# Exercise 1: Life tables and Kaplan-Meier estimates of survival

## (a) Hand calculation: Life table and Kaplan-Meier estimates of survival

Using hand calculation (i.e., using a spreadsheet program or pen, paper, and a calculator) estimate the cause-specific survivor function for the sample of 35 patients diagnosed with colon carcinoma (see the table below) using both the Kaplan-Meier method (up to at least 30 months) and the actuarial method (at least the first 5 annual intervals).

In the lectures we estimated the observed survivor function (i.e. all deaths were considered to be events) using the Kaplan-Meier and actuarial methods; your task is to estimate the cause-specific survivor function (only deaths due to colon carcinoma are considered events) using the same data. The next page includes some hints to help you get started.

### Actuarial approach

We suggest you start with the actuarial approach. Your task is to construct a life table with the following structure.

interval [u,v) | l | d | w | l′ | p | S(u) | S(v)
--- | --- | --- | --- | --- | --- | --- | ---
[0-1) | 35 |  |  |  |  | 1.000 |
[1-2) |  |  |  |  |  |  |
[2-3) |  |  |  |  |  |  |
[3-4) |  |  |  |  |  |  |
[4-5) |  |  |  |  |  |  |
[5-6) |  |  |  |  |  |  |

We have already entered l (the number of people alive) at the start of the first interval. The next step is to add the number who experienced the event (d) and the number censored (w) during the interval. From l, d, and w you will then be able to calculate l′ (the effective number at risk), followed by p (conditional probability of surviving the interval) and finally S(t), the cumulative probability of surviving from time zero until the end of the interval.

### Kaplan-Meier approach

To estimate survival using the Kaplan-Meier approach you will find it easiest to add a line to the table at each and every time there is an event or censoring. We should use time in months. The first time at which there is an event or censoring is time equal to 2 months. The trick is what to do when there are both events and censorings at the same time.

time t | # at risk | d | w | p | S(t)
--- | --- | --- | --- | --- | ---
2 | 35 |  |  |  |
3 |  |  |  |  |
5 |  |  |  |  |
7 |  |  |  |  |
8 |  |  |  |  |
9 |  |  |  |  |
11 |  |  |  |  |
… |  |  |  |  |

## (b) Using R to validate the hand calculations done in part 1 (a)

We will now use R to reproduce the same analyses done by hand calculation in section 1.1 although you can do this part without having done the hand calculations, since this question also serves as an introduction to survival analysis using R. Our aim is to estimate the cause-specific survivor function for the sample of 35 patients diagnosed with colon carcinoma using both the Kaplan-Meier method and the actuarial method. In the lectures we estimated the all-cause survivor function (i.e. all deaths were considered to be events) using the Kaplan-Meier and actuarial methods whereas we will now estimate the cause-specific survivor function (only deaths due to colon carcinoma are considered events).

Life tables are available using the `lifetab` function from the `KMsurv` package on CRAN. We have written a small wrapper `lifetab2` which allows for a `Surv` object and a dataset. The following command will give the actuarial estimates:

A listing of the Kaplan-Meier estimates and a graph are obtained as follows

We can also use the `survminer` package to get elegant survival plots based on `ggplot2`: