Introduction

Meta-analysis is a statistical technique that synthesizes results from independent studies addressing a common research question by computing a combined effect size. By aggregating information, it enhances statistical power. However, a critical prerequisite is the assessment of heterogeneity—variation in effect sizes across studies—which may threaten the validity of pooled conclusions if not appropriately modeled.

A fundamental approach to modeling heterogeneity in meta-analysis is the random-effects model. Let yi denote the observed effect size from study i (for i = 1, ..., N). It is assumed that:

y_i = θ_i + e_i,    e_i ~ N(0, v_i)

Here, θ_i is the (unknown) true effect in study i, and e_i is the sampling error, with known within-study variance v_i. Under standard assumptions, the observed effect sizes y_i are unbiased and normally distributed estimates of θ_i. To model between-study heterogeneity, the true effects θ_i are assumed to vary around a common mean μ:

θ_i = μ + u_i,    u_i ~ N(0, τ²)

where τ² represents the between-study variance. To account for potential sources of heterogeneity, study-level characteristics (moderators) can be incorporated into a mixed-effects meta-regression model:

θ_i = β₀ + β₁ x_i1 + ... + β_p x_ip + u_i,    u_i ~ N(0, τ²)

where x_ij denotes the value of the jth moderator for study i, β_j the corresponding regression coefficient, and τ² captures the residual heterogeneity not explained by the moderators.

These models are special cases of linear mixed-effects models with known heteroscedastic sampling variances. Estimation typically follows a two-step procedure: A. Estimate the between-study variance τ²; B. Estimate μ or β = (β₀, ..., β_p) using weighted least squares with weights: w_i = 1 / (v_i + τ²). Standard errors and confidence intervals are computed assuming normality. Hypothesis tests such as H₀: τ² = 0 are commonly performed using Cochran’s Q-test.

Application in Microbiome Research

Advances in next-generation sequencing have revolutionized microbiome research. Meta-analyses in this field allow synthesis of microbiota–disease associations, but they are particularly prone to heterogeneity due to differences in sequencing technologies, bioinformatics pipelines, sample types, population characteristics. In areas such as gut, respiratory or oral microbiomes, where fewer studies are available and study characteristics vary widely, the resulting high dimensionality and limited sample size pose analytical challenges. Standard meta-regression models may be inadequate in these cases. To address this, researchers have applied multivariate data analysis (e.g., PCA for quantitative variables, MCA for categorical variables, and FAMD for mixed data), or Bayesian penalized regression methods to select informative moderators in high-dimensional settings.

Main activities:

• Investigate traditional mixed-effects models with a limited number of moderators using the metafor R package;
• Explore Bayesian penalized meta-regression approaches using the pema R package;
• Assess the feasibility of implementing non-Bayesian penalized meta-regression using, for example, the CVXR R package;
• Apply the studied methods to reanalyze recent meta-analyses on human microbiome alpha-diversity and disease associations, incorporating study-level characteristics as moderators to better understand heterogeneity.

References

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