Power Analysis Guide

Power Analysis Guide

Calculate appropriate sample sizes for your study using power analysis, understand effect sizes, and avoid underpowered or wastefully overpowered designs.

Core Concepts

The Four Parameters of Power Analysis

Every power analysis involves four interrelated quantities. Fix any three to solve for the fourth:

Error Types

Effect Size Conventions

Cohen's d (Two-Group Comparison)

Copy code
d = (M1 - M2) / SD_pooled

Correlation (r)

Cohen's f (ANOVA)

Odds Ratio (Logistic Regression)

Power Analysis in Python (statsmodels)

Two-Sample t-Test

pythonCopy code
from statsmodels.stats.power import TTestIndPower

analysis = TTestIndPower()

# Solve for sample size
n = analysis.solve_power(
    effect_size=0.5,    # Cohen's d = medium
    alpha=0.05,         # Significance level
    power=0.80,         # 80% power
    ratio=1.0,          # Equal group sizes
    alternative='two-sided'
)
print(f"Required N per group: {int(n) + 1}")  # Output: 64

# Solve for power (given N)
power = analysis.solve_power(
    effect_size=0.5,
    alpha=0.05,
    nobs1=50,
    ratio=1.0,
    alternative='two-sided'
)
print(f"Power with N=50 per group: {power:.3f}")  # Output: 0.697

Paired t-Test

pythonCopy code
from statsmodels.stats.power import TTestPower

analysis = TTestPower()
n = analysis.solve_power(
    effect_size=0.3,    # Small-medium effect
    alpha=0.05,
    power=0.80,
    alternative='two-sided'
)
print(f"Required N (paired): {int(n) + 1}")  # Output: 90

One-Way ANOVA

pythonCopy code
from statsmodels.stats.power import FTestAnovaPower

analysis = FTestAnovaPower()
n = analysis.solve_power(
    effect_size=0.25,   # Cohen's f = medium
    alpha=0.05,
    power=0.80,
    k_groups=4          # Number of groups
)
print(f"Required N per group: {int(n) + 1}")  # Output: 45

Chi-Square Test

pythonCopy code
from statsmodels.stats.power import GofChisquarePower

analysis = GofChisquarePower()
n = analysis.solve_power(
    effect_size=0.3,    # Cohen's w = medium
    alpha=0.05,
    power=0.80,
    n_bins=4            # Degrees of freedom + 1
)
print(f"Required total N: {int(n) + 1}")

Multiple Regression

pythonCopy code
from statsmodels.stats.power import FTestPower

analysis = FTestPower()
# For R-squared: convert to f2 = R2 / (1 - R2)
r_squared = 0.10  # Expected R-squared for the model
f2 = r_squared / (1 - r_squared)  # f2 = 0.111

n = analysis.solve_power(
    effect_size=f2,
    alpha=0.05,
    power=0.80,
    df_num=5            # Number of predictors
)
# n returned is df_denom; total N = n + df_num + 1
total_n = int(n) + 5 + 1
print(f"Required total N: {total_n}")

Power Analysis in R (pwr Package)

rCopy code
library(pwr)

# Two-sample t-test
result <- pwr.t.test(d = 0.5, sig.level = 0.05, power = 0.80,
                     type = "two.sample", alternative = "two.sided")
cat("N per group:", ceiling(result$n), "\n")

# Correlation test
result <- pwr.r.test(r = 0.3, sig.level = 0.05, power = 0.80,
                     alternative = "two.sided")
cat("Total N:", ceiling(result$n), "\n")

# One-way ANOVA (4 groups)
result <- pwr.anova.test(k = 4, f = 0.25, sig.level = 0.05, power = 0.80)
cat("N per group:", ceiling(result$n), "\n")

# Chi-square test
result <- pwr.chisq.test(w = 0.3, df = 3, sig.level = 0.05, power = 0.80)
cat("Total N:", ceiling(result$N), "\n")

# Plot power curve
result <- pwr.t.test(d = 0.5, sig.level = 0.05, power = NULL,
                     n = seq(10, 200, by = 5))
plot(result)

Using G*Power (Desktop Application)

G*Power (gpower.hhu.de) is a free, widely-used GUI application for power analysis:

1. Select test family: t-tests, F-tests, chi-square, z-tests, exact tests
2. Select statistical test: e.g., "Means: Difference between two independent means (two groups)"
3. Select type of analysis: A priori (compute N), Post hoc (compute power), Sensitivity (compute detectable effect)
4. Input parameters: Effect size, alpha, power, allocation ratio
5. Calculate: Click "Calculate" to get the result
6. Plot: Use "X-Y plot for a range of values" to visualize power curves

Practical Recommendations

Choosing Effect Sizes

Do NOT blindly use Cohen's conventions. Instead:

1. Literature review: Find effect sizes reported in similar studies
2. Pilot data: Run a small pilot study to estimate the effect
3. Smallest effect of interest (SESOI): What is the smallest effect that would be practically meaningful?
4. Meta-analyses: Use pooled effect sizes from meta-analyses in your area

Common Mistakes

Reporting Template

Copy code
A priori power analysis was conducted using [G*Power 3.1 / statsmodels / R pwr].
For a [test name] with an expected effect size of [d/r/f = X] (based on
[source: previous study / meta-analysis / pilot data]), alpha = .05, and
power = .80, the required sample size was [N per group / total N]. To account
for an estimated [X]% attrition rate, we recruited [final N] participants.