grad-meta-analysis

Research

Apply meta-analysis to synthesize effect sizes across multiple studies, assess heterogeneity, and evaluate publication bias. Use this skill when the user needs to combine findings from prior research, compare fixed-effect vs random-effects models, compute pooled effect sizes, or when they ask 'what does the overall evidence say', 'how do I combine results across studies', or 'is there publication bias'.

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後設分析 (Meta-Analysis)

Overview

Meta-analysis statistically combines effect sizes from multiple independent studies to produce a pooled estimate with greater precision and generalizability. It quantifies between-study heterogeneity and tests for publication bias, providing a rigorous evidence synthesis that goes beyond narrative literature reviews.

When to Use

Synthesizing quantitative findings from multiple studies on the same research question
Resolving conflicting results across studies
Estimating an overall effect size with tighter confidence intervals
Identifying moderators that explain heterogeneity across studies

When NOT to Use

Studies are too heterogeneous in constructs, measures, or populations to combine meaningfully
Fewer than 5 studies are available (pooled estimates become unreliable)
Primary studies have fundamentally different research designs (mixing RCTs with observational)
The research question is qualitative or conceptual rather than quantitative

Assumptions

IRON LAW: A meta-analysis is only as good as the studies it includes —
garbage in, garbage out. Publication bias inflates pooled effect sizes
because non-significant findings go unpublished.

Key assumptions:

Studies estimate the same underlying construct (conceptual homogeneity)
Effect sizes are statistically independent (one effect per study, or use multilevel models)
Study-level moderators are coded reliably and without bias
The search strategy captures the relevant population of studies (no systematic omission)

Methodology

Step 1 — Extract and Code Effect Sizes

Convert study findings to a common effect size metric (Cohen's d, Hedges' g, r, OR). Code study-level moderators (sample size, design, context). See references/ for conversion formulas.

Step 2 — Choose Fixed-Effect vs Random-Effects Model

Fixed-effect assumes one true effect; random-effects assumes effects vary across studies. If studies span different populations or contexts, random-effects is almost always appropriate.

Step 3 — Assess Heterogeneity

Compute Q statistic (test of homogeneity), I² (proportion of variance due to heterogeneity), and τ² (between-study variance). I² > 75% indicates substantial heterogeneity warranting moderator analysis.

Step 4 — Test for Publication Bias and Report

Use funnel plot, Egger's regression test, and trim-and-fill method. Report pooled effect, CI, prediction interval, and results of bias assessment.

Output Format

## Meta-Analysis: [Research Question]

### Study Inclusion
| Criterion | Value |
|-----------|-------|
| Studies included (k) | xx |
| Total sample size (N) | xxxx |
| Effect size metric | [d / r / OR] |

### Pooled Effect Size
| Model | Effect | 95% CI | z | p-value |
|-------|--------|--------|---|---------|
| Fixed-effect | x.xx | [x.xx, x.xx] | x.xx | x.xx |
| Random-effects | x.xx | [x.xx, x.xx] | x.xx | x.xx |

### Heterogeneity
| Statistic | Value | Interpretation |
|-----------|-------|----------------|
| Q | x.xx (p = x.xx) | [significant/not] |
| I² | x.xx% | [low/moderate/high] |
| τ² | x.xx | [between-study variance] |

### Publication Bias
| Test | Result | Interpretation |
|------|--------|----------------|
| Funnel plot | [symmetric/asymmetric] | [bias suspected?] |
| Egger's test | p = x.xx | [significant?] |
| Trim-and-fill | adjusted effect = x.xx | [studies imputed: x] |

### Limitations
- [Note any assumption violations]

Gotchas

Combining apples and oranges: statistically possible but conceptually meaningless if constructs differ
Random-effects models give more weight to small studies, which are often lower quality
I² depends on precision of included studies; low I² with imprecise studies does not mean homogeneity
Funnel plot asymmetry can be caused by factors other than publication bias (small-study effects)
File-drawer problem: unpublished null results are systematically missing
Moderator analyses with many subgroups and few studies per subgroup are underpowered and unreliable

References

Borenstein, M., Hedges, L. V., Higgins, J. P. T., & Rothstein, H. R. (2009). Introduction to Meta-Analysis. Wiley.
Higgins, J. P. T., & Thompson, S. G. (2002). Quantifying heterogeneity in a meta-analysis. Statistics in Medicine, 21(11), 1539-1558.
Rothstein, H. R., Sutton, A. J., & Borenstein, M. (2005). Publication Bias in Meta-Analysis. Wiley.