TABLE 2.

Simulation results based on 1000 simulations with n₀ = 300, n₁ = n₃ = 50, the validation sample size is n_V = 400, the full cohort size is N=3000

α	β1	γ1	Methods	${\widehat{\beta }}_1$			${\widehat{\gamma }}_1$
				Mean	VAR	$\widehat{\text{VAR}}$	Mean	VAR	$\widehat{\text{VAR}}$	SRE
1.0	0	0	ξSRS	0.000	0.0034	0.0034	−0.001	0.0036	0.0034	0.69
			ξR	0.000	0.0024	0.0025	0.000	0.0025	0.0025	1.00
			ξIPW	0.001	0.0021	0.0020	0.000	0.0025	0.0023	1.00
			ξAIPW	0.001	0.0021	0.0020	0.000	0.0025	0.0023	1.00
			ξSPML	0.001	0.0018	0.0018	0.000	0.0021	0.0021	1.19
		0.5	ξSRS	−0.004	0.0034	0.0034	0.496	0.0034	0.0034	0.65
			ξR	−0.001	0.0023	0.0025	0.500	0.0024	0.0025	0.92
			ξIPW	−0.001	0.0020	0.0020	0.498	0.0022	0.0023	1.00
			ξAIPW	−0.001	0.0013	0.0014	0.500	0.0014	0.0014	1.57
			ξSPML	0.001	0.0012	0.0012	0.498	0.0013	0.0014	1.69
	0.5	0	ξSRS	0.501	0.0032	0.0034	0.001	0.0034	0.0034	0.62
			ξR	0.499	0.0024	0.0025	0.000	0.0025	0.0025	0.84
			ξIPW	0.502	0.0018	0.0022	0.001	0.0021	0.0024	1.00
			ξAIPW	0.502	0.0014	0.0015	0.000	0.0016	0.0015	1.31
			ξSPML	0.500	0.0014	0.0015	0.001	0.0014	0.0014	1.50
		0.5	ξSRS	0.500	0.0032	0.0034	0.498	0.0032	0.0033	0.66
			ξR	0.499	0.0025	0.0025	0.499	0.0025	0.0025	0.84
			ξIPW	0.502	0.0018	0.0022	0.501	0.0021	0.0024	1.00
			ξAIPW	0.503	0.0017	0.0017	0.502	0.0019	0.0018	1.11
			ξSPML	0.500	0.0016	0.0016	0.499	0.0016	0.0018	1.31
1.5	0.5	0	ξSRS	0.500	0.0036	0.0034	0.001	0.0035	0.0034	0.74
			ξR	0.498	0.0025	0.0025	−0.001	0.0026	0.0025	1.00
			ξIPW	0.501	0.0023	0.0022	0.000	0.0026	0.0024	1.00
			ξAIPW	0.502	0.0018	0.0016	0.000	0.0018	0.0017	1.44
			ξSPML	0.500	0.0014	0.0014	0.002	0.0014	0.0014	1.86
		0.5	ξSRS	0.499	0.0035	0.0034	0.499	0.0034	0.0034	0.68
			ξR	0.502	0.0026	0.0025	0.501	0.0027	0.0025	0.85
			ξIPW	0.500	0.0021	0.0022	0.500	0.0023	0.0024	1.00
			ξAIPW	0.501	0.0018	0.0018	0.501	0.0019	0.0019	1.21
			ξSPML	0.499	0.0014	0.0015	0.499	0.0017	0.0017	1.35

Abbreviation: SRE, sample relative efficiency. Results are based on the model Y₁=β₀+β₁X+β₂Z+e, Y₂=γ₀+γ₁X+γ₂Z+ε, where X~N(0, 1), Z~Bernoulli(0.45) and (e, ε) follow a bivariate normal distribution with $\mathit{var}\left(e\right)={\sigma }_1^2$, $\mathit{var}\left(\epsilon \right)={\sigma }_2^2$, cov(e,ε)=ρσ₁σ₂; the true parameter values are β₀=1, β₂=−0.5, γ₀=1, γ₂=−0.5, σ₁=σ₂=1, ρ=0.8. 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_SRS denotes the regression estimator based on simple random sample (SRS) portion of the validation sample. ξ_R denotes the regression estimator from an SRS of the same size as the validation sample. ξ_IPW denotes the estimate from our inverse probability weighted (IPW) estimating equation. ξ_AIPW denotes the estimate from augmented IPW (AIPW) estimating equation. ξ_SPMI is a semiparametric maximum likelihood (SPML) estimator similar to Jiang et al,²⁰ which models (Y₁, Y₂) parametrically using a bivariate normal distribution.