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"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)  ### Proportions and rates

We have entitled the next section 'mortality' rather than 'mortality rate'. This is because several of the measures given below are proportions rather than true rates.

• 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 better medical and veterinary journals now prefer that the term rate is only used to describe a change with time. Nevertheless, some of the proportions below are so often described as 'rates' that we have given the synonyms where appropriate. Moreover, ecologists use the terms finite (or multiplication) rates and instantaneous rates to distinguish between proportions and 'true' rates - for more discussion on finite and instantaneous rates see the related topic (above). Here we begin with a measure that is neither a proportion nor a rate - namely the number of deaths.   ### Mortality

#### Number of deaths

The number of deaths in the population is the simplest measure of its mortality. For human populations, governments usually keep detailed records of the number and timing of deaths. But it can be more difficult to determine the number of deaths for a livestock population, and it may be difficult or impossible for wildlife or insect pest populations. The number of (recorded) deaths is analogous to the number of cases when considering measures of disease frequency.

Even though the total number of human deaths may be recorded fairly accurately (except in war situations), it is much more difficult to obtaining an accurate count of the number of deaths from a specific cause. This is because it is dependent on accurate diagnosis of the disease or condition the individual was suffering from - an issue discussed in the More Information page on measures of disease frequency. Apparent changes in the number of deaths from a particular disease may result from a change in the sensitivity of the reporting, rather than a change in the actual number of deaths.

If the population size is reasonably stable, the number of deaths is directly related to the mortality rate (defined under mortality rate below). This will usually be true for human populations, but again much less so for wildlife and (especially) insect pest populations.

#### Cumulative mortality and cumulative survival

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. It is analogous to the cumulative incidence (the proportion of new cases of a disease), with deaths replacing cases as the numerator.

 Cumulative mortality   = Number of deaths over a period of time Population size at start of period

Cumulative mortality is sometimes qualified in relation to the cause of death. Hence all-cause cumulative mortality is the proportion of individuals dying from any cause over a given period and cause-specific cumulative mortality is the proportion of individuals dying from a specific disease over a given period.

As with cumulative incidence, the length of the period cannot be changed by simply dividing or multiplying the cumulative mortality by the length of the period required - hence a cumulative mortality of 20% over 1 month does not give a cumulative mortality of 120% over 6 months. Instead use the following formula:

 C1 = 1 - [1 - C2](t1 / t2)
where:
• C1 is the cumulative mortality you wish to estimate
• C2 is the cumulative incidence you have measured
• t1 is the time period for mortality you wish to estimate the mortality,
• t2 is the time period over which you measured the mortality.

The cumulative survival is the proportion of individuals alive at the start of a period that remain alive for a specified length of time. It is the complement of cumulative mortality - hence, if the cumulative mortality of piglets after 3 months is 11%, the cumulative survival is 89%.

#### Other proportional measures

These two terms are generally only used by medical and veterinary researchers.

The case fatality is the proportion of infected individuals (cases) that die from a specified disease.

 Case fatality = Total number of deaths from a disease Number of cases of that disease

Case fatality must be interpreted with caution. An episode of illness does not last for a fixed time interval - it varies depending on many factors. The time period used for estimating case fatality is usually the period of medical care. Hence the case fatality is dependent on the length of stay in hospital. Survival analysis, with follow up over a long period, provides a much better measure of mortality from a particular disease.

The proportional mortality is the proportion of deaths that are caused by a specific disease.

 Proportional mortality = Total number of deaths from a specific disease Total number of deaths

Be especially cautious with this measure. It is routinely misinterpreted by the media. A 10% proportional mortality from MSR in hospital wards (in other words 10% of all deaths result from MSR) is likely to be (wrongly) reported as a 10% (cumulative) mortality of all hospital patients.

#### Mortality rate

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. It is analogous to the incidence rate, with deaths replacing cases as the numerator. Again we may refer to a cause-specific mortality rate, or the all-cause mortality rate.

 Mortality rate = Number of deaths over a period of time Total of time at risk of death of all individuals

The mortality rate as calculated above provides an estimate of the hazard function (also termed the instantaneous mortality rate) - which is the theoretical limit approached as the time interval approaches zero (see related topic ).

As with the incidence rate, we can use the average number at risk multiplied by the time period, to approximate sum of time at risk. When the exact time of death is unknown, the mortality rate has to be estimated in this way. This estimate of the mortality rate is sometimes called the crude mortality rate (or crude death rate).

 Crude mortality rate = Number of deaths over a period of time Average number at risk × Time period

This method is commonly used to work out age-specific death rates for use in life tables (see below. )

Epidemiologists distinguish a number of different age-specific death rates for human populations:

• The maternal mortality rate is the number of maternal deaths due to childbearing per 1000 live births.
• The perinatal mortality rate is the sum of neonatal deaths and stillbirths per 1000 live births.
• The infant mortality rate is the death rate of children during the first year of life per 1000 live births. The term is also used in veterinary studies but there is no general agreement on the period of time involved.
• The child mortality rate is the death rate of children less than 5 years old of life per 1000 live births.

Because the observed mortality rate of a population depends critically on its age distribution, mortality rates are often standardized by age in the same way as other event rates.    ### Natality #### Number of births

The same comments apply here as for the number of deaths. For human populations, detailed records are usually kept of the number of births and when they occurred, but this is less true for livestock populations - and not the case at all for natural populations.
Two terms are often used to describe reproductive capacity:
• Fecundity (or physiological natality) is the potential level of reproduction of a population - under ideal conditions. This is what is measured (for example) in laboratory studies of an insect pest. For humans, fecundity is one birth per 10 months per female of child-bearing age
• Fertility (or realized natality) is the actual level of performance - based on the numbers born. For humans the fertility rate is the number of births per year per woman of child-bearing age. The fertility of humans is much lower than their fecundity. Many factors can reduce fecundity to the observed fertility level, including behavioural factors, abortions, and still-births.

#### Natality rate

A precise estimate of the natality rate (or birth rate) could be calculated by dividing the number of births by the sum of time at risk of giving birth over all individuals. But in practice an approximation is nearly always used, namely the number of births divided by the average size of the population multiplied by the time period. This gives the crude natality rate (or crude birth rate). For human populations it is commonly expressed per 1000 of the population.

 Crude natality rate = Number of births over a period of time Average number of individuals × Time period

The same method can be used to work out an age-specific birth rate, except that estimates are usually confined to the female part of the population. Hence the age-specific birth rate is obtained by dividing the number of female offspring by the average size of the population - of a specified age-group of females. ### Life-tables

Life and fertility tables provide a standardized way of tabulating, summarizing and displaying mortality and natality data. Life tables were first used to describe mortality rates in human populations, but have since been widely used for many animal and plant populations. There are two general approaches to obtaining life table data:

#### Cohort life-tables

The simplest method is to follow a cohort of individuals through time from birth (or any other convenient starting point ) to death. This is called a cohort life table (or generation life table or horizontal life-table or age-specific life table). A similar schedule of births in the population is called a fertility table. In the simplest situation observations start on all individuals at the same calendar time. Alternatively individuals may enter the study at the same event (for example birth) but at different calendar times. Usually it is assumed that there is no separate effect of calendar time so that individuals starting at different times can be grouped together. However this may not be the case especially if there are any seasonal or long-term changes in mortality rates.

If we are recording time to death there are three possible outcomes for individuals in a study:

1. The event has occurred - and we have recorded the time up to that event.
2. The event has not occurred - and the individual has survived for the full period of the study.
3. The individual has withdrawn, or otherwise been lost from the study, without the event having been observed.
Survivors and withdrawals are commonly described as censored data points.

It may not be viable to follow cohorts of invertebrate populations, especially if the organisms are very mobile. For these the commonest approach is mark-release-recapture - as described in the related topic. Individuals are marked in some way, and then attempts made to recapture them. If this is done on several occasions it is possible to estimate not only population size, but also population loss and gain rates. An alternative approach, to follow-up a cohort of mobile animals, is some form of restraint (for example within a cage or by tethering). However, there are obvious problems in deciding how applicable the results are to the natural situation.

#### Static life-tables

The alternative is to take a cross-sectional sample to determine the numbers alive in each age group, and also determine the number of deaths in each age group during a given time period. This provides a static life table (or current life table, stationary life table or vertical life-table or time specific life table). For human populations information on the numbers of deaths in each age group is usually available from government records. For animal and plant populations such data can be obtained from age at death, providing individuals that have recently died can be aged, and are equally available to the sampling method.

Another approach to obtain a static life table is to estimate the mortality rates from the relative numbers in each age group. However, this method is only valid if a number of assumptions are made. The most important is that there is a stable age distribution - in other words the relative frequencies of each age class do not change over time. This will only be the case if birth and death rates are constant. For the simplest version of this method the population must also be stationary - in other words, the birth rate must equal the death rate, so that the population is neither increasing nor decreasing. These assumptions are very improbable for most populations including insect vectors of disease which is what this method is mainly used for.

Cohort and static life tables only give the same results for a given population if birth and death rates are constant. In practice they give different results because birth and death rates vary considerably over time in almost all populations whether human or animal.

The approach to life table analysis depends on whether we only use the number of events occurring within set intervals, or whether we utilise the precise timing of the event for each individual. The first of these termed standard life tables is the longer established method, but the newer Kaplan-Meier approach is now more heavily used, especially in medical research.