Meta-Analysis with Order-Constrained Study Effects

Source: R/meta_ordered.R

Computes the Bayes factor for the hypothesis that the true study effects in a random-effects meta-analysis are all positive or negative.

meta_ordered(
  y,
  SE,
  labels,
  data,
  d = prior("norm", c(mean = 0, sd = 0.3), lower = 0),
  tau = prior("invgamma", c(shape = 1, scale = 0.15)),
  prior = c(1, 1, 1, 1),
  logml = "integrate",
  summarize = "stan",
  ci = 0.95,
  rel.tol = .Machine$double.eps^0.3,
  logml_iter = 5000,
  iter = 5000,
  silent_stan = TRUE,
  ...
)

Arguments

y

effect size per study. Can be provided as (1) a numeric vector, (2) the quoted or unquoted name of the variable in data, or (3) a formula to include discrete or continuous moderator variables.

SE

standard error of effect size for each study. Can be a numeric vector or the quoted or unquoted name of the variable in data

labels

optional: character values with study labels. Can be a character vector or the quoted or unquoted name of the variable in data

data

data frame containing the variables for effect size y, standard error SE, labels, and moderators per study.

d

prior distribution on the average effect size d. The prior probability density function is defined via prior.

tau

prior distribution on the between-study heterogeneity tau (i.e., the standard deviation of the study effect sizes dstudy in a random-effects meta-analysis. A (nonnegative) prior probability density function is defined via prior.

prior

prior probabilities over models (possibly unnormalized) in the order c(fixed_H0, fixed_H1, ordered_H1, random_H1). Note that the model random_H0 is not included in the comparison.

logml

how to estimate the log-marginal likelihood: either by numerical integration ("integrate") or by bridge sampling using MCMC/Stan samples ("stan"). To obtain high precision with logml="stan", many MCMC samples are required (e.g., logml_iter=10000, warmup=1000).

summarize

how to estimate parameter summaries (mean, median, SD, etc.): Either by numerical integration (summarize = "integrate") or based on MCMC/Stan samples (summarize = "stan").

ci

probability for the credibility/highest-density intervals.

rel.tol

relative tolerance used for numerical integration using integrate. Use rel.tol=.Machine$double.eps for maximal precision (however, this might be slow).

logml_iter

number of iterations (per chain) from the posterior distribution of d and tau. The samples are used for computing the marginal likelihood of the random-effects model with bridge sampling (if logml="stan") and for obtaining parameter estimates (if summarize="stan"). Note that the argument iter=2000 controls the number of iterations for estimation of the random-effect parameters per study in random-effects meta-analysis.

iter

number of MCMC iterations for the random-effects meta-analysis. Needs to be larger than usual to estimate the probability of all random effects being ordered (i.e., positive or negative).

silent_stan

whether to suppress the Stan progress bar.

...

further arguments passed to rstan::sampling (see stanmodel-method-sampling). Relevant MCMC settings concern the number of warmup samples that are discarded (warmup=500), the total number of iterations per chain (iter=2000), the number of MCMC chains (chains=4), whether multiple cores should be used (cores=4), and control arguments that make the sampling in Stan more robust, for instance: control=list(adapt_delta=.97).

Details

Usually, in random-effects meta-analysis,the study-specific random-effects are allowed to be both negative or positive even when the prior on the overall effect size d is truncated to be positive). In contrast, the function meta_ordered fits and tests a model in which the random effects are forced to be either all positive or all negative. The direction of the study-specific random-effects is defined via the prior on the mode of the truncated normal distribution d. For instance, d=prior("norm", c(0,.5), lower=0) means that all random-effects are positive (not just the overall mean effect size).

The posterior summary statistics of the overall effect size in the model ordered refer to the the average/mean of the study-specific effect sizes (as implied by the fitted truncated normal distribution) and not to the location parameter d of the truncated normal distribution (which is only the mode, not the expected value of a truncated normal distribution).

The Bayes factor for the order-constrained model is computed using the encompassing Bayes factor. Since many posterior samples are required for this approach, the default number of MCMC iterations for meta_ordered is iter=5000 per chain.

References

Haaf, J. M., & Rouder, J. N. (2018). Some do and some don’t? Accounting for variability of individual difference structures. Psychonomic Bulletin & Review, 26, 772–789. doi:10.3758/s13423-018-1522-x

See also

meta_bma, meta_random

Examples

# \donttest{
### Bayesian Meta-Analysis with Order Constraints (H1: d>0)
data(towels)
set.seed(123)
mo <- meta_ordered(logOR, SE, study, towels,
  d = prior("norm", c(mean = 0, sd = .3), lower = 0)
)
#> Warning: There were 13 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems
mo
#> ### Bayesian Meta-Analysis with Order Constraints ###
#>    null:    d = 0
#>    fixed:   d ~ 'norm' (mean=0, sd=0.3) truncated to the interval [0,Inf].
#>    ordered: d      ~ 'norm' (mean=0, sd=0.3) truncated to the interval [0,Inf].
#>             tau    ~ 'invgamma' (shape=1, scale=0.15) with support on the interval [0,Inf].
#>             dstudy ~ Normal(d,tau) truncated to [0,Inf]
#>    random:  d      ~ 'norm' (mean=0, sd=0.3) truncated to the interval [0,Inf].
#>             tau    ~ 'invgamma' (shape=1, scale=0.15) with support on the interval [0,Inf].
#>             dstudy ~ Normal(d,tau)
#> 
#> # Bayes factors:
#>            (denominator)
#> (numerator) null  fixed ordered random
#>     null     1.0 0.0419  0.0563 0.0975
#>     fixed   23.9 1.0000  1.3443 2.3259
#>     ordered 17.8 0.7439  1.0000 1.7302
#>     random  10.3 0.4299  0.5780 1.0000
#> 
#> # Model posterior probabilities:
#>         prior posterior logml
#> null     0.25    0.0189 -5.58
#> fixed    0.25    0.4513 -2.40
#> ordered  0.25    0.3357 -2.70
#> random   0.25    0.1940 -3.25
#> 
#> # Posterior summary statistics of average effect size:
#>          mean    sd  2.5%   50% 97.5% hpd95_lower hpd95_upper  n_eff Rhat
#> fixed   0.214 0.076 0.070 0.214 0.363       0.067       0.360 3372.3    1
#> ordered 0.233 0.074 0.106 0.227 0.394       0.098       0.382     NA   NA
#> random  0.194 0.089 0.029 0.192 0.375       0.021       0.362 9998.3    1
plot_posterior(mo)

# }