Accounting for Clustering in Trial Data

I'll help you calculate the design effect (DE) and effective sample size (ESS) for your clustered or cluster-randomized trial data. These are key adjustments for accounting for intra-cluster correlation, which reduces the precision of estimates compared to independent sampling.

Key Concepts

• Design Effect (DE): Measures how much the clustering inflates the variance of an estimate relative to simple random sampling. It's typically greater than 1, indicating reduced efficiency.


• Formula: \( DE = 1 + (m - 1) \times ICC \)


• \( m \): Average cluster size (total sample size \( n \) divided by number of clusters \( k \), i.e., \( m = n / k \)).


• \( ICC \): Intraclass correlation coefficient (measure of within-cluster similarity; usually between 0 and 1, often small like 0.01–0.05 in trials).


• Effective Sample Size (ESS): The sample size your clustered data "behaves like" under simple random sampling. It accounts for the DE.


• Formula: \( ESS = n / DE \), where \( n \) is your total sample size.

Step 1: Gather Your Data

• Total sample size (\( n \)): Number of individuals/units.


• Number of clusters (\( k \)): Number of groups (e.g., schools, villages).


• Average cluster size (\( m = n / k \)).


• ICC: Either pre-known or estimated from your outcome variable.


• Outcome data: For estimating ICC, you need the values of your primary outcome (e.g., a continuous variable like test scores) per individual within clusters.

Step 2: Estimate ICC (If Needed)

Example Dataset

Estimating ICC in R

# Load data (replace with your data frame)
data <- data.frame(
  cluster = rep(1:4, each=10),
  outcome = c(120,122,118,125,123,  # Cluster 1
              130,128,132,129,131,  # Cluster 2
              115,112,114,116,113,  # Cluster 3
              140,142,138,141,139)  # Cluster 4 (partial; extend as needed)
)
# Method 1: Using irr package (simple for continuous outcomes)
library(irr)
icc_result <- icc(ratings = data$outcome, raters = as.factor(data$cluster), type = "agreement")
icc_value <- icc_result$value  # This gives the ICC estimate
print(icc_value)  # Example output: ~0.15 (varies with data)
# Method 2: Using lme4 for mixed model (more robust)
library(lme4)
model <- lmer(outcome ~ (1 | cluster), data = data)
var_between <- as.data.frame(VarCorr(model))$vcov[1]  # Cluster variance
var_within <- sigma(model)^2  # Residual variance
total_var <- var_between + var_within
icc_value <- var_between / total_var
print(icc_value)

Estimating ICC in Python

import pandas as pd
import pingouin as pg
from statsmodels.stats.anova import anova_lm
from statsmodels.formula.api import ols
# Load data (replace with your DataFrame)
data = pd.DataFrame({
    'cluster': [1]*10 + [2]*10 + [3]*10 + [4]*10,  # Adjust sizes as needed
    'outcome': [120,122,118,125,123, 130,128,132,129,131,
                115,112,114,116,113, 140,142,138,141,139]  # Partial example
})
# Method 1: Pingouin ICC (for agreement)
icc_value = pg.intraclass_corr(data=data, targets='outcome', raters='cluster', ratings=None)
print(icc_value['ICC'])  # Example: ~0.15
# Method 2: ANOVA-based
model = ols('outcome ~ C(cluster)', data=data).fit()
anova_results = anova_lm(model)
msb = anova_results['sum_sq']['C(cluster)'] / anova_results['df']['C(cluster)']  # Between MS
msw = anova_results['sum_sq']['Residual'] / anova_results['df']['Residual']     # Within MS
k = data['cluster'].nunique()
m = len(data) / k
icc_value = (msb - msw) / (msb + (m - 1) * msw)
print(icc_value)

In this example, ICC might come out around 0.15 (depending on exact data—yours will differ).

Step 3: Calculate Design Effect and Effective Sample Size

• n = 40


• k = 4


• m = 40 / 4 = 10


• ICC = 0.15 (from above)

n <- 40
k <- 4
m <- n / k
icc <- 0.15  # From estimation
de <- 1 + (m - 1) * icc
ess <- n / de
print(paste("Design Effect:", round(de, 3)))  # e.g., 2.350
print(paste("Effective Sample Size:", round(ess, 1)))  # e.g., 17.0

n = 40
k = 4
m = n / k
icc = 0.15
de = 1 + (m - 1) * icc
ess = n / de
print(f"Design Effect: {de:.3f}")  # e.g., 2.350
print(f"Effective Sample Size: {ess:.1f}")  # e.g., 17.0

• Interpretation: Your 40-unit sample is as informative as ~17 independent units. Use ESS for power calculations or sample size planning.

Additional Tips

• For Binary/Proportion Outcomes: ICC estimation is similar but use logistic mixed models (e.g., glmer in R). DE adjusts variance of proportions.


• Unequal Cluster Sizes: Use weighted averages for m, or simulate via mixed models.


• In Analysis: Incorporate clustering with mixed-effects models (e.g., lmer in R, MixedLM in Python) or GEE. For sample size, inflate required n by DE.


• If You Share Data: Upload a CSV/summary (n, k, ICC, or raw outcomes), and I'll compute exactly.


• Resources: Donner & Klar's Design and Analysis of Cluster Randomization Trials in Health Research for deeper reading.