Multiple Systems Estimation for Sparse Capture Data: Inferential Challenges When There Are Nonoverlapping Lists

Figures & data

Table 1 Comparison of the performance of standard approaches using glm with the method set out in this article.

Dataset	Pair	Standard			Proposed
		Estimate	SE	p-value	Estimate	p-value
Netherlands	I:K	$-20.79$	5778	0.997	$-\infty$	$9.1\times {10}^{-4}$
	K:R	–19.96	2783	0.994	$-\infty$	$2.1\times {10}^{-5}$
UK	LA:GP	–19.08	5350	0.997	$-\infty$	0.13
	LA:NCA	$-19.19$	7968	0.998	$-\infty$	0.30

NOTE: The quantity estimated is the parameter α_ij for the nonoverlapping pair (i, j) under consideration. The model fitted includes all two-list parameters.

Table 2 An artificial dataset with three lists.

Cases observed only on one list		Cases observed on exactly 2 lists
List	Number	Lists	Number
A	40	A&B	6
B	30
C	20

NOTE: In this dataset, there are no cases with capture histories AC, BC, or ABC.

Table 3 Summary of linear programming output and test result for all possible choices of two-list effects to include in the model.

Two-list parameters included	Test result	$s_{\text{max}}$
None	No error	1.2
${\alpha }_{\text{AB}}$	Nonexistent MLE	0
${\alpha }_{\text{AC}}$	No error	3
${\alpha }_{\text{BC}}$	No error	3
${\alpha }_{\text{AB}},{\alpha }_{\text{AC}}$	Nonexistent MLE	0
${\alpha }_{\text{AB}},{\alpha }_{\text{BC}}$	Nonexistent MLE	0
${\alpha }_{\text{AC}},{\alpha }_{\text{BC}}$	No error	6
${\alpha }_{\text{AB}},{\alpha }_{\text{AC}},{\alpha }_{\text{BC}}$	Unidentifiable	6

NOTE: For the model containing all three two-list effects, there are finite values of the Poisson means ${\mu }_{\mathit{\omega }}$ that maximize the likelihood, so $s_{\text{max}}>0$, but these do not correspond to unique values of parameters in the model.

Table 4 Victims related to modern slavery and trafficking in New Orleans.

Cases observed		Cases observed on		Cases observed on
only on one list		exactly 2 lists		exactly 3 lists
List	Number	Lists	Number	Lists	Number
A	25	A&C	1	A&C&G	1
B	5	A&D	2	A&D&E	1
C	70	A&E	1
D	33	B&F	1
E	6	C&D	1
F	6	C&E	1
G	6	C&G	1
H	21	D&E	2
		E&H	1

NOTE: Numbers of cases on each possible combination of lists, leaving out combinations for which no cases were observed. For reasons of confidentiality the lists are labelled uninformatively.

Table 5 Victims related to human trafficking in the Western site of a research study in the USA.

Cases observed		Cases observed on		Cases observed on
only on one list		exactly 2 lists		exactly 3 lists
List	Number	Lists	Number	Lists	Number
A	52	A&C	4	A&C&E	1
B	90	A&D	2	B&C&D	1
C	114	A&E	5
D	45	B&C	6
E	21	B&D	1
		D&E	3

NOTE: Numbers of cases on each possible combination of lists, leaving out combinations for which no cases were observed. For reasons of confidentiality the lists are labeled uninformatively.