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Power Analysis

Power Analysis

Definition

Core Statement

Power Analysis determines the probability of detecting a true effect if one exists. Statistical Power (1−β) is the probability of correctly rejecting a false H0. Power analysis is essential for planning studies with adequate sample sizes.

Purpose

  1. Plan Sample Size: Calculate n needed to detect an expected effect.
  2. Evaluate Study Design: Assess if a study has reasonable power.
  3. Interpret Non-Significant Results: Low power means non-significance may simply reflect insufficient data.

When to Use

Use Power Analysis When...

  • Before a study: Calculate required n (a priori).
  • After a study: Understand if non-significance could be due to low power (post-hoc, controversial).
  • Grant proposals and ethics: Justify sample size.

Theoretical Background

The Four Components

Power analysis involves four interrelated quantities. Given three, you can solve for the fourth.

Component Symbol Description
Sample Size n Number of observations.
Effect Size d, r, etc. Magnitude of the effect.
Alpha (α) 0.05 Significance level (Type I error rate).
Power (1−β) 0.80 Probability of detecting a true effect.

Effect Sizes (Cohen's Conventions)

Effect Small Medium Large
Cohen's d (mean diff) 0.2 0.5 0.8
Cohen's f (ANOVA) 0.1 0.25 0.4
r (correlation) 0.1 0.3 0.5

Power Tradeoffs

Standard Power Target

80% Power (β=0.20) is the conventional minimum. Some fields prefer 90%.

Assumptions

Limitations

Pitfalls

  1. Effect Size Uncertainty: If your effect size guess is wrong, sample size will be off.
  2. Post-Hoc Power is Flawed: Calculating power after a study using observed effect size is circular and uninformative.

Python Implementation

from statsmodels.stats.power import TTestIndPower

# Calculate Required Sample Size (Per Group)
# Given: effect size = 0.5 (medium), alpha = 0.05, power = 0.80
analysis = TTestIndPower()
n_per_group = analysis.solve_power(effect_size=0.5, alpha=0.05, power=0.8, alternative='two-sided')
print(f"Required n per group: {n_per_group:.0f}")

# Calculate Power Given n
power = analysis.solve_power(effect_size=0.5, nobs1=50, alpha=0.05, alternative='two-sided')
print(f"Power with n=50: {power:.2f}")

R Implementation

library(pwr)

# T-Test: Calculate n for medium effect (d=0.5), power=0.8
result <- pwr.t.test(d = 0.5, power = 0.8, sig.level = 0.05, type = "two.sample")
print(result)

# ANOVA: Calculate n for medium effect (f=0.25), k=3 groups
pwr.anova.test(k = 3, f = 0.25, power = 0.8, sig.level = 0.05)

# Correlation: n for detecting r=0.3
pwr.r.test(r = 0.3, power = 0.8, sig.level = 0.05)

Interpretation Guide

Output Interpretation
Required n = 64 per group You need 128 total participants (64 per arm) to detect a medium effect with 80% power.
Power = 0.45 Only a 45% chance of detecting the effect. Study is underpowered.