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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