Help for package MetaStan

Version:

1.0.0

Title:

Bayesian Meta-Analysis via 'Stan'

Description:

Performs Bayesian meta-analysis, meta-regression and model-based meta-analysis using 'Stan'. Includes binomial-normal hierarchical models and option to use weakly informative priors for the heterogeneity parameter and the treatment effect parameter which are described in Guenhan, Roever, and Friede (2020) <doi:10.1002/jrsm.1370>.

License:

GPL (≥ 3)

Encoding:

UTF-8

Date:

2021-12-23

LazyData:

true

ByteCompile:

true

Depends:

R (≥ 3.5.0)

Imports:

Rcpp (≥ 0.12.17), methods, rstan (≥ 2.18.1), rstantools (≥ 1.5.0), loo (≥ 2.0), forestplot (≥ 1.6), metafor (≥ 2.0-0), HDInterval, coda

Suggests:

testthat, knitr, rmarkdown, ggplot2, shinystan, vdiffr

LinkingTo:

StanHeaders (≥ 2.18.0), rstan (≥ 2.18.1), BH (≥ 1.66.0-1), Rcpp (≥ 0.12.17), RcppEigen (≥ 0.3.3.4.0), RcppParallel (≥ 5.0.1),

SystemRequirements:

GNU make

NeedsCompilation:

yes

RoxygenNote:

7.1.2

URL:

https://github.com/gunhanb/MetaStan

BugReports:

https://github.com/gunhanb/MetaStan/issues

VignetteBuilder:

knitr

Packaged:

2022-01-22 10:25:53 UTC; M305767

Author:

Burak Kuersad Guenhan

[aut, cre], Christian Roever

[aut], Trustees of Columbia University [cph] (src/init.cpp, tools/make_cpp.R, R/stanmodels.R)

Maintainer:

Burak Kuersad Guenhan <burakgunhan@gmail.com>

Repository:

CRAN

Date/Publication:

2022-01-22 12:22:46 UTC

The 'MetaStan' package.

Description

Fitting Bayesian meta-analysis models via Rstan.

Details

To fit meta-analysis models using frequentist methods, there are many R packages available including 'metafor'. On the other hand, Bayesian estimation methods such as Markov chain Monte Carlo (MCMC) are very attractive for meta-analysis, especially because they can be used to fit more complicated models. These include binomial-normal hierarchical models and beta-binomial models which are based on the exact distributional assumptions unlike (commonly used) normal-normal hierarchical model. Another advantage of Bayesian methods to be able to use informative prior distributions for example to regularize heterogeneity estimates in case of low number of studies. Thus, we developed 'MetaStan' which uses Stan (a modern MCMC engine) to fit several pairwise meta-analysis models including binomial-normal hierarchical model and beta-binomial model. This package is also the accompanying package of Guenhan et al (2020). Another important functionality of the package is the model-based meta-analysis models.

Author(s)

Burak Kuersad Guenhan <burak.gunhan@med.uni-goettingen.de>

References

Stan Development Team (2018). RStan: the R interface to Stan. R package version 2.17.3. http://mc-stan.org

Günhan, B and Röver, C and Friede, T (2020). Random-effects meta-analysis of few studies involving rare events. Research Synthesis Methods. doi = 10.1002/jrsm.1370.

Fitting a model-based meta-analysis model using Stan

Description

'MBMA_stan' fits a model-based meta-analysis model using Stan.

Usage

MBMA_stan(
  data = NULL,
  likelihood = NULL,
  dose_response = "emax",
  mu_prior = c(0, 10),
  Emax_prior = c(0, 100),
  alpha_prior = c(0, 100),
  tau_prior = 0.5,
  tau_prior_dist = "half-normal",
  ED50_prior = c(-2.5, 1.8),
  ED50_prior_dist = "functional",
  gamma_prior = c(1, 2),
  Pred_doses,
  re = TRUE,
  ncp = TRUE,
  chains = 4,
  iter = 2000,
  warmup = 1000,
  adapt_delta = 0.95,
  ...
)

Arguments

data

An object of 'create_MBMA_dat'.

likelihood

A string specifying the likelihood of distributions defining the statistical model. Options include "normal", "binomial", and "Poisson".

dose_response

A string specifying the function defining the dose-response model. Options include "linear", "log-linear", "emax", and "sigmoidal".

mu_prior

A numerical vector specifying the parameter of the normal prior density for baseline risks, first value is parameter for mean, second is for variance. Default is c(0, 10).

Emax_prior

A numerical vector specifying the parameter of the normal prior density for Emax parameter, first value is parameter for mean, second is for standard deviation. Default is c(0, 10). Needed for emax and sigmoidal models.

alpha_prior

A numerical vector specifying the parameter of the normal prior density for the alpha parameter, first value is parameter for mean, second is for variance. Default is c(0, 10). Needed for linear and linear log-dose models.

tau_prior

A numerical value specifying the standard dev. of the prior density for heterogenety stdev. Default is 0.5.

tau_prior_dist

A string specifying the prior density for the heterogeneity standard deviation, option is 'half-normal' for half-normal prior, 'uniform' for uniform prior, 'half-cauchy' for half-cauchy prior.

ED50_prior

A numerical vector specifying the parameter of the normal prior density for ED50 parameter, first value is parameter for mean, second is for standard deviation. Default is c(0, 10). Needed for emax and sigmoidal models.

ED50_prior_dist

A string specifying the prior density for the ED50 parameter, 'functional' is for a functional uniform prior, 'half-normal' for uniform prior, 'half-cauchy' for half-cauchy prior.

gamma_prior

A numerical vector specifying the parameter of the normal prior density for gamma parameter, first value is parameter for mean, second is for standard deviation. Default is c(1, 2). Needed for sigmoidal model.

Pred_doses

A numerical vector specifying the doses which prediction will be made.

re

A string specifying whether random-effects are included to the model. When 'FALSE', the model corresponds to a fixed-effects model. The default is 'TRUE'.

ncp

A string specifying whether to use a non-centered parametrization. The default is 'TRUE'.

chains

A positive integer specifying the number of Markov chains. The default is 4.

iter

A positive integer specifying the number of iterations for each chain (including warmup). The default is 2000.

warmup

A positive integer specifying the number of warmup (aka burnin) iterations per chain. The default is 1000.

adapt_delta

A numerical value specfying the target average proposal acceptance probability for adaptation. See Stan manual for details. Default is 0.95. In general you should not need to change adapt_delta unless you see a warning message about divergent transitions, in which case you can increase adapt_delta from the default to a value closer to 1 (e.g. from 0.95 to 0.99, or from 0.99 to 0.999, etc).

...

Further arguments passed to or from other methods.

Value

an object of class 'stanfit' returned by 'rstan::sampling'

References

Boucher M, et al. The many flavors of model-based meta-analysis: Part I-Introduction and landmark data. CPT: Pharmacometrics and Systems Pharmacology. 2016;5:54-64.

Guenhan BK, Roever C, Friede T. MetaStan: An R package for meta-analysis and model-based meta-analysis using Stan. In preparation.

Mawdsley D, et al. Model-based network meta-analysis: A framework for evidence synthesis of clinical trial data. CPT: Pharmacometrics and Systems Pharmacology. 2016;5:393-401.

Zhang J, et al. (2014). Network meta-analysis of randomized clinical trials: Reporting the proper summaries. Clinical Trials. 11(2), 246–262.

Dias S, et al. Absolute or relative effects? Arm-based synthesis of trial data. Research Synthesis Methods. 2016;7:23–28.

Examples

## Not run: 
## Load the dataset
data('dat.Eletriptan', package = "MetaStan")
datMBMA = create_MetaStan_dat(dat = dat.Eletriptan,
                              armVars = c(dose = "d",
                                          responders = "r",
                                          sampleSize = "n"),
                              nArmsVar = "nd")

MBMA.Emax  <- MBMA_stan(data = datMBMA,
                        likelihood = "binomial",
                        dose_response = "emax",
                        Pred_doses = seq(0, 80, length.out = 11),
                        mu_prior = c(0, 100),
                        Emax_prior = c(0, 100),
                        tau_prior_dist = "half-normal",
                        tau_prior = 0.5)
plot(MBMA.Emax) + ggplot2::xlab("Doses (mg)") + ggplot2::ylab("response probabilities")


## End(Not run)

Compare MBMA fits using LOO-IC

Description

Takes a vector of MBMA_stan fits and give the model comparison results based on LOO-IC criteria. This is useful to compare different dose-response models. The function depends on loo_compare function from loo package.

Usage

compare_MBMA(model_list, digits = 2, ...)

Arguments

model_list

A vector of MBMA_stan object.

digits

An integer indicating the number of decimal places.

...

Further arguments passed to or from other methods.

References

Vehtari, A, A Gelman, and J Gabry (Sept. 2017). "Practical Bayesian model evaluation using leave one-out cross-validation and WAIC." In: Statistics and Computing 27.5, pp. 1413–1432.

Convert contrast-based dataset to arm-based dataset (deprecated)

Description

convert_data_arm creates a dataframe to fit a meta-analysis model using meta_Stan function.

Usage

convert_data_arm(nt, nc, pt, pc, pub, data = NULL)

Arguments

nt

Number of subjects in treatment arm

nc

Number of subjects in control arm

pt

Number of events in treatment arm

pc

Number of events in treatment arm

pub

The corresponding publication

data

Optional data frame containing the variables given to the arguments above.

Value

A dataframe object

Examples

## Create arm-based dataset
data('dat.Crins2014', package = "MetaStan")
dat_converted <- convert_data_arm(exp.total, cont.total,
                                  exp.AR.events, cont.AR.events,
                                  publication, data = dat.Crins2014)

Prepare model-based meta-analysis dataset for Stan.

Description

create_MetaStan_dat converts datasets in the one-study-per-row format to one-arm-per-row format,

Usage

create_MetaStan_dat(
  dat = NULL,
  armVars = c(dose = "d", responders = "r", sampleSize = "n"),
  nArmsVar = "nd"
)

Arguments

dat

Data in one-study-per-row format.

armVars

Vector of per-arm variables The name of each component will be the column name in the resulting dataset.

nArmsVar

Variable holding the number of arms for each study.

Details

The resulting data.frame can be used as data argument in MBMA_stan.

Value

A data frame with the generated columns.

Author(s)

Burak Kuersad Guenhan, burak.gunhan@med.uni-goettingen.de and Gert van Valkenhoef

Examples

## Not run: 
data('dat.Eletriptan')
## Create the dataset suitable for MBMA_stan
EletriptanDat <- create_MetaStan_dat(dat = dat.Eletriptan,
                                     armVars = c(dose = "d",
                                                 responders = "r",
                                                 sampleSize = "n"),
                                     nArmsVar = 'nd')
## Check that the data are correct
print(EletriptanDat)

## End(Not run)

Trials investigating effectiveness of the BCG vaccine against TB

Description

A dataset containing the results from 13 trials examining the efficacy of Bacillus Calmette-Guerin (BCG) vaccine against tuberculosis (TB).

Usage

dat.Berkey1995

Format

A data frame with following coloumns

Trial: Trial number
r1: number of TB events in treatment arm
n1: number of subjects in treatment arm
r2: number of TB events in control arm
n2: number of subjects in control arm
Latitude: absolute latitude of the study location
publication: author and date

Source

Berkey, C.S., Hoaglin, D.C., Mosteller, F. and Colditz, G.A., 1995. A random-effects regression model for meta-analysis. Statistics in medicine, 14(4), pp.395-411

PARESTHESIA RATES WITH TOPIRAMATE IN MIGRAINE PROPHYLAXIS TRIALS

Description

Numbers of patients and events (paresthesia rates) in experimental and control groups of 6 studies. It is in one-study-per-row format.

Usage

dat.Boucher2016

Format

A data frame with following coloumns

d1: dose (mg) in the first arm (placebo)
r1: number of events in the first arm (placebo)
n1: number of patients in the first arm (placebo)
d2: dose (mg) in the second arm
r2: number of events in the second arm
n2: number of patients in the second arm
d3: dose (mg) in the third arm
r3: number of events in the third arm
n3: number of patients in the third arm
d4: dose (mg) in the fourth arm
r4: number of events in the fourth arm
n4: number of patients in the fourth arm
nd: number of treatment arms

Source

Boucher M, and Bennets M. The Many Flavors of Model-Based Meta-Analysis: Part I - Introduction and Landmark Data. CPT Pharmacometrics Syst. Pharmacol. (2016) 5, 54-64; doi:10.1002/psp4.12041

PARESTHESIA RATES WITH TOPIRAMATE IN MIGRAINE PROPHYLAXIS TRIALS

Description

Numbers of patients and events (paresthesia rates) in experimental and control groups of 6 studies. It is in one-study-per-row format. Only the arm with 200 mg is included.

Usage

dat.Boucher2016.pairwise

Format

A data frame with following coloumns

study: Study ID
duration: Durtaion of the study
r1: number of events in the first arm (placebo)
n1: number of patients in the first arm (placebo)
r2: number of events in the second arm
n2: number of patients in the second arm

Source

Boucher M, and Bennets M. The Many Flavors of Model-Based Meta-Analysis: Part I - Introduction and Landmark Data. CPT Pharmacometrics Syst. Pharmacol. (2016) 5, 54-64; doi:10.1002/psp4.12041

Pediatric liver transplant example data

Description

Numbers of cases and events (PTLDs or deaths) in experimental and control groups of six studies.

Usage

dat.Crins2014

Format

A data frame with following columns

publication: publication identifier (first author and publication year)
year: publication year
randomized: randomization status (y/n)
control.type: type of control group ("concurrent" or "historical")
comparison: type of comparison ("IL-2RA only", "delayed CNI", or "no/low steroids")
followup: t follow-up time in months
exp.AR.events: number of AR events in experimental group
exp.PTLD.events: number of PTLD events in experimental group
exp.deaths: number of deaths in experimental group
exp.total: number of patients in experimental group
exp.SRR.events: number of SRR events in experimental group
cont.AR.events: number of AR events in control group
cont.SRR.events: number of SRR events in control group
cont.PTLD.events: number of PTLD events in control group
cont.deaths: number of deaths in control group
cont.total: number of patients in control group
r2: number of AR events in experimental group
n1: number of patients in control group
n2: number of patients in experimental group
r1: number of AR events in control group

Source

N.D. Crins, C. Roever, A.D. Goralczyk, T. Friede. Interleukin-2 receptor antagonists for pediatric liver transplant recipients: A systematic review and meta-analysis of controlled studies. Pediatric Transplantation, 18(8):839-850, 2014.

Migraine pain relief example (Eletriptan)

Description

Numbers of patients and events (headcahe free at 2 hours) in experimental and control groups of 12 studies. It is in one-study-per-row format.

Usage

dat.Eletriptan

Format

A data frame with following coloumns

ID: trial ID
d1: dose (mg) in the first arm (placebo)
r1: number of events in the first arm (placebo)
n1: number of patients in the first arm (placebo)
d2: dose (mg) in the second arm
r2: number of events in the second arm
n2: number of patients in the second arm
d3: dose (mg) in the third arm
r3: number of events in the third arm
n3: number of patients in the third arm
d4: dose (mg) in the fourth arm
r4: number of events in the fourth arm
n4: number of patients in the fourth arm
nd: number of treatment arms

Source

Thorlund, K., Mills, E., Wu, P., Ramos, E., Chatterjee, A., Druyts, E., and Goadsby, P. (2014). Comparative efficacy of triptans for the abortive treatment of migraine: A multiple treatment comparison meta-analysis. Cephalalgia, 34(4):258-267.

Plot a forest plot

Description

Takes a meta_stan object which is obtained by function meta_stan and plot a forestplot, showing individual estimates along with their 95 percent credible intervals, resulting effect estimate and prediction interval.

Usage

forest_plot(
  x = NULL,
  labels = NULL,
  digits = 2,
  boxsize = 0.3,
  heterogeneity = TRUE,
  col,
  ...
)

Arguments

x

A meta_stan object.

labels

Optional vector with labels for the studies (publication author/year).

digits

A numerical value specifying the number of significant digits to be shown. Default is 2.

boxsize

A numerical value specifying the box size. Default is 0.3.

heterogeneity

A logical value to include heterogeneity estimates (DEFAULT = TRUE)

col

A function specifying the colors. See forestplot::fpColors for details.

...

Further arguments passed to or from other methods.

Value

The return value is invisible NULL.

Author(s)

Christian Roever and Burak Kuersad Guenhan

Source

This function is based foresplot function from foresplot R package.

Examples

## Not run: 
data('dat.Crins2014', package = "MetaStan")
dat_long <- create_MetaStan_dat(dat = dat.Crins2014,
                                    armVars = c(responders = "r", sampleSize = "n"))
bnhm.Crins  <- meta_stan(data = dat_long, likelihood = "binomial",
                         mu_prior = c(0, 10), theta_prior = c(0, 100),
                         tau_prior =  0.5)
forest_plot(bnhm.Crins, xlab = "log-OR", labels = dat.Crins2014$publication)


## End(Not run)

Fitting a meta-analysis model using Stan

Description

'meta_stan' fits a meta-analysis model using Stan.

Usage

meta_stan(
  data = NULL,
  likelihood = NULL,
  mu_prior = c(0, 10),
  theta_prior = NULL,
  tau_prior = 0.5,
  tau_prior_dist = "half-normal",
  beta_prior = c(0, 100),
  delta = NULL,
  param = "Smith",
  re = TRUE,
  ncp = TRUE,
  interval.type = "shortest",
  mreg = FALSE,
  cov = NULL,
  chains = 4,
  iter = 2000,
  warmup = 1000,
  adapt_delta = 0.95,
  ...
)

Arguments

data

Data frame created by 'create_MetaStan_dat'

likelihood

A string specifying the likelihood function defining the statistical model. Options include 'normal', 'binomial', and 'Poisson'.

mu_prior

A numerical vector specifying the parameter of the normal prior density for baseline risks, first value is parameter for mean, second is for variance. Default is c(0, 10).

theta_prior

A numerical vector specifying the parameter of the normal prior density for treatment effect estimate, first value is parameter for mean, second is for variance. Default is NULL.

tau_prior

A numerical value specifying the standard dev. of the prior density for heterogeneity stdev. Default is 0.5.

tau_prior_dist

A string specifying the prior density for the heterogeneity standard deviation, option is 'half-normal' for half-normal prior, 'uniform' for uniform prior, 'half-cauchy' for half-cauchy prior.

beta_prior

A numerical vector specifying the parameter of the normal prior density for beta coefficients in a meta-regression model, first value is parameter for mean, second is for variance. Default is c(0, 100).

delta

A numerical value specifying the upper bound of the a priori interval for treatment effect on odds ratio scale (Guenhan et al (2020)). This is used to calculate a normal weakly informative prior. for theta. Thus when this argument is specified, 'theta' should be left empty. Default is NULL.

param

Paramteriztaion used. The default is the 'Smith' model suggested by Smith et al (1995). The alternative is 'Higgins' is the common meta-analysis model (Simmonds and Higgins, 2014).

re

A string specifying whether random-effects are included to the model. When 'FALSE', the model corresponds to a fixed-effects model. The default is 'TRUE'.

ncp

A string specifying whether to use a non-centered parametrization. The default is 'TRUE'.

interval.type

A string specifying the type of interval estimate. Options include shortest credible interval 'shortest' (default) and qui-tailed credible interval 'central'.

mreg

A string specifying whether to fit a meta-regression model. The default is 'FALSE'.

cov

A numeric vector or matrix specifying trial-level covariates (in each row). This is needed when 'mreg = TRUE'.

chains

A positive integer specifying the number of Markov chains. The default is 4.

iter

A positive integer specifying the number of iterations for each chain (including warmup). The default is 2000.

warmup

A positive integer specifying the number of warmup (aka burnin) iterations per chain. The default is 1000.

adapt_delta

A numerical value specifying the target average proposal acceptance probability for adaptation. See Stan manual for details. Default is 0.95. In general you should not need to change adapt_delta unless you see a warning message about divergent transitions, in which case you can increase adapt_delta from the default to a value closer to 1 (e.g. from 0.95 to 0.99, or from 0.99 to 0.999, etc).

...

Further arguments passed to or from other methods.

Value

an object of class 'MetaStan'.

References

Guenhan BK, Roever C, Friede T. MetaStan: An R package for meta-analysis and model-based meta-analysis using Stan. In preparation.

Guenhan BK, Roever C, Friede T. Random-effects meta-analysis of few studies involving rare events Resarch Synthesis Methods 2020; doi:10.1002/jrsm.1370.

Jackson D, Law M, Stijnen T, Viechtbauer W, White IR. A comparison of 7 random-effects models for meta-analyses that estimate the summary odds ratio. Stat Med 2018;37:1059–1085.

Kuss O. Statistical methods for meta-analyses including information from studies without any events-add nothing to nothing and succeed nevertheless, Stat Med, 2015; 4; 1097–1116, doi: 10.1002/sim.6383.

Examples

## Not run: 

## TB dataset
data('dat.Berkey1995', package = "MetaStan")
## Fitting a Binomial-Normal Hierarchical model using WIP priors
dat_MetaStan <- create_MetaStan_dat(dat = dat.Berkey1995,
                                    armVars = c(responders = "r", sampleSize = "n"))

 ma.stan  <- meta_stan(data = dat_MetaStan,
                           likelihood = "binomial",
                           mu_prior = c(0, 10),
                           theta_prior = c(0, 100),
                           tau_prior = 0.5,
                           tau_prior_dist = "half-normal")
print(ma.stan)
forest_plot(ma.stan)


meta.reg.stan  <- meta_stan(data = dat_MetaStan,
                           likelihood = "binomial",
                           mu_prior = c(0, 10),
                           theta_prior = c(0, 100),
                           tau_prior = 0.5,
                           tau_prior_dist = "half-normal",
                           mreg = TRUE,
                           cov = dat.Berkey1995$Latitude)

print(meta.reg.stan)

## End(Not run)

Plot a dose-response plot

Description

Takes a MBMA_stan object which is obtained by function MBMA_stan and plot a dose-response plot, showing observed event probabilities and the estimated dose-response function with pointwise 95

Usage

## S3 method for class 'MBMA_stan'
plot(x = MBMA.stan, ...)

Arguments

x

A MBMA_stan object.

...

Further arguments passed to ggplot.

Value

The return value is invisible NULL.

Author(s)

Christian Roever and Burak Kuersad Guenhan

Source

This function uses ggplot function from ggplot2 R package.

Examples

## Not run: 
data('dat.Eletriptan', package = "MetaStan")
datMBMA = create_MetaStan_dat(dat = dat.Eletriptan,
                              armVars = c(dose = "d",
                                          responders = "r",
                                          sampleSize = "n"),
                              nArmsVar = "nd")

MBMA.Emax  <- MBMA_stan(data = datMBMA,
                        likelihood = "binomial",
                        dose_response = "emax",
                        Pred_doses = seq(0, 80, length.out = 11),
                        mu_prior = c(0, 100),
                        Emax_prior = c(0, 100),
                        tau_prior_dist = "half-normal",
                        tau_prior = 0.5)
plot(MBMA.Emax) + ggplot2::xlab("Doses (mg)") + ggplot2::ylab("response probabilities")


## End(Not run)

Print MBMA object

Description

Takes an MBMA_stan object which is obtained by function MBMA_stan and print the model and data information such as model type used in the model.

Usage

## S3 method for class 'MBMA_stan'
print(x, digits = 2, ...)

Arguments

x

A MBMA_stan object.

digits

An integer indicating the number of decimal places.

...

Further arguments passed to or from other methods.

Value

The return value is invisible NULL

Print meta_stan object

Description

Takes an meta_stan object which is obtained by function meta_stan and print the model and data information such as model type used in the model.

Usage

## S3 method for class 'meta_stan'
print(x, digits = 2, ...)

Arguments

x

A meta_stan object.

digits

An integer indicating the number of decimal places.

...

Further arguments passed to or from other methods.

Value

The return value is invisible NULL

The 'MetaStan' package.

Description

Details

Author(s)

References

Fitting a model-based meta-analysis model using Stan

Description

Usage

Arguments

Value

References

Examples

Compare MBMA fits using LOO-IC

Description

Usage

Arguments

References

See Also

Convert contrast-based dataset to arm-based dataset (deprecated)

Description

Usage

Arguments

Value

Examples

Prepare model-based meta-analysis dataset for Stan.

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Trials investigating effectiveness of the BCG vaccine against TB

Description

Usage

Format

Source

PARESTHESIA RATES WITH TOPIRAMATE IN MIGRAINE PROPHYLAXIS TRIALS

Description

Usage

Format

Source

PARESTHESIA RATES WITH TOPIRAMATE IN MIGRAINE PROPHYLAXIS TRIALS

Description

Usage

Format

Source

Pediatric liver transplant example data

Description

Usage

Format

Source

Migraine pain relief example (Eletriptan)

Description

Usage

Format

Source

Plot a forest plot

Description

Usage

Arguments

Value

Author(s)

Source

See Also

Examples

Fitting a meta-analysis model using Stan

Description

Usage

Arguments

Value

References

Examples

Plot a dose-response plot

Description

Usage

Arguments

Value

Author(s)