Constructing Life Tables

Objectives:

Construct a life table for humans based on data gathered from a local cemetery.  Calculate a survivorship curve.

Background Information:

A life table shows for each person, at each age, what the probability is that they will die before their next birthday.  You will use the date of birth and date of death on each gravestone to calculate the age of each person at death.  This can then be used to construct a life table and then calculate a survivorship curve.  Survivorship curves are used to show what percent of the population we expect to live to a certain age.  These are used to calculate things such as average life expectancy of humans.

 Questions About Survivorship Curves

There are a variety of questions that we might examine with life table data. The most obvious question that can be asked is whether the survival curves of males and females are the same or different. Evolutionary theory allows us to predict which sex should have a survival advantage. Reproduction in humans is thought to be much more costly for females because they have to carry their babies through pregnancy and then go through a period of lactation when they produce energetically expensive milk. So females of child-bearing age (roughly from 15-20 to 45-50) should suffer higher mortality (and thus lower survival) than males of similar age. Social or political influences may also cause sexual differences in survival curves. For example, some cemeteries date back to the Civil War, and World Wars I and II. During a war, mortality should be higher and thus survival curves lower, especially for men due to the historical bias against women for combat assignments.

Another question that we might ask about life-table from local cemeteries is whether different historical periods have had different survival curves. For example, over the past hundred years there has been a dramatic increase in the average life expectancy of an adult in western societies because of improved health attention, health care, and medical services. We should expect to see an increase in survival curves over historical time. The "health care" hypothesis predicts that cemeteries with older mean ages should have lower survival curves. So, in all, we have three hypotheses that we should be able to discriminate by their different predictions. And you should be able to think of more hypotheses.

 Methods

We will take you to a cemetery where we will gather data on the birth of individuals for two time periods, 1840-1849 and 1850-1899. Work in groups of about two students. One of you can record the data, and the other can work with a calculator to find out how old each individual was when they died. The lab groups should divide up the area of the cemetery so that all areas are sampled, but none are sampled more than once. Be sure to keep a running tally of: sex of the individual, age at death, and year of birth. To determine sex, examine the individual's name. Some names are obviously male or female. For names that could be either (e.g., Leslie), keep an "unknown sex" category. Work as quickly as you can, but also be careful to show respect for where you are. Do not walk across graves! Stick to the pathways.

 Data Analysis

Back in the lab we will use this information to construct an age pyramid for 1876, allowing us to characterize the population.  Total up the number of deaths at each age class for everyone in your lab for the two different time periods. Remember to keep the sexes separate. Then total the number of individuals in your "male"  and "female" samples.  The number of individuals in each age class may be plotted as a histogram, forming a "pyramid."  Age is placed on the vertical axis and the number, or the proportion, of individuals in each age class is plotted that a symmetrical, pyramidal graph results.  Age pyramids such as these are useful for comparing populations from different sites, or the same population at different times of the year or from year to year.  Construct an age pyramid for the year 1876.  Do this by: 

1. Excluding people who either died before or were born after 1876.
2. Recalculate the ages of the remaining individuals by subtracting their birth dates from 1876.  For example, a woman born in 1832 would be 44 years old in 1876.
3. Calculate the percentage of individuals, male or female, in each group in 1876 (following the example above), by dividing the number of people in each age-sex class by the total number of people in our hypothetical 1876 population.
4. Graph the ages of the population in ten year intervals by the percent of the population within each age-sex class.  An example is shown below (Figure 1) based on data in Table 1.  Note that some graphics software may not allow the traditional pyramidal format.

Table 1.  Number of males and females in each age class for 1876.

 
Figure 1.  Age structure diagram of Austin population in 1876.  Data collected from tombstones in Oakwood Cemetery, Austin, Texas.

 
Next construct a life table for both males and females living during the year 1876.  You may choose to create these tables using MS Excel.  An example follows:

In a life table (Table 2), various statistics are compiled for each age class, or cohort (designated as x).  Data are commonly collected as numbers of individuals in each age class.  L_x is the number of individuals in age class x.  It is assumed that L_x is the number alive at the  middle of age class x (for example, in the above table, 33 individuals are assumed to be 5 years old, even though the true ages of the 33 might range between 0 and 10 years old).

We designate  l_x as the number of individuals alive at the beginning of age class x.  Thus,  L_x may be defined as 

 L_x = ( l_x +  l_x+1)/2  and,

 l_x = 2L_x -  l_x+1

For example, in Table 2, L₀ = 33, L₁ = 16, L₂ = 9, L₃ = 4, L₄ = 1, L₅ = 0.

Since L₅ = 0, we can set l₅ = 0.  Then by applying the above equation,

l₅ = 0
l₄ = 2(1) - 0 = 2
l₃ = 2(4) - 2 = 8 - 2 = 6
l₂ = 2(9) - 6 = 18 - 6 = 12
l₁ = 2(16) - 12 = 32 - 12 = 20
l₀ = 2(33) - 20 = 66 - 20 = 46.

The number of individuals in the population that die during interval x is:

d_x = l_x -  l_x+1

Note that the sum of the d_x values must equal l₀; in our example 46.

The age-specific mortality rate (q_x) is the proportion of individuals at the start of age interval x who die during that age interval:

q_x = d_x / l_x

also expressed as the probability of an individual dying during the interval.  The age-specific survival rate (s_x) for age interval x is the proportion of individuals alive at the start of the interval who do not die during the interval period.  In other words, s_x is the probability of surviving age interval x:

s_x = 1 - q_x

We can calculate age-specific life expectancy (commonly done for human populations) as follows.  Let us define T_x as the number of time units left for all individuals to live from age x onward; this is obtained by summing L_x values as follows: 

T_x = L_x + T_x+1

and expressing T_x in time units; so

T₄ = L₄ = 1 yr
T₃ = L₃ + T₄ = 4 + 1 = 5 yr
T₂ = L₂ + T₃ = 9 + 5 = 14 yr
T₁ = L₁ + T₂ = 16 + 14 = 30 yr
T₀ = L₀ + T₁ = 33 + 30 = 63 yr

Then, the life expectancy for an individual of age x is 

e_x = T_x/l_x

Life expectancy represents the average additional length of time that an individual will live, once it has reached age x.  To compare different populations, the numbers dying (d_x) or surviving (l_x) are often expressed as number per 100 or per 1000 individuals entering the population at age 0; that is l₀ is set to 100 or 1000, and all other life-table entries are expressed relative to this value.  For the example in our table, we have l₀ = 46 and l₁ = 20.  If we set l₀ = 100, then we would have an  l₁ value of (20/46)(100) = 43.5.  In other words, for every 100 individuals born into the population, 43.5 survive to age 1.

Various types of graphs may be constructed from life table data, including mortality rate curves, life expectancy curves and survivorship curves.  Most widely used among ecologists, the survivorship curve is prepared by plotting the of the number of survivors against age (sometimes this done on semilogarithmic paper).  From this graphical presentation, three basic types of curves are recognizable, as discussed in lecture.

We will construct percentage survivorship curves by age for both males and females living during each of the two time periods, 1800-1849 and 1850-1899.  Using the data sheet for the survivorship, determine the number surviving to each age for males and females.  Calculate the age class, number that died, percent surviving.  Using these data, plot the survivorship curve for each gender (male or female) for each time period.   What can you say about survival and life expectancies during these two time periods?

To calculate percentage survivorship by age:
Calculate cumulative number of dead at each age class from total # of individuals counted for each gender (male or female).

Example: 
Total # females counted in survey = 1000.
Total deaths before age 10 = 50.
For age class #1 (0 - 9 years), total percent surviving 1000 - 50 = 950/1000 = 95%.
Total deaths in 10-19 age class = 50.
For age class #2, total percent surviving:
Subtract Age class 1 (50 deaths) + Age class 2 (50 deaths) from total no. of individuals counted: 1000 - 100 =900 (or 90%).

Example:

Figure 2.  Survivorship curve for females living between 1800-1849.

Questions

1. Take a close look at your survivorship curves. How might you explain their general form or appearance? Are factors in the ecological or social environment important to the shape of survivorship curves?

2.  Are there "wiggles" in your survivorship curves? What sorts of factors, real or artifact, might cause variations in survival schedules? How would you go about testing the influence of such factors.

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