TABLE 3.

Simulation results when ε is not normal. The full cohort size is N=3000, (n₀, n₁, n₃)=(200, 100, 100)

α	β1	γ1	Methods	${\widehat{\gamma }}_1$
				Mean	VAR	$\widehat{\text{VAR}}$	CI
1.0	0	0	ξIPW	0.003	0.0038	0.0036	0.941
			ξAIPW	0.003	0.0039	0.0036	0.926
			ξSPML	0.001	0.0027	0.0026	0.960
		0.5	ξIPW	0.499	0.0038	0.0037	0.946
			ξAIPW	0.502	0.0037	0.0035	0.932
			ξSPML	0.493	0.0055	0.0041	0.871
	0.5	0	ξIPW	0.002	0.0036	0.0034	0.933
			ξAIPW	0.001	0.0032	0.0030	0.930
			ξSPML	−0.001	0.0021	0.0021	0.953
		0.5	ξIPW	0.501	0.0031	0.0032	0.954
			ξAIPW	0.505	0.0026	0.0027	0.940
			ξSPML	0.494	0.0038	0.0029	0.899

Results are based on the model Y₁=β₀+β₁X+β₂Z+e, Y₂=γ₀+γ₁X+γ₁Z+ε, where X~N(0, 1), Z~Bernoulli(0.45), and e~N(0, 1), ε is a gamma distribution with shape parameter 2, rate parameter 1, normalized to have mean 0 and variance 1. The true parameter values are β₀=1, β₂=−0.5, γ₀=1, γ₂=−0.5. The cutoff points for the outcome-dependent sampling design are ${\mu }_{Y_1}-a{\sigma }_{Y_1}$ and ${\mu }_{Y_1}+a{\sigma }_{Y_1}$. ξ_IPW denotes the estimate from our inverse probability weighted (IPW) estimating equation; ξaipw denotes the estimate from augmented IPW (AIPW) estimating equation; ξ_SPML is a semiparametric maximum likelihood (SPML) estimator similar to Jiang et al,²⁰ which models (Y₁, Y₂) parametrically using a bivariate normal distribution.