القائمة
Sample size and power analysis with effect size estimation and Type I/II error control
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| name | power-analysis |
| category | statistics |
| discipline | general |
| description | Sample size and power analysis with effect size estimation and Type I/II error control |
Protocol for a priori sample size calculation, effect size estimation, and power determination for common study designs. Ensures studies are adequately powered to detect clinically or scientifically meaningful effects.
Statistical power is the probability of correctly rejecting a false null hypothesis (1 - beta).
Four interrelated quantities (knowing any three determines the fourth):
| Parameter | Symbol | Description | Typical Value |
|---|---|---|---|
| Significance level | alpha | Probability of Type I error (false positive) | 0.05 |
| Power | 1 - beta | Probability of detecting a true effect | 0.80 or 0.90 |
| Effect size | d, r, f, w, etc. | Magnitude of the effect to detect | Study-specific |
| Sample size | N or n | Number of participants needed | Calculated |
The most critical and often most difficult step. Sources for effect size estimates (in order of preference):
| Effect Size Measure | Small | Medium | Large | Used With |
|---|---|---|---|---|
| Cohen's d | 0.2 | 0.5 | 0.8 | Two-group mean comparisons |
| Cohen's f | 0.10 | 0.25 | 0.40 | ANOVA |
| Cohen's w | 0.10 | 0.30 | 0.50 | Chi-square tests |
| Pearson r | 0.10 | 0.30 | 0.50 | Correlation |
| Cohen's f-squared | 0.02 | 0.15 | 0.35 | Multiple regression |
| Odds Ratio | 1.5 | 2.5 | 4.3 | Logistic regression |
| Hazard Ratio | 1.3 | 1.7 | 2.5 | Survival analysis |
import numpy as np
# Cohen's d to r
def d_to_r(d):
return d / np.sqrt(d**2 + 4)
# r to Cohen's d
def r_to_d(r):
return 2 * r / np.sqrt(1 - r**2)
# Cohen's d to f (for ANOVA with 2 groups)
def d_to_f(d):
return d / 2
# Odds ratio to Cohen's d
def or_to_d(odds_ratio):
return np.log(odds_ratio) * np.sqrt(3) / np.pi
# Cohen's d to odds ratio
def d_to_or(d):
return np.exp(d * np.pi / np.sqrt(3))
# Eta-squared to Cohen's f
def eta_sq_to_f(eta_sq):
return np.sqrt(eta_sq / (1 - eta_sq))
from statsmodels.stats.power import TTestIndPower
analysis = TTestIndPower()
# Parameters
effect_size = 0.5 # Cohen's d
alpha = 0.05 # significance level
power = 0.80 # desired power
ratio = 1.0 # n2/n1 ratio (1.0 = equal groups)
# Calculate required sample size per group
n_per_group = analysis.solve_power(
effect_size=effect_size,
alpha=alpha,
power=power,
ratio=ratio,
alternative='two-sided'
)
print(f"Required n per group: {np.ceil(n_per_group):.0f}")
print(f"Total N: {np.ceil(n_per_group) * 2:.0f}")
# Generate power curve
import matplotlib.pyplot as plt
sample_sizes = np.arange(10, 200)
powers = analysis.power(effect_size=effect_size, nobs1=sample_sizes,
alpha=alpha, ratio=ratio, alternative='two-sided')
plt.figure(figsize=(8, 5))
plt.plot(sample_sizes, powers)
plt.axhline(y=0.80, color='r', linestyle='--', label='Power = 0.80')
plt.axvline(x=n_per_group, color='g', linestyle='--', label=f'n = {np.ceil(n_per_group):.0f}')
plt.xlabel('Sample Size per Group')
plt.ylabel('Power')
plt.title(f'Power Curve (d={effect_size}, alpha={alpha})')
plt.legend()
plt.grid(True, alpha=0.3)
plt.savefig('power_curve_ttest.png', dpi=150, bbox_inches='tight')
from statsmodels.stats.power import TTestPower
analysis = TTestPower()
effect_size = 0.5 # Cohen's d for paired differences
alpha = 0.05
power = 0.80
n = analysis.solve_power(
effect_size=effect_size,
alpha=alpha,
power=power,
alternative='two-sided'
)
print(f"Required n (pairs): {np.ceil(n):.0f}")
from statsmodels.stats.power import GofChisquarePower
# For chi-square goodness of fit
analysis = GofChisquarePower()
effect_size = 0.3 # Cohen's w
alpha = 0.05
power = 0.80
n_bins = 4 # degrees of freedom + 1
n = analysis.solve_power(
effect_size=effect_size,
alpha=alpha,
power=power,
n_bins=n_bins
)
print(f"Required total N: {np.ceil(n):.0f}")
# Alternative: using proportion difference (2x2 table)
from statsmodels.stats.power import NormalIndPower
# Two proportions
p1 = 0.30 # expected proportion in group 1
p2 = 0.45 # expected proportion in group 2
# Cohen's h for two proportions
h = 2 * np.arcsin(np.sqrt(p1)) - 2 * np.arcsin(np.sqrt(p2))
analysis = NormalIndPower()
n_per_group = analysis.solve_power(
effect_size=abs(h),
alpha=alpha,
power=power,
ratio=1.0,
alternative='two-sided'
)
print(f"Required n per group (proportions): {np.ceil(n_per_group):.0f}")
from statsmodels.stats.power import FTestAnovaPower
analysis = FTestAnovaPower()
effect_size = 0.25 # Cohen's f
alpha = 0.05
power = 0.80
k_groups = 3 # number of groups
n_per_group = analysis.solve_power(
effect_size=effect_size,
alpha=alpha,
power=power,
k_groups=k_groups
)
print(f"Required n per group: {np.ceil(n_per_group):.0f}")
print(f"Total N: {np.ceil(n_per_group) * k_groups:.0f}")
from statsmodels.stats.power import NormalIndPower
import numpy as np
def power_correlation(r, alpha=0.05, power=0.80):
"""Sample size for testing Pearson correlation != 0.
Uses Fisher's z transformation approach.
"""
z_alpha = stats.norm.ppf(1 - alpha/2)
z_beta = stats.norm.ppf(power)
z_r = 0.5 * np.log((1 + r) / (1 - r)) # Fisher's z
n = ((z_alpha + z_beta) / z_r) ** 2 + 3
return int(np.ceil(n))
from scipy import stats
r = 0.30 # expected correlation
n = power_correlation(r, alpha=0.05, power=0.80)
print(f"Required N for r={r}: {n}")
# Multiple effect sizes
for r in [0.10, 0.20, 0.30, 0.40, 0.50]:
n = power_correlation(r)
print(f"r = {r:.2f}: N = {n}")
from statsmodels.stats.power import FTestPower
analysis = FTestPower()
# Parameters
f2 = 0.15 # Cohen's f-squared (medium effect)
alpha = 0.05
power = 0.80
n_predictors = 5 # number of predictors in full model
n_tested = 2 # number of predictors being tested
# df_num = n_tested (predictors being tested)
# df_denom = N - n_predictors - 1
# Iterative solution
from scipy.optimize import brentq
def power_equation(n):
df_num = n_tested
df_denom = n - n_predictors - 1
if df_denom <= 0:
return -1
noncentrality = f2 * n
power_calc = analysis.power(
effect_size=np.sqrt(f2),
df_num=df_num,
df_denom=df_denom,
alpha=alpha
)
return power_calc - power
n_required = int(np.ceil(brentq(power_equation, n_predictors + 2, 10000)))
print(f"Required N for {n_predictors} predictors (f2={f2}): {n_required}")
def adjust_for_dropout(n_calculated, dropout_rate):
"""Inflate sample size to account for expected dropout."""
return int(np.ceil(n_calculated / (1 - dropout_rate)))
n_base = 64 # calculated sample size per group
dropout = 0.20 # 20% expected dropout
n_adjusted = adjust_for_dropout(n_base, dropout)
print(f"Adjusted n per group (for {dropout*100:.0f}% dropout): {n_adjusted}")
When randomization ratios are not 1:1 (e.g., 2:1 treatment:control):
from statsmodels.stats.power import TTestIndPower
analysis = TTestIndPower()
# 2:1 allocation ratio
n_smaller_group = analysis.solve_power(
effect_size=0.5,
alpha=0.05,
power=0.80,
ratio=2.0, # n_treatment / n_control
alternative='two-sided'
)
print(f"Control group n: {np.ceil(n_smaller_group):.0f}")
print(f"Treatment group n: {np.ceil(n_smaller_group * 2):.0f}")
print(f"Total N: {np.ceil(n_smaller_group) + np.ceil(n_smaller_group * 2):.0f}")
When testing multiple primary outcomes, adjust alpha for multiplicity:
# Bonferroni-adjusted alpha
n_outcomes = 3
alpha_adjusted = 0.05 / n_outcomes
n_per_group = TTestIndPower().solve_power(
effect_size=0.5,
alpha=alpha_adjusted,
power=0.80,
alternative='two-sided'
)
print(f"Adjusted alpha: {alpha_adjusted:.4f}")
print(f"Required n per group (Bonferroni-adjusted): {np.ceil(n_per_group):.0f}")
Pre-computed sample sizes per group for two-sample t-test, alpha=0.05, two-sided:
| Cohen's d | Power 0.80 | Power 0.90 | Power 0.95 |
|---|---|---|---|
| 0.2 (small) | 394 | 527 | 651 |
| 0.3 | 176 | 235 | 290 |
| 0.4 | 100 | 133 | 164 |
| 0.5 (medium) | 64 | 86 | 105 |
| 0.6 | 45 | 60 | 74 |
| 0.7 | 33 | 44 | 55 |
| 0.8 (large) | 26 | 34 | 42 |
| 1.0 | 17 | 23 | 28 |
| 1.2 | 12 | 16 | 20 |
Post-hoc (observed) power analysis using the observed effect size is generally discouraged because:
Acceptable alternatives for negative studies:
# Sensitivity analysis: what effect could we detect?
from statsmodels.stats.power import TTestIndPower
analysis = TTestIndPower()
actual_n = 50 # actual sample size per group
detectable_d = analysis.solve_power(
nobs1=actual_n,
alpha=0.05,
power=0.80,
alternative='two-sided'
)
print(f"With n={actual_n} per group, the study could detect d >= {detectable_d:.3f} at 80% power")