1. Dynamics
Mortality rates improved substantially over the last 100 years and will hopefully continue to improve in the future. This video explains how to account for evolutions in mortality.
2. Multiple dataframes with mortality rates
You have been working with mortality rates applicable to one particular year, but mortality data are published regularly. For each year or period you have a table with mortality rates at your disposal. You'll now use the combined data.
3. The evolution of mortality rates over time
Let us picture the combined information as a matrix. In this matrix mortality rates are indexed with age x in the rows and period t in the columns. qxt is the mortality rate applicable to an individual who is x-year-old in year t. For example, Eden Hazard's current mortality rate is the q for a 27-year-old in year 2018.
4. The evolution of mortality rates in Belgium over time
This animation plots the mortality rates of males in Belgium from 1950 to 2015. Age x is on the x-axis and the logarithm of the mortality rate is on the y-axis. Each line represents one year of data. The animation illustrates how mortality rates clearly improved and thus: decreased over time.
5. The period approach
Using the matrix of mortality rates you can calculate survival probabilities of an x-year old using only the rates from period t. It is important to stress that index t which refers to period t is kept constant in the multiplication shown here. This is the vertical way, also called the period approach.
6. The cohort approach
However, it is more clever to anticipate evolutions in mortality rates by using the data in a diagonal way! Indeed, an individual who is x-year-old in year t, will be x+1 years old in year t+1 and so on. Therefore, when calculating the k-year survival probability of an x-year-old you should multiply pxt with px+1,t+1 and so on. That is: you multiply the diagonal survival probabilities applicable to a newborn from cohort or birth year t-x.
On the one hand, this approach gives a more fair picture of the evolution of mortality rates over time. But on the other hand, the diagonal way of thinking typically requires a projection of mortality rates. That is: mortality rates for future periods t beyond the last observed period! This is a very challenging topic, widely studied in research.
7. Life tables over time in R
In R you load a dataframe with the mortality rates from period 1841 until 2015. Such tables are available from the Human Mortality Database. The dataframe displays a new variable: year. Note that the difference between year and age equals the considered cohort. For example, a 0-year-old in year 1841 was born in 1841, a 1-year-old in year 1841 was born in 1840 and so on.
8. The period survival probabilities of a famous Belgian in R
Jacques Brel was a famous Belgian singer-songwriter. Born in 1929 he died at age 49. Using the mortality rates of 1929, what is the probability that a newborn dies at exact age 49? The subset() function takes a data frame as first argument and a logical expression as second argument to subset the data. Variable names are directly accessible in the second argument. You extract the mortality rates applicable to year 1929 and multiply the one-year survival probabilities for a 0 up to an 48-year-old. That's the probability that a newborn survives to age 49. Multiplying with q49 gives the probability to die at exact age 49.
9. The cohort survival probabilities of a famous Belgian in R
You now repeat the calculation with the mortality rates applicable to the birth year of Brel. You subset the life table by extracting all observations for which year minus age equals 1929. You use the qx from this new data set and repeat the calculation. The resulting probability is smaller, since mortality improved.
10. Let's practice!
That's exciting stuff; your turn!