The never-listed cell of a three-list capture under no three-way interaction

With the saturated log-linear model the never-listed cell is free (Research.P1.three_list_saturated_fits); under the conventional restriction \(u_{123} = 0\) it is the closed form \(m_{000} = m_{100} m_{010} m_{001} m_{111} / (m_{110} m_{101} m_{011})\) (Research.P1.Interaction.missing_cell_identified), and with a three-way interaction \(t\) the same ratio equals \(m_{000} e^{t}\) (missing_cell_with_interaction), so the dark figure moves by exactly \(e^{-t}\) per unit of the interaction the analyst assumes away.

Usage

morie_dark_figure_no_interaction(cells, interaction = NULL)

Arguments

cells

Named counts n100, n010, n001, n110, n101, n011, n111 (list memberships as the three digits).

interaction

Optional vector of three-way interaction values \(t\) at which the implied never-listed cell is also reported.

Value

A list with missing (the never-listed cell under \(u_{123} = 0\)), observed, total, and, with interaction, a data frame sensitivity of interaction, missing and total.

References

Fienberg SE (1972). The multiple recapture census for closed populations and incomplete 2^k contingency tables. Biometrika 59(3).

Examples

morie_dark_figure_no_interaction(c(n100 = 120, n010 = 80, n001 = 60, n110 = 30, n101 = 24,
                                   n011 = 16, n111 = 8), interaction = c(-0.5, 0, 0.5))
#> $missing
#> [1] 400
#> 
#> $observed
#> [1] 338
#> 
#> $total
#> [1] 738
#> 
#> $theorem
#> [1] "Research.P1.Interaction.missing_cell_identified"
#> 
#> $sensitivity
#>   interaction  missing    total
#> 1        -0.5 659.4885 997.4885
#> 2         0.0 400.0000 738.0000
#> 3         0.5 242.6123 580.6123
#>