## Introduction

It is clear that the fields of experimental and evolutionary demography are growing (Curtsinger *et al*., 1995; Vaupel *et al*., 1998), and well-defined techniques for identifying differences in phenotypic and genetic patterns of age-specific mortality are needed. Frequently, a mathematical model is fit to observed mortality rates, and hypotheses concerning parameter values among treatment populations are investigated (Fukui *et al*., 1993; Nusbaum *et al*., 1996). There are several considerations to keep in mind when using this approach. First, we require an objective method for choosing an adequate model for the data. Historically, the Gompertz model, which predicts an exponential (log-linear) increase in mortality rates with age, was used almost exclusively for mortality analysis. Often this model was fit to data that were clearly not linear on the log scale (for a discussion see Promislow *et al*., 1997), and recent observations of more complex mortality patterns suggest non-Gompertzian dynamics are the norm rather than the exception (Curtsinger *et al*., 1995; Pletcher & Curtsinger, 1998).

After choosing the appropriate model, we need an efficient method for estimating the parameters of that model from the data. Traditionally, Gompertz parameters were estimated by linear regression of log-mortality rates on age (Hughes & Charlesworth, 1994; Orr & Sohal, 1994; Stearns & Kaiser, 1996). Parameters estimated in this way can be highly biased – small samples result in large over-estimates of the initial mortality parameter and under-estimates of the rate parameter (Mueller *et al*., 1995; Promislow *et al*., 1997; Pletcher, unpublished results). Moreover, variation in sample sizes among populations can generate apparent differences in parameters values (Promislow *et al*., 1997). Although various forms of nonlinear regression have been suggested for fitting mortality models (Wilson, 1994; Eakin *et al*., 1995; Hughes, 1995), well-defined techniques for objectively comparing the fit of different models from this method are lacking.

In many cases evolutionary biologists are also interested in testing hypotheses about the parameter values from two or more treatment populations. For example, Tatar *et al*. (1993) were interested in whether the differences in longevity between populations of bean beetles allowed to vary in reproductive effort resulted from changes in the rate of ageing (senescence) or from a proportional decrease in mortality at all ages. This amounts to asking whether treatment populations differ in certain parameters of a mortality model.

It is the goal of this paper to discuss statistical techniques – based on the ideas of maximum likelihood – for analysing mortality data using mathematical models. Maximum likelihood provides a simple and powerful framework for mortality analysis that consists of: (i) objectively choosing a mortality model that adequately describes the data, (ii) estimating the parameters of that model and (iii) testing hypotheses about differences in parameter values among treatment populations. The methods are then used to investigate how large mortality experiments should be to answer questions about the mortality patterns of experimental populations. A set of computer packages that provide easy implementation of the ideas presented here is available for IBM (DOS and Windows) and UNIX environments and are freely available from the author.