Generating unbiased correlated random survival rates for stochastic population models

Abstract

A problem for population modellers in undertaking quantitative population viability analysis has been restricting survival rates to the unit interval and retaining the specified mean and standard deviation. In order to overcome this problem, we have developed a general method for generating correlated survival rates. The method consists of applying the transformation Φ x , the cumulative distribution function of the standard normal distribution, to suitably correlated normal variates X 1, X 2, …, X k . The means, standard deviations and correlation coefficients of X 1, X 2, …, X k are calculated so that the transformed results have the desired means, standard deviations and correlations for the survival rates. This method allows the full spectrum of correlations among different ages/stages to be considered, as well as the flexibility of a range of differently shaped distributions. The method proposed may generally apply to any function that satisfies the definition of a cumulative distribution function for which a derivative and inverse exist. A case study using the Φ x transform exemplifies these advantages, especially in comparison with the typical process of limiting random deviates to the unit interval currently in use. The case study emphasises the point that the limiting of random deviates may lead to an under-estimation of the specified variation and, as a result, simulated risk curves exhibiting a smaller range in possible outcomes compared with our method. This has particular importance when developing population viability analysis models to assess the risk of extinction or quasiextinction of a population.