	0.95	0.05
		14	0	$- 0.01$	0.16	0.15	0.95	0.05
2. Validity under the null, causal exposures counteract	WQS^h	4	0	0.32	0.08	0.08	0.02	0.98
		9	0	0.41	0.11	0.11	0.04	0.96
		14	0	0.46	0.14	0.14	0.09	0.91
	Q-gcompⁱ	4	0	0.00	0.09	0.09	0.95	0.05
		9	0	0.00	0.13	0.13	0.96	0.04
		14	0	$- 0.01$	0.16	0.16	0.96	0.04
3. Validity under single nonnull effect	WQS^h	4	0.25	0.07	0.07	0.07	0.83	1.00
		9	0.25	0.15	0.10	0.10	0.67	0.98
		14	0.25	0.21	0.14	0.13	0.57	0.94
	Q-gcompⁱ	4	0.25	0.00	0.08	0.08	0.94	0.88
		9	0.25	0.00	0.12	0.12	0.95	0.52
		14	0.25	$- 0.01$	0.16	0.15	0.95	0.36
4. Validity under all nonnull effects with directional homogeneity	WQS^h	4	0.25	$- 0.06$	0.10	0.09	0.87	0.58
		9	0.25	$- 0.09$	0.13	0.13	0.87	0.26
		14	0.25	$- 0.10$	0.17	0.15	0.88	0.19
	Q-gcompⁱ	4	0.25	0.00	0.08	0.08	0.95	0.86
		9	0.25	0.00	0.12	0.12	0.95	0.55
		14	0.25	$- 0.01$	0.15	0.15	0.95	0.37

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

^{^a}

Total number of exposures in the model.

^{^b}

True value of $ψ$ , the net effect of the exposure mixture.

^{^c}

Estimate of $ψ$ minus the true value.

^{^d}

Standard deviation of the bias across 1,000 iterations.

^{^e}

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

^{^f}

Proportion of simulations in which the estimated 95% confidence interval contained the truth.

^{^g}

Power when the effect is nonnull (scenarios 3 and 4); otherwise (in scenarios 1 and 2), it is the type 1 error rate (false rejection of null), which should equal alpha (0.05 here) under a valid test.

^{^h}

Weighted quantile sum regression (R package gWQS defaults).

^ⁱ

Quantile g-computation (R package qgcomp defaults).