Table 1. EPS performance and comparison with MICE on the simulation study (true β₂ = 0.5, 100 simulated data sets).

	Adjustment/imputation	Posterior mean for β₂	Bias	RMSE	CI95 width	CI95 coverage
Scenario 1: M are available in all areas
	Direct adj	0.50	0.00	0.02	0.054	0.90
	EPS adj	0.50	0.00	0.02	0.062	0.95
Naïve case: Ignoring M
	NA	0.78	0.28	0.28	0.049	0.00
Scenario 2: M are only available in some areas
Case 2.1: Analysis on i ∈ S	EPS adj	0.51	0.01	0.03	0.087	0.89
Case 2.2 : Analysis on	MICE	0.54	0.04	0.05	0.107	0.65
i ∈ S ∪ I	EPS imput	0.50	0.00	0.03	0.084	0.90
Scenario 3: M are NOT directly available, but m are available in some areas
extbfSample size n = 5
Case 3.1.1: Analysis on i ∈ S	EPS adj	0.59	0.08	0.09	0.094	0.16
Case 3.2.1: Analysis on	MICE	0.62	0.12	0.14	0.132	0.20
i ∈ S ∪ I	EPS imput	0.57	0.07	0.08	0.087	0.32
Sample size n = 10
Case 3.1.2: Analysis on i ∈ S	EPS adj	0.59	0.09	0.10	0.094	0.47
Case 3.2.2: Analysis on	MICE	0.62	0.12	0.14	0.129	0.51
i ∈ S ∪ I	EPS imput	0.57	0.07	0.08	0.086	0.60
Sample size n = 20
Case 3.1.3: Analysis on i ∈ S	EPS adj	0.59	0.09	0.10	0.094	0.52
Case 3.2.3: Analysis on	MICE	0.60	0.10	0.11	0.128	0.53
i ∈ S ∪ I	EPS imput	0.57	0.07	0.08	0.085	0.65
Sample size n = 100
Case 3.1.4: Analysis on i ∈ S	EPS adj	0.52	0.02	0.04	0.088	0.78
Case 3.2.4: Analysis on	MICE	0.56	0.06	0.06	0.119	0.56
i ∈ S ∪ I	EPS imput	0.51	0.01	0.03	0.083	0.85

Table 1. EPS performance and comparison with MICE on the simulation study (true β2 = 0.5, 100 simulated data sets).