Rct Power Calculations

Power = 80% is conventional. Higher power (90%) is better when study is expensive or one-shot.

import numpy as np
from scipy.stats import norm

# Parameters
alpha = 0.05
power = 0.80
mde = 0.25  # Effect size in SD units (realistic for social programs)
p = 0.50  # Treatment allocation

# Base calculation
z_alpha = norm.ppf(1 - alpha/2)
z_power = norm.ppf(power)
n_base = ((z_alpha + z_power)**2) / (p * (1-p) * mde**2)

print(f"COMPREHENSIVE POWER ANALYSIS")
print(f"=" * 60)
print(f"Parameters: α={alpha}, power={power}, MDE={mde} SD")
print(f"\n1. Base sample size: {int(np.ceil(n_base))}")

# Adjustment 1: Clustering
icc = 0.05
cluster_size = 30
design_effect = 1 + (cluster_size - 1) * icc
n_after_clustering = n_base * design_effect

print(f"\n2. Clustering adjustment:")
print(f"   ICC={icc}, avg cluster size={cluster_size}")
print(f"   Design effect: {design_effect:.2f}")
print(f"   After clustering: {int(np.ceil(n_after_clustering))}")

# Adjustment 2: Non-compliance
compliance_rate = 0.70
n_after_compliance = n_after_clustering / (compliance_rate**2)

print(f"\n3. Non-compliance adjustment:")
print(f"   Expected compliance: {compliance_rate:.0%}")
print(f"   Inflation factor: {1/compliance_rate**2:.2f}")
print(f"   After compliance: {int(np.ceil(n_after_compliance))}")

# Adjustment 3: Attrition
attrition_rate = 0.20
n_final = n_after_compliance / (1 - attrition_rate)

print(f"\n4. Attrition adjustment:")
print(f"   Expected attrition: {attrition_rate:.0%}")
print(f"   Inflation factor: {1/(1-attrition_rate):.2f}")
print(f"   FINAL REQUIRED N: {int(np.ceil(n_final))}")

print(f"\n" + "=" * 60)
print(f"Total inflation: {n_final/n_base:.2f}x base sample size")
print(f"Baseline N needed: {int(np.ceil(n_final))}")
print(f"Endline N expected: {int(np.ceil(n_final * (1-attrition_rate)))}")

return int(np.ceil(n_final)),

</code_template>

</implementation_pattern>

<examples>

<example context="cluster_rct_power" difficulty="intermediate">
<description>Calculate power for cluster-randomized trial with design effect</description>
<code>
```python
@app.cell
def cluster_rct_power():
    # Cluster RCT power accounting for design effect
    # Shows how clustering can reduce power by 2-4x

    import numpy as np
    from scipy.stats import norm

    alpha = 0.05
    power = 0.80
    mde = 0.30

    # Individual randomization (baseline)
    z_a = norm.ppf(1 - alpha/2)
    z_p = norm.ppf(power)
    n_individual = ((z_a + z_p)**2) / (0.25 * mde**2)

    print(f"CLUSTER vs. INDIVIDUAL RANDOMIZATION")
    print(f"=" * 60)
    print(f"Individual randomization: {int(np.ceil(n_individual))} participants")

    # Cluster randomization scenarios
    scenarios = [
        {"icc": 0.05, "cluster_size": 20},
        {"icc": 0.05, "cluster_size": 50},
        {"icc": 0.10, "cluster_size": 30},
        {"icc": 0.20, "cluster_size": 30},
    ]

    for scenario in scenarios:
        icc = scenario["icc"]
        m = scenario["cluster_size"]
        de = 1 + (m - 1) * icc
        n_cluster = n_individual * de
        n_clusters_needed = int(np.ceil(n_cluster / m))

        print(f"\nICC={icc}, cluster size={m}:")
        print(f"  Design effect: {de:.2f}")
        print(f"  Total N needed: {int(np.ceil(n_cluster))}")
        print(f"  Clusters needed: {n_clusters_needed}")
        print(f"  Inflation: {de:.2f}x individual design")

    print("\nKey lesson: Higher ICC or larger clusters → much larger N needed")

    return ()

Then multiply by:

• Design effect if clustering: 1+(m-1)×ICC
• 1/compliance² for non-compliance
• 1/(1-attrition_rate) for attrition

Do NOT: proceed with underpowered study and hope for best.