brycewang-stanford/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:

| Parameter | Symbol | Definition | Typical Value | |-----------|--------|-----------|---------------| | Effect size | d, r, f, etc. | Magnitude of the phenomenon you expect to detect | Varies by field | | Significance level (alpha) | alpha | Probability of Type I error (false positive) | 0.05 | | Statistical power (1 - beta) | 1 - beta | Probability of detecting a true effect | 0.80 or 0.90 | | Sample size | N | Number of observations needed | Solve for this |

Error Types

| | H0 is true (no effect) | H0 is false (effect exists) | |---|---|---| | Reject H0 | Type I error (alpha) | Correct (power = 1 - beta) | | Fail to reject H0 | Correct (1 - alpha) | Type II error (beta) |

Effect Size Conventions

Cohen's d (Two-Group Comparison)

d = (M1 - M2) / SD_pooled

| Size | Cohen's d | Interpretation | |------|-----------|---------------| | Small | 0.2 | Subtle, may need large N to detect | | Medium | 0.5 | Noticeable, typical in social sciences | | Large | 0.8 | Obvious, often visible without statistics |

Correlation (r)

| Size | r | r-squared | |------|---|-----------| | Small | 0.1 | 1% variance explained | | Medium | 0.3 | 9% variance explained | | Large | 0.5 | 25% variance explained |

Cohen's f (ANOVA)

| Size | f | Equivalent eta-squared | |------|---|----------------------| | Small | 0.10 | 0.01 | | Medium | 0.25 | 0.06 | | Large | 0.40 | 0.14 |

Odds Ratio (Logistic Regression)

| Size | OR | |------|-----| | Small | 1.5 | | Medium | 2.5 | | Large | 4.0 |

Power Analysis in Python (statsmodels)

Two-Sample t-Test

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

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

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

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

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)

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

| Mistake | Problem | Solution | |---------|---------|----------| | Post hoc power analysis | Circular and uninformative after data collection | Only do a priori power analysis | | Using Cohen's "medium" by default | May be unrealistic for your field | Base on literature or SESOI | | Ignoring attrition | Actual N may be lower than planned | Inflate N by 10-20% for expected dropout | | Forgetting multiple comparisons | Bonferroni corrections reduce power | Adjust alpha for the number of tests | | Not reporting power analysis | Reviewers cannot evaluate adequacy | Always report in Methods section |

Reporting Template

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.