Incorporating variance uncertainty into a power analysis of monitoring designs

Power calculations usually assume that the components of the population variance are known, but it is frequently the case that they are estimated using data from a pilot study. Imprecision in the estimates is then ignored and a single value for power is generated. We present a method that incorporates the error in the estimates of any number of variance components into the power calculations. We show that, by sampling values for the variance components from the residual likelihood function of the pilot data, our method can approximate the distribution of powers expected given the uncertainty in the variance components. Alternative summary measures of power can then be derived: we strongly recommend treating a minimum acceptable power as a quality standard and summarizing power in terms of the probability that this quality standard is attained. The method is illustrated by application to counts of common guillemots (Uria aalge ) on the Isle of May in Scotland to assess the power of detecting long-term trends in abundance using a model for random variation with seven parameters.