4. Concluding remarks

Stephen Ash

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The paper provided the conditions for SDR to be equivalent to SD2 and showed how they are equivalent when the sample size is both smaller and larger than the chosen Hadamard matrix. When a smaller Hadamard matrix $H_A$ is used and replicates are only derived from $H_A,$ the paper showed how the reduced set of replicates provides a reasonable approximation of the SD2 estimator. The empirical examples indicated that using a reduced set of replicates is reasonable since decreasing the number of replicates does not increase the bias of the estimates. Additionally, we saw that using many connected loops reduces the impact of the squared difference between the first and last unit in the sample. Since SD1 usually has larger biases and RMSEs than SD2, SDR estimators that use more rather than fewer connected loops will have smaller biases and RMSEs than SDR estimators.

Acknowledgements

The author would like to thank David Hornick and Brian Dumbacher for their review of the early draft and the referees and the editor for their comments that helped refine and clarify the paper.

Appendix

Table A1
SDR simulation results
Table summary
This table displays the SDR simulation results. The information is grouped by population (appearing as row headers), $k_A$ ,RA, Expected Relative Bias by # Replicates , Relative Mean Squared Errors, Coverage Ratios(appearing as column headers).

Population	$k_A$	RA	Expected Relative Bias by # Replicates				Relative Mean Squared Errors				Coverage Ratios
			16	32	48	64	16	32	48	64	16	32	48	64
A1	4	1	0.010	0.009	0.009	0.009	0.176	0.091	0.066	0.054	93	94	94	94
		2	0.010	0.010	0.010	0.009	0.176	0.095	0.064	0.048	92	94	94	95
	16	1	0.009	0.008	0.010	0.009	0.141	0.080	0.059	0.048	93	94	94	95
		2	0.009	0.010	0.010	0.009	0.194	0.096	0.065	0.049	92	94	94	95
	64	1 or 2	0.009	0.009	0.010	0.009	0.194	0.096	0.064	0.049	92	94	94	94
A2	4	1	-0.696	-0.840	-0.888	-0.907	0.485	0.706	0.789	0.823	62	45	38	35
		2	-0.538	-0.768	-0.845	-0.883	0.290	0.590	0.714	0.780	77	54	45	39
	16	1	0.113	-0.270	-0.500	-0.615	0.013	0.073	0.250	0.378	100	97	80	100
		2	1.302	0.152	-0.231	-0.423	1.695	0.023	0.054	0.179	100	100	99	100
	64	1 or 2	1.302	1.379	1.404	1.417	1.695	1.901	1.972	2.008	100	100	100	100
A3	4	1	0.049	0.031	0.025	0.021	0.195	0.095	0.068	0.054	93	94	94	95
		2	0.070	0.040	0.030	0.025	0.222	0.103	0.067	0.050	93	94	94	95
	16	1	0.155	0.105	0.075	0.060	0.207	0.106	0.070	0.055	95	95	95	95
		2	0.314	0.163	0.112	0.086	0.374	0.144	0.085	0.061	96	95	95	95
	64	1 or 2	0.314	0.324	0.327	0.327	0.374	0.245	0.199	0.176	96	97	97	97
A4	4	1	0.040	0.023	0.017	0.014	0.192	0.104	0.077	0.063	93	94	94	94
		2	0.060	0.030	0.021	0.017	0.217	0.110	0.075	0.058	93	94	94	95
	16	1	0.144	0.095	0.066	0.052	0.208	0.109	0.077	0.063	95	95	95	95
		2	0.291	0.146	0.098	0.075	0.357	0.144	0.090	0.067	96	95	95	95
	64	1 or 2	0.291	0.299	0.303	0.305	0.357	0.232	0.191	0.170	96	97	97	97
A5	4	1	0.063	0.063	0.063	0.065	0.192	0.106	0.076	0.063	94	94	95	95
		2	0.068	0.066	0.066	0.065	0.217	0.111	0.075	0.057	93	94	95	95
	16	1	0.063	0.063	0.063	0.065	0.161	0.093	0.068	0.057	94	95	95	95
		2	0.065	0.067	0.066	0.066	0.214	0.111	0.075	0.056	93	94	95	95
	64	1 or 2	0.065	0.066	0.066	0.065	0.214	0.110	0.074	0.056	93	94	95	95
A6	4	1	0.093	0.092	0.093	0.094	0.211	0.117	0.088	0.072	94	95	95	95
		2	0.092	0.096	0.095	0.094	0.229	0.120	0.086	0.067	94	95	95	95
	16	1	0.099	0.095	0.094	0.094	0.185	0.107	0.080	0.067	94	95	95	95
		2	0.093	0.094	0.094	0.093	0.226	0.117	0.085	0.067	94	95	95	95
	64	1 or 2	0.093	0.096	0.095	0.095	0.226	0.118	0.084	0.066	94	95	95	95
A7	4	1	0.105	0.069	0.112	0.253	0.219	0.106	0.091	0.143	94	95	95	97
		2	0.004	0.004	0.073	0.310	0.187	0.098	0.079	0.175	92	94	95	97
	16	1	0.177	0.168	0.462	0.847	0.229	0.137	0.351	0.828	95	96	98	99
		2	0.002	0.003	0.027	1.248	0.187	0.097	0.065	1.689	92	94	95	100
	64	1 or 2	0.002	0.003	0.030	0.115	0.187	0.097	0.065	0.062	92	94	95	96

Table A2
Comparison methods simulation results
Table summary
This table displays the comparison methods simulation results. The information is grouped by population (appearing as row headers), Expected Relative Bias
by # Replicates, Relative Mean Squared
Errors, Coverage Ratios, (appearing as column headers).

Population	Expected Relative Bias by # Replicates			Relative Mean Squared Errors			Coverage Ratios
	SD1	SD2	SRSWOR	SD1	SD2	SRSWOR	SD1	SD2	SRSWOR
A1	0.009	0.009	-0.001	0.049	0.049	0.032	94	94	97
A2	-0.960	1.417	25.317	0.921	2.008	640.916	23	100	100
A3	0.015	0.327	3.462	0.049	0.176	12.203	94	97	100
A4	0.006	0.305	3.284	0.057	0.170	11.109	94	97	100
A5	0.064	0.065	0.055	0.056	0.056	0.039	95	95	97
A6	0.093	0.095	0.084	0.065	0.066	0.046	95	95	98
A7	0.112	0.115	20.641	0.063	0.062	427.141	96	96	100

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