Randomized Controlled Trials — Design and Analysis

A randomized controlled trial (RCT) is the gold standard for establishing causal relationships because randomization ensures that treatment and control groups are comparable in expectation.

Key Components of an RCT

| Component | Description |

|-----------|------------|

| Randomization | Random assignment to treatment/control |

| Control group | Receives placebo or standard treatment |

| Blinding | Participants/researchers unaware of assignment |

| Sample size | Determined by power analysis |

| Pre-registration | Specify analysis plan before data collection |

Why Randomization Works

Treatment Effects in RCTs

With randomization, the naive comparison identifies the ATE.

Sample Size and Power

Types of Analysis

Intention-to-Treat (ITT)

| Advantage | Disadvantage |

|-----------|-------------|

| Preserves randomization | May underestimate effect |

| Handles non-compliance | Diluted by non-adherence |

| Clinically relevant | |

Per-Protocol Analysis

Analyze only participants who fully complied with the protocol. May introduce bias if non-compliance is related to outcomes.

Blinding

| Type | Who is blinded | Purpose |

|------|---------------|---------|

| Single-blind | Participants | Reduces placebo effect |

| Double-blind | Participants + researchers | Reduces observer bias |

| Triple-blind | Participants + researchers + analysts | Reduces analysis bias |

Common Pitfalls

CONSORT Flow

A well-reported RCT follows the CONSORT guidelines:

1. Enrollment: How many were assessed and randomized?

2. Allocation: How many assigned to each group?

3. Follow-up: How many lost to follow-up?

4. Analysis: How many included in final analysis?

Python Implementation

import numpy as np

import pandas as pd

from scipy import stats

import matplotlib.pyplot as plt



np.random.seed(42)



# Simulate RCT

n = 500

X1 = np.random.randn(n)  # Age

X2 = np.random.binomial(1, 0.5, n)  # Gender



# Randomization

T = np.random.binomial(1, 0.5, n)



# Outcome (true ATE = 3.0)

Y0 = 50 + 0.5*X1 + 2*X2 + np.random.randn(n)*10

Y1 = Y0 + 3.0

Y = T * Y1 + (1 - T) * Y0



df = pd.DataFrame({'Y': Y, 'T': T, 'age': X1, 'gender': X2})



# Check balance (should be balanced due to randomization)

treat = df[df['T']==1]

control = df[df['T']==0]

print("Balance check:")

print(f"Age: treat={treat['age'].mean():.2f}, control={control['age'].mean():.2f}")

print(f"Gender: treat={treat['gender'].mean():.2f}, control={control['gender'].mean():.2f}")



# Two-sample t-test

t_stat, p_val = stats.ttest_ind(treat['Y'], control['Y'])

print(f"\nTreatment effect: {treat['Y'].mean() - control['Y'].mean():.2f}")

print(f"95% CI: [{treat['Y'].mean()-control['Y'].mean()-1.96*10*np.sqrt(2/n):.2f}, "

      f"{treat['Y'].mean()-control['Y'].mean()+1.96*10*np.sqrt(2/n):.2f}]")

print(f"p-value: {p_val:.4f}")



# Power analysis

from statsmodels.stats.power import TTestIndPower

power_analysis = TTestIndPower()

power = power_analysis.power(effect_size=3.0/10, nobs1=250, ratio=1.0, alpha=0.05)

print(f"\nPower: {power:.3f}")

Worked Example

Key Takeaways

Related Topics

• See Causal Inference for the potential outcomes framework

• See Propensity Score Matching for when randomization is not possible

• See Hypothesis Testing for p-values and significance