TABLE 3.

Multiple imputation in two-stage analysis with continuous surrogates when Z = ηX for independent η ∼ Γ(4, 4)

		Estimation performance
					MI		MIR			Empirical powera
$\left({\beta }_0,{\delta }_0\right)$	Criterion	MLE	Raking	RC	Boot	Bayes	Boot	Bayes	Abs corra	MP test	Lin. test
(1, 0)	$\sqrt{\text{MSE}}$	0.018	0.030	0.216	0.099	0.094	0.029	0.029	-	0.048	0.056
	Bias	0.006	0.001	0.215	0.097	0.092	0.002	0.002
	$\sqrt{\text{Var}}$	0.017	0.030	0.013	0.018	0.018	0.029	0.029
(1.045,−0.068)	$\sqrt{\text{MSE}}$	0.040	0.030	0.227	0.111	0.106	0.029	0.029	0.585	0.149	0.062
	Bias	0.036	0.001	0.227	0.109	0.104	0.002	0.002
	$\sqrt{\text{Var}}$	0.018	0.030	0.013	0.018	0.018	0.029	0.029
(1.087, −0.131)	$\sqrt{\text{MSE}}$	0.068	0.031	0.239	0.123	0.117	0.030	0.030	0.584	0.427	0.075
	Bias	0.065	0.001	0.238	0.121	0.116	0.002	0.002
	$\sqrt{\text{Var}}$	0.018	0.031	0.013	0.018	0.018	0.030	0.030
(1.127, −0.191)	$\sqrt{\text{MSE}}$	0.095	0.032	0.249	0.134	0.128	0.031	0.031	0.585	0.697	0.099
	Bias	0.093	0.001	0.249	0.133	0.127	0.002	0.002
	$\sqrt{\text{Var}}$	0.018	0.032	0.014	0.018	0.018	0.030	0.031
(1.165, −0.247)	$\sqrt{\text{MSE}}$	0.121	0.032	0.259	0.144	0.139	0.031	0.031	0.583	0.890	0.136
	Bias	0.119	0.001	0.259	0.143	0.138	0.002	0.002
	$\sqrt{\text{Var}}$	0.019	0.032	0.014	0.019	0.019	0.031	0.031
(1.200, −0.3)	$\sqrt{\text{MSE}}$	0.146	0.033	0.269	0.155	0.149	0.032	0.032	0.580	0.967	0.179
	Bias	0.145	0.001	0.268	0.154	0.148	0.003	0.002
	$\sqrt{\text{Var}}$	0.019	0.033	0.014	0.019	0.019	0.032	0.032

Note: We compare relative performance of the semiparametric efficient maximum likelihood (MLE), standard raking, regression calibration (RC), multiple imputations using (MI) either the wild bootstrap or Bayesian approach, and the proposed multiple imputation with raking (MIR) estimators for a two-phase design with cohort size N = 5000, phase 2 subset $\left|S_2\right|=750$ in average, M = 100 imputations, and 1000 Monte Carlo runs. We report the root-mean squared error ($\sqrt{\text{MSE}}$) for β= 1, its bias and variance decomposition (10), and the empirical power to reject the nearly true model (12) through the most powerful (MP) test and the goodness-of-fit test of linear fits.^42,43

^aThe absolute value of the correlation between ${\widehat{\beta }}_{\text{MLE}}-{\widehat{\beta }}_{\text{Raking}}$ and $\log Q_N-\log P_N$, where P_N and Q_N are likelihood functions at ${\theta }_0=\left({\alpha }_0,{\beta }_0,{\delta }_0\right)$ and ${\theta }^{\ast }=(\alpha ,\beta )$, respectively.