Equivalence design (Outcome binary): Help

Aim: To Calculate Sample Size for Clinical trial: Equivalence 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 (out of 1) (e.g. 25% = 0.25)

P₂ = = Estimated proportion of success in control group (out of 1) (e.g. 15% = 0.15)

δ= Equivalence margin (Out of 1) (e.g. 5% = 0.05).

r= Controls to Cases ratio

Z _1-α = the standard normal deviate corresponding the confidence level

β = 1 – Power (Out of 1) (e.g. 80% = 0.8)

Z _1-β/2 = the standard normal deviate corresponding to 1- β/2.

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 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 success rate in test group (P₁) is below 70%, then test drug will be considered as inferior.

If success rate test group (P₁) is more than 90%, then control drug (P₂) will be considered as inferior.

Hence, in equivalence studies we test two separate hypotheses. 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 a known therapy has success rate of 60%, and a new therapy is expected to give success rate of 55%. We want to test the equivalence of new therapy with known therapy. How much sample size shall be required, if equivalence margin is decided as 10%, confidence level of 95% and power of 80% (equal sample size in both the groups).

Solution:

Here

P₁ = 55%, P₂ = 60%, δ = 10, confidence level = 95%, power = 80%, r = 1.

After putting these values, we get required sample size in each group = 1670.

Please note that the equivalence limit (δ) has to be more than the absolute difference between P₁ and P₂.

Mathematically it is possible to calculate the sample size using given formula, even if δ is less than absolute difference between P₁ and P₂. However, this will give erroneous results, because even with infinite sample size we can not prove equivalence, if δ < |P₁ - P₂|.

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.

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.