How many offenders, victims or incidents no list records, from the
overlap of two or three lists (police records, hospital records, a
survey). With two lists the Lincoln-Petersen estimate n1 n2 / m
assumes the lists are independent; when being on one list changes the
odds of being on the other by at most a factor kappa, the truth lies
within [n1 n2 / (kappa m), kappa n1 n2 / m] (lower, upper), and no
data can narrow that box. Chapman's estimate `(n1 + 1)(n2 + 1) / (m + 1)
1` is nearly unbiased, with Seber's standard error.
Arguments
- x
c(n1, n2, m), the sizes of two lists and their overlap (named so or in that order), or the seven cells of three lists, named by membership:"100","010","001","110","101","011","111". Counts from 0.- kappa
The largest factor by which being on one list changes the odds of being on the other: one number from 1 (two lists).
- int
NULL, or three-way log interactions at which to give the missing cell (three lists).
Value
An object of class crcap, a list: type ("two" or
"three"); est, a named vector (two lists: petersen, lower,
upper, chapman, se, floor; three: observed, missing,
total); pairs (three lists: lists, n1, n2, m, floor,
petersen, chapman) and sens (with int: int, missing,
total), otherwise NULL.
Details
With three lists the seven observed cells identify the never-listed cell
only under an assumption on the three-way interaction of a log-linear
model: with none, it is m100 m010 m001 m111 / (m110 m101 m011), and a
three-way log interaction t multiplies it by exp(-t); int gives
that sensitivity. The pairwise estimates are given beside it, each at
least the number seen on either of its two lists (floor). See
proof("crcap").
References
Chapman, D. G. (1951). Some properties of the hypergeometric distribution with applications to zoological sample censuses. University of California Publications in Statistics, 1, 131-160.
Seber, G. A. F. (1970). The effects of trap response on tag recapture estimates. Biometrics, 26(1), 13-22. doi:10.2307/2529040
Fienberg, S. E. (1972). The multiple recapture census for closed populations and incomplete 2^k contingency tables. Biometrika, 59(3), 591-603. doi:10.1093/biomet/59.3.591
Examples
crcap(c(n1 = 500, n2 = 400, m = 80), kappa = 1.5)
#> crcap: two lists
#> petersen lower upper chapman se floor
#> 2500.0000 1666.6667 3750.0000 2479.2593 224.0262 820.0000
crcap(c("100" = 120, "010" = 90, "001" = 60, "110" = 30, "101" = 20,
"011" = 15, "111" = 8), int = c(-0.5, 0, 0.5))
#> crcap: three lists
#> observed missing total
#> 343 576 919
#> lists n1 n2 m floor petersen chapman
#> 1-2 178 143 38 283 669.8421 659.9231
#> 1-3 178 103 28 253 654.7857 640.9310
#> 2-3 143 103 23 223 640.3913 623.0000
#> int missing total
#> -0.5 949.6635 1292.6635
#> 0.0 576.0000 919.0000
#> 0.5 349.3617 692.3617