---
id: beita6969/ScienceClaw/meta-analysis
version: "cb99e00f"
license: MIT
install: manual
updated: 2026-06-08
---

# meta-analysis — meta-analysis synthesizes findings across studies by computing pooled effect sizes, testing for heterogeneity, and detecting publication bias through forest and funnel plots. It supports multiple effect size types including odds ratios, risk ratios, standardized mean differences, and hazard ratios, with random-effects modeling and bias diagnostics built in.

Publisher: beita6969 · Stars: 869 · Updated: 2026-06-08

Install (manual): `git clone https://github.com/beita6969/ScienceClaw`

## SKILL.md

# 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

```python
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'
    }
```

### Forest Plot

```python
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches

def forest_plot(effects, ci_lower, ci_upper, study_names, pooled, pooled_ci,
                xlabel='Effect Size', title='Forest Plot', output_path='forest_plot.png'):
    """Generate a publication-quality forest plot."""
    k = len(effects)
    fig, ax = plt.subplots(figsize=(8, max(4, k * 0.4 + 2)))

    y_positions = list(range(k, 0, -1))

    # Individual studies
    for i, y in enumerate(y_positions):
        ax.plot(effects[i], y, 'ks', markersize=8)
        ax.plot([ci_lower[i], ci_upper[i]], [y, y], 'k-', linewidth=1.5)

    # Pooled estimate (diamond)
    diamond_y = 0
    diamond_half_h = 0.3
    diamond = plt.Polygon([
        [pooled_ci[0], diamond_y],
        [pooled, diamond_y + diamond_half_h],
        [pooled_ci[1], diamond_y],
        [pooled, diamond_y - diamond_half_h]
    ], closed=True, facecolor='steelblue', edgecolor='black')
    ax.add_patch(diamond)

    # Reference line at null effect
    ax.axvline(x=0, color='gray', linestyle='--', linewidth=0.8)

    # Labels
    yticks = y_positions + [diamond_y]
    ylabels = study_names + ['Pooled']
    ax.set_yticks(yticks)
    ax.set_yticklabels(ylabels)
    ax.set_xlabel(xlabel)
    ax.set_title(title)
    ax.set_ylim(-1, k + 1.5)

    fig.tight_layout()
    fig.savefig(output_path, dpi=300, bbox_inches='tight')
    print(f"Forest plot saved: {output_path}")
    return fig
```

### Funnel Plot (Publication Bias)

```python
def funnel_plot(effects, se_values, pooled_effect,
                xlabel='Effect Size', output_path='funnel_plot.png'):
    """Generate a funnel plot to assess publication bias."""
    fig, ax = plt.subplots(figsize=(6, 5))

    ax.scatter(effects, se_values, c='black', s=30, zorder=3)

    # Pseudo-confidence region
    se_range = np.linspace(0.001, max(se_values) * 1.1, 100)
    ci_low = pooled_effect - 1.96 * se_range
    ci_high = pooled_effect + 1.96 * se_range
    ax.fill_betweenx(se_range, ci_low, ci_high, alpha=0.1, color='gray')
    ax.axvline(pooled_effect, color='red', linestyle='--', linewidth=1)

    ax.set_xlabel(xlabel)
    ax.set_ylabel('Standard Error')
    ax.set_title('Funnel Plot')
    ax.invert_yaxis()  # Convention: smaller SE at top

    fig.tight_layout()
    fig.savefig(output_path, dpi=300, bbox_inches='tight')
    print(f"Funnel plot saved: {output_path}")
    return fig
```

## Publication Bias Tests

### Egger's Regression Test
```python
def egger_test(effects, se_values):
    """Egger's test for funnel plot asymmetry."""
    precision = 1.0 / np.array(se_values)
    standardized = np.array(effects) / np.array(se_values)
    slope, intercept, r, p, se = stats.linregress(precision, standardized)
    return {'intercept': intercept, 'se': se, 'p_value': p,
            'interpretation': 'Significant asymmetry' if p < 0.10 else 'No significant asymmetry'}
```

### Begg's Rank Correlation Test
```python
def begg_test(effects, variances):
    """Begg-Mazumdar rank correlation test."""
    standardized = effects / np.sqrt(variances)
    tau, p = stats.kendalltau(standardized, variances)
    return {'tau': tau, 'p_value': p}
```

### Trim-and-Fill Method
- Identifies and imputes missing studies from funnel plot asymmetry
- Re-estimates the pooled effect including imputed studies
- Use `statsmodels` or `metafor` (R) for implementation

## Heterogeneity Interpretation

| I² Value | Interpretation |
|----------|---------------|
| 0-25% | Low heterogeneity |
| 25-50% | Moderate heterogeneity |
| 50-75% | Substantial heterogeneity |
| 75-100% | Considerable heterogeneity |

When I² > 50%, investigate sources:
1. **Subgroup analysis**: Split by study design, population, intervention dose
2. **Meta-regression**: Model effect size as function of study-level covariates
3. **Sensitivity analysis**: Leave-one-out, exclude high risk-of-bias studies

## Reporting Standards

Follow PRISMA 2020 for reporting meta-analyses. Include:
1. Number of studies (k) and total participants (N)
2. Pooled effect size with 95% CI
3. Heterogeneity: Q statistic (df, p), I², tau²
4. Model type: fixed-effect vs. random-effects with justification
5. Publication bias assessment results
6. Forest plot and funnel plot as figures

## Best Practices

1. Use random-effects model by default (studies rarely share a true common effect)
2. Always report both Q and I² for heterogeneity
3. Log-transform ORs and RRs before pooling; back-transform for reporting
4. Use Hedges' g rather than Cohen's d for small-sample correction
5. Minimum 5-10 studies for reliable funnel plot interpretation
6. Never fabricate study data or effect sizes

## Zero-Hallucination Rule

- ALL study-level data must come from tool results or user-provided data
- NEVER generate fictional study names, sample sizes, or effect sizes
- If insufficient data for meta-analysis, say so explicitly


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