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In my master thesis, I need to determine and calculate the number of cases for median time to event. The method is according to Brookmeyer & Crowley, 1982. My question is: How can I determine the sample size according to Brookmeyer? So determine the number of cases for median time to event. How can I define the equation for N? I know how to calculate the confidence interval, but my problem, how do I determine the case number theoretically for this.

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"Designing the trial with different characteristics: planning a single arm study without historical control. How can I determine the sample size N and what method is the best", this is my plan. Assuming "Median Time to event "PFS" ". I want to determine the sample size N and then calculate it, that's why I thought that I can clearly use or find a formula for N. I firmly assume that the survival time is exponentially distributed I want to see with it: 1- Sample size based on distributional assumptions? 2- No implementation available? How to derive p-value? Thanks for further help best regards

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    $\begingroup$ Please edit the question to say more about why you "need to" do this. Are you designing a study and determining how many participants N you need? If so, how close an estimate of median survival do you need? Almost any N can give you an estimate of median survival so long as you have some cases with events after that time. The precision of the estimate will depend on the sample size. Do you have any information of the overall shape of the survival curve? Might an exponential survival curve be close enough? Please add that information to the question, as comments can be overlooked or lost. $\endgroup$
    – EdM
    Jun 20 '21 at 15:07
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    $\begingroup$ You can only compute the median from an empirical survival curve estimate such as the Kaplan-Meier estimator if more than 1/2 of subjects suffered the event, roughly speaking. And roughly speaking the sample size needed in this case will be similar (but somewhat greater than) the sample size needed to estimate the median using the sample quantile, with a given precision. Look at formulas for that. The needed sample size will be fairly large. $\endgroup$ Jun 20 '21 at 15:16
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    $\begingroup$ It is almost not fair to assume an exponential distribution. The median has an extreme dependence on the shape of distribution assumed. It is true that the sample size for the median in an exponential distribution will give you a lower bound for the sample size but I wouldn't take it too seriously. To get this lower bound, the variance of log $\lambda$ is the reciprocal of the number of events. The median is a simple function of $\lambda$; solve for the number of events that gives a multiplicative margin of error in estimating the median of say 1.25. Backsolve for $n$. $\endgroup$ Jun 21 '21 at 11:37
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    $\begingroup$ @FrankHarrell Thank you for your answer. You are right: It is almost not fair to assume an exponential distribution. But I want to try the sample size based on distributional assumptions and see with simulation how it looks after that. I am also working on the transformation of Kaplan Meier and variance of survival function thanks to the new paper Nagashima, K ;2020 [doi.org/10.1002/pst.2090]. But I wondered which formula to use to determine the sample size. Are we getting it implicitly by estimating the median? $\endgroup$ Jun 21 '21 at 12:07
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    $\begingroup$ Another useful approach might be a simple simulation of a 2-parameter Weibull distribution which has a closed-form expression for the median. $\endgroup$ Jun 21 '21 at 14:20
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How can I determine the sample size according to Brookmeyer? So determine the number of cases for median time to event. How can I define the equation for N?

There is no single equation for the number of cases, N. With even a few cases you can get an estimate of the median survival--it just might not be a very good estimate.

Say you have 3 cases all with known event times. Put them into increasing time order. The middle event time is an estimate of the median survival time. The problem: there is 1 chance in 4 that all 3 event times are either all above or all below the true median, as each case has probability 1/2 of being on one particular side of the true median. Thus there's a good chance that the entire range of your 3 event times has completely missed the true median.

So the question isn't just how to "determine the number of cases for median time to event." It's how to determine the number of cases needed to estimate the median time to event with a desired level of precision. The level of precision is typically taken to be the width of the desired confidence interval.

If you aren't assuming any particular distribution of event times, then you need a non-parametric estimate of that confidence interval. That's what Brookmeyer and Crowley provide in the situation with right-censored event times. It's simplest, however, to start with all cases having events and then see what changes when censoring comes into play.

This answer shows how to get a non-parametric estimate from the order statistics of a sample, the rank-order numbers of the observations. You put the event times into increasing order, and use the properties of the binomial distribution to set the upper and lower limits of the (time-ordered) case numbers that provide the desired confidence for containing the true median. Then the event-time values corresponding to those cases give you the confidence interval for the median estimate.

As Frank Harrell says in comments, those non-parametric confidence intervals can be pretty wide. In the first example under "Discussion" in the answer linked above, if you have 10 ordered event times then the times between the 3rd and the 8th case only gives you an 89% confidence interval even though that interval includes 6 of your 10 events. If you took multiple samples of N = 10, all with events, and used the 3rd and 8th event times as the confidence interval for the median, you would still miss the true median in 11% of those samples. For the larger sample sizes needed to get better precision, you can use the normal approximation to the binomial distribution.

Brookmeyer and Crowley extend this type of analysis to a situation where some event times are right censored. In that situation, those censored cases provides information that helps reduce the variance of the median estimate and the corresponding confidence-interval width. Just how much the censored cases help, however, depends on the details of the distributions of censoring and event times: each censored case only helps at event times up to its own censoring time. So the relative distribution of events and censorings matters.

That's why, if you can't assume a particular parametric distribution for the event times, you are best off simulating both event times and censoring times under different reasonable assumptions, and then seeing how well the Brookmeyer-Crowley or other non-parametric estimates of median survival work at different sample sizes N. The CRAN survival task view lists many tools for simulating survival data. The simsurv package, for example, can simulate event times from standard distributions or a user-defined distribution; you would then have to impose censoring upon the results.

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