### GIGAMIND

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113 Math

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113.020.020.40 Survival analysis - Kaplan-Meier Model

# Kaplan-Meier Model

aka: Product Limit Method

- Downward stepped graph from left to right
- Y-axis = Pct chance of survial
- X-axis = Time
- At
`t=0`

you have a 100% chance of surviving. As you increase in time you have lower probability of surviving. - KM is non-parametric. E.g. it is a step function that steps down

This is an example of how to calculate the Kaplan-Meier:

Given the following data set:

```
Time (months) Died (1=Yes, 0=No)
------------- ------------------
2 1
3 0
6 1
6 1
7 1
10 0
15 1
15 1
16 1
27 1
30 1
32 1
```

For each row:

```
Time Risk # Died HAZ I-HAZ SURV = S(t)
---- ---- ------ --- ----- -----------
0 12 0 0/12 12/12 1 = 100%
2 12 1 1/12 11/12 (11/12) = 0.917
6 10 2 2/10 8/10 0.917(8/10) = 0.734
7 8 1 1/8 7/8 0.734(8/10) = 0.642
15 6 2 2/6 4/6 0.642(4/6) = 0.428
16 4 1 1/4 3/4 0.428(3/4) = 0.321
27 3 1 1/3 2/3 0.321(2/3) = 0.214
30 2 1 1/2 1/2 0.214(1/2) = 0.107
32 1 1 1/1 0/1 0.107(0) = 0
```

A couple of things to note about this methodology:

- The person at 3 months who was censored was simply not included - both excluded from the table, and also excluded from the hazard.

To draw a graph for this:

- Y-axis = `S(t)`

- X-axis = Time (months)

- It manifests as a chart that starts at 100%, high on the left side, then zigs downward to the right toward 0% at the bottom right.

- Easy to see that "median survival" is 15 months - 50% of people survive past this, and 50% do not.

## LOG-RANK TEST

A way of testing whether there is a significant difference between two survival curves

113.020.020.40.10 Kaplan-Meier model - Log Rank Test

##### Source:

- Me