Statistical Methods for Estimating Ratios and Products in Ecological Studies

Abstract

There are many examples in the ecological literature in which population parameters are estimated using the ratio or product of two random variables. The statistical aspects of such problems have not been adequately addressed in the applied literature. This paper deals with point estimation, confidence intervals, tests of hypotheses, and determination of sample sizes for the product θ = μ x · μ y and the ratio ρ = μ x /μ y of two means with independent samples. More specifically, if $\bar X$ and $\bar Y$ denote the two sample means, the authors discuss the bias and standard errors associated with $\bar X$$\bar Y$ and $\bar X$/$\bar Y$ and use the large sample behavior of P and R to obtain confidence intervals for θ and ρ respectively. Standard errors and confidence intervals are also found using the jackknife and the bootstrap. In addition, these techniques produce alternative estimators for the ratio. Finally, optimal sample sizes are derived for achieving desired levels of precision for either P or R . The authors provide a cost for each type of observation and determine the total sample size and the allocation of the total sample size to either minimize variance for a fixed cost or to minimize cost for a fixed variance. Two examples arising from the estimation of larval densities and hatching densities of gypsy moths are considered in detail.