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# Measures of mortality and natality: Use and misuse

## (proportions and rates, cumulative mortality, case fatality, mortality rate, natality, life tables, fertility tables, age distribution)

Statistics courses, especially for biologists, assume formulae = understanding and teach how to do  statistics, but largely ignore what those procedures assume,  and how their results mislead when those assumptions are unreasonable. The resulting misuse is, shall we say, predictable...

### Use and Misuse

A proportion is a ratio in which the numerator is a subset of the denominator. A rate describes the rate of change in the numerator with respect to the denominator with time usually included in the denominator. The cumulative mortality (also termed mortality risk or finite mortality rate) is the proportion of individuals alive at the start of a period that die over that period. The case fatality is the proportion of infected individuals (cases) that die from a specified disease. The proportional mortality is the proportion of deaths that are caused by a specific disease. The mortality rate (or death rate or mortality density) is the probability of an individual dying per unit time. A precise estimate is obtained by dividing the number of deaths in an interval by the (total) sum of time at risk over all individuals. The crude mortality rate is obtained by dividing the number of deaths in an interval by (average number at risk over period × time period). Similarly the crude natality rate is obtained by dividing the number of births in an interval by (average number of individuals × time period).

The better medical and veterinary journals usually now insist that the term rate is only used to describe a change with time. However, confusion in the literature in the past means that terminology is still highly unstable and in the papers we reviewed  we found 'mortality rate' misused to mean cumulative mortality, and sometimes even case fatality. It does not help that ecologists usually use the terms finite and instantaneous mortality rates to refer to what epidemiologists call cumulative mortality and mortality rate respectively. The important thing is to define the terms you are using - do not assume that others will understand your terminology. Care must also be taken in the way rates are combined - combining rates of different age groups must be done using a weighted mean.

Ferocious debates often arise in the medical literature over small changes in mortality over time, or small differences between population groups or countries. Yet such differences may be fully explained by confounding factors. Any comparison of mortality rates between populations obviously requires age standardization as well as other adjustments depending on the populations being compared. Measurement error is less often considered and can sometimes explain quite large differences. Case fatality is often misinterpreted simply because there are several ways in which it can change. With births there is a need to consider exactly what one is looking at, whether only (recorded) live births, or births plus abortions.

Whilst medical and veterinary researchers usually use direct methods to measure mortality, this is more difficult for wildlife researchers, and often impossible for ecologists studying insect pests. Mark-release-recapture may provide a means to follow a cohort  of a mobile organism. Alternatively the organisms may be restrained in some way, although such methods are liable to bias. Possible biases are often not fully evaluated in the literature. This is especially true when mortality rate is estimated from the age distribution, and most such estimates have little value.

### What the statisticians say

Pfeiffer (2002) provides a very useful introduction to epidemiology including a clear distinction between proportions and rates. Rothman & Greenland (eds)(1998) set down the basics about proportions and rates for medical epidemiologists in Chapter 3 - although you may find Rothman (2002) more accessible. Woodward (2004) describes analytical methods for both cohort and current life tables. Dohoo (2003) and Thrusfield (2005) cover mortality statistics for veterinary epidemiologists.

Christensen et al. (1994), Hurd & Kaneene (1990) and Thurmond (1986), look at choice of measures of mortality in veterinary research. Lettink & Armstrong (2003) provide a useful introduction to the use of mark-recapture for monitoring threatened species. Amstrup et al. (eds)(2007) is a relatively new practical handbook covering all recent developments in the field of mark recapture. McCallum (1999) covers a wide range of techniques for estimating mortality and natality rates. Williams et al. (2002) , Krebs (1999) , Seber (1982) and Caughley (1977) are all standard texts on quantitative population ecology. Begon (1979) provided a good (albeit outdated) introductory text on mark-recapture methods.

Zens & Peart (2003) look at new methods to analyse mortality rates at the population level. Seber & Schwarz (2002) and Schwarz & Seber (1998) provide authoritative and comprehensive accounts of recent developments and future directions in estimating population abundance. Sheil et al. (1995) review the difference between finite and instantaneous measures of mortality - although the finite rate is misleadingly termed the annual mortality rate, and the instantaneous rate is denoted by l, a symbol normally used for the finite rate of increase. Van Sickle & Phelps (1988) estimate tsetse fly mortality rates from age distributions after correcting for rate of increase.

Wikipedia provides sections on mark-release-recapture , mortality rate , case fatality , birth rate , survivorship curves and life tables. Beware! - several of these Wikipedia references still refer to proportions as rates (e.g. case fatality) Various universities such as University of Texas provide details on constructing life tables.