#### Sample size for a survival data model (2004-05-13)

I got an email from Japan recently with an interesting question. By the way, one of the fun things about my web pages is that I get emails from all over the world. I can only offer a bit of help by email, and it can't really replace a face-to-face meeting. Still, I'm glad to spend a few minutes trying discussing general principles and offering a reference or two. People are very grateful for the time, and I've had an offer for free room and board should I ever visit Rome. I also am getting a copy of Opera News on CD in thanks for a bit of advice I provided.  [Conflict of interest alert: should I ever publish any research on opera, I will have to disclose this non-monetary compensation.]

The question was about an analysis of mortality for children under 5 years of age. There were 1992 patients and 272 of them died. I was asked if this was an adequate sample size and whether there was a problem because the median survival time was unavailable for some of the subgroups.

The issue of median survival times is an easy one to discuss. A group lacks a median survival time if the survival curve never dips below 50%. So lack of a median is actually good news for that particular subgroup. You can still have a lot of precision in a survival model even when you cannot compute the median survival for some or all groups.

Sample size is a tricky problem, and it is especially tricky for survival data models. As a rough rule, it is the number of deaths rather than the number of patients being studied that is most important. But patients who do not die will still contribute to precision, up to the point at which they drop out of the study. The words "drop out" imply that the person left the study, but it could also be someone who switches to a different therapy that makes them ineligible for continuing follow-up. Early dropouts are of very little use, then, and a patient who drops out before the first death contributes absolutely nothing to the precision of the findings. Patients who stay in the study for a long time are very valuable.

A formal power calculation would have to look at how many patients start the study at the very beginning, the rate at which additional patients join during the study (accrual rate), the rate at which patients drop out of the study (attrition rate), and the rate at which they die (mortality rate). You can make some simplifying assumptions, but generally, power calculations for a survival data model are a mess. Here are some references.

1. Sample size calculations for the log rank test: a Gompertz model approach. A. B. Cantor. J Clin Epidemiol 1992: 45(10); 1131-6. [Medline]
2. Planning the size and duration of a clinical trial studying the time to some critical event. S. L. George, M. M. Desu. J Chronic Dis 1974: 27(1); 15-24. [Medline]
3. A simple method of sample size calculation for unequal-sample-size designs that use the logrank or t-test. F. Y. Hsieh. Stat Med 1987: 6(5); 577-81. [Medline]
4. Sample-size calculations for the Cox proportional hazards regression model with nonbinary covariates. F. Y. Hsieh, P. W. Lavori. Control Clin Trials 2000: 21(6); 552-60. [Medline]
5. Sample Sizes Based on the Log-Rank Statistic in Complex Clinical Trials. Edward Lakatos. Biometrics 1988: 44(1); 229-241. [Medline]
6. Design and Analysis of Multiarm Clincial Trials with Survival Endpoints. Ping-Yu Lin, Steve Dahlberg. Controlled Clinical Trials 1995: 16(2); 119-130. [Medline]
7. Sample size determination under an exponential model in the presence of a confounder and type I censoring. K. J. Lui. Control Clin Trials 1992: 13(6); 446-58. [Medline]
8. Sample size requirements for comparing time-to-failure among k treatment groups. R. W. Makuch, R. M. Simon. J Chronic Dis 1982: 35(11); 861-7. [Medline]
9. Sample size calculation for complex clinical trials with survival endpoints. J. H. Shih. Control Clin Trials 1995: 16(6); 395-407. [Medline]

You could also use the rule of 50 as a rough guide to whether the sample size is adequate. It's a concept that I first learned about from the Statistical Rules of Thumb web site http://www.vanbelle.org/.

Unfortunately, I can't find the details of the rule of 50 anymore. The derivation is quite simple, and I have all the details at

Would the rule of fifty apply to survival data? I think so, but I would have to do the math.

Another rule to consider is that you want to have 10 to 15 observations per independent variable. There's a couple of provisos to this statement. For survival analysis or logistic regression it is probably 10 to 15 events (deaths). Also it is not the number of independent variables in your final model, but the number that you examined along the way to the final model.

This rule of 10 to 15 observations per variable is widely quoted, but as far as I can tell, there is only one study which validated the rule. Frank Harell showed that stepwise logistic regression provides unstable and poorly reproduced findings when the number of independent variables is more than 10% of the number of outcomes.

Regression modeling strategies for improved prognostic prediction. Frank E. Harrell. Statistics in Medicine 1984: 3143-152.

This page was written by Steve Simon while working at Children's Mercy Hospital. Although I do not hold the copyright for this material, I am reproducing it here as a service, as it is no longer available on the Children's Mercy Hospital website. Need more information? I have a page with general help resources. You can also browse for pages similar to this one at Category: Sample size justification.