# Example comparing software for evaluating sample sizes for log rank test

We give a worked examples comparing the total number of participants required using the Schoenfeld (1983) formula, WINPEPI and web calculator software based upon Machin et al.

Using the formula of Schoenfeld (1983) for 80% power and a type I error of 5% and comparing equal sized groups we have z(0.05/2) = 1.96 and z(0.8) = 0.84.

Number of amyloid events = $$ 4 (1.96 + 0.84)^{2 } / [log (log(0.75)/log(0.37))]^{ 2 }$$ = 20.39.

In particular for comparing equally sized groups from Collett (1983)

The total number of people required = 1-([S(1)+S(0)]/2) x number of events

(with the number of deaths worked our using WINPEPI or the Schoenfeld (1983) formula)

where S(0) and S(1) are the two survival probabilities being compared at a particular time.

[1/[1 - (0.75+0.37)/2]] x 20.39 = 47 participants needed in total to compare survival rates of 0.75 and 0.37 with 80% power and a type I error rate of 5%.

Using Compare2 in WINPEPI with a hazard ratio of log(0.75)/log(0.37)=0.29 and 80% power, type I error of 5% and equal sized group gives the required number of amyloid events as 26 for the log rank test. Multiplying by a factor of [1/[1 - (0.75+0.37)/2]] gives a total 59 participants required. The web calculator inputting survival rates of 0.75 and 0.37 gives a total of 60 required.