. Author manuscript; available in PMC: 2014 Feb 26.

Published in final edited form as: Ann Appl Stat. 2013 Mar;7(1):543–569. doi: 10.1214/12-AOAS599

Table 4.

Simulation results under exposure enriched sampling with all E = 1 in phase I data are selected in phase II for both cases and controls. We consider two association scenarios 1) G₁⊥E, G₁⊥G₂, and G₂⊥E association, 2) G₁⊥E, G₁ ~ G₂, and G₂ ~ E. The results are based on 1,000 replicated datasets, each with 1,000 cases and 1,000 controls in phase I and 600 cases and 600 controls in phase II. The approaches listed, TPFB, TPFB_emp, WL, PL, UML, CML, and EB where each represents Two-phase full Bayes (with empirically obtained prior variance), Weighted likelihood, Pseudolikeliohod, Unconstrained Maximum likelihood, Constrained Maximum Likelihood, and Empirical Bayes respectively. The CML imposes G₁-E and G₁–G₂ independence, however, no constraints on G₂-E association. We set (β_E, β_G₁G₂, β_G₁E, β_G₂E) = (−1.5, 0, log(2), log(2)) for all scenarios. The rows with smallest two sum(MSE) are in bold.

		G₁⊥E, G₁⊥G₂, G₂⊥E					G₁⊥E, G₁~G₂, G₂~E

Stratified sampling (a)^†		E	G₁ × G₂	G₁ × E	G₂ × E	Sum (MSE)^¶	E	G₁ × G₂	G₁ × E	G₂ × E	Sum (MSE)^¶
			(λ_G₁G₂, λ_G₁E, λ_G₂E) = (0, 0, 0)				(λ_G₁G₂, λ_G₁E, λ_G₂E) = (log(2), 0, log(1.5))
TPFB	Bias	−0.024	−0.017	0.020	−0.017		−0.056	0.166	−0.022	0.119
	(MSE)	(0.093)	(0.044)	(0.120)	(0.135)	(0.392)	(0.117)	(0.081)	(0.091)	(0.122)	(0.411)
TPFB_emp	Bias	0.007	−0.019	−0.021	−0.062		−0.033	0.043	−0.029	0.026
	(MSE)	(0.089)	(0.025)	(0.111)	(0.126)	(0.351)	(0.113)	(0.064)	(0.091)	(0.120)	(0.388)
WL	Bias	−0.038	−0.025	0.043	0.009		−0.038	0.011	0.011	0.006
	(MSE)	(0.099)	(0.058)	(0.144)	(0.157)	(0.458)	(0.105)	(0.057)	(0.101)	(0.121)	(0.384)
PL	Bias	−0.038	−0.026	0.043	0.009		−0.038	0.011	0.011	0.006
	(MSE)	(0.098)	(0.056)	(0.144)	(0.157)	(0.455)	(0.105)	(0.056)	(0.101)	(0.121)	(0.383)
UML	Bias	−0.093	−0.026	0.043	0.009		−0.096	0.011	0.011	0.006
	(MSE)	(0.110)	(0.056)	(0.144)	(0.157)	(0.467)	(0.116)	(0.056)	(0.101)	(0.121)	(0.394)
CML	Bias	−0.085	−0.020	0.026	0.003		−0.100	0.700	0.011	0.009
	(MSE)	(0.099)	(0.025)	(0.083)	(0.155)	(0.362)	(0.112)	(0.520)	(0.070)	(0.116)	(0.818)
EB	Bias	−0.087	−0.025	0.036	0.004		−0.099	0.089	0.010	0.008
	(MSE)	(0.099)	(0.036)	(0.099)	(0.155)	(0.389)	(0.112)	(0.069)	(0.075)	(0.116)	(0.392)

^†

All subjects with E = 1 in case and control are sub-sampled for phase II.

^¶

The combined MSEs as summed over all four parameters.

TPFB uses the informative prior N (0, 10⁻²) on G-G and G-E associations in the model (2.2)

TPFB_emp uses the prior N (0, θ̂²) on G-G and G-E associations in the model (2.2) where θ̂² is empirically estimated G-G or G-E association parameter under controls.