Compromise Power Analysis

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?”

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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?”

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3. How It Works

1. Define effect size (δ) of interest.
2. Provide the available sample size (n).
3. Set a desired ratio of Type I to Type II error rates (often 1:1).
   • Example: α = β, or α = 0.5β, etc.
4. The analysis finds values of α and β that satisfy the constraint.

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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.

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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

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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).

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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?”

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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.