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# Trial Designs—Non-inferiority vs. Superiority vs. Equivalence

The primary purpose of a clinical trial is to address a scientific hypothesis. To address a hypothesis, different statistical methods are used depending on the type of question to be answered. Most often the hypothesis is related to the effect of one treatment as compared to another. For example, one trial could compare the effectiveness of a new antibiotic to that of an older antibiotic. Yet the specific comparison to be used will depend on the hypothesis to be addressed. Let’s use this two antibiotic example for this discussion.

We can construct 3 possible hypothesis to be addressed when comparing these two antibiotics. The three hypothesis are:

1. The New Antibiotic is at least as good as the Old Antibiotic.
2. The New Antibiotic is better than the Old Antibiotic.
3. The New Antibioitic is equivalent to the Old Antibiotic.

While each of these hypothesis may seem similar, they are slightly different scientific questions, thus each requires a slightly different statistical test. For the first hypothesis (“at least as good as”), the New Antibiotic must be as good as the Old Antibiotic, but it can also be better than the Old Antibioitic. In mathematical terms:

1. $New Antibiotic \ge Old Antibiotic$

For the second hypothesis (“better than”), the New Antibiotic can no longer be equivalent to the Old Antibiotic. In mathematical terms:

(ダイアルインID：2）$New Antibiotic \textgreater Old Antibiotic$

The last hypothesis (“equivalent to”) indicates that the New Antibiotic cannot be worse than or better than the Old Antibiotic. In mathematical terms:

(ダイアルインID：3）$\frac{New Antibiotic}{Old Antibiotic} = 1\pm \alpha$

As you can see, each hypothesis contains a slightly different mathematical arrangement. These mathematical expressions have been given “lay names” by satisticians in an effort to describe the math to non-statisticians; these names are:

1. Non-inferiority
• $New Antibiotic \ge Old Antibiotic$
2. Superiority
• $New Antibiotic \textgreater Old Antibiotic$
3. Bioequivalence
• $\frac{New Antibiotic}{Old Antibiotic} = 1\pm \alpha$

Of these three comparisons, the non-inferiority has the largest range of successful trial outcomes (equivalence or superiority). Thus a calculated sample size for a non-inferiority trial is usually the smallest of the three hypothesis. The superiority comparison is a subset of the non-inferiority and will have a sample size that is similar to the non-inferiority or a sample size that is much larger. As the expected difference between the two treatments decreases, the sample size will increase, often dramatically. Finally, the bioequivalence comparison is the most restrictive because it requires that the two treatments be identical within some acceptable range defined by α (normally ±20%). In general a bioequivalence trial will have a sample size that is larger than a non-inferiority trial.

So which is best to use? It all depends on which scientific question you are trying to answer. All three study types are useful in the development of drugs. Non-inferiority studies are used to show that a minimum level of efficacy has been achieved. In comparison studies with a current therapy, non-inferiority is used to demonstrate that the new therapy provides at least the same benefit to the patient. Superiority trials are always used when comparisons are made to placebo or vehicle treatments. In these studies, it is critical that the effect in the treatment group be clearly superior to any effects in the placebo groups. Failure to demonstrate superiority over vehicle suggests that the drug is not effective. Superiority trials are also used for marketing purposes (“our drug is better than your drug” studies). Bioequivalence trials are used to show that a new treatment is identical (within an acceptable range) to a current treatment. This is used in the registration and approval of generic drugs that are shown to be bioequivalent to their branded reference drugs.

In the end, ask yourself which hypothesis am I trying to address, then use the appropriate study design. Best of luck!

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### 筆者について

By: Nathan Teuscher
Dr. Teuscher has been involved in clinical pharmacology and pharmacometrics work since 2002. He holds a PhD in Pharmaceutical Sciences from the University of Michigan and has held leadership roles at biotechnology companies, contract research organizations, and mid-sized pharmaceutical companies. Prior to joining Certara, Dr. Teuscher was an active consultant for companies and authored the Learn PKPD blog for many years. At Certara, Dr. Teuscher developed the software training department, led the software development of Phoenix, and now works as a pharmacometrics consultant. He specializes in developing fit-for-purpose models to support drug development efforts at all stages of clinical development. He has worked in multiple therapeutic areas including immunology, oncology, metabolic disorders, neurology, pulmonary, and more. Dr. Teuscher is passionate about helping scientists leverage data to aid in establishing the safety and efficacy of therapeutics.