meta-analysis-guide

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

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'
    )