An Improper Method of Randomization in Experimental Design

The mark-recapture method was devised by Petersen in 1896 Petersen, C. G. J. 1896. “The Yearly Immigration of Young Plaice Into the Limfjord From the German Sea,”. Report of Danish Biological Station, 6: 1–48.  [Google Scholar] to estimate the number of fish migrating into the Limfjord, and independently by Lincoln in 1930 Lincoln, F. C. 1930. “Calculating Waterfowl Abundance on the Basis of Banding Returns,”. United States Department of Agriculture Circular, 118: 1–4.  [Google Scholar] to estimate waterfowl abundance. The technique can be applied to any search for a finite number of items by two or more people or agents, allowing the number of searched-for items to be estimated. This ubiquitous problem appears in fields from ecology and epidemiology, through to mathematics, social sciences, and computing. Here, we exactly calculate the moments of the hypergeometric distribution associated with this longstanding problem, confirming that widely used estimates conjectured in 1951 are often too small. Our Bayesian approach highlights how different search strategies will modify the estimates. The estimates are applied to several examples. For some published applications, substantial errors are found to result from using the Chapman or Lincoln–Petersen estimates. Supplementary materials for this article are available online.