InfluentialPoints.com
Biology, images, analysis, design...
Use/Abuse Principles How To Related
"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)

 

 

Simple measures of mortality

Before we look at examples of cohort and static life tables, we will first give a worked example showing how the different measures of mortality are calculated.

Worked example

We will use a similar example to that used when looking at measures of disease frequency, except that now we have a number of deaths occurring as a result of the infections. Eight pigs are observed over twelve weeks.

AnimalWeekTime at
risk (weeks)
 1  2  3  4  5  6  7  8  9 101112
1      infecteddied9
2            12
3 infecteddied4
4            12
5   infected12
6            12
7            12
8    infecteddied7
Total time at risk:76

  • Number of deaths
    There were 3 deaths over the 12 week period.

  • Cumulative mortality
    The cumulative mortality for the 12 week period was 3/8 which is 0.375.

  • Case fatality
    There are 4 cases of which 3 died, so the case fatality was 3/4 = 0.75. But note that the animal that was infected, but remained alive, may have died shortly afterwards. The case fatality is very dependent on the duration of observations.

  • Mortality rate
    Using the exact method, we add up the number of weeks at risk to give a total of 76 pig weeks. The mortality rate is then given by 3/76 which is 0.039 per pig week.

    Using the approximate method, we estimate the average number at risk over the period by adding the number at risk at the start and end of the period, and dividing by two. In this case we have 8 present at the start, 5 at the end, so the average number present is 6.5. The crude mortality rate is then given by 3/(6.512) = 0.038 per pig week.

    We can also estimate the instantaneous mortality rate per week from the cumulative mortality over 12 weeks as [- ln (1 -cumulative survival)]/12 = [-ln(.625)]/12 = 0.039.

 

Standard cohort life table

Worked example

The table below is just the first part of a standard life table of a cohort of cattle after first calving (data from Dohoo & Martin (1984)). The first five columns should be self-explanatory, but we then move to a set of three functions - the first concerning survival and the latter two concerning mortality:

Standard Life Table
Time at start of ith interval (t) Number at start of interval (ni) Number dying during interval (ei) Proportion dying during interval (qi) Proportion surviving during interval (pi) Cumulative survival function (Si) Probability density function (fi) Hazard function (hi)
    ni − ni + 1 ei 1 − qi Π pi Siqi ei
ni bi bi(ni − ½ei)
0 2009 55 0.0274 0.9726 0.9726 0.000888 0.000925
30 1954 28 0.0143 0.9857 0.9587 0.000458 0.000481
60 1926 32 0.0166 0.9834 0.9428 0.000522 0.000558
90 1894 26 0.0137 0.9863 0.9299 0.000426 0.000461
120 1868 24 0.0128 0.9872 0.9180 0.000393 0.000431
150 1844 28 0.0152 0.9848 0.9039 0.000458 0.000510
180 1816 25 0.0138 0.9862 0.8915 0.000409 0.000462
210 1791 29 0.0162 0.9838 0.8771 0.000473 0.000544
240 1762 21 0.0119 0.9881 0.8667 0.000344 0.000400

 

So what exactly are these three 'functions'?

  • The cumulative survival function (S) (or cumulative proportion surviving) is the proportion of the starting population still surviving at the start of the next interval. The importance of this statistic should be obvious - it is often used to compare the outcome of clinical trials. If we look at the first interval (0-30 days), the proportion surviving to the start of the next interval is 0.9726 or 97.26%.

  • The probability density function (F) (or unconditional mortality rate) is the probability per unit time that an individual alive at time 0 will die in a given interval. It is estimated by simply multiplying two probabilities - the probability of surviving up to the start of the next interval (S) and the probability of dying in that interval (q). We then divide by the number of time units in the interval to give the estimate as per unit time (in this case per day). If we plot the probability density function against time we get an event density curve which represents the frequency distribution of the survival times.

  • The hazard function (h) (or conditional mortality rate or hazard rate) is the probability per unit time that an individual that has survived to the start of the interval will die in that interval. We calculate it in a similar way to the cumulative mortality (q) except for two aspects. First we divide the number of deaths by the number alive in the middle of the interval rather than at the start of the interval - this is estimated by subtracting half the number of deaths in the interval from the number at the start of the interval. We then again divide by the number of time units in the interval to give the estimate as per unit time.

{Fig. 1}
U17func.gif

Note especially the interval to which the particular rates apply. This affects how we plot the cumulative survival function (S) against time, known as a survival plot.

The value of S is plotted against the starting time of the next interval, shown here typically as a step function. The hazard function (shown in the second figure) is also plotted as a step function but against the starting time of the same interval. The arrows show that the times where there is a larger-than-average drop in the proportion surviving follow the peaks in the hazard function.

So why do we work out cumulative survival by multiplying together all the interval survival probabilities, rather than just by dividing the number surviving by the original population? The answer is that by doing it this way we can correct our estimates for censored observations.


 

Cohort life table with censoring

Worked example

We correct our estimates by subtracting half the number of censored individuals from the number of individuals at the start of each interval. This then gives an 'effective number' during the interval (n'i) which is used for subsequent calculations:

Standard Life Table with Censoring
Time at start of ith interval (t) Number at start of interval (ni) Number lost to follow-up during interval (wi) Effective number during interval (n'i) Number dying during interval (ei) Proportion dying during interval (qi) Proportion surviving during interval (pi) Cumulative survival (Si)
      = ni − w/2   = ei / n'i = 1 − qi = Π pi
0 2009 0 2009 55 0.0274 0.9726 0.9726
30 1954 3 1952.5 25 0.0128 0.9872 0.9602
60 1926 4 1924 26 0.0135 0.9865 0.9472
90 1894 6 1891 20 0.0106 0.9894 0.9372
120 1868 4 1866 20 0.0107 0.9893 0.9272
150 1844 3 1842.5 25 0.0136 0.9864 0.9146

We are making a number of assumptions here which we need to clarify:

  • Censored individuals do not differ from those that continue in the study as regards mortality rates.
  • The risk of death per interval subdivision is constant within an interval. This does not matter if there are no censored observations, but is necessary for our simple method of obtaining the effective population to be valid. This may well not be true if the interval is long relative to the total lifetime.
  • Likewise withdrawals must occur uniformly through the interval for us to correct the number at risk by subtracting half the number of censored individuals. Again the shorter the interval, the better, if this is to be true.

There is one obvious way you can make the interval shorter, and also make more efficient use of the data. That is to deal with each event/death individually and use a variable interval length set by the time between individual deaths. This is known as the Kaplan-Meier (or product-limit) approach.

In this approach the event times themselves define the length of the intervals at which the cumulative survival probability (S) is calculated. This ensures that the interval times used are the shortest possible and we use all the available data (usually a good idea!).

 

Kaplan-Meier life table

Worked example


We will use the Kaplan-Meier approach to look data on the development of encephalopathy in patients treated for sleeping sickness with melarsoprol (data are based on but not identical to study of Burri et al.). For now we will just consider results from one treatment group - those treated with the new schedule. Note that each time in the table is assumed to be just before the time of the event. Hence when we calculate the proportion affected at 3 days, we divide the number of events on day 3 (1) by the number present (250) just before the event occurs.

Kaplan-Meier Life Table
Time (t) No. at time t (nt) No. events (et) Proportion affected (qt) Proportion not affected (pt) Cumulative survival function (St) Probability density function (fi) Hazard function (hi)
      et 1 − qt Π pt Stqt et
nt bt btnt
0 250 0 0 1 1 0.000000 0.000000
3 250 1 0.004000 0.996000 0.996000 0.003984 0.004000
4 249 2 0.008032 0.991968 0.988000 0.002645 0.002677
7 247 1 0.004049 0.995951 0.984000 0.003984 0.004049
8 246 3 0.012195 0.987805 0.972000 0.011854 0.012195
9 243 1 0.004115 0.995885 0.968000 0.003984 0.004115
10 242 4 0.016529 0.983471 0.952000 0.015736 0.016529
11 238 1 0.004202 0.995798 0.948000 0.000996 0.001050
15 237 1 0.004219 0.995781 0.944000 0.000443 0.000469

{Fig. 2}
U17bur01.gif
Since the intervals are now much shorter, we simply deduct censored individuals from the number at time t. In this table we have no censored individuals. But this would not have been the case if there had been any deaths not related to encephalopathy. For example, if one person had died from unrelated causes on day 5 the number of day 7 would have been reduced to 246 rather than 247. If censoring occurs on the same day as an event (say on day 7) it is assumed to occur after the event.

The survival plot derived from this data is known as a Kaplan-Meier plot. It is similar to our previous survival plot except that changes no longer occur at regular intervals (= sampling period) but whenever there is an event.

 

Fertility tables

Worked example of a fertility table

For a fertility table we start the table with a column giving the proportion of females surviving to the mid-point of the interval. The next column gives the number of female offspring new column added, the mi column, which details the age-specific fertility. We then multiply these two columns together to give the total number of females born in each age category. The sum of this column is the net reproductive 'rate' (Ro):

Survivorship and fertility table
Time
at start
of ith
interval
(t)
Proportion of
females surviving
to midpoint
of interval
(Li)
Number of
female offspring
per female
during interval
(mi)
Total number
of females born
in each interval
(Vi)
Product of
Vi and time
at midpoint
of interval (x)
 
 
    Limi Vix
0 0.713 0 0 0
10 0.373 2.755 1.028 15.42
20 0.267 3.978 1.062 26.55
30 0.187 4.012 0.750 26.25
40 0.120 2.346 0.282 12.69
50 0.053 0.325 0.017 0.935
      Σ = 3.139 = Ro Σ = 81.845

The net reproductive rate is not a true rate, but a multiplication factor - in fact the multiplication factor per generation. It will equal the number of females in generation (n+1) divided by the number in generation n.

But how long is a generation?

The cohort generation time (Tc) is defined as the mean length of time elapsing between the birth of the parents, and the birth of the offspring. It is estimated by dividing the sum of the Vix column (81.45) by the net reproductive rate (3.139) to give 26.1 weeks.

In summary, the population should increase by a factor of 3.139 times every 26.1 weeks. We look at how the net reproductive rate is related to the population rate of increase, per unit time, in the Related Topic on population change.

 

Static life table

A static life table is derived from the age structure of a single sample of a population, at a particular time. The age structure is taken to reflect the fate of a cohort of animals born at time 0. As we have pointed out above, this is only true if there is a stable age distribution and the population is stationary. Providing these conditions are met, mortality rates can be calculated in the same way as with cohort life tables.

Worked example

The data represent a random sample taken from an insect population. The age distribution of the sample is therefore assumed to be representative of that of the population.

Static Life Table
Age
category
(days)(t)
Frequency
in sample
(ni)
Number
of deaths
(di)
Cumulative
mortality
(qi)
Mortality
rate
(hi)
    ni − ni+1
di
ni
di
bi(ni+ni+1)/2
0-9 85 49 0.576 0.0810
10-19 36 12 0.333 0.0400
20-29 24 12 0.500 0.0667
30-39 12 5 0.417 0.0526
40-49 7 6 0.857 0.1500
50-59 1 - - -

The mortality rate per day appears to be rather high initially (0.081/indiv/day), then varies between 0.0404 and 0.0667/indiv/day, before increasing in later life to 0.15/indiv/day.

Note that, because we do not to know the exact ages, the estimates of cumulative mortality and mortality rate are a little biased in this type of life table. Instead we only know the number within rather wide age ranges. However it seems this error of age classification does not affect the estimate too much - provided the mortality does not change too abruptly from one age class to the next.

 

In the example above, survival appears to vary between age groups. But this may just be a result of sampling error. In some insect populations it may be valid to assume that survival (or at least adult survival) is constant over the different age categories. In this case there are various ways to obtain a pooled estimate of the mortality rate. A weighted mean is one possibility - in this case giving an average mortality rate of 0.070. But another method is more widely used, especially in medical entomology.

Providing there is a stable age distribution, and the population is stationary, the age structure will reflect the fate of a cohort of animals born at time 0. Hence we can represent the total sample in the following way:

Total in sample  =  No + NoS + NoS2+ NoS3....+ NoS
Where:
  • No = the number in the youngest age group,
  • S is the cumulative survival per age interval.

It is then straightforward to show that S is given by the proportion that age groups older than No make up of the population.

Worked example

Using the above example, there are 80 individuals in the youngest age class out of a total of 165 individuals. Hence S is given by 80/165 = 0.485

The daily cumulative survival is the bth root of S, where b is the length of the age interval. This equals 0.930 in this example.

The daily instantaneous mortality rate = − ln (daily cumulative survival) = 0.072

The big problem with this approach is that a stable age distribution is very rarely achieved in natural populations, and a stationary population is even rarer! When a population is increasing, survivorship will be underestimated, and mortality rates will be overestimated. The opposite bias is produced when the population is decreasing.

It is sometimes argued that, despite the biases, a static life table is better than no life table, especially since they are relatively easy to construct. But this is not good advice. The methods given here should only be used when:

  1. A stable age distribution has been demonstrated (by examining several sequential age distributions).
  2. A stationary population has been demonstrated (by examining several sequential population estimates).

In fact it is now possible to correct an estimate, depending on whether the population is increasing or decreasing. There are also other methods which require neither a stable age distribution nor a stationary population. References for these methods are given below.