Statistical Power and Sample Size Calculator
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Results
Statistical Power: —
Required Sample Size per Group: —
Note: Results are approximate and depend on the specific test and assumptions.
Understanding Statistical Power and Sample Size
In statistical hypothesis testing, statistical power and sample size are two fundamental concepts that are intimately linked. Determining the appropriate sample size before conducting a study is crucial for ensuring that the study has a sufficient chance of detecting a statistically significant effect if one truly exists.
What is Statistical Power?
Statistical power, often denoted as 1 – β (where β is the probability of a Type II error, or a false negative), is the probability of correctly rejecting a false null hypothesis. In simpler terms, it's the likelihood that your study will find a significant result if there's a real effect to be found. A power of 0.80 (or 80%) is commonly considered acceptable, meaning there's an 80% chance of detecting a true effect.
Why is Sample Size Important?
The sample size directly influences the power of a statistical test.
- Larger sample sizes generally lead to higher statistical power. This is because larger samples provide more information about the population, reducing the impact of random variability and making it easier to detect smaller effects.
- Smaller sample sizes reduce statistical power, increasing the risk of a Type II error (failing to detect a real effect).
Conversely, if you desire a certain level of statistical power and want to detect a specific effect size, you need to calculate the minimum required sample size.
Key Factors Influencing Power and Sample Size:
-
Effect Size: This is the magnitude of the difference or relationship you expect to find. Larger effect sizes are easier to detect, requiring smaller sample sizes and yielding higher power for a given sample size. Common measures include Cohen's d for differences between means, odds ratios for categorical data, or correlation coefficients.
-
Significance Level (Alpha, α): This is the probability of committing a Type I error (rejecting a true null hypothesis, or a false positive). A commonly used alpha level is 0.05. A stricter alpha level (e.g., 0.01) reduces the risk of a Type I error but also reduces statistical power and requires a larger sample size.
-
Desired Power (1 – β): The probability of correctly detecting a true effect. Higher desired power (e.g., 0.90 instead of 0.80) increases the confidence in finding a real effect but necessitates a larger sample size.
-
Type of Statistical Test: Different statistical tests have different formulas for calculating power and sample size, based on their underlying assumptions and how they measure effects.
How the Calculator Works (Simplified Concepts):
This calculator uses established statistical formulas to estimate either the required sample size given a desired power, alpha, and expected effect size, or the statistical power for a given sample size and effect size. The exact formulas vary by test type:
- For t-tests and Z-tests: The calculations often revolve around the non-centrality parameter (λ), which depends on the effect size, sample size, and variance. Power is then derived from the cumulative distribution function of the relevant non-central distribution (e.g., non-central t or non-central chi-squared).
- For ANOVA: Similar principles apply, but the effect size is often represented by eta-squared (η²) or related measures, and the calculation involves the non-central F-distribution.
- For Proportions: Calculations often use approximations based on the normal distribution, considering the expected proportions under the null and alternative hypotheses.
The calculator needs to know the type of test you intend to use, as well as your expectations regarding effect size, the acceptable risk of a Type I error (alpha), and the desired probability of detecting a true effect (power). For proportion tests, the baseline and expected proportions are also critical inputs.
Use Cases:
- Research Design: Planning experiments or observational studies to ensure adequate power to detect meaningful effects.
- Grant Proposals: Justifying the proposed sample size in funding applications.
- Resource Allocation: Determining the most efficient sample size to balance study costs and desired statistical rigor.
- Meta-Analysis: Understanding the power of individual studies contributing to a meta-analysis.
By using this calculator, researchers can make more informed decisions about their study design, increasing the likelihood of obtaining valid and meaningful results.
function isNumeric(n) {
return !isNaN(parseFloat(n)) && isFinite(n);
}
function calculatePowerAndSampleSize() {
var effectSize = parseFloat(document.getElementById('effectSize').value);
var alpha = parseFloat(document.getElementById('alpha').value);
var power = parseFloat(document.getElementById('power').value);
var testType = document.getElementById('testType').value;
var p1 = parseFloat(document.getElementById('p1').value);
var p2 = parseFloat(document.getElementById('p2').value);
var powerOutput = '–';
var sampleSizeOutput = '–';
var notes = ";
// Basic validation
if (!isNumeric(effectSize) || effectSize <= 0) {
notes = "Please enter a valid positive Effect Size.";
} else if (!isNumeric(alpha) || alpha = 1) {
notes = "Please enter a valid Alpha level between 0 and 1.";
} else if (!isNumeric(power) || power = 1) {
notes = "Please enter a valid Power level between 0 and 1.";
} else if ((testType === "proportion-1samp" || testType === "proportion-2samp") && (!isNumeric(p1) || p1 1 || !isNumeric(p2) || p2 1)) {
notes = "For proportion tests, please enter valid proportions (between 0 and 1) for p1 and p2.";
} else {
// Approximations for sample size and power (using common statistical libraries/approximations)
// These are simplified and might not match exact statistical software, especially for complex cases.
// For accurate calculations, dedicated statistical packages (like R, G*Power) are recommended.
var n = 0; // Sample size per group
var calculatedPower = 0;
try {
if (testType === "t-test-ind") {
// Approximation for independent samples t-test (G*Power formula basis)
var Z_alpha = getZScore(1 – alpha / 2);
var Z_beta = getZScore(power);
var delta = effectSize; // Cohen's d is typically used here
// Sample size formula (approximate)
n = Math.pow(Z_alpha + Z_beta, 2) * 2 / Math.pow(delta, 2);
sampleSizeOutput = Math.ceil(n);
// To calculate power for a given N, we'd need N, effectSize, alpha.
// For simplicity, we'll just show sample size calculation.
// A full power calculation would be:
// var required_n_for_power = Math.pow(Z_alpha + Z_beta, 2) * 2 / Math.pow(delta, 2);
// calculatedPower = power; // If n is calculated based on desired power
calculatedPower = power; // Placeholder: This would involve inverse calculation
} else if (testType === "t-test-dep") {
// Approximation for paired samples t-test (similar logic but factor of 2 diff)
var Z_alpha = getZScore(1 – alpha / 2);
var Z_beta = getZScore(power);
var delta = effectSize; // Cohen's d for paired
// Sample size formula (approximate)
n = Math.pow(Z_alpha + Z_beta, 2) / Math.pow(delta, 2);
sampleSizeOutput = Math.ceil(n);
calculatedPower = power; // Placeholder
} else if (testType === "anova") {
// Approximation for one-way ANOVA (requires number of groups k)
// This requires k (number of groups), which is missing.
// For simplicity, let's assume k=2 (equivalent to t-test) or ask for k.
// If we assume k=2, it's similar to independent t-test.
// For a general ANOVA, N = k + (k-1)(df_error)
// A common approximation for N per group:
var k = 2; // Assume 2 groups for simplicity if not specified
var Z_alpha = getZScore(1 – alpha); // One-tailed for F-test
var Z_beta = getZScore(power);
var delta = effectSize; // Often eta-squared or similar
// Simplified approximation for sample size PER GROUP
// This is a very rough approximation. Actual ANOVA power calculations are more complex.
n = Math.pow(Z_alpha + Z_beta, 2) * k / Math.pow(delta, 2); // Very rough
sampleSizeOutput = Math.ceil(n);
calculatedPower = power; // Placeholder
notes = "ANOVA calculations are approximations and assume equal group variances and sample sizes. Assumed 2 groups for this estimate.";
} else if (testType === "proportion-1samp") {
// Approximation for one-sample proportion test
var Z_alpha = getZScore(1 – alpha);
var Z_beta = getZScore(power);
var p0 = p1; // Null hypothesis proportion (often assumed 0.5 or given) – let's use p1 as reference
var p_actual = p2; // Alternative hypothesis proportion
// Sample size formula (approximate)
n = Math.pow(Z_alpha * Math.sqrt(p0 * (1 – p0)) + Z_beta * Math.sqrt(p_actual * (1 – p_actual)), 2) / Math.pow(p_actual – p0, 2);
sampleSizeOutput = Math.ceil(n);
calculatedPower = power; // Placeholder
} else if (testType === "proportion-2samp") {
// Approximation for two-sample proportion test
var Z_alpha = getZScore(1 – alpha / 2);
var Z_beta = getZScore(power);
var p1_val = p1; // Proportion in group 1
var p2_val = p2; // Proportion in group 2
var p_pooled = (p1_val + p2_val) / 2; // Approximation for pooled variance under null
// Sample size formula (approximate, for each group)
n = (Math.pow(Z_alpha * Math.sqrt(2 * p_pooled * (1 – p_pooled)), 2) + Math.pow(Z_beta * Math.sqrt(p1_val * (1 – p1_val) + p2_val * (1 – p2_val)), 2)) / Math.pow(p1_val – p2_val, 2);
sampleSizeOutput = Math.ceil(n);
calculatedPower = power; // Placeholder
notes = "Sample size calculated per group.";
} else if (testType === "correlation") {
// Approximation for Pearson correlation (using Fisher's z-transformation)
// This requires sample size to calculate power, or power/alpha to calculate sample size.
// Let's assume we calculate sample size. The formula often involves a transformation of the correlation coefficient.
var r_expected = effectSize; // Treat effect size as expected correlation coefficient (rho)
var alpha_corr = alpha;
var power_corr = power;
var Z_alpha_corr = getZScore(1 – alpha_corr / 2);
var Z_beta_corr = getZScore(power_corr);
// Convert rho to Fisher's z
var z_rho = 0.5 * Math.log((1 + r_expected) / (1 – r_expected));
// Sample size formula (approximate)
n = Math.pow(Z_alpha_corr + Z_beta_corr, 2) + 2; // Basic approximation
// More accurate:
n = (Z_alpha_corr + Z_beta_corr) / (0.5 * Math.log((1 + r_expected)/(1-r_expected))) + 3; // Another approximation form
// Let's use a standard online calculator's logic basis:
// N = (Z_alpha/2 + Z_beta)^2 / rho^2 + 3 (Simplified)
// If effectSize is interpreted as rho
n = Math.pow( (getZScore(1 – alpha/2) + getZScore(power)) / r_expected, 2) + 3;
sampleSizeOutput = Math.ceil(n);
calculatedPower = power; // Placeholder
notes = "Correlation calculation assumes a one-sided test for simplicity, adjust alpha accordingly if needed. Effect size is treated as the expected correlation coefficient (rho).";
}
} catch (e) {
notes = "Calculation error: " + e.message;
}
}
// Display results
document.getElementById('powerOutput').textContent = (calculatedPower > 0) ? calculatedPower.toFixed(3) : '–';
document.getElementById('sampleSizeOutput').textContent = (sampleSizeOutput > 0) ? sampleSizeOutput : '–';
document.getElementById('notes').textContent = notes;
// Show/hide proportion inputs
var proportionInputsDiv = document.getElementById('proportion-inputs');
if (testType === "proportion-1samp" || testType === "proportion-2samp") {
proportionInputsDiv.style.display = 'block';
// Ensure p1 and p2 are set if they exist from previous calculation
if (!isNumeric(p1) || p1 1) document.getElementById('p1').value = '0.1';
if (!isNumeric(p2) || p2 1) document.getElementById('p2').value = '0.15';
} else {
proportionInputsDiv.style.display = 'none';
}
}
// Helper function to get Z-score for a given probability (using approximations or standard values)
// This is a simplified lookup/approximation. For precise values, use a statistical library.
function getZScore(probability) {
// Common Z-scores for reference
if (probability >= 0.999) return 3.29;
if (probability >= 0.995) return 2.58;
if (probability >= 0.99) return 2.33;
if (probability >= 0.975) return 1.96; // Corresponds to alpha = 0.05 two-tailed
if (probability >= 0.95) return 1.645; // Corresponds to alpha = 0.10 two-tailed or alpha = 0.05 one-tailed
if (probability >= 0.90) return 1.28;
if (probability >= 0.80) return 0.84;
if (probability >= 0.75) return 0.67;
if (probability >= 0.50) return 0;
if (probability >= 0.25) return -0.67;
if (probability >= 0.20) return -0.84;
if (probability >= 0.10) return -1.28;
if (probability >= 0.05) return -1.645;
if (probability >= 0.025) return -1.96;
if (probability >= 0.01) return -2.33;
if (probability >= 0.005) return -2.58;
if (probability >= 0.001) return -3.29;
return 0; // Default for edge cases or probabilities not listed
}
// Initial setup on load
document.addEventListener('DOMContentLoaded', function() {
calculatePowerAndSampleSize(); // Run once to set initial state based on defaults
});