meta-analysis

Meta-Analysis

Quantitative synthesis of results from multiple studies. Calculates pooled effect sizes, assesses heterogeneity, detects publication bias, and generates forest and funnel plots.

When to Use

• "Combine these study results into a meta-analysis"
• "Calculate the pooled odds ratio from these trials"
• "Create a forest plot of these effect sizes"
• "Test for publication bias with a funnel plot"
• "What's the heterogeneity (I²) across these studies?"
• "Run a random-effects meta-analysis"

When NOT to Use

• Designing a systematic review protocol (use systematic-review)
• Searching for studies (use literature-search)
• Single-study statistical analysis (use statsmodels-stats)
• Narrative literature review (use paper-writing)

Effect Size Types

Outcome Type	Effect Size	Formula	Use When
Continuous	SMD (Cohen's d / Hedges' g)	$(M_1 - M_2) / S_p$	Comparing means across studies with different scales
Continuous	Mean Difference (MD)	$M_1 - M_2$	Same outcome measure across all studies
Binary	Odds Ratio (OR)	$(a \times d) / (b \times c)$	Case-control studies, binary outcomes
Binary	Risk Ratio (RR)	$(a/(a+b)) / (c/(c+d))$	Cohort studies, clinical trials
Binary	Risk Difference (RD)	$R_1 - R_2$	Absolute risk reduction
Time-to-event	Hazard Ratio (HR)	From Cox model	Survival analysis
Correlation	Fisher's z	$0.5 \ln((1+r)/(1-r))$	Correlation studies

Core Analysis with Python

Random-Effects Meta-Analysis

import numpy as np
from scipy import stats

def meta_analysis_random_effects(effects, variances, study_names=None):
    """
    DerSimonian-Laird random-effects meta-analysis.

    Args:
        effects: array of effect sizes (log-OR, SMD, etc.)
        variances: array of within-study variances
        study_names: optional list of study labels

    Returns:
        dict with pooled estimate, CI, heterogeneity stats
    """
    effects = np.array(effects, dtype=float)
    variances = np.array(variances, dtype=float)
    k = len(effects)

    # Fixed-effect weights
    w_fe = 1.0 / variances
    pooled_fe = np.sum(w_fe * effects) / np.sum(w_fe)

    # Cochran's Q
    Q = np.sum(w_fe * (effects - pooled_fe) ** 2)
    df = k - 1
    p_heterogeneity = 1 - stats.chi2.cdf(Q, df)

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

    # I-squared
    I2 = max(0, (Q - df) / Q * 100) if Q > 0 else 0

    # Random-effects weights
    w_re = 1.0 / (variances + tau2)
    pooled_re = np.sum(w_re * effects) / np.sum(w_re)
    se_pooled = np.sqrt(1.0 / np.sum(w_re))

    ci_lower = pooled_re - 1.96 * se_pooled
    ci_upper = pooled_re + 1.96 * se_pooled
    z = pooled_re / se_pooled
    p_value = 2 * (1 - stats.norm.cdf(abs(z)))

    return {
        'pooled_effect': pooled_re,
        'se': se_pooled,
        'ci_lower': ci_lower,
        'ci_upper': ci_upper,
        'z': z,
        'p_value': p_value,
        'tau2': tau2,
        'I2': I2,
        'Q': Q,
        'Q_df': df,
        'Q_p': p_heterogeneity,
        'k': k,
        'model': 'DerSimonian-Laird random-effects'
    }