### Abstract

- Top of page
- Abstract
- Introduction
- Background
- Analysing mortality data
- Simulations
- Discussion
- Acknowledgments
- References
- Appendix

Demographic studies focusing on age-specific mortality rates are becoming increasingly common throughout the fields of life-history evolution, ecology and biogerontology. Well-defined statistical techniques for quantifying patterns of mortality within a cohort and identifying differences in age-specific mortality among cohorts are needed. Here I discuss using maximum likelihood (ML) statistical methods to estimate the parameters of mathematical models, which are used to describe the change in mortality with age. ML provides a convenient and powerful framework for choosing an adequate mortality model, estimating model parameters and testing hypotheses about differences in parameters among experimental or ecological treatments. Simulations suggest that experiments designed to estimate age-specific mortality should involve at least 100-500 individuals per cohort per treatment. Significant bias in the estimation of model parameters is introduced when the mortality model is misspecified and samples are too small to detect the true mortality pattern. Furthermore, the lack of simple and efficient procedures for comparing different mortality models has forced the use of the Gompertz model, which specifies an exponentially increasing mortality with age, and which may not apply to the majority of experimental systems.

### Introduction

- Top of page
- Abstract
- Introduction
- Background
- Analysing mortality data
- Simulations
- Discussion
- Acknowledgments
- References
- Appendix

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.

### Discussion

- Top of page
- Abstract
- Introduction
- Background
- Analysing mortality data
- Simulations
- Discussion
- Acknowledgments
- References
- Appendix

In this paper I have discussed the benefits of a maximum likelihood approach to the analysis of age-specific mortality data. I have extended the standard uses of ML (i.e. parameter estimation) to include a hierarchical testing scheme that allows the choice of a mathematical mortality model that adequately fits the observed data (see also Fukui *et al*., 1993). Further, I have provided the means for applying likelihood ratio tests to hypotheses about the parameter values for different experimental or ecological treatments. All of the methods presented in the paper are implemented in an easy to use software package (IBM PC compatible and UNIX), which makes the maximum likelihood analyses easily accessible.

The use of the ML techniques described in this paper will result in a number of improvements in the design and analysis of experiments investigating mortality patterns. First, ML provides better parameter estimates that are more consistent and less influenced by technical aspects of the experimental design such as sample size than those from other methods. Promislow *et al*. (1997) discuss how published estimates of mortality parameters may be highly biased due to a combination of incorrect statistical methods and small sample sizes.

Second, the range of models presented here will result in greater documentation of different patterns of age-specific mortality. Brooks *et al*. (1994) reported that mortality rates increased with age in *C. elegans* according to the Gompertz model. Subsequent analysis (Vaupel *et al*., 1994) showed a significant deceleration in mortality rates at older ages. It is unknown how many studies fail to report levelling off simply because of the inability to properly test for it. Moreover, because of its complexity and difficulty in fitting, it is likely the Gompertz–Makeham model is applicable to a much wider range of data than is currently realized.

Despite the difference in the *c* and λ parameters in this model, there have not been any attempts to use the different models on real data to assess the relative contribution of each to mortality patterns from different types of populations (Finch, 1990). Identifying a significant Makeham term not only serves to provide insight into the levels of extrinsic mortality but also to remove the effects of age-independent mortality that can bias the estimates of mortality parameters (Table 2).

Third, the simulations suggest that researchers interested in understanding mortality patterns should venture to obtain sample sizes of at least 100–500 individuals per cohort per experimental treatment. A similar conclusion was presented by Service *et al*. (1998b) for parameter estimation of single populations and hypothesis testing using ANOVA methods. Shouman & Witten (1995) used simulations to illustrate the large variance associated with ML parameter estimates when sample sizes are small (<100). Perhaps the most crucial problem with small samples is the inability to detect non-Gompertzian mortality dynamics such as age-independent early mortality and mortality deceleration at older ages (Figs 1 and 2). Mortality plateaus are now established as a real and repeatable result in laboratory experiments (Curtsinger *et al*., 1995; Pletcher & Curtsinger, 1998; Service *et al*., 1998a), and some degree of random (age-independent) mortality is to be expected in any ecological study. Until these aspects of mortality are recognized and accounted for, estimates of the rate of senescence (γ) or the initial mortality rate (λ) are questionable.

Finally, the need for big samples is exacerbated by variation among individuals. This suggests that laboratory mortality experiments would be better off using fewer cohorts with larger numbers of individuals. Variation introduced within cohorts by different environments (such as different vials in *Drosophila* studies) could significantly reduce the statistical power to detect specific mortality patterns. Whether this source of variation can be reduced by ’pooling’ parameter estimates from two or more replicate cohorts is unknown at this time. This observation points out the biggest limitation of these methods as developed thus far, the inability to properly account for complex experimental designs – hierarchical or nested designs for example. Unfortunately, this is not an easy problem. It involves maximizing the marginal likelihood of the distribution of deaths after accounting for (integrating over) the assumed distribution of the model parameters within a treatment. Since there are often a large number of replicates (e.g. vials) within each treatment, Markov Chain Monte Carlo maximization is required (Geyer, 1995; C. Geyer personal communication). This work is in progress.

The question of how much power a specific experimental design provides is always a difficult one. Statistical power depends not only on sample size but also on the actual values of the mortality parameters under investigation. Simulations with parameters ranging from ½ to twice the values reported here produce very similar results. Therefore, the sample sizes I suggest are directly relevant to a number of experimental systems, including *Drosophila melanogaster*, *Caenorhabditis elegans*, and *Callosobruchus maculatus*. In general, it is impossible to provide precise advice about how large an average mortality experiment should be, but preliminary estimates of mortality parameters coupled with the power estimates offered here can be used for guidance.

Even larger samples may be needed to address questions about truly age-specific phenomena. The estimation of mortality patterns by ML is based on the distribution of deaths at all ages. Nonparametric statistical techniques designed to examine differences in mortality at specific ages are not well developed. Whether there are transient, localized differences in mortality between experimental treatments (see Pletcher *et al*., 1998) is an open and difficult question and will likely require much larger samples than those suggested here.