Median time is not the same as 50% percentile from survival analysis

[Raed Bahelah]

Dear all,

I am performing a survival analysis and getting percentiles for time-to-event. I checked the summary statistics first for the time variable and then performed the survival analysis but noticed a discrepancy between the median (and 25% percentile) time obtained using the summary statistics and the 50% percentile from the survival output?? Please see the below output.

Summary statistics output

Quantiles (Definition 5)
Level	Quantile
100% Max	103.26361
99%	86.24466
95%	59.77987
90%	50.87614
75% Q3	28.30468
50% Median	14.83408
25% Q1	8.13164
10%	3.77834
5%	1.65918
1%	0.00000
0% Min	0.00000

Survival analysis output

Quartile Estimates
Percent	Point Estimate	95% Confidence Interval
		Transform	[Lower	Upper)
75	.	LOGLOG	52.831	.
50	35.089	LOGLOG	24.444	52.831
25	15.015	LOGLOG	11.959	19.319

Thanks for your clarification.

Raed

[Marc Schwartz]

Hi,

Median survival is defined as the time 't' where the survivorship probability function crosses 0.5. Essentially, the time 't' where the probability of survival is 0.5.

Plot your survival analyses and draw a horizontal line at 0.5 to see where the plotted function crosses 0.5. In your output below, presumably it should be around 35 of whatever time unit you are using.

It is not the simple median of the time to event variable as a continuous measure if that is what your summary table represents.

You can define the median of the time interval as a continuous variable, while the median survival is undefined, if the survivorship function never crosses 0.5. 

One exception to the above would be if you had no censored observations (all patients had the event). In that case they should be close, since the median would be the time at which 50% of your sample has not yet observed the event of interest. 

Regards,

Marc Schwartz

[Raed Bahelah]

Thanks Mark for your detailed email. Yes, the summary stat I'm showing are for the time to event as a continuous variable. I'm having censored observations. You explained in your email but can you (or anyone else) kindly explain in more details why the two should not be the same in case of censored observations? Is that because one represents continuous value while the other a probability? So, in the first case, we say that 25% of the sample failed at 8 months, while in the second case (survival) we say 25% of the sample failed at 15 months? I'm sure I'm missing something in the interpretation!

Thanks for any help.

Raed

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[Marc Schwartz]

Hi Raed,

In the case of censored observations, it is a probability defined via the underlying method used (e.g. Kaplan-Meier versus parametric), so you account for the contributions of the censored observations up to the time point where they are censored.

If you have censored observations and the univariate quantile summary of your continuous time variable is the followup time interval for **all** observations, both those with the event and those censored, then it simply the Xth percentile of the follow up interval for all observations. It is not the Xth percentile of the time to the event. 

You can run the summary on only the subset of those observations that had the event to get the Xth percentile of the time to event, but then you are not considering the contributions of the patients who are censored, which is what the survival analysis is doing and what makes survival analyses arguably unique. Presumably, this is what you have below.

That you can have a survival curve that never gets below 0.5 in the presence of censored observations, thus not have a median survival time defined, should be a trigger for understanding that there are differences between survival probabilities and simple descriptive statistics of a continuous time to event variable. :-)

It might be prudent to Google survival analyses and take time to look at some of the methods used and how censored observations are handled, perhaps easiest for the KM method. If your sample is small enough, you could work through the formula on your data manually, which might provide further insight.

Regards,

Marc

[Raed Bahelah]

Marc,

Thanks a lot for your email. That was helpful.

Best,

Raed