Experimental Design Guide

A skill for designing rigorous experiments using formal Design of Experiments (DOE) methodology. Covers factorial designs, fractional factorials, response surface methods, and optimal design strategies for scientific research.

Fundamental Principles

Fisher's Three Principles

1. Randomization: Assign experimental units to treatments randomly to eliminate systematic bias
2. Replication: Include enough replicates to estimate experimental error and ensure statistical power
3. Blocking: Group similar experimental units to reduce nuisance variability

Sample Size and Power Analysis

from scipy import stats
import numpy as np

def power_analysis_ttest(effect_size: float, alpha: float = 0.05,
                          power: float = 0.80, ratio: float = 1.0) -> dict:
    """
    Calculate required sample size for a two-sample t-test.

    Args:
        effect_size: Cohen's d (expected effect size)
        alpha: Significance level
        power: Desired statistical power
        ratio: Ratio of n2/n1 (for unequal groups)
    """
    from statsmodels.stats.power import TTestIndPower
    analysis = TTestIndPower()
    n1 = analysis.solve_power(
        effect_size=effect_size,
        alpha=alpha,
        power=power,
        ratio=ratio,
        alternative='two-sided'
    )

    return {
        'n_per_group': int(np.ceil(n1)),
        'total_n': int(np.ceil(n1) + np.ceil(n1 * ratio)),
        'effect_size_d': effect_size,
        'alpha': alpha,
        'power': power,
        'interpretation': (
            f"Need {int(np.ceil(n1))} per group "
            f"(total N = {int(np.ceil(n1) + np.ceil(n1 * ratio))}) "
            f"to detect d = {effect_size} with {power*100:.0f}% power."
        )
    }

# Example: medium effect size
result = power_analysis_ttest(effect_size=0.5, alpha=0.05, power=0.80)
print(result['interpretation'])

Full Factorial Designs

2^k Factorial Design

import itertools
import pandas as pd

def create_factorial_design(factors: dict, replicates: int = 3) -> pd.DataFrame:
    """
    Create a full factorial experimental design.

    Args:
        factors: Dict mapping factor names to lists of levels
                 e.g., {'Temperature': [60, 80], 'Pressure': [1, 2], 'Catalyst': ['A', 'B']}
        replicates: Number of replicates per combination
    """
    factor_names = list(factors.keys())
    factor_levels = list(factors.values())

    # Generate all combinations
    combinations = list(itertools.product(*factor_levels))

    # Create design matrix with replicates
    rows = []
    run_order = 0
    for rep in range(replicates):
        for combo in combinations:
            run_order += 1
            row = {'Run': run_order, 'Replicate': rep + 1}
            for name, value in zip(factor_names, combo):
                row[name] = value
            row['Response'] = None  # To be filled with experimental data
            rows.append(row)

    design = pd.DataFrame(rows)

    # Randomize run order
    design = design.sample(frac=1, random_state=42).reset_index(drop=True)
    design['RandomizedRun'] = range(1, len(design) + 1)

    print(f"Design summary:")
    print(f"  Factors: {len(factors)}")
    print(f"  Levels per factor: {[len(v) for v in factors.values()]}")
    print(f"  Total treatments: {len(combinations)}")
    print(f"  Replicates: {replicates}")
    print(f"  Total runs: {len(design)}")

    return design

# Example: 2^3 factorial
design = create_factorial_design({
    'Temperature': [60, 80],
    'Pressure': [1, 2],
    'Catalyst': ['A', 'B']
}, replicates=3)