The estimation of density dependence using census data from several sites

The discussion of density dependence in the ecological literature has tended to concentrate on significance testing, whereas estimation is often of more importance. In this paper we use simulation to investigate the bias and precision of a number of methods for estimating density dependence using census data from several sites. The estimation methods examined were a modification of an existing maximum likelihood approach, a regression method, and a new approach using restricted (or residual) maximum likelihood (REML). Simulations indicated that the REML method produced the most accurate estimates, with negligible bias for most parameter combinations. A further advantage of the REML method is that it can be easily implemented using standard statistical software. Using a second series of simulations we investigated the relationship between accuracy of estimation and sample size for the REML method. The results indicated that using more than one site gave substantial improvement in accuracy, but that using more than five sites gave little further improvement unless in excess of 10 years of data were available for each site. Where a standard error is required for the density dependence estimate we suggest using bootstrapping at the site level. Where this is not possible, a parametric bootstrap or a randomisation test may be used instead. The REML method is demonstrated using bag totals of red-legged partridges Alectoris rufa and grey partridges Perdix perdix for shoots on estates in East Anglia in the United Kingdom. This paper shows that combining information from several sites can give improved estimation of density dependence, particularly if REML estimation is adopted.