Aim: To calculate statistical power gained with given sample Size for Clinical trial: Non-inferiority design (Outcome variable - dichotomous)
Formula Used
N= Samle size in test group = N1
P1 = Proportion of success in test group (out of 1) (e.g. 25% = 0.25) = Successful outcomes in test group / Sample Size in test group
P2 = Proportion of success in control group (out of 1) (e.g. 15% = 0.15) =Successful outcomes in control group / Sample Size in control group
δ= Non-inferiority margin (Out of 1) (e.g. 5% = 0.05). Has to be more than the difference between P2 and P1.
r= Sample size in Control group : Sample Size in Cases group = N2 / N1
Z1-α = the standard normal deviate corresponding the confidence level
Φ(x)=P(Z ≤ x). It is the cumulative distribution function (CDF) of normal distribution. Simply, it is the area of the standard normal curve towards left side of x.
What is non-inferiority margin (δ)?
It is the clinically acceptable difference between P2 and P1 to define non-inferiority.
In other words, even if P1 is less than P2, but the difference is less than δ, then P1 can be considered as non-inferior to P2.
Null hypothesis and alternate hypotheis in non-inferiority design are as follows.
H0: P2 - P1 > δ (implying P1 is inferior)
The alternate hypothesis will be
P2 - P1 < = δ (implying P1 is not inferior to P2; difference between P2 and P1 is less than the clinically acceptable non-inferiority margin )
If null hypothesis is rejected, then alternate hypothesis of non-inferiority is accepted.
Example:
A new drug is to be tested for non-inferiority with a known (control) drug. The success rates of this test drug and known drug are be 60 % and 65%, respectively, in a non-inferiority trial conducted with 200 participants in each group.. How much was the power to detect non-inferiority, if non-inferiority margin is decided as 10%, confidence level of 95%.
Solution:
Here
P1 = 60% so exact number of successful outcomes = 120, P2 = 65% so exact number of successful outcomes = 130, δ = 10%, confidence level = 95%,N 1 = N2 = 200.
δ = 10%, which is more than P2 - P1.
After putting these values, we get power = 27.07%.
Please note that the non-inferiority limit (δ) has to be more than the difference between P2 and P1.
Mathematically it is possible to calculate the power using given formula, even if δ is less than difference between P2 and P1. However, this will give erroneous results, because even with infinite sample size we can not prove non-inferiority, if δ < P2 - P1 and power will always be 0.
Shein-Chung Chow, Jun Shao, Hansheng Wang. Sample Size Calculations in Clinical Research Second Ed. Chapman and Hall/CRC Biostatistics Series 2008.