Aim: To Calculate Sample Size for Clinical trial: Superiority design (Outcome variable - dichotomous)
Formula Used
Above formula gives sample size for first group (N1).
Sample size for second group is calculated as N2 = N1 * r
P1 = Estimated proportion of success in test group (test drug / therapy) (out of 1) (e.g. 25% = 0.25)
P2 = Estimated proportion of success in control group (Known drug / thearpy etc) (out of 1) (e.g. 15% = 0.15)
δ = Superiority margin (Out of 1) (e.g. 5% = 0.05). Has to be less than the difference between P2 and P1.
r = Controls to Cases ratio
Z1-α = the standard normal deviate corresponding the confidence level
Z 1-β = the standard normal deviate corresponding to 1- β.
What is superiority margin (δ)?
It is the margin by which test arm success proportion should be more than control arm success proportion, to define superiority.
In other words, only if P1 is more than P2 by a margin of δ, superiority of P1 is defined.
Null hypothesis and alternate hypotheis in superiority design are as follows.
H0: P1 - P2 < = δ (implying P1 is not superior)
The alternate hypothesis will be
P1 - P2 > δ (implying P1 is superior to P2; difference between P1 and P2 is more than the superiority margin )
If null hypothesis is rejected, then alternate hypothesis of superiority is accepted.
Example:
Success rate of a known drug is 65%. A new drug is to be tested for superiority with this known drug. The success rate of this test drug is expected to be 75 %. How much sample size shall be required, if superiority margin is decided as 5%, confidence level of 95% and power of 80%.
Solution:
Here
P1 = 75%, P2 = 65%, δ = 5%, confidence level = 95%, power = 80%
δ = 5%, which is less than P1 - P2.
After putting these values, we get required sample size in each group = 1026.
Please note that the superiority margin (δ) has to be less than the difference between P1 and P2.
Mathematically it is possible to calculate the sample size using given formula, even if δ is more than the difference between P1 and P2. However, this will give erroneous results, because even with infinite sample size we can not prove superiority, if δ > P1 - P2.
Because, if the difference between success rates is less than the superiority limit, then it is already assumed that test arm success rate is already below the success rate of control (known) arm + δ. In this situation, whatever sample size we take, it will not be possible to reject the null hypothesis.
Shein-Chung Chow, Jun Shao, Hansheng Wang. Sample Size Calculations in Clinical Research Second Ed. Chapman and Hall/CRC Biostatistics Series 2008.
Xiaofeng Wang, Xinge Ji. Sample Size Formulas for Different Study Designs Supplement Document for Wang, X. and Ji, X., 2020. Sample size estimation in clinical research:
from randomized controlled trials to observational studies. Chest, 158(1), pp.S12-S20.