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name meta-analysis-psych description Use this Skill for psychology meta-analysis: Cohen's d, Hedges' g, random-effects pooling, forest plot, publication bias (Egger, p-curve, PET-PEESE), and sensitivity analysis.
tags ["psychology","meta-analysis","effect-sizes","forest-plot","publication-bias","p-curve"] version 1.0.0 authors [{"name":"awesome-rosetta-skills contributors","github":"@xjtulyc"}] license MIT platforms ["claude-code","codex","gemini-cli","cursor"] dependencies {"python":["numpy>=1.23","scipy>=1.9","pandas>=1.5","matplotlib>=3.6"]} last_updated 2026-03-18 status stable
Meta-Analysis for Psychology
TL;DR — Complete meta-analysis pipeline in pure Python/NumPy: Cohen's d
and Hedges' g from raw statistics, DerSimonian-Laird random-effects pooling,
I² heterogeneity, forest plot with summary diamond, funnel plot, Egger test,
trim-and-fill, PET-PEESE regression correction, and p-curve analysis.
When to Use
Use this Skill when you need to:
Pool effect sizes from multiple independent studies on the same research question
Quantify and visualize heterogeneity across studies (I², τ²)
Detect and correct for publication bias (Egger test, trim-and-fill, PET-PEESE)
Assess whether p-values are consistent with a true effect (p-curve analysis)
Generate publication-quality forest and funnel plots
Run sensitivity analyses (leave-one-out, cumulative meta-analysis)
Background
Effect Size Measures
Cohen's d (raw mean difference standardized by pooled SD):
d = (M1 - M2) / SD_pooled
Hedges' g (small-sample bias correction for d):
J = 1 - 3 / (4 * (n1 + n2 - 2) - 1)
g = J × d
Var(g) = (n1 + n2) / (n1 × n2) + g² / (2 × (n1 + n2))
Random-Effects Model (DerSimonian-Laird)
Under the random-effects model, true effect sizes θi vary:
θi = θ + ui + εi where ui ~ N(0, τ²), εi ~ N(0, vi)
Between-study variance τ² is estimated by the DerSimonian-Laird method:
Q = Σ wi(gi - g_FE)² (Cochran's Q statistic)
τ² = max(0, (Q - (k-1)) / (Σwi - Σwi²/Σwi))
wi* = 1 / (vi + τ²) (RE weights)
g_RE = Σ(wi* × gi) / Σwi*
I² = (Q - (k-1)) / Q × 100% (% of total variance due to heterogeneity)
Publication Bias Methods
Method Description Funnel plot Asymmetry suggests missing small-n null studies Egger test Regress std normal deviate on precision; intercept ≠ 0 = bias Trim-and-fill Impute mirror studies to restore funnel symmetry PET-PEESE Regress g on SE (PET) or SE² (PEESE); intercept = bias-corrected effect p-curve Distribution of p-values below .05; right-skew = real effect
Environment Setup
conda create -n meta_env python=3.11 -y
conda activate meta_env
pip install numpy>=1.23 scipy>=1.9 pandas>=1.5 matplotlib>=3.6
python -c "import numpy, scipy, pandas, matplotlib; print('All OK')"
Core Workflow
Step 1 — Effect Size Computation import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.patches as mpatches
from scipy import stats
from typing import Optional , Dict , List , Tuple
def cohens_d_from_means (
m1: float , m2: float ,
sd1: float , sd2: float ,
n1: int , n2: int ,
) -> Dict :
"""
Compute Cohen's d and Hedges' g from group means and SDs.
Args:
m1, m2: Group means.
sd1, sd2: Group standard deviations.
n1, n2: Sample sizes.
Returns:
Dict with d, g, var_g, se_g, 95% CI.
"""
sd_pooled = np.sqrt(((n1 - 1 ) * sd1**2 + (n2 - 1 ) * sd2**2 ) / (n1 + n2 - 2 ))
d = (m1 - m2) / sd_pooled
df_total = n1 + n2 - 2
J = 1 - 3 / (4 * df_total - 1 )
g = J * d
var_g = (n1 + n2) / (n1 * n2) + g**2 / (2 * (n1 + n2))
se_g = np.sqrt(var_g)
return {
"d" : round (d, 4 ),
"g" : round (g, 4 ),
"J" : round (J, 4 ),
"var_g" : round (var_g, 6 ),
"se_g" : round (se_g, 4 ),
"ci_lo" : round (g - 1.96 * se_g, 4 ),
"ci_hi" : round (g + 1.96 * se_g, 4 ),
"n1" : n1, "n2" : n2,
}
def cohens_d_from_t (
t: float , n1: int , n2: int ,
) -> Dict :
"""
Compute Cohen's d from an independent-samples t-statistic.
Args:
t: t-statistic (signed).
n1, n2: Group sample sizes.
Returns:
Dict with d, g, var_g, se_g, 95% CI.
"""
d = t * np.sqrt((n1 + n2) / (n1 * n2))
df_total = n1 + n2 - 2
J = 1 - 3 / (4 * df_total - 1 )
g = J * d
var_g = (n1 + n2) / (n1 * n2) + g**2 / (2 * (n1 + n2))
se_g = np.sqrt(var_g)
return {
"d" : round (d, 4 ), "g" : round (g, 4 ), "J" : round (J, 4 ),
"var_g" : round (var_g, 6 ), "se_g" : round (se_g, 4 ),
"ci_lo" : round (g - 1.96 * se_g, 4 ), "ci_hi" : round (g + 1.96 * se_g, 4 ),
"n1" : n1, "n2" : n2,
}
def cohens_d_from_F (
F: float , n1: int , n2: int ,
) -> Dict :
"""
Compute Cohen's d from a one-df F-ratio (two-group comparison).
Args:
F: F-statistic (F = t²).
n1, n2: Group sample sizes.
Returns:
Dict with d, g, var_g, se_g, 95% CI.
"""
t = np.sqrt(F)
return cohens_d_from_t(t, n1, n2)
Step 2 — Random-Effects Meta-Analysis def run_meta_analysis (
df: pd.DataFrame,
g_col: str = "g" ,
var_col: str = "var_g" ,
study_col: str = "study" ,
) -> Dict :
"""
DerSimonian-Laird random-effects meta-analysis.
Args:
df: DataFrame with one row per study.
g_col: Column of Hedges' g effect sizes.
var_col: Column of effect size variances.
study_col: Column of study labels.
Returns:
Dict with pooled estimate, heterogeneity statistics, and weights.
"""
g = df[g_col].values
v = df[var_col].values
k = len (g)
w_FE = 1 / v
g_FE = np.sum (w_FE * g) / np.sum (w_FE)
Q = np.sum (w_FE * (g - g_FE) ** 2 )
c = np.sum (w_FE) - np.sum (w_FE ** 2 ) / np.sum (w_FE)
tau2 = max (0.0 , (Q - (k - 1 )) / c)
I2 = max (0.0 , (Q - (k - 1 )) / Q * 100 ) if Q > 0 else 0.0
w_RE = 1 / (v + tau2)
g_RE = np.sum (w_RE * g) / np.sum (w_RE)
var_RE = 1 / np.sum (w_RE)
se_RE = np.sqrt(var_RE)
z_RE = g_RE / se_RE
p_RE = 2 * stats.norm.sf(abs (z_RE))
ci_lo = g_RE - 1.96 * se_RE
ci_hi = g_RE + 1.96 * se_RE
result = {
"k" : k,
"g_RE" : round (g_RE, 4 ),
"se_RE" : round (se_RE, 4 ),
"ci_lo" : round (ci_lo, 4 ),
"ci_hi" : round (ci_hi, 4 ),
"z" : round (z_RE, 3 ),
"p" : round (p_RE, 4 ),
"Q" : round (Q, 3 ),
"Q_df" : k - 1 ,
"Q_p" : round (float (stats.chi2.sf(Q, df=k - 1 )), 4 ),
"I2" : round (I2, 1 ),
"tau2" : round (tau2, 6 ),
"tau" : round (np.sqrt(tau2), 4 ),
"w_RE" : w_RE,
"g_FE" : round (g_FE, 4 ),
}
print (
f"Random-effects meta-analysis (k={k} studies):\n"
f" g = {g_RE:.3 f} [{ci_lo:.3 f} , {ci_hi:.3 f} ], p = {p_RE:.4 f} \n"
f" Q({k-1 } ) = {Q:.2 f} , p = {result['Q_p' ]:.4 f} \n"
f" I² = {I2:.1 f} %, τ = {result['tau' ]:.4 f} \n"
f" Heterogeneity: "
+ ("low" if I2 < 25 else "moderate" if I2 < 75 else "high" )
)
return result
Step 3 — Forest Plot def forest_plot (
df: pd.DataFrame,
meta_result: Dict ,
g_col: str = "g" ,
ci_lo_col: str = "ci_lo" ,
ci_hi_col: str = "ci_hi" ,
study_col: str = "study" ,
sort_by: str = "g" ,
output_path: Optional [str ] = None ,
) -> plt.Figure:
"""
Generate a forest plot with study CIs and summary diamond.
Args:
df: Study-level DataFrame (one row per study).
meta_result: Output from run_meta_analysis().
g_col: Column of effect sizes.
ci_lo_col: Column of lower 95% CIs.
ci_hi_col: Column of upper 95% CIs.
study_col: Column of study labels.
sort_by: Sort studies by 'g' (default) or 'year'.
output_path: Optional path to save figure.
Returns:
Matplotlib Figure.
"""
if sort_by == "g" :
df = df.sort_values(g_col).reset_index(drop=True )
k = len (df)
g_RE = meta_result["g_RE" ]
ci_lo_RE = meta_result["ci_lo" ]
ci_hi_RE = meta_result["ci_hi" ]
fig, ax = plt.subplots(figsize=(10 , max (6 , k * 0.45 + 2 )))
y_positions = list (range (k, 0 , -1 ))
for i, (_, row) in enumerate (df.iterrows()):
y = y_positions[i]
g = row[g_col]
lo = row[ci_lo_col]
hi = row[ci_hi_col]
ci_width = hi - lo
w = 1 / (hi - lo) ** 2 if (hi - lo) > 0 else 1
ms = min (max (4 , w * 0.5 ), 16 )
ax.plot([lo, hi], [y, y], color="black" , linewidth=1 )
ax.plot(g, y, "s" , color="steelblue" , markersize=ms, zorder=5 )
ax.text(-0.05 + min (df[ci_lo_col]) - 0.3 , y,
row[study_col], ha="right" , va="center" , fontsize=8 )
ax.text(max (df[ci_hi_col]) + 0.3 , y,
f"{g:.2 f} [{lo:.2 f} , {hi:.2 f} ]" ,
ha="left" , va="center" , fontsize=7 , color="dimgray" )
y_summary = 0
diamond_x = [ci_lo_RE, g_RE, ci_hi_RE, g_RE]
diamond_y = [y_summary, y_summary + 0.4 , y_summary, y_summary - 0.4 ]
ax.fill(diamond_x, diamond_y, color="crimson" , alpha=0.85 , zorder=6 )
ax.text(max (df[ci_hi_col]) + 0.3 , y_summary,
f"{g_RE:.2 f} [{ci_lo_RE:.2 f} , {ci_hi_RE:.2 f} ]" ,
ha="left" , va="center" , fontsize=8 , fontweight="bold" , color="crimson" )
ax.axvline(0 , color="gray" , linestyle="--" , linewidth=1 )
ax.set_yticks([])
ax.set_xlabel("Hedges' g" )
ax.set_title(
f"Forest Plot — {k} Studies\n"
f"g = {g_RE:.3 f} , I² = {meta_result['I2' ]:.1 f} %, "
f"τ = {meta_result['tau' ]:.3 f} "
)
ax.set_xlim(min (df[ci_lo_col]) - 1.0 , max (df[ci_hi_col]) + 1.5 )
fig.tight_layout()
if output_path:
fig.savefig(output_path, dpi=150 )
plt.show()
return fig
Advanced Usage
Publication Bias: Egger Test, Trim-and-Fill, PET-PEESE def egger_test (
df: pd.DataFrame,
g_col: str = "g" ,
var_col: str = "var_g" ,
) -> Dict :
"""
Egger's test for funnel plot asymmetry (publication bias).
Regresses standardized effect sizes on precision:
z = g / se (standard normal deviate)
w = 1 / se (precision)
Regression: z = a + b × w
Intercept a ≠ 0 indicates asymmetry.
Args:
df: Study DataFrame.
g_col: Effect size column.
var_col: Variance column.
Returns:
Dict with intercept, SE, t-stat, p-value.
"""
g = df[g_col].values
v = df[var_col].values
se = np.sqrt(v)
z = g / se
prec = 1 / se
slope, intercept, r, p_val, se_intercept = stats.linregress(prec, z)
t_stat = intercept / se_intercept
df_resid = len (g) - 2
p_intercept = 2 * stats.t.sf(abs (t_stat), df=df_resid)
result = {
"intercept" : round (intercept, 4 ),
"se_intercept" : round (se_intercept, 4 ),
"t_stat" : round (t_stat, 3 ),
"p_value" : round (p_intercept, 4 ),
"significant" : p_intercept < 0.05 ,
"slope" : round (slope, 4 ),
}
print (
f"Egger test: intercept = {intercept:.3 f} (SE = {se_intercept:.3 f} ), "
f"t = {t_stat:.3 f} , p = {p_intercept:.4 f} "
+ ("← SIGNIFICANT bias" if p_intercept < 0.05 else "← non-significant" )
)
return result
def pet_peese (
df: pd.DataFrame,
g_col: str = "g" ,
var_col: str = "var_g" ,
alpha_fat: float = 0.10 ,
) -> Dict :
"""
PET-PEESE bias correction (Stanley & Doucouliagos, 2014).
FAT-PET: g_i = β0 + β1 × SE_i + ε_i
β0 = bias-corrected effect when SE = 0 (no bias)
PEESE: g_i = β0 + β1 × SE_i² + ε_i
Use if FAT-PET β0 is significant.
Args:
df: Study DataFrame.
g_col: Effect size column.
var_col: Variance column.
alpha_fat: Significance threshold for FAT-PET intercept (default 0.10).
Returns:
Dict with PET estimate, PEESE estimate, and recommendations.
"""
g = df[g_col].values
v = df[var_col].values
se = np.sqrt(v)
w = 1 / v
def wls_intercept (x_vals, y_vals, weights ):
"""WLS regression — return intercept and p-value."""
W = np.diag(weights)
X = np.column_stack([np.ones(len (y_vals)), x_vals])
XWX_inv = np.linalg.inv(X.T @ W @ X)
coef = XWX_inv @ X.T @ W @ y_vals
residuals = y_vals - X @ coef
sigma2 = np.sum (weights * residuals**2 ) / (len (y_vals) - 2 )
var_coef = sigma2 * XWX_inv
se_intercept = np.sqrt(var_coef[0 , 0 ])
t_stat = coef[0 ] / se_intercept
p_val = 2 * stats.t.sf(abs (t_stat), df=len (y_vals) - 2 )
return coef[0 ], se_intercept, t_stat, p_val
pet_b0, pet_se, pet_t, pet_p = wls_intercept(se, g, w)
peese_b0, peese_se, peese_t, peese_p = wls_intercept(se**2 , g, w)
if pet_p < alpha_fat:
recommended = "PEESE"
recommended_estimate = peese_b0
else :
recommended = "PET (effect non-significant)"
recommended_estimate = pet_b0
result = {
"PET_estimate" : round (pet_b0, 4 ),
"PET_se" : round (pet_se, 4 ),
"PET_p" : round (pet_p, 4 ),
"PEESE_estimate" : round (peese_b0, 4 ),
"PEESE_se" : round (peese_se, 4 ),
"PEESE_p" : round (peese_p, 4 ),
"recommended_method" : recommended,
"corrected_estimate" : round (recommended_estimate, 4 ),
}
print (
f"PET-PEESE:\n"
f" PET: β0 = {pet_b0:.4 f} (SE = {pet_se:.4 f} ), p = {pet_p:.4 f} \n"
f" PEESE: β0 = {peese_b0:.4 f} (SE = {peese_se:.4 f} ), p = {peese_p:.4 f} \n"
f" Recommended: {recommended} → corrected estimate = {recommended_estimate:.4 f} "
)
return result
def p_curve_analysis (
p_values: np.ndarray,
alpha: float = 0.05 ,
) -> Dict :
"""
P-curve analysis for evidential value assessment.
Under H0 (no true effect), p-values are uniform on [0, alpha].
Under H1 (true effect), p-values are right-skewed (more very small p's).
The p-curve plots conditional distribution of p-values < alpha.
Args:
p_values: Array of p-values from studies (all p < alpha are included).
alpha: Significance threshold.
Returns:
Dict with binomial test for right-skew and uniform comparison.
"""
sig_p = p_values[p_values < alpha]
k_sig = len (sig_p)
if k_sig < 3 :
print ("Warning: fewer than 3 significant p-values — p-curve unreliable." )
return {"warning" : "insufficient significant studies" , "k_sig" : k_sig}
expected_prop = 0.5
observed_prop = np.mean(sig_p < alpha / 2 )
binom_result = stats.binom_test(
int (observed_prop * k_sig), k_sig, p=expected_prop, alternative="greater"
) if hasattr (stats, "binom_test" ) else stats.binomtest(
int (observed_prop * k_sig), k_sig, p=expected_prop, alternative="greater"
).pvalue
result = {
"k_sig" : k_sig,
"observed_prop_below_half_alpha" : round (observed_prop, 4 ),
"expected_prop_H0" : expected_prop,
"p_right_skew" : round (float (binom_result), 4 ),
"evidential_value" : float (binom_result) < 0.05 ,
}
print (
f"P-curve (k_sig={k_sig} ):\n"
f" Prop p < {alpha/2 :.3 f} : {observed_prop:.3 f} (expected under H0: {expected_prop} )\n"
f" Right-skew test p = {float (binom_result):.4 f} "
+ ("← evidential value present" if float (binom_result) < 0.05
else "← insufficient evidential value" )
)
return result
Troubleshooting Problem Likely Cause Solution τ² is 0 (homogeneous) Low heterogeneity; Q < k-1 Report as-is; fixed-effects model appropriate I² near 100% Very heterogeneous studies Report sub-group analyses; reconsider pooling Egger test false positive with large k Multiple comparisons Use Egger only as exploratory; also check funnel PET-PEESE gives impossible estimate Small-study effect inflated d Report with caution; use sensitivity analysis p-curve requires binom_test Older scipy version Use scipy.stats.binomtest (scipy >= 1.7) Forest plot labels overlap Many studies Reduce fontsize or increase figure height Negative CI lower bound for g Very small studies Expected; report accurately
External Resources
DerSimonian, R., & Laird, N. (1986). Meta-analysis in clinical trials.
Controlled Clinical Trials , 7(3), 177–188.
Hedges, L. V., & Olkin, I. (1985). Statistical Methods for Meta-Analysis.
Egger, M., et al. (1997). Bias in meta-analysis detected by funnel plot.
BMJ , 315, 629–634.
Stanley, T. D., & Doucouliagos, H. (2014). Meta-regression approximations.
Research Synthesis Methods , 5(1), 60–78.
Simonsohn, U., Nelson, L. D., & Simmons, J. P. (2014). p-Curve.
Journal of Experimental Psychology: General , 143(2), 534–547.
metafor R package: https://www.metafor-project.org/
Examples
Example 1 — Full Meta-Analysis Pipeline with Forest Plot
studies_data = [
{"study" : "Smith 2015" , "m1" : 52.3 , "sd1" : 9.2 , "n1" : 30 , "m2" : 47.1 , "sd2" : 8.8 , "n2" : 30 },
{"study" : "Jones 2016" , "m1" : 61.2 , "sd1" : 11.0 , "n1" : 45 , "m2" : 57.8 , "sd2" : 10.5 , "n2" : 45 },
{"study" : "Brown 2017" , "m1" : 55.0 , "sd1" : 8.5 , "n1" : 25 , "m2" : 49.3 , "sd2" : 9.0 , "n2" : 25 },
{"study" : "Garcia 2018" , "m1" : 70.1 , "sd1" : 13.2 , "n1" : 60 , "m2" : 66.4 , "sd2" : 12.8 , "n2" : 60 },
{"study" : "Patel 2019" , "m1" : 48.5 , "sd1" : 7.9 , "n1" : 20 , "m2" : 44.2 , "sd2" : 8.1 , "n2" : 20 },
{"study" : "Chen 2020" , "m1" : 65.3 , "sd1" : 10.8 , "n1" : 35 , "m2" : 60.7 , "sd2" : 11.2 , "n2" : 35 },
{"study" : "Wilson 2021" , "m1" : 58.9 , "sd1" : 9.7 , "n1" : 40 , "m2" : 53.2 , "sd2" : 9.3 , "n2" : 40 },
{"study" : "Rodriguez 2022" , "m1" : 53.1 , "sd1" : 8.3 , "n1" : 28 , "m2" : 48.7 , "sd2" : 8.0 , "n2" : 28 },
]
rows = []
for s in studies_data:
es = cohens_d_from_means(
s["m1" ], s["m2" ], s["sd1" ], s["sd2" ], s["n1" ], s["n2" ]
)
rows.append({
"study" : s["study" ],
"n1" : s["n1" ], "n2" : s["n2" ],
"g" : es["g" ], "var_g" : es["var_g" ],
"ci_lo" : es["ci_lo" ], "ci_hi" : es["ci_hi" ],
})
df_meta = pd.DataFrame(rows)
print ("Effect sizes:" )
print (df_meta[["study" , "g" , "ci_lo" , "ci_hi" ]].round (3 ))
meta_res = run_meta_analysis(df_meta)
forest_plot(df_meta, meta_res, output_path="forest_plot.png" )
egger_res = egger_test(df_meta)
pet_res = pet_peese(df_meta)
Example 2 — Funnel Plot, Egger Test, and P-Curve
fig, ax = plt.subplots(figsize=(7 , 6 ))
se = np.sqrt(df_meta["var_g" ].values)
ax.scatter(df_meta["g" ].values, se, alpha=0.7 , color="steelblue" , s=60 )
ax.axvline(meta_res["g_RE" ], color="crimson" , linestyle="--" , linewidth=1.5 , label="g_RE" )
ax.axvline(0 , color="gray" , linestyle=":" , linewidth=1 )
ax.invert_yaxis()
ax.set_xlabel("Hedges' g" )
ax.set_ylabel("Standard Error (SE)" )
ax.set_title("Funnel Plot" )
ax.legend()
fig.tight_layout()
plt.savefig("funnel_plot.png" , dpi=150 )
plt.show()
p_vals_list = []
for _, row in df_meta.iterrows():
t_stat = row["g" ] / np.sqrt(row["var_g" ])
df_test = row["n1" ] + row["n2" ] - 2
p_val = 2 * stats.t.sf(abs (t_stat), df=df_test)
p_vals_list.append(p_val)
p_vals = np.array(p_vals_list)
print (f"\nP-values: {np.round (p_vals, 4 )} " )
p_curve_res = p_curve_analysis(p_vals, alpha=0.05 )
sig_p = p_vals[p_vals < 0.05 ]
fig2, ax2 = plt.subplots(figsize=(8 , 4 ))
bins = np.linspace(0 , 0.05 , 6 )
ax2.hist(sig_p, bins=bins, color="steelblue" , alpha=0.7 , edgecolor="white" , label="Observed" )
ax2.axhline(len (sig_p) / 5 , color="crimson" , linestyle="--" ,
linewidth=2 , label="Expected under H0 (uniform)" )
ax2.set_xlabel("p-value" )
ax2.set_ylabel("Frequency" )
ax2.set_title("P-Curve" )
ax2.legend()
fig2.tight_layout()
plt.savefig("p_curve.png" , dpi=150 )
plt.show()
print ("Meta-analysis complete." )
Changelog Version Date Change 1.0.0 2026-03-18 Initial release — Hedges' g, DL random-effects, forest/funnel plots, Egger, PET-PEESE, p-curve
