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)

Analyzing Factorial Experiments

import statsmodels.api as sm
from statsmodels.formula.api import ols

def analyze_factorial(df: pd.DataFrame, response: str,
                       factors: list[str]) -> dict:
    """
    Analyze a factorial experiment using ANOVA.
    """
    # Build formula with all main effects and interactions
    main_effects = ' + '.join([f'C({f})' for f in factors])
    interactions = ' + '.join([f'C({f1}):C({f2})'
                               for i, f1 in enumerate(factors)
                               for f2 in factors[i+1:]])
    formula = f'{response} ~ {main_effects} + {interactions}'

    model = ols(formula, data=df).fit()
    anova_table = sm.stats.anova_lm(model, typ=2)

    # Effect sizes (eta-squared)
    ss_total = anova_table['sum_sq'].sum()
    anova_table['eta_sq'] = anova_table['sum_sq'] / ss_total

    return {
        'anova_table': anova_table,
        'r_squared': model.rsquared,
        'significant_effects': anova_table[anova_table['PR(>F)'] < 0.05].index.tolist()
    }

Fractional Factorial Designs

When a full factorial has too many runs:

def fractional_factorial_2k(k: int, resolution: int = 3) -> pd.DataFrame:
    """
    Generate a 2^(k-p) fractional factorial design.

    Args:
        k: Number of factors
        resolution: Design resolution (III, IV, or V)
    """
    from pyDOE2 import fracfact

    # Resolution III: 2^(k-p) where p minimizes runs
    # Common designs:
    # 2^(3-1) = 4 runs (Resolution III)
    # 2^(4-1) = 8 runs (Resolution IV)
    # 2^(5-2) = 8 runs (Resolution III)
    # 2^(7-4) = 8 runs (Resolution III, Plackett-Burman)

    design = fracfact(f'a b c {"d" if k >= 4 else ""} {"e" if k >= 5 else ""}')
    df = pd.DataFrame(design, columns=[f'Factor_{i+1}' for i in range(design.shape[1])])

    print(f"Fractional factorial: {len(df)} runs for {k} factors")
    return df

Response Surface Methodology (RSM)

Central Composite Design

def central_composite_design(factor_ranges: dict) -> pd.DataFrame:
    """
    Create a Central Composite Design for response surface optimization.
    """
    from pyDOE2 import ccdesign

    k = len(factor_ranges)
    design_coded = ccdesign(k, center=(4,), alpha='orthogonal', face='circumscribed')

    factor_names = list(factor_ranges.keys())
    df = pd.DataFrame(design_coded, columns=factor_names)

    # Convert from coded (-1, +1) to natural units
    for name, (low, high) in factor_ranges.items():
        center = (high + low) / 2
        half_range = (high - low) / 2
        df[name] = center + df[name] * half_range

    return df

# Example: optimize a chemical reaction
design = central_composite_design({
    'Temperature_C': [50, 90],
    'pH': [5, 9],
    'Time_min': [10, 60]
})

Randomization and Blinding

• Single-blind: Participants do not know their treatment assignment
• Double-blind: Neither participants nor experimenters know assignments
• Allocation concealment: Assignment sequence is hidden until the moment of assignment

For computer-generated randomization, always record and report the random seed used. Use block randomization to ensure balanced groups when enrollment is sequential.

Reporting Checklist

Follow CONSORT (clinical trials), ARRIVE (animal studies), or STROBE (observational) guidelines:

• State the primary and secondary outcomes before analysis
• Report all planned analyses, including non-significant results
• Describe randomization method and any deviations from protocol
• Include sample size justification with power analysis parameters