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Meta-analysis (evidence synthesis)

The output half of a systematic review. Once you have extracted an effect size and its standard error from each study, sp.meta_analysis pools them — fixed-effect and random-effects — and reports the heterogeneity statistics, a prediction interval, Egger's test for small-study effects, and forest / funnel plots [@dersimonian1986meta; @higgins2002quantifying; @egger1997bias].

This is summary-data meta-analysis: you pass per-study effects and SEs (log odds ratios, mean differences, log hazard ratios, …). It does not fit individual-participant-data models.

1. One call

import statspai as sp

# five studies' log odds ratios and their standard errors
effects = [0.10, 0.25, -0.05, 0.30, 0.15]
ses     = [0.05, 0.10,  0.08, 0.12, 0.06]

m = sp.meta_analysis(effects, ses, labels=["Trial A", "B", "C", "D", "E"])
print(m.summary())

The summary reports both models — you do not have to choose blind:

  • Fixed-effect (inverse-variance) — assumes one common true effect.
  • Random-effects (DerSimonian-Laird) — assumes the true effect varies across studies; wider CI, and the model StatsPAI reports by default (method="DL").

Switch the headline model with method="fixed".

2. Heterogeneity — is pooling even sensible?

m.q, m.q_pvalue     # Cochran's Q and its chi-square p-value
m.i2                # I^2 (fraction of variation due to heterogeneity)
m.tau2              # between-study variance estimate
m.prediction_interval   # where a *future* study's true effect is expected

A high I^2 (say > 50%) or a small Q p-value warns that a single pooled number hides real between-study differences — report the random-effects estimate and the prediction interval, and consider a subgroup or meta-regression analysis. The prediction interval is typically much wider than the confidence interval of the pooled mean, which is the honest way to convey heterogeneity.

3. Publication bias / small-study effects

egg = m.egger_test()        # {'intercept', 'se', 't', 'p_value', 'df'}
print(egg["p_value"])

m.funnel_plot()             # visual asymmetry check

Egger's test regresses the standard normal deviate on precision; a non-zero intercept flags funnel-plot asymmetry (often small-study or publication bias). Treat it as a screen, not proof — asymmetry has many possible causes.

4. The forest plot

m.forest_plot()             # per-study CIs + pooled diamond

Notes & limitations

  • DerSimonian-Laird is the classic random-effects estimator; for very few studies or rare events, REML / Paule-Mandel tau^2 and Hartung-Knapp CIs are more conservative and are on the roadmap.
  • Effect sizes must be supplied on an additive scale (log-transform ratio measures before pooling, then exponentiate the pooled estimate for reporting).

Where to next