Aim: To calculate statistical power with given Sample Size for Clinical trial: Equivalence design (Outcome variable - Ratio)
Formula Used
N = Sample size in test arm = N1
μT = Estimated mean in test (experimental) group (test drug / therapy)
μC = Estimated mean in control group (Known drug / thearpy etc)
SDT = Estimated standard deviation in test (experimental) group (test drug / therapy)
SDC = Estimated Estimated standard deviation in control group (Known drug / thearpy etc)
δ= Equivalence margin
r= Saple size in Control group : Sample Size in Cases group = N2 / N1
Z 1-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 equivalence margin?
The "equivalance" of two drugs does not strictly mean that they are exactly equal in efficacy.
It indicates that the efficacies of these two drugs are very close, close enough so that no one is considered as superior or inferior to other. Or in practical sence " close enough" to say that difference between them is clinically acceptable.
How to quantify this concept of "close enough"? This is conceptalized by the term "equivalence margin". So if the absolute difference between mean of outcome variable of these two drugs are equal to or less than this equivalece margin, then we can say that these drugs / therapies are equivalent.
For example, if mean of outcome variable in control group (μC) is 80 and equivalence margin (δ) is set at 10, then to define equivalence of a test drug, test drug should not have mean of outcome variable (μT) below 70 or above 90.
If mean of outcome variable in test (μT) is below 70, then test drug will be considered as inferior.
If mean of outcome variable in test (μT)is more than 90, then known drug will be considered as inferior.
Hence, in equivalence studies we test two separate hypothesis. Whether known drug is inferior to test drug OR whether test drug is inferior to known drug.
Hence, Null hypothesis in equivalence trial is that test drug is inferior to known drug OR known drug is inferior to test drug.
Which can be written as
H0: μC - μT > δ (implying μT is inferior)
OR
μT - μC > δ (implying μC is inferior) > δ
The alternate hypothesis will be
μT - μC < = | δ | (implying μT and μC are equivalent)
If null hypothesis is rejected, then alternate hypothesis of equivalence is accepted.
By now, you must have realised that, equivalence test requires testing of two one sided hypothesis (TOST).
Example:
In an equivalence trial, mean reduction in SBP with a known antihypertensive drug is 30 ± 6 mm of Hg. A new antihypertensive drug is tested for equivalence with this known drug. The mean reduction in SBP with this test drug is 25 ± 6 mm of Hg. How much was the power of the study to detect equivalence, if sample size in each group is 30, equivalence margin is decided as 10 mm of Hg, confidence level of 95%.
Solution:
Here
μT = 25, μC = 30, SDT = SDC = 6, δ = 10, confidence level = 95%, N1 = NT = 30.
After putting these values, we get required sample size in each group = 88.65%
Please note that the equivalence limit (δ) has to be more than the absolute difference between μT and μC.
Mathematically it is possible to calculate the power using given formula, even if δ is less than absolute difference between μT and μC. However, this will give erroneous results, because even with infinite sample size we can not prove equivalence, if δ < |μT - μC|, so power will be always 0.