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Calculates the required sample size per group for a two-group statistical comparison using Cohen's d effect size, significance level, statistical power, and standard deviation.

Purpose

This function performs statistical power analysis to determine the minimum number of participants needed in each group of a two-group study design. It uses the standard formula for sample size calculation based on effect size (Cohen's d), desired significance level (alpha), statistical power, and population standard deviation. This is commonly used in experimental design and A/B testing to ensure adequate statistical power before conducting a study.

Source Code

def calculate_sample_size(effect_size, alpha, power, sd):
    """
    Calculate required sample size for two-group comparison
    effect_size: Cohen's d
    alpha: significance level (default 0.05)
    power: statistical power (default 0.80)
    sd: standard deviation
    """
    from scipy.stats import norm
    
    z_alpha = norm.ppf(1 - alpha/2)  # Two-tailed test
    z_beta = norm.ppf(power)
    
    # Sample size per group
    n = 2 * ((z_alpha + z_beta) ** 2) * (sd ** 2) / ((effect_size * sd) ** 2)
    
    return np.ceil(n)

Parameters

Parameter Details

effect_size: Cohen's d effect size - a standardized measure of the difference between two groups. Typical values: 0.2 (small), 0.5 (medium), 0.8 (large). Must be a positive number greater than 0.

alpha: Significance level (Type I error rate) for the statistical test. Commonly set to 0.05 (5%). Must be between 0 and 1. The function uses a two-tailed test, so this value is divided by 2 internally.

power: Statistical power (1 - Type II error rate) - the probability of detecting an effect if it exists. Commonly set to 0.80 (80%) or 0.90 (90%). Must be between 0 and 1.

sd: Standard deviation of the population or expected standard deviation of the outcome variable. Must be a positive number greater than 0. Should be in the same units as the effect size.

Return Value

Returns a numpy scalar (float64) representing the required sample size per group, rounded up to the nearest whole number using np.ceil(). This is the minimum number of participants needed in each of the two groups to achieve the specified statistical power at the given significance level.

Dependencies

• scipy
• numpy

Required Imports

import numpy as np
from scipy.stats import norm

Conditional/Optional Imports

These imports are only needed under specific conditions:

from scipy.stats import norm

Usage Example

import numpy as np
from scipy.stats import norm

def calculate_sample_size(effect_size, alpha, power, sd):
    from scipy.stats import norm
    z_alpha = norm.ppf(1 - alpha/2)
    z_beta = norm.ppf(power)
    n = 2 * ((z_alpha + z_beta) ** 2) * (sd ** 2) / ((effect_size * sd) ** 2)
    return np.ceil(n)

# Example: Calculate sample size for medium effect size
effect_size = 0.5  # Cohen's d (medium effect)
alpha = 0.05  # 5% significance level
power = 0.80  # 80% power
sd = 1.0  # Standard deviation

sample_size = calculate_sample_size(effect_size, alpha, power, sd)
print(f"Required sample size per group: {int(sample_size)}")
# Output: Required sample size per group: 64

Best Practices

• Ensure that the effect_size and sd parameters are in compatible units and scales
• The function assumes equal sample sizes in both groups (balanced design)
• The calculation is based on a two-tailed test; adjust if one-tailed test is needed
• Always round up the result (which the function does automatically) to ensure adequate power
• Validate that alpha is typically 0.05 or 0.01, and power is typically 0.80 or 0.90
• The standard deviation (sd) should be estimated from pilot data or literature when possible
• Note that the formula simplifies because Cohen's d already incorporates sd, so the sd terms cancel out in the calculation
• Consider using a slightly higher sample size than calculated to account for potential dropouts or missing data

Tags

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