Conduct Bayesian Regularized Meta-Analysis

This function conducts Bayesian regularized meta-regression (Van Lissa & Van Erp, 2021). It uses the stan function rstan::sampling to fit the model. A lasso or horseshoe prior is used to shrink the regression coefficients of irrelevant moderators towards zero. See Details.

Usage

brma(x, ...)

# S3 method for class 'formula'
brma(
  formula,
  data,
  vi = "vi",
  study = NULL,
  method = "hs",
  standardize = TRUE,
  prior = switch(method, lasso = c(df = 1, scale = 1), hs = c(df = 1, df_global = 1,
    df_slab = 4, scale_global = 1, scale_slab = 2, relevant_pars = NULL)),
  mute_stan = TRUE,
  ...
)

# Default S3 method
brma(
  x,
  y,
  vi,
  study = NULL,
  method = "hs",
  standardize,
  prior,
  mute_stan = TRUE,
  intercept,
  ...
)

Arguments

x: An k x m numeric matrix, where k is the number of effect sizes and m is the number of moderators.
...: Additional arguments passed on to rstan::sampling(). Use this, e.g., to override default arguments of that function.
formula: An object of class formula (or one that can be coerced to that class), see lm.
data: Either a data.frame containing the variables in the model, see lm, or a list of multiple imputed data.frames, or an object returned by mice.
vi: Character. Name of the column in the data that contains the variances of the effect sizes. This column will be removed from the data prior to analysis. Defaults to "vi".
study: Character. Name of the column in the data that contains the study id. Use this when the data includes multiple effect sizes per study. This column can be a vector of integers, or a factor. This column will be removed from the data prior to analysis. See Details for more information about analyzing dependent data.
method: Character, indicating the type of regularizing prior to use. Supports one of c("hs", "lasso"), see Details. Defaults to "hs".
standardize: Either a logical argument or a list. If standardize is logical, it controls whether all predictors are standardized prior to analysis or not. Parameter estimates are restored to the predictors' original scale. Alternatively, users can provide a list to standardize to gain more control over the standardization process. In this case, it is assumed that the standardization has already taken place. This list must have two elements: list(center = c(mean(X1) , mean(X2), mean(X...)), scale = c(sd(X1), sd(X2), sd(X...))). It is used only to restore parameter estimates to the original scale of the predictors. This is useful, e.g., to standardize continuous and dichotomous variables separately. Defaults to TRUE, which is recommended so that shrinking affects all parameters similarly.
prior: Numeric vector, specifying the prior to use. Note that the different methods require this vector to contain specific named elements.
mute_stan: Logical, indicating whether mute all 'Stan' output or not.
y: A numeric vector of k effect sizes.
intercept: Logical, indicating whether or not an intercept should be included in the model.

Value

A list object of class brma, with the following structure:

list(
  fit          # An object of class stanfit, for compatibility with rstan
  coefficients # A numeric matrix with parameter estimates; these are
               # interpreted as regression coefficients, except tau2 and tau,
               # which are interpreted as the residual variance and standard
               # deviation, respectively.
  formula      # The formula used to estimate the model
  terms        # The predictor terms in the formula
  X            # Numeric matrix of moderator variables
  Y            # Numeric vector with effect sizes
  vi           # Numeric vector with effect size variances
  tau2         # Numeric, estimated tau2
  R2           # Numeric, estimated heterogeneity explained by the moderators
  k            # Numeric, number of effect sizes
  study        # Numeric vector with study id numbers
)

Details

The Bayesian regularized meta-analysis algorithm (Van Lissa & Van Erp, 2021) penalizes meta-regression coefficients either via the lasso prior (Park & Casella, 2008) or the regularized horseshoe prior (Piironen & Vehtari, 2017).

lasso: The Bayesian equivalent of the lasso penalty is obtained when placing independent Laplace (i.e., double exponential) priors on the regression coefficients centered around zero. The scale of the Laplace priors is determined by a global scale parameter scale, which defaults to 1 and an inverse-tuning parameter \(\frac{1}{\lambda}\) which is given a chi-square prior governed by a degrees of freedom parameter df (defaults to 1). If standardize = TRUE, shrinkage will affect all coefficients equally and it is not necessary to adapt the scale parameter. Increasing the df parameter will allow larger values for the inverse-tuning parameter, leading to less shrinkage.
hs: One issue with the lasso prior is that it has relatively light tails. As a result, not only does the lasso have the desirable behavior of pulling small coefficients to zero, it also results in too much shrinkage of large coefficients. An alternative prior that improves upon this shrinkage pattern is the horseshoe prior (Carvalho, Polson & Scott, 2010). The horseshoe prior has an infinitely large spike at zero, thereby pulling small coefficients toward zero but in addition has fat tails, which allow substantial coefficients to escape the shrinkage. The regularized horseshoe is an extension of the horseshoe prior that allows the inclusion of prior information regarding the number of relevant predictors and can be more numerically stable in certain cases (Piironen & Vehtari, 2017). The regularized horseshoe has a global shrinkage parameter that influences all coefficients similarly and local shrinkage parameters that enable flexible shrinkage patterns for each coefficient separately. The local shrinkage parameters are given a Student's t prior with a default df parameter of 1. Larger values for df result in lighter tails and a prior that is no longer strictly a horseshoe prior. However, increasing df slightly might be necessary to avoid divergent transitions in Stan (see also https://mc-stan.org/misc/warnings.html). Similarly, the degrees of freedom for the Student's t prior on the global shrinkage parameter df_global can be increased from the default of 1 to, for example, 3 if divergent transitions occur although the resulting prior is then strictly no longer a horseshoe. The scale for the Student's t prior on the global shrinkage parameter scale_global defaults to 1 and can be decreased to achieve more shrinkage. Moreover, if prior information regarding the number of relevant moderators is available, it is recommended to include this information via the relevant_pars argument by setting it to the expected number of relevant moderators. When relevant_pars is specified, scale_global is ignored and instead based on the available prior information. Contrary to the horseshoe prior, the regularized horseshoe applies additional regularization on large coefficients which is governed by a Student's t prior with a scale_slab defaulting to 2 and df_slab defaulting to 4. This additional regularization ensures at least some shrinkage of large coefficients to avoid any sampling problems.

References

Van Lissa, C. J., van Erp, S., & Clapper, E. B. (2023). Selecting relevant moderators with Bayesian regularized meta-regression. Research Synthesis Methods. doi:10.31234/osf.io/6phs5

Park, T., & Casella, G. (2008). The Bayesian Lasso. Journal of the American Statistical Association, 103(482), 681–686. doi:10.1198/016214508000000337

Carvalho, C. M., Polson, N. G., & Scott, J. G. (2010). The horseshoe estimator for sparse signals. Biometrika, 97(2), 465–480. doi:10.1093/biomet/asq017

Piironen, J., & Vehtari, A. (2017). Sparsity information and regularization in the horseshoe and other shrinkage priors. Electronic Journal of Statistics, 11(2). https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-11/issue-2/Sparsity-information-and-regularization-in-the-horseshoe-and-other-shrinkage/10.1214/17-EJS1337SI.pdf

Examples

data("curry")
df <- curry[c(1:5, 50:55), c("d", "vi", "sex", "age", "donorcode")]
suppressWarnings({res <- brma(d~., data = df, iter = 10)})