Table 2.

Population size estimation results from five different simulation scenarios and four capture-recapture modeling frameworks.

Scenario 1: L₁=20%; L₂=25%; L₃=30%
Model	Bias	RMSE	# of times selected (%)	CI includes N (%)	Average CI width
1. Independence^†	0.3	52.8	332 (66.4)	475 (95)	205.7
2. L₁*L₂	−0.2	61.3	50 (10)	470 (94)	235.0
3. L₁*L₃	0.3	61.0	50 (10)	479 (95.8)	245.3
4. L₂*L₃	−3.9	66.6	43 (8.6)	476 (95.2)	266.7
5. L₁L₂, L₁L₃	−1.6	76.3	6 (1.2)	472 (94.4)	302.1
6. L₁L₂, L₂L₃	−9.3	88.9	7 (1.4)	473 (94.6)	347.3
7. L₁L₃, L₂L₃	−11.4	92.9	12 (2.4)	479 (95.8)	386.4
8. L₁L₃, L₂L₃, L₁*L₂	−30.2	195.0	0	469 (93.8)	752.8
9. DGA	0.7	54.9	NA	481 (96.2)	225.6
10. SparseMSE	0.5	53.0	NA	475 (95)	298.9
11. ${\hat{N}}_{0}$	−18.8	196.5	NA	489 (97.8)	1117
12. $\hat{N}$	−1.1	52.5	NA	473 (94.6)	211.9
13. ${\hat{N}}_{1}$	−0.1	69.6	NA	481 (96.2)	281.3
*Scenario 2: L₁=20%; L₂=25%; L₃=24% + (30%L₁)**
Model	Bias	RMSE	# of times selected (%)	CI includes N (%)	Average CI width
1. Independence	193.5	196.9	0	3 (0.6)	134.0
2. L₁*L₂	232.5	235.4	0	1 (0.2)	130.8
3. L₁*L₃^†	−0.6	66.2	371 (74.2)	479 (95.8)	271.7
4. L₂*L₃	275.3	277.3	0	0	124.8
5. L₁L₂, L₁L₃	−3.4	87.8	56 (11.2)	471 (94.2)	355.8
6. L₁L₂, L₂L₃	333.8	335.0	6 (1.2)	0	99.4
7. L₁L₃, L₂L₃	−17.7	120.6	67 (13.4)	482 (96.4)	517.7
8. L₁L₃, L₂L₃, L₁*L₂	−30.1	195.8	0	475 (95)	789.0
9. DGA	0.2	71.0	NA	487 (97.4)	332.1
10. SparseMSE	0.2	67.8	NA	166 (33)	206.5
11. ${\hat{N}}_{0}$	−22.7	154.1	NA	492 (98.4)	716.7
12. $\hat{N}$	191.5	194.9	NA	4 (0.8)	136.7
13. ${\hat{N}}_{1}$	136.8	146.1	NA	173 (34.6)	201.5
*Scenario 3: L₁=20%; L₂=25%; L₃=25% + (45%L₁) + (20%L₁L₂)**
Model	Bias	RMSE	# of times selected (%)	CI includes N (%)	Average CI width
1. Independence	249.3	251.6	0	0	98.0
2. L₁*L₂	297.4	299.1	0	0	89.9
3. L₁*L₃	−1.6	70.5	27 (5.4)	478 (95.6)	267.6
4. L₂*L₃	286.4	288.5	0	0	81.0
5. L₁L₂, L₁L₃	−6.5	101.0	8 (1.6)	476 (95.2)	347.2
6. L₁L₂, L₂L₃	356.6	357.8	0	0	58.6
7. L₁L₃, L₂L₃	−9.70e+9	1.39e+11	465 (93)	134 (26.8)	ND**
8. L₁L₃, L₂L₃, L₁*L₂	−6.32e+11	8.29e+12	0	120 (24)	ND**
9. DGA	225.1	316.4	NA	499 (99.8)	869.3
10. SparseMSE	−661.7	2,539.3	NA	473 (94.6)	254.7
11. ${\hat{N}}_{0}$	−1,054.2	1,290.1	NA	236 (47.2)	16,677.2
12. $\hat{N}$	219.4	222.4	NA	2 (0.4)	136.7
13. ${\hat{N}}_{1}$	32.1	68.2	NA	448 (89.6)	235.7
*Scenario 4: U=50%; L₁=40%U; L₂=50%U; L₃=30%*
Model	Bias	RMSE	# of times selected (%)	CI includes N (%)	Average CI width
1. Independence	183.8	187.6	0	7 (1.4)	137.5
2. L₁*L₂^†	−9.8	66.3	372 (74.4)	481 (96.2)	266.9
3. L₁*L₃	237.9	240.9	0	0	133.5
4. L₂*L₃	266.0	268.4	0	0	128.9
5. L₁L₂, L₁L₃	−16.8	95.0	61 (12.2)	479 (95.8)	379.3
6. L₁L₂, L₂L₃	−22.3	116.8	66 (13.2)	477 (95.4)	464.0
7. L₁L₃, L₂L₃	347.1	348.3	1 (0.2)	0	92.5
8. L₁L₃, L₂L₃, L₁*L₂	−42.5	185.7	0	481 (96.2)	771.7
9. DGA	−10.8	72.4	NA	488 (97.6)	320.2
10. SparseMSE	−9.8	66.3	NA	498 (99.6)	248.2
11. ${\hat{N}}_{0}$	−30.7	162.6	NA	490 (98)	479.9
12. $\hat{N}$	196.7	200.0	NA	5 (1)	135.3
13. ${\hat{N}}_{1}$	139.3	148.3	NA	168 (33.6)	204.5
*Scenario 5: U=50%; L₁=2% + (20%U); L₂=3% + (20%U); L₃=6%*
Model	Bias	RMSE	# of times selected (%)	CI includes N (%)	Average CI width
1. Independence	257.7	273.0	76 (15.2)	168 (33.6)	356.6
2. L₁*L₂^†	−67.0	287.2	253 (50.6)	ND**	ND**
3. L₁*L₃	300.9	314.5	42 (8.4)	ND**	ND**
4. L₂*L₃	306.9	320.6	41 (8.2)	ND**	ND**
5. L₁L₂, L₁L₃	−7.11e+9	1.59e+11	28 (5.6)	ND**	ND**
6. L₁L₂, L₂L₃	−3.53e+10	4.96e+11	27 (5.4)	ND**	ND**
7. L₁L₃, L₂L₃	365.9	377.1	33 (6.6)	ND**	ND**
8. L₁L₃, L₂L₃, L₁*L₂	−2.09e+11	2.69e+12		ND**	ND**
9. DGA	312.7	319.7	NA	0	338.7
10. SparseMSE	258.6	296.1	NA	144 (28.8)	359.1
11. ${\hat{N}}_{0}$	−454.6	3,373.5	NA	411 (82.2)	67,275.0
12. $\hat{N}$	206.8	229.4	NA	279 (55.8)	426.2
13. ${\hat{N}}_{1}$	175.4	217.4	NA	382 (76.4)	536.6

L₁= List 1; L₂= List 2; L₃= List 3; U= binary third variable; CI= confidence interval; DGA= Decomposable Graph Approach; ${\hat{N}}_{0}$ = Sample Coverage model assuming independence; $\hat{N}$ = Sample Coverage model allowing list dependence and sufficient sample coverage fraction (>=55%); ${\hat{N}}_{1}$ = Sample Coverage model estimating lower(upper) bound estimate due to insufficient sample coverage fraction (<55%); RMSE = Root Mean Squared Error

N=1,000 (the true population size)

^†

Indicates the correct log-linear model for each scenario

ND**=Not Defined. Upper limit of confidence interval not defined in simulations; therefore, average confidence interval width could not be calculated.

Bias = ( $E ⌈ {\hat{N}}_{i} ⌉ - N$ ); RMSE = $\sqrt{\frac{1}{m} \sum_{i = 1}^{m} {(N - {\hat{N}}_{i})}^{2}}$ ); where ${\hat{N}}_{i}$ is the estimated population size from simulation i, m is the number of simulations, and N is the true population size.