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Publication bias occurs when studies with statistically significant or large effects are more likely to be published than null or small-effect studies, causing the pooled estimate to be inflated. The pipeline assesses this risk at three levels: visual inspection with contour-enhanced funnel plots, asymmetry testing with Doi plots, and formal adjustment with the Copas selection model. All three approaches are applied to the overall dataset and to each predefined subgroup.

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Why assess publication bias

Small-study effects — a pattern where smaller studies report larger effects — are the most common empirical signal of publication bias. When many small studies with high prevalence estimates are published but comparable studies with low estimates go unreported, the pooled prevalence will be biased upward. Detecting and adjusting for this bias is especially important in prevalence meta-analyses, where the literature may be dominated by studies conducted in high-prevalence settings that are more likely to attract attention and reach publication.

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Contour-enhanced funnel plots

The vizFunnel() function generates contour-enhanced funnel plots for each model. Internally it calls meta::funnel() with type = "contour", common = TRUE, and random = TRUE, overlaying significance contours at conventional thresholds (p < 0.01, 0.05, 0.10). The plot displays each study’s effect estimate against a measure of its precision (standard error). Asymmetry in the funnel — a gap in one corner of the plot — suggests that studies of a particular size and direction are missing from the literature. Contour shading helps distinguish asymmetry driven by publication bias (missing studies tend to fall outside significance contours) from other small-study effects (missing studies fall inside contours). 
# Overall funnel plot — generated automatically as a pipeline target
targets::tar_load(plt_funnel_prop)

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Doi plots

The pipeline prefers Doi plots over standard funnel plots for formal asymmetry assessment because funnel plots are susceptible to subjective visual interpretation. A Doi plot re-expresses each study’s contribution as a function of its absolute normal deviate (LFK index), making asymmetry easier to quantify and reducing analyst-dependent judgment. Doi plots are generated with metasens::doiplot(): 
targets::tar_load(mod_prop)
metasens::doiplot(mod_prop)  # Recommended over funnel plot (doi: 10.1111/jebm.70063)
The LFK index summarizes asymmetry numerically: values within ±1 suggest no asymmetry, ±1 to ±2 suggest minor asymmetry, and values beyond ±2 indicate major asymmetry.

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Copas selection model

The Copas selection model provides a formal adjustment for publication bias by jointly modeling the effect sizes and the probability that each study was published. Publication probability is treated as a function of the study’s standard error and a selection parameter; the model traces how the pooled estimate changes as increasingly strict selection is assumed, letting you evaluate how sensitive conclusions are to potential bias. The applyCopas() function in src/R/meta-analysis.R wraps metasens::copas(): 
applyCopas <- function(obj, varname = NULL, ...) {
  require("metasens")
  require("meta")

  tbl <- obj

  if (grepl(x = obj, "meta") |> any()) {
    mod_copas <- metasens::copas(obj, backtransf = TRUE, ...)
    return(mod_copas)
  } else if (is.null(varname)) {
    mod <- fitMetaprop(obj)
    mod_copas <- applyCopas(mod)
    return(mod_copas)
  }

  groups <- tbl[[varname]] |> unique() |> na.omit()
  sub_tbls <- lapply(groups, \(group) tbl %>% subset(.[[varname]] == group))
  sub_metas <- lapply(sub_tbls, \(sub_tbl) fitMetaprop(sub_tbl))
  mod_copas <- lapply(sub_metas, applyCopas)

  return(mod_copas)
}
The function handles three input types:
  • A fitted meta object — calls metasens::copas() directly with backtransf = TRUE so results are reported on the original probability scale.
  • A data frame with no varname — fits the overall meta-analysis first, then applies Copas.
  • A data frame with a varname — splits the data into per-group subsets, fits a separate meta-analysis for each group, and returns a list of Copas objects.
The pipeline applies this function to the overall model and to each subgroup: 
tar_target(mod_copas_prop,     applyCopas(mod_prop))

# Subgroup Copas targets (generated by tar_map):
# mod_copas_subgroup_Age, mod_copas_subgroup_Continent, etc.
tar_target(mod_copas_subgroup, applyCopas(tbl_clean, varname = colname))

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Accessing Copas results

targets::tar_load(mod_copas_prop)
summary(mod_copas_prop)
plot(mod_copas_prop)
plot() produces a panel of diagnostics: the contour plot of the selection parameter space, the adjusted pooled estimate as a function of the selection parameter, and a test for residual selection bias. summary() prints the adjusted estimate alongside the unadjusted estimate for direct comparison. For subgroup Copas objects (which are lists), iterate over the list: 
targets::tar_load(mod_copas_subgroup_Continent)
lapply(mod_copas_subgroup_Continent, summary)

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Interpreting the Copas model

The key question is whether the Copas-adjusted estimate differs meaningfully from the unadjusted pooled estimate. A large divergence — especially if the adjusted estimate is substantially lower — indicates that publication bias may be inflating the pooled prevalence. Conversely, if the adjusted estimate tracks the unadjusted estimate across the full range of selection parameters and the test for residual selection bias is non-significant, publication bias is unlikely to materially influence the conclusions. In this pipeline’s analysis, Copas-adjusted estimates for the overall dataset and most subgroups closely matched their unadjusted counterparts, supporting the robustness of the pooled prevalence findings.

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References

  • Copas selection model: Copas, J. B., & Shi, J. Q. (2001). A sensitivity analysis for publication bias in systematic reviews. Statistical Methods in Medical Research, 10(4), 251–265. doi:10.1177/096228020101000402
  • Copas model evaluation (preferability over trim-and-fill): Moreno, S. G., et al. (2009). Empirical evaluation suggests Copas selection model preferable to trim-and-fill method for selection bias in meta-analysis. Journal of Clinical Epidemiology, 62(5), 1234–1241. doi:10.1016/j.jclinepi.2008.10.010
  • Doi plot recommendation: Furuya-Kanamori, L., et al. (2025). Examining and interpreting Doi plot asymmetry in meta-analyses of randomized controlled trials. Journal of Evidence-Based Medicine. doi:10.1111/jebm.70063

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