- The project is out-of-sync -- use `renv::status()` for details.Survival Analysis
Introduction to the analysis of time-to-event data
June 21, 2025
Survival analysis?
Survival analysis is studied distinctly from other topics since survival data have some important distinguishing features.
Nonetheless, our tour of survival analysis will (re)visit many of the topics we have already learned considered so far
- estimation
- one-sample inference
- two-sample inference
- regression
Survival analysis is the area of (bio)statistics that deals with time-to-event data.
Survival analysis?
“Survival” analysis isn’t restricted to analyzing time until death: Applicable whenever the time until study subjects experience a relevant endpoint is of interest.
The terms “failure” and “event” are used interchangeably for the endpoint of interest.
Example: Researchers record time from diagnosis until death for a sample of patients diagnosed with laryngeal cancer.
- How would you summarize the survival of these patients?
- Which of the summary measures discussed thus far seem appropriate or applicable?
Summarizing survival data
What if the mean or median survival time were reported?
- The sample mean is not robust to outliers: Did everyone live a little bit longer or just a few people live a lot longer?
- The median is just a summary measure: If considering your own prognosis, you’d want to know more!
What if survival rate at
- How to choose the cutoff time
- Also, still just a single summary measure…
- How to choose the cutoff time
Measuring survival times
Denote the measured time by the RV
Time is recorded relative to when a person enters the study, not according to the calendar.
- A study unit reaches
- Time is relative to a given unit’s own entry into the study: It doesn’t matter how far along others are
- A study unit reaches
Measuring survival times
The probability of an individual in our population surviving past a given time
Here, we are interested in inference about a function, not just a single population value (e.g.,
Can we estimate
- If only it was as simple as it sounds…
When data go missing: Censoring
Analyses of survival times usually include censored/missing data; valid inference with missing data is a topic of ongoing research in statistics – generally, assumptions are required.
Time-to-event data exhibit a particular form of missingness: units are lost to follow-up (or the study ends) prior to their experiencing the event of interest; this is called right-censoring.
Censored data provide partial information: we don’t know how long a patient lived, but we know they lived at least as long as the time elapsed prior to being lost to follow-up.
The censoring problem
Why would a person be lost to follow-up? They could have…
- moved to another city
- withdrawn from the study
- died of a different cause
- still be in the study without an event at time of analysis
Critical assumption: Censoring is non-informative—that is, assume that being lost to follow-up is unrelated to prognosis. (Is this plausible?)
If this assumption can’t be made, inference becomes more complicated and requires strong(er) assumptions.
The censoring problem
Counterexample: Researchers administer a new chemotherapy drug to 10 cancer patients to assess survival while on the drug.
- 5 patients can’t tolerate the side effects, drop out of study
- Those who dropped out were likely more ill—thus, shorter survival times disproportionately removed from the sample (bias?)
If non-informative censoring were assumed, the drug would probably appear falsely impressive.
Estimating the survival curve
The Kaplan-Meier estimator (Kaplan and Meier 1958) gives an estimate of
For specific times
Probability of surviving to time
What if we applied this recursive trick repeatedly?
Estimating the survival curve
Survival function
Without censoring:
Note: a patient alive but censored at
- That patient was not eligible to die during the interval from
- When a patient (study unit) is ineligible to die during a given interval, we say that that patient is not in the risk set
- That patient was not eligible to die during the interval from
Estimating the survival curve
Under censoring, we estimate
Denominator counts those at risk for the event at time
With independent censoring, we can invoke the logic of conditional probability:
Estimating the survival curve
For convenience, define the following quantities
Now, notice that
Estimating the survival curve
From this, at observed times
- Estimated survival curve
- Estimated survival curve
Estimating the survival curve: Worked example
Consider
year fail censored 2 7 2 4 16 5 6 19 8 8 14 7 10 11 4 12 5 2
Computing estimates survival probabilities with the KM estimator just applies the formula:
- and so on…
The Kaplan-Meier Estimator
Estimated survival curves – in action!
Inference for the Kaplan-Meier estimator
Estimating
Using a log-transformation for this improves upon the normal approximation. A step-by-step sketch:
- Take logarithm of
- Confidence interval (CI) for
- Convert back to CI for
- Take logarithm of
Inference for the Kaplan-Meier estimator
- At
- So, at a given time
- So, the
- In our example:
Inference for the Kaplan-Meier estimator
- Then, applying Greenwood’s formula,
- And the
Hypothesis testing: The log-rank test
In addition to estimating survival, we may want to compare two groups’ survival functions formally (i.e., via a hypothesis test).
Rather than testing for differences in a summary measure (e.g., mean in groups 1 vs. 2), we compare the entire survival curves.
- The test for this is called the Log-Rank Test.
- But what does it mean to say “two curves are the same”?
To precisely define what we’re testing, we need to define a new function called the hazard function,
Interlude: The hazard function
The hazard function
Equivalently, this is the instantaneous conditional event rate
In discrete time, this is the probability of an event occurring at time
Interlude: Hazard to survival (and back again…)
Special relationships of the survival function
See Freedman (2008) for an overview of these critical relationships.
The log-rank test
Hazard functions are natural constructs for handling censoring.
Formulate null and alternative hypotheses to test whether two groups have different survival functions based on the hazard:
The log-rank test subdivides time into
Regression (is all you need?)
For continuous outcomes (
For binary outcomes (
What about for survival data? Any ideas?
- What is the outcome that we observe?
- A failure time (
- Not quite: The minimum of these, i.e.,
- What is the outcome that we observe?
The Cox model: Regression (is all you need?)
Survival curves don’t necessarily follow a given distribution…
- Sometimes a distribution will be assumed, but inference can be seriously distorted if the distribution is not correct.
- Is the survival time really exponential, Weibull, etc.?
Cox’s proportional hazards model is a special semiparametric regression heavily used in survival analysis (Cox 1972).
Assumption: hazard
The Cox model: Regression (is all you need?)
In this regression model,
Given hazard rate
Proportional hazards: Unit change in covariate
We’ve said nothing about the baseline hazard rate
References
Bland, J Martin, and Douglas G Altman. 2004. “The Logrank Test.” The BMJ 328 (7447): 1073. https://doi.org/10.1136/bmj.328.7447.1073.
Cox, David R. 1972. “Regression Models and Life-Tables.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 34 (2): 187–202. https://doi.org/10.1111/j.2517-6161.1972.tb00899.x.
Freedman, David A. 2008. “Survival Analysis: A Primer.” The American Statistician 62 (2): 110–19. https://doi.org/10.1198/000313008X298439.
Kaplan, Edward L, and Paul Meier. 1958. “Nonparametric Estimation from Incomplete Observations.” Journal of the American Statistical Association 53 (282): 457–81.
9
1
›
This component is an instance of the CodeMirror interactive text editor. The editor has been configured so that the Tab key controls the indentation of code. To move focus away from the editor, press the Escape key, and then press the Tab key directly after it. Escape and then Shift-Tab can also be used to move focus backwards.
'citation()' on how to cite R or R packages in publi
cations.
Type 'demo()' for some demos, 'help()' for on-line h
elp, or
'help.start()' for an HTML browser interface to help
.
Type 'q()' to quit R.
>
WebR is downloading, please wait...
R version 4.4.1 (2024-06-14) -- "Race for Your Life"
Copyright (C) 2024 The R Foundation for Statistical Computing
Platform: wasm32-unknown-emscripten (32-bit)
R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
>
WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW
'citation()' on how to cite R or R packages in publi
cations.
Type 'demo()' for some demos, 'help()' for on-line h
elp, or
'help.start()' for an HTML browser interface to help
.
Type 'q()' to quit R.
>
HST 190: Introduction to Biostatistics
Survival Analysis Introduction to the analysis of time-to-event data Nima Hejazi nhejazi@hsph.harvard.edu Harvard Biostatistics June 21, 2025