Calculating median survival time in control group for sample size calculation I am confused about some maths in the help for calculating sample size for time to event data in the PS Sample Size calculator.
One of the fields you need to enter is the m1 paramter, which is listed in the help as 'the median survival time on control treatment'. The help goes on to say
"If you do not have a direct estimate of m1, proceed as follows. Let p be the probability that a control subject survives until some time t. Then we can estimate m1 by"
$m1 = t\log_e(1/2)/log_e(p)$
So for a toy problem say we have a trial with a 2-year maximum follow-up, 6 month uniform enrollment period, hazard of 0.1 per 1-person year for the treatment group, hazard of 0.2 per 1-person year for the control group, drop out hazard 0.1. per 1-person year, alpha of 0.025 (1 sided), power of 0.9 (default, beta = 0.1).
Based on the above equation the median survival time for the control m1, subbing 2 for t and 0.2 for p should be
$m1 = 2\log_e(1/2)/log_e(0.2) = 2*log(0.5)/log(0.2) = 2*-0.69/-1.61 = 0.86$
However in the nSurvival function from the gsDesign package in R, which should perform the same analysis when we run the function like so
ss <- nSurvival(lambda1 = 0.2, ## hazard rate placebo
                lambda = 0.1,  ## hazard rate treatment
                eta = 0.1,     ## equal dropout rate for both groups
                Ts = 2,        ## maximum study duration
                Tr = 0.5,      ## accrual duration
                sided = 1,
                alpha = 0.025,
                ratio = 1) 

ss

We get the following output
Fixed design, two-arm trial with time-to-event
outcome (Lachin and Foulkes, 1986).
Study duration (fixed):          Ts=2
Accrual duration (fixed):        Tr=0.5
Uniform accrual:              entry="unif"
Control median:      log(2)/lambda1=3.5
Experimental median: log(2)/lambda2=6.9
Censoring median:        log(2)/eta=6.9
Control failure rate:       lambda1=0.2
Experimental failure rate:  lambda2=0.1
Censoring rate:                 eta=0.1
Power:                 100*(1-beta)=90%
Type I error (1-sided):   100*alpha=2.5%
Equal randomization:          ratio=1
Sample size based on hazard ratio=0.5 (type="rr")
Sample size (computed):           n=430
Events required (computed): nEvents=91

Especially notice the line of the output that states the control median as log(2)/lambda1 where lambda1 is the hazard in the control group (i.e 0.2). The value for this equation is 3.5. Very different from my value of 0.86. However, when I enter this value of 3.5 into the m1 field in the PS software it returns an estimated required sample size of n=201 for each arm, 402 in total, quite similar to the estimate from the gsDesign package.
So my question is where am I going wrong when I calculate the median survival time for the control subject based on the PS equation? Why am I getting such different results (0.86 vs 3.5) and why are the two equations different in gsDesign and the PS software?

The method for estimating median survival (based on an underlying exponential survival curve) specifies:

Let p be the probability that a control subject survives until some time t. (Emphasis added)

You expect only a "hazard of 0.2 per 1-person year for the control group."  Yet you did the following:

subbing 2 for t and 0.2 for p

which is inconsistent with the method in two ways.
First, the probability of survival to 1 year is 0.8, not 0.2. Second, you used 2 years for the time, while that's the the estimated survival for 1 year. It's not surprising that your estimate of median survival was less than 1 year, contrary to your assumption of 80% survival at 1 year.
I get a median survival of 3.1 years based on yourstated assumptions. Quick rough sanity check: at 0.8 survival per year, survival over three years should be approximately $0.8^3 = 0.51$.
