Two-source (capture-recapture) count with a dependence box

Two lists of the same events (police records and a survey, hospital or NGO register) of sizes n1 and n2 share m events. Under independence the true count is the Lincoln-Petersen ratio \(n_1 n_2 / m\). Independence is rarely credible: events recorded by the police are often more (or less) likely to reach a second list. With the dependence factor \(\theta\) only known to lie in \([1/\kappa, \kappa]\), the true count lies in \([n_1 n_2/(\kappa m),\ \kappa n_1 n_2/m]\) and both ends are attainable (Research.P1.TwoSource.petersen_bounds).

Usage

morie_dark_figure_two_source(n1, n2, m, kappa = 1, chapman = FALSE)

Arguments

n1, n2

Sizes of the two lists; positive.

m

Number of events on both lists; positive and at most min(n1, n2).

kappa

Largest credible dependence factor, at least 1; kappa = 1 asserts independence.

chapman

If TRUE, also return Chapman's small-sample version \((n_1+1)(n_2+1)/(m+1) - 1\) of the point estimate.

Value

A list with point (Lincoln-Petersen), lower, upper, kappa, theorem, and chapman when requested.

References

Petersen CGJ (1896); Lincoln FC (1930); Chapman DG (1951). Bird SM, King R (2018). Multiple systems estimation (or capture-recapture estimation) to inform public policy. Annual Review of Statistics and Its Application 5, 95-118.

Examples

# police records 400, survey 250, 80 in both
morie_dark_figure_two_source(400, 250, 80)              # independence: 1250
#> $point
#> [1] 1250
#> 
#> $lower
#> [1] 1250
#> 
#> $upper
#> [1] 1250
#> 
#> $kappa
#> [1] 1
#> 
#> $theorem
#> [1] "Research.P1.TwoSource.lincoln_petersen"
#> 
morie_dark_figure_two_source(400, 250, 80, kappa = 2)   # [625, 2500]
#> $point
#> [1] 1250
#> 
#> $lower
#> [1] 625
#> 
#> $upper
#> [1] 2500
#> 
#> $kappa
#> [1] 2
#> 
#> $theorem
#> [1] "Research.P1.TwoSource.petersen_bounds"
#>