Measures used in survival analysis
1. The survival function
In our last video, we learned about why we need special methods for survival analysis.2. Remember the example
In particular, we learned about censoring. Usually, in regression, we think about densities and distribution functions. In survival analysis, we are often interested in what we call the survival curve or survival function.3. Survival analysis questions
The reason for the popularity of the survivor function is that it essentially answers the main questions in survival analysis: For example, with the survival function, I can answer questions like: What is the probability that a breast cancer patient survives longer than 5 years? What is the typical waiting time for a cab? Out of 100 unemployed people, how many do we expect to have a job again after 2 months? This is the reason why, in this course, we want to focus on this powerful function.4. Survival function
The survival function is so popular because it has such a straightforward interpretation. In the survival context, the survival function gives the probability to survive beyond a time point small t. If you are familiar with the cumulative distribution function: the survival function is just 1 minus the cumulative distribution function. The survival function is a function over time and for any point in time you can say how probable it is to survive longer than that point in time.5. Survival function
For example, if we talk about survival, the survival function can tell me, for any time point t, what the probability is to survive longer than t. Or for the cab example, the survival function tells me the probability that the cab takes more than t minutes to arrive. So for this example curve here, the probability that it takes more than 2 minutes is almost certain - almost 1, but we are quite sure that it will not take longer than 6 minutes. Note that I am using different scales to explain the two examples. In the survival example, the time on the x-axis is in years and here in the cab example, the x-axis is in minutes.6. Survival function
We can also look at the survival function from the other direction, by fixing a certain quantile. Most popular is looking at the median. The dashed line shows that the median duration time - so the time corresponding to the 50% quantile - is 3-point-7. So for our two examples, a possible interpretation would be: The median survival time is 3-point-7 years. Or in the cab example: The median time until the cab arrives is 3-point-7 minutes. That means that half the cabs take less than or equal to 3-point-7 minutes to arrive at your house and the other half take more than 3-point-7 minutes.7. Survival function
The survival curve also gives us the percentage of durations taking longer than t. If we look at t=4, we see that the survival probability is 0-point-37, in other words, 37%. For the survival example, that means that 37% of all patients survive longer than 4 years and 73% (that is 100-37) die within the first 4 years. In the cab example, we could say that out of 100 cabs, 37 take more than 4 minutes to arrive.8. Let's practice!
Now let's practice interpreting survival curves.Create Your Free Account
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