1. Definition
- Compromise Power Analysis is a method for determining an appropriate balance between:
- Type I error (α) → false positives, and
- Type II error (β) → false negatives,
given a fixed sample size (n).
Instead of fixing α and power (1 – β) ahead of time (like in a priori analysis), you already know n and ask:
- “Given my sample size, what α and β trade-off makes sense?”
2. Why It’s Useful
- Sometimes you can’t change n (e.g., limited participants, budget, or historical dataset).
- A priori analysis would tell you “you need 500 subjects” but you only have 200.
- Compromise analysis answers: “With 200 subjects, what α and β should I set to have a balanced test?”
3. How It Works
- Define effect size (δ) of interest.
- Provide the available sample size (n).
- Set a desired ratio of Type I to Type II error rates (often 1:1).
- Example: α = β, or α = 0.5β, etc.
- The analysis finds values of α and β that satisfy the constraint.
4. Example
Suppose:
- Effect size = medium (Cohen’s d = 0.5).
- Available n = 40 (20 per group).
- Want α and β to be equal (α = β).
Compromise power analysis might yield:
- α = 0.12
- β = 0.12 (so power = 0.88)
Interpretation: Instead of forcing α = 0.05, you accept a higher Type I error rate (12%) to balance with Type II error, given your small sample.
5. Comparison with Other Analyses
| Feature | A Priori | Post Hoc | Compromise |
|---|---|---|---|
| Input | Effect size, α, power | Observed n & effect size | Effect size, available n, error ratio |
| Output | Required n | Achieved power | Appropriate α & β |
| Use | Planning sample size | Explaining sensitivity after study | Balancing errors when n is fixed |
6. Limitations
- Less commonly reported in papers (journals prefer α = 0.05 convention).
- May be harder to justify to non-technical audiences.
- Still depends on assumed effect size (which might be uncertain).
7. Key Takeaway
- A priori: “How many subjects do I need?”
- Post hoc: “Given what I saw, what was the power?”
- Compromise: “I only have this many subjects — how should I balance α and β?”
In short:
Compromise power analysis is used when your sample size is fixed, and you want to find a reasonable balance between Type I and Type II error rates instead of strictly fixing α at 0.05. It’s a pragmatic approach when data collection is limited.