## Introduction

Age-specific mortality schedules lie at the heart of models that explore an enormous range of biological phenomena, from basic calculations of Darwinian fitness (Fisher, 1930) to the evolution of ageing (Hamilton, 1966; Charlesworth, 1994), from explorations of population dynamics (Caswell, 1989) to predictive analyses for conservation strategies (Lande, 1988a,b). But the estimation of age-specific mortality rates brings with it a variety of statistical challenges, and unless we address these challenges in our experiments, we are likely to draw biased conclusions.

In the following work, we describe how studies of mortality rates that use insufficient sample sizes or inappropriate statistical models may bias conclusions about many contemporary issues in evolutionary research. First, studies of ageing in laboratory animals have claimed that genetic factors extend life span by slowing the rate of senescence. We will argue here that when one considers the effect of sample size, evidence for changes in baseline mortality is strong, but evidence for changes in rates of ageing may turn out to be weaker than previously thought. Second, evolutionary theory predicts that the onset of senescence should coincide with the onset of reproduction. Here we discuss how sample size affects our ability to determine when senescence begins. Third, both field and laboratory studies provide evidence for a trade-off between reproduction and survival. We would like to know whether increases in reproduction actually affect the rate of ageing. Again, insufficient sample sizes can bias our estimates of cost of reproduction. Fourth, Medawar's widely accepted ‘mutation accumulation’ model for the evolution of senescence (Medawar, 1952) makes certain predictions about variation in age-specific mortality rates among genotypes. Only recently have biologists begun to test this model by asking how genetic variance for mortality rates changes with age. Results from these studies are critical to our understanding of evolutionary models of ageing. But sampling error may bias our results in favour of the model we set out to test.

In each case, we focus in particular on the influence of sample size on mortality estimates at the youngest adult ages. At early ages, mortality rates are low and so are relatively prone to sampling error. However, models by Hamilton and others (Lewontin, 1965; Hamilton, 1966; Abrams, 1991; Charlesworth, 1994) show that it is at these early ages that fitness is most influenced by variation in life history characters. From an evolutionary perspective, early age mortality rates are extremely important, but are also those most likely to be estimated incorrectly, leading to biased estimates of mortality patterns over the life-span of a cohort. Early age fecundity is also a critical component of lifetime fitness. However, the statistical nature of age-specific mortality and fecundity are qualitatively different. We confine our discussion here to estimates of mortality rates, whose estimates lead to particular statistical challenges.

Before examining the specific problems outlined above, we will provide some conceptual background, including an explanation of mortality rates and how they are estimated, and we introduce the statistical challenges of below-threshold mortality. This brief statistical discussion will then motivate our subsequent discussion of the effect of below-threshold mortality on specific biological problems.