# Bayesian meta-analysis to combine aggregated and individual participant data for cross design synthesis. This function performers a Bayesian cross design synthesis. The function fits a hierarchical meta-regression model based on a bivariate random effects model. ```r hmr( data, two.by.two = TRUE, dataIPD, re = "normal", link = "logit", mean.mu.1 = 0, mean.mu.2 = 0, mean.mu.phi = 0, sd.mu.1 = 1, sd.mu.2 = 1, sd.mu.phi = 1, sigma.1.upper = 5, sigma.2.upper = 5, sigma.beta.upper = 5, mean.Fisher.rho = 0, sd.Fisher.rho = 1/sqrt(2), df = 4, df.estimate = FALSE, df.lower = 3, df.upper = 20, split.w = FALSE, nr.chains = 2, nr.iterations = 10000, nr.adapt = 1000, nr.burnin = 1000, nr.thin = 1 ) ``` ## Arguments - `data`: Aggregated data results: a data frame where the first four columns containing the number of events in the control group (yc), the number of patients in the control group (nc), the number of events in the treatment group (yt) and the number of patients in the treatment group (nt). If two.by.two = TRUE a data frame where each line contains the trial results with column names: yc, nc, yt, nt. - `two.by.two`: If TRUE indicates that the trial results are with names: yc, nc, yt, nt. - `dataIPD`: Individual participant data: a data frame where the first column is the outcome variable and the other columns represent individual participant charachteristics. - `re`: Random effects distribution for the resulting model. Possible values are **normal** for bivariate random effects and **sm** for scale mixtures. - `link`: The link function used in the model. Possible values are **logit**, **cloglog** **probit**. - `mean.mu.1`: Prior mean of baseline risk, default value is 0. - `mean.mu.2`: Prior mean of treatment effect, default value is 0. - `mean.mu.phi`: Prior mean of the bias parameter which measures the difference between the baseline mean mu.1 and the intercept parameter of the logistic regression of the individual participant data. The defalut vaule is 0. - `sd.mu.1`: Prior standard deviation of mu.1, default value is 1. The default prior of mu.1 is a logistic distribution with mean 0 and dispersion 1. The implicit prior for mu.1 in the probability scale is a uniform between 0 and 1. - `sd.mu.2`: Prior standard deviation of mu.2, default value is 1. The default prior of mu.2 is a logistic distribution with mean 0 and dispersion 1. The implicit prior for mu.2 in the probability scale is a uniform between 0 and 1. - `sd.mu.phi`: Prior standard deviation of mu.phi, default value is 1. - `sigma.1.upper`: Upper bound of the uniform prior of sigma.1, default value is 5. - `sigma.2.upper`: Upper bound of the uniform prior of sigma.2, default value is 5. - `sigma.beta.upper`: Upper bound of the uniform prior of sigma.beta, default value is 5. - `mean.Fisher.rho`: Mean of rho in the Fisher scale, default value is 0. - `sd.Fisher.rho`: Standard deviation of rho in the Fisher scale, default value is 1/sqrt(2). - `df`: If de.estimate = FALSE, then df is the degrees of freedom for the scale mixture distribution, default value is 4. - `df.estimate`: Estimate the posterior of df. The default value is FALSE. - `df.lower`: Lower bound of the prior of df. The default value is 3. - `df.upper`: Upper bound of the prior of df. The default value is 30. - `split.w`: Split the w parameter in two independent weights one for each random effect. The default value is FALSE. - `nr.chains`: Number of chains for the MCMC computations, default 5. - `nr.iterations`: Number of iterations after adapting the MCMC, default is 10000. Some models may need more iterations. - `nr.adapt`: Number of iterations in the adaptation process, default is 1000. Some models may need more iterations during adptation. - `nr.burnin`: Number of iteration discarded for burnin period, default is 1000. Some models may need a longer burnin period. - `nr.thin`: Thinning rate, it must be a positive integer, the default value 1. ## Returns This function returns an object of the class "hmr". This object contains the MCMC output of each parameter and hyper-parameter in the model, the data frame used for fitting the model, the link function, type of random effects distribution and the splitting information for conflict of evidence analysis. The results of the object of the class hmr can be extracted with R2jags. In addition a summary, a print and a plot function are implemented for this type of object. ## Details The number of events in the control and treated group are modeled with two conditional Binomial distributions and the random-effects are based on a bivariate scale mixture of Normals. The individual participant data is modeled as a Bayesian logistic regression for participants in the control group. Coefficients in the regression are modeled as exchangeables. The function calculates the implicit hierarchical meta-regression, where the treatment effect is regressed to the baseline risk (rate of events in the control group). The scale mixture weights are used to adjust for internal validity and structural outliers identification. The implicit hierarchical meta-regression is used to predict the treatment effect for subgroups of individual participant data. Computations are done by calling JAGS (Just Another Gibbs Sampler) to perform MCMC (Markov Chain Monte Carlo) sampling and returning an object of the class **mcmc.list**. Installation of JAGS: It is important to note that R 3.3.0 introduced a major change in the use of toolchain for Windows. This new toolchain is incompatible with older packages written in C++. As a consequence, if the installed version of JAGS does not match the R installation, then the rjags package will spontaneously crash. Therefore, if a user works with R version >= 3.3.0, then JAGS must be installed with the installation program JAGS-4.2.0-Rtools33.exe. For users who continue using R 3.2.4 or an earlier version, the installation program for JAGS is the default installer JAGS-4.2.0.exe. ## Examples ```r ## Not run: library(jarbes) data("healing") AD <- healing[, c("y_c", "n_c", "y_t", "n_t")] data("healingipd") IPD <- healingipd[, c("healing.without.amp", "PAD", "neuropathy", "first.ever.lesion", "no.continuous.care", "male", "diab.typ2", "insulin", "HOCHD", "HOS", "CRF", "dialysis", "DNOAP", "smoking.ever", "diabdur", "wagner.class")] mx2 <- hmr(AD, two.by.two = FALSE, dataIPD = IPD, re = "sm", link = "logit", sd.mu.1 = 2, sd.mu.2 = 2, sd.mu.phi = 2, sigma.1.upper = 5, sigma.2.upper = 5, sigma.beta.upper = 5, sd.Fisher.rho = 1.25, df.estimate = FALSE, df.lower = 3, df.upper = 10, nr.chains = 1, nr.iterations = 1500, nr.adapt = 100, nr.thin = 1) print(mx2) # This experiment corresponds to Section 4 in Verde (2018). # # Experiment: Combining aggretated data from RCTs and a single # observational study with individual participant data. # # In this experiment we assess conflict of evidence between the RCTs # and the observational study with a partially identified parameter # mu.phi. # # We run two simulated data: 1) mu.phi = 0.5 which is diffucult to # identify. 2) mu.phi = 2 which can be identify. The simulations are # used to see if the hmr() function can recover mu.phi. # library(MASS) library(ggplot2) library(jarbes) library(gridExtra) library(mcmcplots) # Simulation of the IPD data invlogit <- function (x) { 1/(1 + exp(-x)) } # Data set for mu.phi = 0.5 ......................................... # Parameters values mu.phi.true <- 0.5 beta0 <- mu.1.true + mu.phi.true beta1 <- 2.5 beta2 <- 2 # Regression variables x1 <- rnorm(200) x2 <- rbinom(200, 1, 0.5) # Binary outcome as a function of "b0 + b1 * x1 + b2 * x2" y <- rbinom(200, 1, invlogit(beta0 + beta1 * x1 + beta2 * x2)) # Preparing the plot to visualize the data jitter.binary <- function(a, jitt = 0.05) ifelse(a==0, runif(length(a), 0, jitt), runif(length(a), 1-jitt, 1)) plot(x1, jitter.binary(y), xlab = "x1", ylab = "Success probability") curve(invlogit(beta0 + beta1*x), from = -2.5, to = 2.5, add = TRUE, col ="blue", lwd = 2) curve(invlogit(beta0 + beta1*x + beta2), from = -2.5, to = 2.5, add = TRUE, col ="red", lwd =2) legend("bottomright", c("b2 = 0", "b2 = 2"), col = c("blue", "red"), lwd = 2, lty = 1) noise <- rnorm(100*20) dim(noise) <- c(100, 20) n.names <- paste(rep("x", 20), seq(3, 22), sep="") colnames(noise) <- n.names data.IPD <- data.frame(y, x1, x2, noise) # Application of HMR ........................................... res.s2 <- hmr(AD.s1, two.by.two = FALSE, dataIPD = data.IPD, sd.mu.1 = 2, sd.mu.2 = 2, sd.mu.phi = 2, sigma.1.upper = 5, sigma.2.upper = 5, sd.Fisher.rho = 1.5) print(res.s2) # Data set for mu.phi = 2 ...................................... # Parameters values mu.phi.true <- 2 beta0 <- mu.1.true + mu.phi.true beta1 <- 2.5 beta2 <- 2 # Regression variables x1 <- rnorm(200) x2 <- rbinom(200, 1, 0.5) # Binary outcome as a function of "b0 + b1 * x1 + b2 * x2" y <- rbinom(200, 1, invlogit(beta0 + beta1 * x1 + beta2 * x2)) # Preparing the plot to visualize the data jitter.binary <- function(a, jitt = 0.05) ifelse(a==0, runif(length(a), 0, jitt), runif(length(a), 1-jitt, 1)) plot(x1, jitter.binary(y), xlab = "x1", ylab = "Success probability") curve(invlogit(beta0 + beta1*x), from = -2.5, to = 2.5, add = TRUE, col ="blue", lwd = 2) curve(invlogit(beta0 + beta1*x + beta2), from = -2.5, to = 2.5, add = TRUE, col ="red", lwd =2) legend("bottomright", c("b2 = 0", "b2 = 2"), col = c("blue", "red"), lwd = 2, lty = 1) noise <- rnorm(100*20) dim(noise) <- c(100, 20) n.names <- paste(rep("x", 20), seq(3, 22), sep="") colnames(noise) <- n.names data.IPD <- data.frame(y, x1, x2, noise) # Application of HMR ................................................ res.s3 <- hmr(AD.s1, two.by.two = FALSE, dataIPD = data.IPD, sd.mu.1 = 2, sd.mu.2 = 2, sd.mu.phi = 2, sigma.1.upper = 5, sigma.2.upper = 5, sd.Fisher.rho = 1.5 ) print(res.s3) # Posteriors for mu.phi ............................ attach.jags(res.s2) mu.phi.0.5 <- mu.phi df.phi.05 <- data.frame(x = mu.phi.0.5) attach.jags(res.s3) mu.phi.1 <- mu.phi df.phi.1 <- data.frame(x = mu.phi.1) p1 <- ggplot(df.phi.05, aes(x=x))+ xlab(expression(mu[phi])) + ylab("Posterior distribution")+ xlim(c(-7,7))+ geom_histogram(aes(y=..density..),fill = "royalblue", colour = "black", alpha= 0.4, bins=60) + geom_vline(xintercept = 0.64, colour = "black", size = 1.7, lty = 2)+ geom_vline(xintercept = 0.5, colour = "black", size = 1.7, lty = 1)+ stat_function(fun = dlogis, n = 101, args = list(location = 0, scale = 1), size = 1.5) + theme_bw() p2 <- ggplot(df.phi.1, aes(x=x))+ xlab(expression(mu[phi])) + ylab("Posterior distribution")+ xlim(c(-7,7))+ geom_histogram(aes(y=..density..),fill = "royalblue", colour = "black", alpha= 0.4, bins=60) + geom_vline(xintercept = 2.2, colour = "black", size = 1.7, lty = 2)+ geom_vline(xintercept = 2, colour = "black", size = 1.7, lty = 1)+ stat_function(fun = dlogis, n = 101, args = list(location = 0, scale = 1), size = 1.5) + theme_bw() grid.arrange(p1, p2, ncol = 2, nrow = 1) # Cater plots for regression coefficients ........................... var.names <- names(data.IPD[-1]) v <- paste("beta", names(data.IPD[-1]), sep = ".") mcmc.x <- as.rjags.mcmc(res.s2$BUGSoutput$sims.matrix) mcmc.x.2 <- as.mcmc.rjags(res.s2) mcmc.x.3 <- as.mcmc.rjags(res.s3) greek.names <- paste(paste("beta[",1:22, sep=""),"]", sep="") par.names <- paste(paste("beta.IPD[",1:22, sep=""),"]", sep="") caterplot(mcmc.x.2, parms = par.names, col = "black", lty = 1, labels = greek.names, greek = T, labels.loc="axis", cex =0.7, style = "plain",reorder = F, denstrip = F) caterplot(mcmc.x.3, parms = par.names, col = "grey", lty = 2, labels = greek.names, greek = T, labels.loc="axis", cex =0.7, style = "plain",reorder = F, denstrip = F, add = TRUE, collapse=TRUE, cat.shift=-0.5) abline(v=0, lty = 2, lwd = 2) abline(v =2, lty = 2, lwd = 2) abline(v =2.5, lty = 2, lwd = 2) # End of the examples. ## End(Not run) ``` ## References Verde, P.E, Ohmann, C., Icks, A. and Morbach, S. (2016) Bayesian evidence synthesis and combining randomized and nonrandomized results: a case study in diabetes. Statistics in Medicine. Volume 35, Issue 10, 10 May 2016, Pages: 1654 to 1675. Verde, P.E. (2017) The hierarchical meta-regression approach and learning from clinical evidence. Submited to the Biometrical Journal. Verde, P.E. (2018) The Hierarchical Meta-Regression Approach and Learning from Clinical Evidence. Technical report.

hmr function