# Kaplan-Meier for actuaries

In Richards & Macdonald (2024) we advocate that actuaries use the Kaplan-Meier estimate of the survival curve. This is not just because it is an excellent visual communication tool, but also because it is a particularly useful data-quality check.

One issue is that not all software implementations of the Kaplan-Meier estimate account for left-truncated data. However, left-truncation is standard in actuarial work, as lives usually only enter observation well into adult life. To help the actuarial community, Figure 1 shows some R script that calculates the Kaplan-Meier estimate for left-truncated data by age:

Figure 1. R script for Kaplan-Meier estimates of survival curves with left-truncated data.

################################################################################ # # Read in the data and store as a data frame. d = read.csv(file="C:/sandpit/SurvivalInput_94729.csv", header=TRUE) # Calculate the exit ages. d$ExitAge = d$EntryAge + d$TimeObserved # Function to calculate the Kaplan-Meier estimate of the survival curve, tpx. # The argument g is the data frame, which can be a sub-group. KaplanMeier = function(g) { # Find ordered set of unique death ages. ages = as.matrix(sort(unique(g$ExitAge["DEATH"==g$Status]))) # Count deaths at each unique death age. deaths = apply(ages, 1, function(y) sum(y==g$ExitAge & "DEATH"==g$Status)) # Count lives in-force immediately prior to each unique death age. lives = apply(ages, 1, function(y) sum(g$EntryAge<y & g$ExitAge>=y)) return(list(ages=c(min(g$EntryAge), ages), tpx=c(1, cumprod(1-deaths/lives)))) } # Calculate separate Kaplan-Meier estimates for males and females. Females = KaplanMeier(d["F"==d$Gender,]) Males = KaplanMeier(d["M"==d$Gender,]) # Plot the Kaplan-Meier step functions. plot(Females$ages, Females$tpx, type="S", main="Kaplan-Meier estimate", xlab="Age", ylab="Survival probability", col="red", lwd=2) lines(Males$ages, Males$tpx, type="S", col="blue", lwd=2) legend("topright", col=c("red", "blue"), lty=c(1, 1), lwd=c(2, 2), bty="n", legend=c("Females", "Males"), cex=1.5) ################################################################################

Readers can scrape the R script in Figure 1 and use it directly with Longevitas input files (but remember to change the filename!). For those without access to Longevitas, see the example file on the right. After running the script on the example file, you should get something like Figure 2 in your R console:

Figure 2. Kaplan-Meier estimates of the survival curves for males and females.

**References: **

Macdonald, A. S., Richards. S. J. and Currie, I. D. (2018). Modelling Mortality with Actuarial Applications. Cambridge University Press, Cambridge.

Richards, S. J. and Macdonald, A. S. (2024) On contemporary mortality models for actuarial use I: practice, *Longevitas working paper*.

## Kaplan-Meier in Longevitas

Longevitas will generate Kaplan-Meier estimates for each sub-group when you set the option in the **Advanced Options** section of the initial modelling screen.

Longevitas automatically deduplicates portfolio records, and calculates the exposure for each record, taking into account left-truncation and right-censoring.

## Example file

SurvivalInput_94729.csv contains example scheme data prepared in accordance with the steps described in Macdonald *et al* (2018, Chapter 2), including validation and deduplication. For the purposes of the R script on the left, the key fields are:

`EntryAge`

, which contains the (left-truncated) entry age in years,`TimeObserved`

, which contains the time in years until death or censoring, and`Status`

, which contains "DEATH" for a death, or 0 for a censored observation.

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