🧮 Compromise Power Analysis

Compromise Power Analysis#

Sizing a study by trading Type I against Type II error at a fixed ratio rather than fixing one.

Important

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What it is#

Compromise power analysis finds a sensible balance between the Type I error \(\alpha\) (false positives) and the Type II error \(\beta\) (false negatives) when the sample size \(n\) is fixed. Unlike a-priori analysis — which fixes \(\alpha\) and power and solves for \(n\) — here you already know \(n\) and ask what \(\alpha\)/\(\beta\) trade-off is reasonable.

Why it’s useful#

Sometimes \(n\) simply cannot change — a limited participant pool, a budget cap, or a historical dataset. A-priori analysis might say “you need 500 subjects” when you have 200; compromise analysis answers the real question: with 200, what :math:`alpha` and :math:`beta` give a balanced test?

How it works#

Specify the effect size \(\delta\), the available \(n\), and a desired ratio of Type I to Type II error (often \(\alpha = \beta\), i.e. a 1:1 ratio). The procedure then solves for the \(\alpha\) and \(\beta\) that satisfy the constraint.

Example#

With a medium effect (Cohen’s \(d = 0.5\)), \(n = 40\) (20 per group), and a requirement that \(\alpha = \beta\), the analysis might return \(\alpha = \beta = 0.12\) (power \(= 0.88\)). You accept a higher false-positive rate (12%) to keep false negatives equally low, given the small sample.

The three power analyses#

A-priori — input effect size, \(\alpha\), power → output required \(n\) (“how many subjects do I need?”).
Post-hoc — input observed \(n\) and effect size → output achieved power (“given what I saw, what was the power?”).
Compromise — input effect size, available \(n\), error ratio → output appropriate \(\alpha\) and \(\beta\) (“with this \(n\), how do I balance the two errors?”).

It is the pragmatic choice when data is scarce, though it is less conventional than the fixed \(\alpha = 0.05\) and depends on an assumed effect size.

Theme: Statistical Inference & Power · All terminology

Hint

Mind map — connected ideas

Type I Error · Post Hoc Power Analysis · A Priori Power Analysis · Statistical Power · Effect Size (δ) · Frequentist

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More in Statistical Inference & Power

A Priori Power Analysis · Chi-square (χ²) Test · Clopper–Pearson Interval · Confidence Intervals (CIs) · Effect Size (δ) · Hypothesis Testing · Kolmogorov–Smirnov (KS) Test · Minimum Detectable Lift (MDL) · P-Value (probability value) · Post Hoc Power Analysis · Power (1 – β) · Power Analysis · Sample size · Significance Level (α)