10. Sample Size and Power I: The Ingredients and the Two Formulas
After this lesson you can
- Name the four ingredients and identify the one that is a clinical judgment
- Define type one and type two errors and power
- Derive the 16 and 21 multipliers from the z values
- Compute a per-arm sample size for a binary and a continuous endpoint
- Explain why an underpowered non-significant result is inconclusive, not negative
Video coming soon
Lessons publish daily from November 17, 2026.
This lesson introduces the four ingredients of a sample-size calculation, defines alpha, beta and power, and reads a power curve. It derives the shortcut multipliers of 16 and 21 and computes per-arm sizes for the fictional Oakhaven binary and continuous endpoints.
Questions and answers
Questions a learner would ask about this lesson. Think through your own answer before you open each one.
- What are the four ingredients of a sample-size calculation?
Show answer
Alpha (the false alarm rate you accept), power (the chance of seeing a real effect), the difference worth detecting, and the variability of the endpoint. Alpha and power are conventions, the variability is an estimate, and the difference worth detecting is the one clinical judgment. It is the smallest difference that would change practice.
- What do alpha, beta and power mean?
Show answer
Alpha is the probability of declaring a difference when there is none, a type one error, set at a two-sided point zero five. Beta is the probability of missing a real effect, a type two error, usually point two or point one. Power is one minus beta, so eighty or ninety percent.
- Where do the shortcut multipliers 16 and 21 come from?
Show answer
For eighty percent power, the z values are 1.96 and 0.84; their sum is 2.80, squared is 7.84, and doubled is 15.68, rounded to 16. For ninety percent power, 1.96 plus 1.28 is 3.24, squared is about 10.5, and doubled is about 21. Each shortcut assumes a two-sided alpha of point zero five.
- How big must the Oakhaven trial be for its primary endpoint?
Show answer
Usual care is expected at 40 percent poor control and the program at 20 percent. The average proportion is 0.30, so the variance term is 0.30 times 0.70, which is 0.21, and the squared difference is 0.20 squared, which is 0.04. At eighty percent power, 16 times 0.21 over 0.04 is 84 per arm; at ninety percent, 21 times 0.21 over 0.04 is 110.25, rounded up to 111 per arm.
- If a trial turns out underpowered and not significant, does that mean the program does not work?
Show answer
No. If the true difference is smaller than planned, power falls, and a non-significant result is inconclusive, not negative. The confidence interval still includes a worthwhile effect. A result is negative only when the interval excludes the effect worth having.
Practice quiz
- Question 1Which ingredient is a clinical judgment rather than a convention or an estimate?
- Question 2Power is defined as:
- Question 3At eighty percent power and two-sided alpha of point zero five, the shortcut multiplier is about:
- Question 4For 40 versus 20 percent poor control at eighty percent power, what is the per-arm sample size?
- Question 5At ninety percent power, the same comparison gives 21 times 0.21 over 0.04, which is 110.25. What per-arm size goes in the protocol?
- Question 6For the symptom score, with a difference of 8 and a standard deviation of 20 at eighty percent power, the per-arm size is:
- Question 7The sample-size formula gives 84 per arm for a two-arm trial. How many participants does the trial need in total, before inflation for loss?
- Question 8Why should a pilot's observed effect not be used as the difference worth detecting?
- Question 9If the true effect is smaller than the one the trial was planned for, the trial's power:
- Question 10On a power curve, what is the height at zero true difference?
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Design of Clinical Studies is education only. Every lesson follows the fictional Oakhaven Adherence Trial with invented numbers; the course uses no real data and cites no real studies.