Meta-Analysis Guide

A skill for conducting rigorous meta-analyses: computing and pooling effect sizes, assessing heterogeneity, evaluating publication bias, and generating forest plots. Follows Cochrane Handbook and PRISMA guidelines.

Effect Size Computation

Common Effect Size Measures

Measure	Use Case	Formula	Interpretation
Cohen's d	Mean difference (2 groups)	(M1 - M2) / S_pooled	0.2 small, 0.5 medium, 0.8 large
Hedges' g	d with small-sample correction	d * J(df)	Preferred over d for small N
Pearson r	Correlation	r	0.1 small, 0.3 medium, 0.5 large
Odds Ratio	Binary outcomes	(ad)/(bc)	1 = no effect
Risk Ratio	Binary outcomes	(a/(a+b))/(c/(c+d))	1 = no effect
SMD	Standardized mean difference	Same as Hedges' g	When scales differ

Computing Effect Sizes in Python

import numpy as np
from dataclasses import dataclass

@dataclass
class EffectSize:
    estimate: float
    variance: float
    se: float
    ci_lower: float
    ci_upper: float
    measure: str

def cohens_d(m1: float, m2: float, sd1: float, sd2: float,
              n1: int, n2: int) -> EffectSize:
    """
    Compute Hedges' g (bias-corrected Cohen's d).
    """
    # Pooled standard deviation
    sd_pooled = np.sqrt(((n1-1)*sd1**2 + (n2-1)*sd2**2) / (n1+n2-2))

    # Cohen's d
    d = (m1 - m2) / sd_pooled

    # Small-sample correction (Hedges' g)
    df = n1 + n2 - 2
    j = 1 - (3 / (4*df - 1))
    g = d * j

    # Variance of g
    var_g = (n1+n2)/(n1*n2) + g**2 / (2*(n1+n2))
    se_g = np.sqrt(var_g)

    return EffectSize(
        estimate=g,
        variance=var_g,
        se=se_g,
        ci_lower=g - 1.96*se_g,
        ci_upper=g + 1.96*se_g,
        measure='Hedges_g'
    )

def odds_ratio(a: int, b: int, c: int, d: int) -> EffectSize:
    """
    Compute log odds ratio from a 2x2 table.
    a=treatment success, b=treatment failure, c=control success, d=control failure
    """
    # Add 0.5 continuity correction if any cell is 0
    if any(x == 0 for x in [a, b, c, d]):
        a, b, c, d = a+0.5, b+0.5, c+0.5, d+0.5

    log_or = np.log((a*d) / (b*c))
    var = 1/a + 1/b + 1/c + 1/d
    se = np.sqrt(var)

    return EffectSize(
        estimate=log_or,
        variance=var,
        se=se,
        ci_lower=log_or - 1.96*se,
        ci_upper=log_or + 1.96*se,
        measure='log_OR'
    )

Fixed-Effect and Random-Effects Models

Inverse-Variance Pooling

def random_effects_meta(effects: list[EffectSize]) -> dict:
    """
    Random-effects meta-analysis using DerSimonian-Laird estimator.
    """
    yi = np.array([e.estimate for e in effects])
    vi = np.array([e.variance for e in effects])
    wi = 1 / vi
    k = len(effects)

    # Fixed-effect estimate
    fe_estimate = np.sum(wi * yi) / np.sum(wi)

    # Q statistic for heterogeneity
    Q = np.sum(wi * (yi - fe_estimate)**2)
    df = k - 1

    # DerSimonian-Laird tau-squared
    C = np.sum(wi) - np.sum(wi**2) / np.sum(wi)
    tau2 = max(0, (Q - df) / C)

    # Random-effects weights
    wi_re = 1 / (vi + tau2)
    re_estimate = np.sum(wi_re * yi) / np.sum(wi_re)
    re_se = np.sqrt(1 / np.sum(wi_re))
    re_ci = (re_estimate - 1.96*re_se, re_estimate + 1.96*re_se)

    # Heterogeneity statistics
    I2 = max(0, (Q - df) / Q * 100) if Q > 0 else 0
    H2 = Q / df if df > 0 else 1

    return {
        'pooled_effect': re_estimate,
        'se': re_se,
        'ci_95': re_ci,
        'tau_squared': tau2,
        'Q_statistic': Q,
        'Q_df': df,
        'Q_pvalue': 1 - stats.chi2.cdf(Q, df),
        'I_squared': I2,
        'H_squared': H2,
        'interpretation': (
            f"I-squared = {I2:.1f}%: "
            + ('low' if I2 < 25 else 'moderate' if I2 < 75 else 'high')
            + ' heterogeneity'
        )
    }

Forest Plot

import matplotlib.pyplot as plt

def forest_plot(studies: list[dict], pooled: dict,
                title: str = 'Forest Plot') -> plt.Figure:
    """
    Create a publication-quality forest plot.

    Args:
        studies: List of dicts with 'name', 'effect', 'ci_lower', 'ci_upper', 'weight'
        pooled: Dict with 'pooled_effect', 'ci_95'
    """
    fig, ax = plt.subplots(figsize=(10, max(6, len(studies)*0.5)))
    k = len(studies)

    for i, study in enumerate(studies):
        y = k - i
        ax.plot([study['ci_lower'], study['ci_upper']], [y, y], 'b-', linewidth=1)
        size = study.get('weight', 5) * 2
        ax.plot(study['effect'], y, 'bs', markersize=max(3, min(size, 15)))
        ax.text(-0.05, y, study['name'], ha='right', va='center', fontsize=9,
                transform=ax.get_yaxis_transform())

    # Pooled estimate (diamond)
    pe = pooled['pooled_effect']
    ci = pooled['ci_95']
    ax.fill([ci[0], pe, ci[1], pe], [0.3, 0.6, 0.3, 0], 'r', alpha=0.7)

    ax.axvline(x=0, color='gray', linestyle='--', linewidth=0.5)
    ax.set_xlabel('Effect Size (Hedges g)')
    ax.set_title(title)
    ax.set_yticks([])
    plt.tight_layout()
    return fig

Publication Bias Assessment

Methods to assess and address publication bias:

1. Funnel plot: Visual inspection for asymmetry
2. Egger's test: Regression test for funnel plot asymmetry (p < 0.10 suggests bias)
3. Trim-and-fill: Imputes missing studies to correct for bias
4. p-curve analysis: Tests whether significant results contain evidential value
5. Selection models: Formally model the publication process (e.g., Vevea-Hedges)

Reporting Standards

Follow PRISMA 2020 guidelines for reporting:

• Report all effect sizes with 95% CIs
• Report Q, I-squared, and tau-squared for heterogeneity
• Include forest plots for all primary outcomes
• Report funnel plots and publication bias tests
• Provide subgroup analyses and sensitivity analyses (leave-one-out)
• Register the protocol on PROSPERO before conducting the review