Three-list capture-recapture: what is and is not identified

With three lists the observed table has seven cells; the count of units on no list is the eighth. The saturated log-linear model (all pairwise interactions and the three-way interaction) reproduces any positive eight-cell table exactly (Research.P1.three_list_saturated_fits), so the missing cell can take any positive value while the seven observed cells are matched (Research.P1.missing_cell_unconstrained). The population size is identified only once the analyst fixes the three-way interaction; the conventional choice sets it to zero, which gives the closed form \(m_{000} = m_{111} m_{100} m_{010} m_{001} / (m_{110} m_{101} m_{011})\). The function reports that estimate, the logical floor (the number of distinct units seen), the two-list Petersen and Chapman estimates for each pair of lists (each proved to respect the floor, Research.P1.petersen_ge_floor, chapman_ge_floor), and the three-way interaction that any candidate missing cell would imply.

Usage

morie_dark_figure_three_list(counts, candidate_missing = NULL)

Arguments

counts

Named vector or list of the seven observed cells with names "100", "010", "001", "110", "101", "011", "111" (digits: on list 1, list 2, list 3); all positive.

candidate_missing

Optional positive values of the missing cell for which the implied three-way interaction is reported.

Value

A list with observed (units seen, the floor), missing_no_three_way, N_no_three_way, pairwise (a data frame of Petersen and Chapman estimates per pair of lists with the floor for that pair), implied_three_way (for candidate_missing) and theorems.

Examples

morie_dark_figure_three_list(c("100" = 120, "010" = 90, "001" = 70,
                               "110" = 40, "101" = 30, "011" = 25, "111" = 15))
#> $observed
#> [1] 390
#> 
#> $missing_no_three_way
#> [1] 378
#> 
#> $N_no_three_way
#> [1] 768
#> 
#> $pairwise
#>   lists  n1  n2  m floor petersen  chapman
#> 1   1-2 205 170 55   320 633.6364 628.0357
#> 2   1-3 205 140 45   300 637.7778 630.4348
#> 3   2-3 170 140 40   270 595.0000 587.0732
#> 
#> $implied_three_way
#> NULL
#> 
#> $theorems
#> [1] "Research.P1.three_list_saturated_fits" 
#> [2] "Research.P1.missing_cell_unconstrained"
#> [3] "Research.P1.petersen_ge_floor"         
#> [4] "Research.P1.chapman_ge_floor"          
#>