Table 2.

Simulation study for linear regression in Section 4.2 with n = 500 and, where applicable, the external study has sample size N = 300 and 2 replicates, while β₀ = 0, β₁ = 0.75. The target parameter, Θ = (θ₁, …, θ₅)^T, where θ_j is the parameter for the j^th category. Displayed are results for the estimation of θ₅ − θ₁. Ext-Int Data is the case that external data are used to estimate the measurement error variance. Int Data is the case that the internal data have 2 replicates, and the Ignore ME estimator ignores the measurement error and is based on the mean of these replicates. Coverage is the coverage rate of nominal 95% confidence intervals. RMSE is the square root of the mean squared error.

		Results Analysis (θ5 − θ1)
Data	Method	mean bias	Mean Estimated Std. Err.	Actual Standard Deviation	RMSE	Coverage
			Homoscedastic ϵ ~ N(0, 1)
X observed		0.004	0.145	0.150	0.150	95.1%
Ext-Int Data
	Our Method	0.013	0.249	0.233	0.233	95.8%
	Ignore ME	−0.814	0.139	0.142	0.826	0.1%
Int Data
	Our method	−0.007	0.176	0.170	0.170	95.3%
	Ignore ME	−0.536	0.142	0.145	0.555	3.7%
			Heteroscedastic ϵ ~ N(0,0.2 + 0.5x2)
X observed		0.004	0.123	0.169	0.169	85.3%
Ext-Int Data
	Our Method	0.011	0.261	0.245	0.245	95.9%
	Ignore ME	−0.814	0.122	0.135	0.825	0.1%
Int Data
	Our Method	−0.010	0.197	0.189	0.189	95.9%
	Ignore ME	−0.537	0.123	0.141	0.555	1.8%