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 success rates 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 success rate in control group (P2) is 80% and equivalence margin (δ) is set at 10%, then to define equivalence of a test drug, test drug should not have success rate (P1) below 70% or above 90%.
If mean of outcome variable in test (P1) is below 70%, then test drug will be considered as inferior.
If mean of outcome variable in test (P1)is more than 90%, then known drug (P2) 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: P2 - P1 > δ (implying p1 is inferior)
OR
P1 - P2 > δ (implying μ2 is inferior) > δ
The alternate hypothesis will be
P1 - P2 < = | δ | (implying P1 and P2 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:
Suppose an equivalence trial is conducted with 200 participants each in test arm and control arm. The results of the trial gave success rate of 70% in test arm and 75% in control arm. How much was the power of the study to detect equivalence between these two treatments, if equivalence margin is decided as 15%, confidence level of 95%.
Solution:
Here
δ = 15, confidence level = 95%, N1 = 200, N2 = 200, P1 = 70% So exact number of successful outcomes in test arm = 140 , P2 = 75% so exact number of succcful outcomes in control arm = 150.
After putting these values, we get power = 45.03%.
Please note that the equivalence limit (δ) has to be more than the absolute difference between percentage success rates in the test group and control group..
Mathematically it is possible to calculate the power using given formula, even if δ is less than absolute difference between percentage success rates in the test group and control group. However, this will give erroneous results, because even with infinite sample size we can not prove equivalence, if δ < |P1 - P2|and power will be always 0.