metaselection

Selection models for meta-analyses of dependent effect sizes

James E. Pustejovsky and Martyna Citkowicz

2025-06-13

Collaborators from the American Institutes for Research

Megha Joshi

Melissa Rodgers

Ryan Williams

Joshua Polanin

David Miller

Acknowledgement

The research reported here was supported, in whole or in part, by the Institute of Education Sciences, U.S. Department of Education, through grant R305D220026 to the American Institutes for Research. The opinions expressed are those of the authors and do not represent the views of the Institute or the U.S. Department of Education.

Selective reporting of primary study results

• Selective reporting occurs if affirmative findings are more likely to be reported and available for inclusion in meta-analysis

• Selective reporting distorts the evidence base available for systematic review/meta-analysis

  • Inflates average effect size estimates from meta-analysis
  • Biases estimates of heterogeneity (Augusteijn, van Aert, and van Assen 2019)

• Concerns about selective reporting span the social, behavioral, and health sciences.

• For a given meta-analysis, we expect strength of selection to depend on

  • Rigor of the systematic review search process.

  • Whether effect sizes are from focal or ancillary analysis.

Many available tools for investigating selective reporting

• Graphical diagnostics

  • Funnel plots
  • Contour-enhanced funnel plots
  • Power-enhanced funnel plots (sunset plots)

• Tests/adjustments for funnel plot asymmetry

  • Trim-and-fill
  • Egger’s regression
  • PET/PEESE
  • Kinked meta-regression

• Selection models

  • Weight-function models
  • Copas models
  • Sensitivity analysis

• p-value diagnostics

  • Test of Excess Significance
  • \(p\)-curve / \(p\)-uniform / \(p\text{-uniform}^*\)

But few that accommodate dependent effect sizes

• Dependent effect sizes are ubiquitous in education and social science meta-analyses.

• We have well-developed methods for modeling dependent effect sizes assuming no selection.

• But only very recent developments for investigating selective reporting in databases with dependent effect sizes (Chen and Pustejovsky 2024).

Selection models have two parts

• Random effects model for the evidence-generating process (before selective reporting): \[T_{ij} \sim N\left(\ \mu, \ \tau^2 + \sigma_{ij}^2 \right)\]

• A model describing \(\text{Pr}(\ T_{ij} \text{ is observed} \ )\) as a function of its \(p\)-value \((p_{ij})\)

Vevea and Hedges (1995) step-function model

Citkowicz and Vevea (2017) beta-function model

A piece-wise normal distribution

Under Vevea and Hedges (1995) step-function model, the distribution of observed effect size estimates is piece-wise normal.

math = require("mathjs")
norm = import('https://unpkg.com/norm-dist@3.1.0/index.js?module')

eta = math.sqrt(tau**2 + sigma**2)
H = 2
alpha = [alpha1, alpha2]
lambda = [1, lambda1, lambda2_ratio * lambda1]
lambda_max = math.max(lambda)

function findlambda(p, alp, lam) {
  var m = 0;
  while (p >= alp[m]) {
    m += 1;
  }
  return lam[m];
}

function findMoments(mu, tau, sigma, alp, lam) {
  let H = alp.length;
  let eta = math.sqrt(tau**2 + sigma**2);
  
  let gamma_h = Array(H+2).fill(null).map((x,i) => {
    if (i==0) {
      return Infinity;
    } else if (i==H+1) {
      return -Infinity;
    } else {
      return sigma * norm.icdf(1 - alp[i-1]);
    }
  });
  
  let c_h = Array(H+2).fill(null).map((x,i) => {
    return (gamma_h[i] - mu) / eta;
  });

  let B_h = Array(H+1).fill(null).map((x,i) => {
    return norm.cdf(c_h[i]) - norm.cdf(c_h[i+1])
  });

  let Ai = 0;
  for (let i = 0; i <= H; i++) {
    Ai += lam[i] * B_h[i];
  }
  
  let psi_h = Array(H+1).fill(null).map((x,i) => {
    return (norm.pdf(c_h[i+1]) - norm.pdf(c_h[i])) / B_h[i]
  }); 

  let psi_top = 0;
  for (let i = 0; i <= H; i++) {
    psi_top += lam[i] * B_h[i] * psi_h[i];
  }
  
  let psi_bar = psi_top / Ai;

  let ET = mu + eta * psi_bar;

  let dc_h = c_h.map((c_val) => {
    if (math.abs(c_val) == Infinity) {
      return 0;
    } else {
      return c_val * norm.pdf(c_val);
    }
  });

  let kappa_h = Array(H+1).fill(null).map((x,i) => {
    return (dc_h[i] - dc_h[i+1]) / B_h[i];
  });

  let kappa_top = 0;
  for (let i = 0; i <= H; i++) {
    kappa_top += lam[i] * B_h[i] * kappa_h[i];
  }
  let kappa_bar = kappa_top / Ai;
  let SDT = eta * math.sqrt(1 - kappa_bar - psi_bar**2);
  
  return ({Ai: Ai, ET: ET, SDT: SDT});
}

moments = findMoments(mu, tau, sigma, alpha, lambda)
Ai_toprint = moments.Ai.toFixed(3)
ET_toprint = moments.ET.toFixed(3)
eta_toprint = eta.toFixed(3)
SDT_toprint = moments.SDT.toFixed(3)

pts = 201

density_dat = Array(pts).fill().map((element, index) => {
  let t = mu - 3 * eta + index * eta * 6 / (pts - 1);
  let p = 1 - norm.cdf(t / sigma);
  let dt = norm.pdf((t - mu) / eta) / eta;
  let lambda_val = findlambda(p, alpha, lambda);
  return ({
    t: t,
    d_unselected: dt,
    d_selected: lambda_val * dt / lambda_max
  })
})

viewof mu = Inputs.range(
  [-2, 2], 
  {value: 0.1, step: 0.01, label: tex`\mu`}
)

viewof tau = Inputs.range(
  [0, 2], 
  {value: 0.15, step: 0.01, label: tex`\tau`}
)

viewof sigma = Inputs.range(
  [0, 1], 
  {value: 0.25, step: 0.01, label: tex`\sigma_i`}
)

viewof alpha1 = Inputs.range(
  [0, 1],
  {value: 0.025, step: 0.005, label: tex`\alpha_1`}
)

viewof alpha2 = Inputs.range(
  [0, 1],
  {value: 0.50, step: 0.005, label: tex`\alpha_2`}
)

viewof lambda1 = Inputs.range(
  [0, 2],
  {value: 1, step: 0.01, label: tex`\lambda_1`}
)

viewof lambda2_ratio = Inputs.range(
  [0, 2],
  {value: 1, step: 0.01, label: tex`\lambda_2 / \lambda_1`}
)

Plot.plot({
  height: 500,
  width: 1000,
  y: {
    grid: false,
    label: "Density"
  },
  x: {
    label: "Effect size estimate (Ti)"
  },   
  marks: [
    Plot.ruleY([0]),
    Plot.ruleX([0]),
    Plot.areaY(density_dat, {x: "t", y: "d_unselected", fillOpacity: 0.3}),
    Plot.areaY(density_dat, {x: "t", y: "d_selected", fill: "blue", fillOpacity: 0.5}),
    Plot.lineY(density_dat, {x: "t", y: "d_selected", stroke: "blue"})
  ]
})

tex`
\begin{aligned}
\mu &= ${mu} & \qquad \sqrt{\tau^2 + \sigma_{ij}} &= ${eta_toprint} \\
\mathbb{E}\left(T_{ij}\right) &= ${ET_toprint}
& \qquad \sqrt{\mathbb{V}\left(T_{ij}\right)} &= ${SDT_toprint} \\ 
\Pr(T_{ij} \text{ is observed}) &= ${Ai_toprint}
\end{aligned}
`

Estimation Strategy

• Model the marginal distribution of observed effects, ignoring the dependence structure

  • Maximum likelihood (composite marginal likelihood)

  • Augmented, reweighted Gaussian likelihood

Two methods of handling dependence

• Cluster-robust variance estimation (sandwich estimators)

• Clustered bootstrap re-sampling

Color priming

Lehmann, Elliot, and Calin-Jageman (2018) reported a systematic review of studies on color-priming, examining whether exposure to the color red influenced attractiveness judgments.

	Mean ES		Heterogeneity Variance
Coef.	Est.	SE	Est.	SE
Overall	0.207	0.0571	0.103	0.0251
Between-Subjects	0.19	0.0642	0.104	0.0256
Within-Subjects	0.273	0.1456	0.104	0.0256

	Mean ES		Heterogeneity Variance
Coef.	Est.	SE	Est.	SE
Overall	0.207	0.0571	0.103	0.0251
Between-Subjects	0.19	0.0642	0.104	0.0256
Within-Subjects	0.273	0.1456	0.104	0.0256

Contour-enhanced funnel plot

Color-priming selection models

library(metaselection)

# load the data
data("dat.lehmann2018", package = "metadat")

# tidy up
dat.lehmann2018$study <- dat.lehmann2018$Full_Citation
dat.lehmann2018$sei <- sqrt(dat.lehmann2018$vi)
dat.lehmann2018$Design <- factor(dat.lehmann2018$Design, levels = c("Between Subjects","Within Subjects"), labels = c("Between","Within"))

# fit a one-step selection model
sel1 <- selection_model(
  yi = yi,                 # effect size est.
  sei = sei,               # standard error
  cluster = study,         # identifier for independent clusters
  data = dat.lehmann2018,  # dataset
  selection_type = "step", # type of selection model
  steps = .025,            # single threshold for step-function
  estimator = "CML",       # estimation method
  bootstrap = "none"       # large-sample sandwich standard errors
)

summary(sel1)

Step Function Model 
 
Call: 
selection_model(data = dat.lehmann2018, yi = yi, sei = sei, cluster = study, 
    selection_type = "step", steps = 0.025, estimator = "CML", 
    bootstrap = "none")

Number of clusters = 41; Number of effects = 81

Steps: 0.025 
Estimator: composite marginal likelihood 
Variance estimator: robust 

Log composite likelihood of selection model: -44.46436
Inverse selection weighted partial log likelihood: 58.35719 

Mean effect estimates:                                                
                                    Large Sample
 Coef. Estimate Std. Error p-value  Lower  Upper
  beta    0.133      0.137   0.333 -0.136  0.402

Heterogeneity estimates:                                                
                                    Large Sample
 Coef. Estimate Std. Error p-value  Lower  Upper
  tau2   0.0811     0.0845     --- 0.0105  0.625

Selection process estimates:
 Step: 0 < p <= 0.025; Studies: 16; Effects: 25                                                 
                                     Large Sample
   Coef. Estimate Std. Error p-value Lower  Upper
 lambda0        1        ---     ---   ---    ---

 Step: 0.025 < p <= 1; Studies: 29; Effects: 56                                                  
                                      Large Sample
   Coef. Estimate Std. Error p-value  Lower  Upper
 lambda1    0.548      0.616   0.593 0.0607   4.96

Selective reporting of non-significant results

selection_plot(sel1)

Now with bootstrapping!

# turn on parallel processing
library(future)
plan(multisession, workers = 8)

set.seed(20250613) # for reproducibility

sel1_boot <- selection_model(
  yi = yi,                    # effect size est.
  sei = sei,                  # standard error
  cluster = study,            # identifier for independent clusters
  data = dat.lehmann2018,     # dataset
  selection_type = "step",    # type of selection model
  steps = .025,               # single threshold for step-function
  estimator = "CML",          # estimation method
  bootstrap = "two-stage",    # recommended type of bootstrapping
  R = 1999,                   # number of bootstrap re-samples
  CI_type = c("large-sample", # keep the large-sample sandwich CI
              "percentile")   # recommended type of bootstrap CI
)

summary(sel1_boot)

Step Function Model with Cluster Bootstrapping 
 
Call: 
selection_model(data = dat.lehmann2018, yi = yi, sei = sei, cluster = study, 
    selection_type = "step", steps = 0.025, estimator = "CML", 
    CI_type = c("large-sample", "percentile"), bootstrap = "two-stage", 
    R = 1999)

Number of clusters = 41; Number of effects = 81

Steps: 0.025 
Estimator: composite marginal likelihood 
Variance estimator: robust 
Bootstrap type: two-stage 
Number of bootstrap replications: 1999 

Log composite likelihood of selection model: -44.46436
Inverse selection weighted partial log likelihood: 58.35719 

Mean effect estimates:                                                                     
                                    Large Sample Percentile Bootstrap
 Coef. Estimate Std. Error p-value  Lower  Upper      Lower     Upper
  beta    0.133      0.137   0.333 -0.136  0.402    -0.0174     0.435

Heterogeneity estimates:                                                                     
                                    Large Sample Percentile Bootstrap
 Coef. Estimate Std. Error p-value  Lower  Upper      Lower     Upper
  tau2   0.0811     0.0845     --- 0.0105  0.625   1.73e-17     0.238

Selection process estimates:
 Step: 0 < p <= 0.025; Studies: 16; Effects: 25                                                                      
                                     Large Sample Percentile Bootstrap
   Coef. Estimate Std. Error p-value Lower  Upper      Lower     Upper
 lambda0        1        ---     ---   ---    ---        ---       ---

 Step: 0.025 < p <= 1; Studies: 29; Effects: 56                                                                       
                                      Large Sample Percentile Bootstrap
   Coef. Estimate Std. Error p-value  Lower  Upper      Lower     Upper
 lambda1    0.548      0.616   0.593 0.0607   4.96     0.0537      2.88

Add a moderator

set.seed(20250613) # for reproducibility

sel1_mod <- selection_model(
  yi = yi,                    # effect size est.
  sei = sei,                  # standard error
  cluster = study,            # identifier for independent clusters
  mean_mods = ~ 0 + Design,   # design type moderator
  data = dat.lehmann2018,     # dataset
  selection_type = "step",    # type of selection model
  steps = .025,               # single threshold for step-function
  estimator = "CML",          # estimation method
  bootstrap = "two-stage",    # recommended type of bootstrapping
  R = 1999,                   # number of bootstrap re-samples
  CI_type = c("large-sample", # keep the large-sample sandwich CI
              "percentile")   # recommended type of bootstrap CI
)

summary(sel1_mod)

Step Function Model with Cluster Bootstrapping 
 
Call: 
selection_model(data = dat.lehmann2018, yi = yi, sei = sei, cluster = study, 
    selection_type = "step", steps = 0.025, mean_mods = ~0 + 
        Design, estimator = "CML", CI_type = c("large-sample", 
        "percentile"), bootstrap = "two-stage", R = 1999)

Number of clusters = 41; Number of effects = 81

Steps: 0.025 
Estimator: composite marginal likelihood 
Variance estimator: robust 
Bootstrap type: two-stage 
Number of bootstrap replications: 1990 

Log composite likelihood of selection model: -44.12226
Inverse selection weighted partial log likelihood: 61.14273 

Mean effect estimates:                                                                                  
                                                 Large Sample Percentile Bootstrap
              Coef. Estimate Std. Error p-value  Lower  Upper      Lower     Upper
 beta_DesignBetween    0.113      0.117   0.333 -0.116  0.343    -0.0484     0.339
  beta_DesignWithin    0.196      0.234   0.400 -0.261  0.654     0.0104     0.985

Heterogeneity estimates:                                                                     
                                    Large Sample Percentile Bootstrap
 Coef. Estimate Std. Error p-value  Lower  Upper      Lower     Upper
  tau2   0.0785      0.081     --- 0.0104  0.593   1.14e-17     0.197

Selection process estimates:
 Step: 0 < p <= 0.025; Studies: 16; Effects: 25                                                                      
                                     Large Sample Percentile Bootstrap
   Coef. Estimate Std. Error p-value Lower  Upper      Lower     Upper
 lambda0        1        ---     ---   ---    ---        ---       ---

 Step: 0.025 < p <= 1; Studies: 29; Effects: 56                                                                       
                                      Large Sample Percentile Bootstrap
   Coef. Estimate Std. Error p-value  Lower  Upper      Lower     Upper
 lambda1    0.533      0.601   0.577 0.0584   4.86      0.042      2.66

Add another step

set.seed(20250613) # for reproducibility

sel2_mod <- selection_model(
  yi = yi,                    # effect size est.
  sei = sei,                  # standard error
  cluster = study,            # identifier for independent clusters
  mean_mods = ~ 0 + Design,   # design type moderator
  data = dat.lehmann2018,     # dataset
  selection_type = "step",    # type of selection model
  steps = c(.025,.500),       # two thresholds for step-function
  estimator = "CML",          # estimation method
  bootstrap = "two-stage",    # recommended type of bootstrapping
  R = 1999,                   # number of bootstrap re-samples
  CI_type = c("large-sample", # keep the large-sample sandwich CI
              "percentile")   # recommended type of bootstrap CI
)

summary(sel2_mod)

Step Function Model with Cluster Bootstrapping 
 
Call: 
selection_model(data = dat.lehmann2018, yi = yi, sei = sei, cluster = study, 
    selection_type = "step", steps = c(0.025, 0.5), mean_mods = ~0 + 
        Design, estimator = "CML", CI_type = c("large-sample", 
        "percentile"), bootstrap = "two-stage", R = 1999)

Number of clusters = 41; Number of effects = 81

Steps: 0.025, 0.5 
Estimator: composite marginal likelihood 
Variance estimator: robust 
Bootstrap type: two-stage 
Number of bootstrap replications: 1990 

Log composite likelihood of selection model: -43.51452
Inverse selection weighted partial log likelihood: 85.42712 

Mean effect estimates:                                                                                  
                                                 Large Sample Percentile Bootstrap
              Coef. Estimate Std. Error p-value  Lower  Upper      Lower     Upper
 beta_DesignBetween   0.0419      0.140   0.765 -0.233  0.317    -0.1706     0.335
  beta_DesignWithin   0.1462      0.247   0.554 -0.338  0.631    -0.0329     0.989

Heterogeneity estimates:                                                                      
                                     Large Sample Percentile Bootstrap
 Coef. Estimate Std. Error p-value   Lower  Upper      Lower     Upper
  tau2   0.0804     0.0881     --- 0.00937   0.69   1.26e-17     0.207

Selection process estimates:
 Step: 0 < p <= 0.025; Studies: 16; Effects: 25                                                                      
                                     Large Sample Percentile Bootstrap
   Coef. Estimate Std. Error p-value Lower  Upper      Lower     Upper
 lambda0        1        ---     ---   ---    ---        ---       ---

 Step: 0.025 < p <= 0.5; Studies: 22; Effects: 33                                                                       
                                      Large Sample Percentile Bootstrap
   Coef. Estimate Std. Error p-value  Lower  Upper      Lower     Upper
 lambda1    0.476      0.595   0.552 0.0412    5.5     0.0364      2.68

 Step: 0.5 < p <= 1; Studies: 17; Effects: 23                                                                       
                                      Large Sample Percentile Bootstrap
   Coef. Estimate Std. Error p-value  Lower  Upper      Lower     Upper
 lambda2    0.307      0.466   0.437 0.0156   6.03     0.0122      3.13

Selective reporting of non-significant results

selection_plot(sel2_mod, draw_boots = FALSE) + 
  ggplot2::coord_cartesian(ylim = c(0,1))

Discussion

• Other supported models:

  • beta-function models (Citkowicz and Vevea 2017)

  • Location-scale meta-regression (Viechtbauer and López‐López 2022)

  • Predictors of selection (Coburn and Vevea 2015)

• Marginal step-function selection models are worth adding to the toolbox (Pustejovsky, Citkowicz, and Joshi 2025).

  • Low bias compared to other selective reporting adjustments (including PET-PEESE)

  • Bias-variance trade-off relative to regular meta-analytic models

  • Two-stage clustered bootstrap percentile confidence intervals work tolerably well

R package metaselection

• Currently available on Github at https://github.com/jepusto/metaselection

• Install using

remotes::install_github("jepusto/metaselection", build_vignettes = TRUE)

• Under active development, suggestions welcome!

References

Augusteijn, Hilde E. M., Robbie C. M. van Aert, and Marcel A. L. M. van Assen. 2019. “The Effect of Publication Bias on the Q Test and Assessment of Heterogeneity.” Psychological Methods 24 (1): 116–34. https://doi.org/10.1037/met0000197.

Chen, Man, and James E. Pustejovsky. 2024. “Adapting Methods for Correcting Selective Reporting Bias in Meta-Analysis of Dependent Effect Sizes.” https://doi.org/10.31222/osf.io/jq52s.

Citkowicz, Martyna, and Jack L Vevea. 2017. “A parsimonious weight function for modeling publication bias.” Psychological Methods 22 (1): 28–41. https://doi.org/10.1037/met0000119.

Coburn, Kathleen M, and Jack L Vevea. 2015. “Publication bias as a function of study characteristics.” Psychological Methods 20 (3): 310–30. https://doi.org/10.1037/met0000046.

Lehmann, Gabrielle K, Andrew J Elliot, and Robert J Calin-Jageman. 2018. “Meta-Analysis of the Effect of Red on Perceived Attractiveness.” Evolutionary Psychology 16 (4): 1474704918802412. https://doi.org/10.1177/1474704918802412.

Pustejovsky, James E., Martyna Citkowicz, and Megha Joshi. 2025. “Estimation and Inference for Step-Function Selection Models in Meta-Analysis with Dependent Effects.” https://doi.org/10.31222/osf.io/qg5x6_v1.

Vevea, Jack L, and Larry V Hedges. 1995. “A General Linear Model for Estimating Effect Size in the Presence of Publication Bias.” Psychometrika 60 (3): 419–35. https://doi.org/10.1007/BF02294384.

Viechtbauer, Wolfgang, and José Antonio López‐López. 2022. “Location‐scale Models for Meta‐analysis.” Research Synthesis Methods, April, jrsm.1562. https://doi.org/10.1002/jrsm.1562.