Performs the arm-based network meta-analysis for binary outcomes when the follow-up time of each study is reported, which estimates the treatment-specific rate, rate ratio between treatments, and their logarithms.
nma.ab.followup(s.id, t.id, event.n, total.n, followup, data, trtname,
param = c("lograte", "logratio", "rank.prob"),
model = "het_cor", prior.type, a = 0.001, b = 0.001,
c = 10, higher.better = FALSE, digits = 4, n.adapt = 5000,
n.iter = 100000, n.burnin = floor(n.iter/2), n.chains = 3,
n.thin = max(1, floor((n.iter - n.burnin)/100000)),
conv.diag = FALSE, trace = NULL, dic = FALSE,
postdens = FALSE, mcmc.samples = FALSE)nma.ab.followup returns a list with estimates of effect measures specified in param. If the argument dic = TRUE, the deviance information criterion (DIC) statistic will be returned in the output list. In addition, if conv.diag = TRUE, a .txt file containing the point estimates of the potential scale reduction factor and their upper confidence limits by Gelman and Rubin (1992) will be saved in users' current working directory. If postdens = TRUE, the posterior densities of treatment-specific absolute risks will be saved as a .pdf file. If trace is specified, the trace plots are saved as .png files.
a numeric or character vector specifying study ID, or the corresponding column name in the argument data.
a numeric or character vector specifying treatment ID, or the corresponding column name in the argument data.
a numeric vector of non-negative integers, specifying number of events in each study's treatment group, or the corresponding column name in the argument data.
a numeric vector of non-negative integers, specifying total number of participants in each study's treatment group, or the corresponding column name in the argument data.
a numeric vector of positive numbers, specifying follow-up times for different studies, or the corresponding column name in the argument data.
an optional data frame containing the dataset of the network meta-analysis. If data is specified, the previous arguments, s.id, t.id, event.n, total.n, and followup, should be specified as the corresponding column names in data; otherwise, the previous arguments use environment variables.
a vector of character strings specifying the treatment names for the corresponding treatment IDs according to their order in t.id. If not specified, t.id is used as treatment names.
a vector of character strings specifying the effect measures to be estimated. The default includes log treatment-specific rate ("lograte"), log rate ratio ("logratio"), and treatment rank probability ("rank.prob"). "lograte" is automatically added into param even if it is not specified. In addition to the default measures, treatment-specific rate ("rate") and rate ratio ("ratio") can be added into the argument param.
a character string specifying which Bayesian hierarchical model to be applied in the arm-based network meta-analysis. This argument can be set as "hom_eqcor", "het_eqcor", or "het_cor" (the default). See "Details" for the models.
prior distribution of variances and/or covariances of random effects. If model is "hom_eqcor" or "het_eqcor", it can be set as "unif" (uniform prior for standard deviations, the default) or "invgamma" (inverse gamma prior for variances). If model is "het_cor", prior.type can be "invwishart" (the default) or "chol". Specifying "invwishart" yields an inverse-Wishart prior for the variance-covariance matrix of random effects; by specifying "chol", non-informative priors are assigned to variance and correlation components using the separation strategy by Cholesky decomposition. See "Details".
positive numbers, specifying the shape and scale parameters of inverse gamma priors for variance(s) of random effects if using prior.type as "invgamma" for model "hom_eqcor" or "het_eqcor". The default values for both parameters are 0.001.
positive number, specifying the upper bound of uniform prior for standard deviation(s) of random effects if using prior.type as "unif" for model "hom_eqcor" or "het_eqcor". The default is 10.
an optional logical value which needs to be specified when estimating the treatment rank probabilities (i.e., "rank.prob" is included in argument param). TRUE indicates that a higher treatment-specific rate implies a better treatment, and vice versa. The default is FALSE.
a positive integer specifying the digits after the decimal point for the effect measure estimates. The default is 4.
the number of iterations for adaptation in Markov chain Monte Carlo (MCMC) algorithm. The default is 5,000. If a warning "adaptation incomplete" appears, users may increase n.adapt. This argument and the following n.iter, n.burnin, n.chains, and n.thin are passed to the functions in R package rjags.
the total number of iterations in each MCMC chain. The default is 100,000.
the number of iterations for burn-in period. The default is n.iter/2.
the number of MCMC chains. The default is 3.
a positive integer specifying the thinning rate. The default is the thinning rate which yields no more than 100,000 iterations remaining in each chain.
a logical value indicating whether to perform MCMC convergence diagnostic. The default is FALSE. If TRUE, n.chains must be greater than 1, and a .txt file, which contains the point estimates of the potential scale reduction factor and their upper confidence limits (see Gelman and Rubin 1992), will be written in users' current working directory.
a vector of character strings of effect measures. The character strings should be selected from those specified in param (except "rank.prob"), and trace plots of the specified effect measures will be saved in users' current working directory. The default is not drawing trace plots (NULL).
a logical value indicating whether to calculate the deviance information criterion (DIC) value. The default is FALSE.
a logical value indicating whether to draw the posterior density plots for treatment-specific rates. If TRUE, a .pdf file containing the density plot will be saved in users' current working directory. The default is FALSE.
a logical value indicating whether to save MCMC posterior samples in the output object. The default is FALSE.
Suppose that a network meta-analysis collects \(I\) studies on \(K\) treatments, where each study investigates a subset of the \(K\) treatments. The outcome is binary, and the follow-up time for each study is reported. Label the studies from \(i = 1\) to \(I\) and the treatments from \(k = 1\) to \(K\). Let \(T_{i}\) be the subset of the \(K\) treatments that is compared in the \(i\)th study. Also, in the \(i\)th study, let \(y_{ik}\) and \(n_{ik}\) be the number of events and the total number of participants in treatment group \(k\). Denote \(f_{i}\) as the follow-up time of the \(i\)th study. The arm-based network meta-analysis model for these settings is constructed as: $$y_{ik} \sim Bin (n_{ik}, p_{ik}) \qquad k \in T_{i};$$ $$cloglog (p_{ik}) = \log (f_{i}) + \log (\lambda_{ik});$$ $$\log (\lambda_{ik}) = \mu_{k} + \nu_{ik};$$ $$(\nu_{i1}, \nu_{i2}, \ldots, \nu_{iK})^{T} \sim N (\boldsymbol{0}, \mathbf{\Sigma}_{K}),$$ where \(cloglot (t) = \log (-\log(1 - t))\) is the complementary log-log link function, and \(\mathbf{\Sigma}_{K}\) is a \(K \times K\) positive definite covariance matrix. The \(\mu_{k}\)'s are treatment-specific fixed effects, and the random effects \(\nu_{ik}\) are correlated within each study with the covariance matrix \(\mathbf{\Sigma}_{K}\).
An unstructured covariance matrix \(\mathbf{\Sigma}_{K}\) in the model above corresponds to model = "het_cor". The inverse-Wishart prior can be assigned to \(\mathbf{\Sigma}_{K}\). Alternatively, using the separation strategy by Cholesky decomposition (prior.type = "chol"), uniform priors \(U(0, c)\) are assigned to the standard deviations in \(\mathbf{\Sigma}_{K}\) and non-informative priors are assigned to the correlation components (Barnard et al., 2000; Lu and Ades, 2009; Wei and Higgins, 2013; Lin and Chu, 2018). Denote \(\sigma_{k}\) as the standard deviation of \(\nu_{ik}\) and \(\mathbf{D} = diag(\sigma_{1}, \ldots, \sigma_{K})\), then the correlation matrix is \(\mathbf{R}_{K} = \mathbf{D}^{-1} \mathbf{\Sigma}_{K} \mathbf{D}^{-1}\). If we assume that all of the off-diagonal elements in \(\mathbf{R}_{K}\) are equal, say to \(\rho\), then this model corresponds to model = "het_eqcor". If we further assume the homogeneity of variances of the random effects, that is, \(\sigma_{k} = \sigma\) for \(k = 1, 2, \ldots, K\), then the model is "hom_eqcor". In addition, for the models "hom_eqcor" and "het_eqcor", setting prior.type as "invgamma" implies using inverse-gamma priors with shape and scale parameters, \(a\) and \(b\), for \(\sigma_{k}^2\) or \(\sigma^2\), and "unif" implies uniform priors \(U(0, c)\) for \(\sigma_{k}\) or \(\sigma\).
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Lin L, Zhang J, Hodges JS, Chu H (2017). "Performing arm-based network meta-analysis in R with the pcnetmeta package." Journal of Statistical Software, 80(5), 1--25. <tools:::Rd_expr_doi("10.18637/jss.v080.i05")>
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Lu G, Ades AE (2009). "Modeling between-trial variance structure in mixed treatment comparisons." Biostatistics, 10(4), 792--805. <tools:::Rd_expr_doi("10.1093/biostatistics/kxp032")>
Spiegelhalter DJ, Best NG, Carlin BP, Van Der Linde A (2002). "Bayesian measures of model complexity and fit." Journal of the Royal Statistical Society, Series B (Statistical Methodology), 64(4), 583--639. <tools:::Rd_expr_doi("10.1111/1467-9868.00353")>
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Zhang J, Carlin BP, Neaton JD, Soon GG, Nie L, Kane R, Virnig BA, Chu H (2014). "Network meta-analysis of randomized clinical trials: Reporting the proper summaries." Clinical Trials, 11(2), 246--262. <tools:::Rd_expr_doi("10.1177/1740774513498322")>
nma.ab.bin, nma.ab.cont, nma.ab.py
#data("diabetes")
# increase n.iter to reach convergence of MCMC
# increase n.adapt to enhance efficiency
#set.seed(1234)
#followup.out <- nma.ab.followup(s.id, t.id, r, n, folup, data = diabetes,
# model = "het_cor", n.adapt = 500, n.iter = 100, n.chains = 1)
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