Aim: To Calculate Sample Size for Clinical trial: Equivalence design (Outcome variable - Ratio)

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What is equivalence margin (δ)?

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 (μ_C) 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 (μ_T) below 70 or above 90.

If mean of outcome variable in test (μ_T) is below 70, then test drug will be considered as inferior.

If mean of outcome variable in test (μ_T)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: μ_C - μ_T > δ (implying μ_T is inferior)
OR
μ_T - μ_C > δ (implying μ_C is inferior) > δ

The alternate hypothesis will be

μ_T - μ_C < = | δ | (implying μ_T and μ_C 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).

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Example:

Mean reduction in SBP with a known antihypertensive drug is 30 ± 6 mm of Hg. A new antihypertensive drug is to be tested for equivalence with this known drug. The mean reduction in SBP with this test drug is expected to be 25 ± 6 mm of Hg. How much sample size shall be required, if equivalence margin is decided as 8 mm of Hg towards both sides of known drug, confidence level of 95% and power of 80%.

Solution:

Here

μ_T = 25, μ_C = 30, SD_T = SD_C = 6, δ = 8, confidence level = 95%, power = 80%

After putting these values, we get required sample size in each group = 69.

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Please note that the equivalence limit (δ) has to be more than the absolute difference between μ_T and μ_C.

Mathematically it is possible to calculate the sample size using given formula, even if δ is less than absolute difference between μ_T and μ_C. However, this will give erroneous results, because even with infinite sample size we can not prove equivalence, if δ < |μ_T - μ_C|.

Because, if absolute difference is more than the equivalence limit, then it is already assumed that one of the two treatment's mean is more than the other by a margin more than acceptable. In this situation, whatever sample size we take, it will not be possible to reject the null hypothesis.