R Tutorial : The survival function

Key Takeaways

in our last video we learned about why we need special methods for survival analysis 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 the reason for the popularity of the survival 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 where what is the typical waiting time for a cab or 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 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 the 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 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 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 a survival example the time on the x axis is in years and here in the calves example the x axis is in minutes 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 3.7 so for our two examples a possible interpretation would be the median survival time is three point seven years or in a cab example the median time until the cab arrives is three point seven minutes that means that half the cabs take less than or equal to three point seven minutes to arrive at your house and the other half takes more than three point seven minutes the survival curve also gives us the percentage of durations taking longer than T if we look at T equal to 4 we see that the survival probability is 0.37 in other words 37% for the survival example that means that 37% of all patients survive longer than four years in 73 percent that is 100 minus 37 die within the first four years in the cab example we could say that out of 100 caps 37 take more than four minutes to arrive now let's practice interpreting survived