# A Bayesian model for a simulated meta-analysis

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This is essentially an addendum to the previous post where I simulated data from multiple RCTs to explore an analytic method to pool data across different studies. In that post, I used the `nlme`

package to conduct a meta-analysis based on individual level data of 12 studies. Here, I am presenting an alternative hierarchical modeling approach that uses the Bayesian package `rstan`

.

### Create the data set

We’ll use the exact same data generating process as described in some detail in the previous post.

library(simstudy) library(rstan) library(data.table) defS <- defData(varname = "a.k", formula = 3, variance = 2, id = "study") defS <- defData(defS, varname = "d.0", formula = 3, dist = "nonrandom") defS <- defData(defS, varname = "v.k", formula = 0, variance = 6, dist= "normal") defS <- defData(defS, varname = "s2.k", formula = 16, variance = .2, dist = "gamma") defS <- defData(defS, varname = "size.study", formula = ".3;.5;.2", dist = "categorical") defS <- defData(defS, varname = "n.study", formula = "(size.study==1) * 20 + (size.study==2) * 40 + (size.study==3) * 60", dist = "poisson") defI <- defDataAdd(varname = "y", formula = "a.k + x * (d.0 + v.k)", variance = "s2.k") RNGkind(kind = "L'Ecuyer-CMRG") set.seed(12764) ds <- genData(12, defS) dc <- genCluster(ds, "study", "n.study", "id", ) dc <- trtAssign(dc, strata = "study", grpName = "x") dc <- addColumns(defI, dc) d.obs <- dc[, .(study, id, x, y)]

### Build the Stan model

There are multiple ways to estimate a `Stan`

model in `R`

, but I choose to build the Stan code directly rather than using the `brms`

or `rstanarm`

packages. In the Stan code, we need to define the data structure, specify the parameters, specify any transformed parameters (which are just a function of the parameters), and then build the model - which includes laying out the prior distributions as well as the likelihood.

In this case, the model is slightly different from what was presented in the context of a mixed effects model. This is the mixed effects model:

\[ y_{ik} = \alpha_k + \delta_k x_{ik} + e_{ik} \\

\\

\delta_k = \delta_0 + v_k \\

e_{ik} \sim N(0, \sigma_k^2), v_k \sim N(0,\tau^2)

\]

In this Bayesian model, things are pretty much the same:

\[ y_{ik} \sim N(\alpha_k + \delta_k x_{ik}, \sigma_k^2) \\

\\

\delta_k \sim N(\Delta, \tau^2)

\]

The key difference is that there are prior distributions on \(\Delta\) and \(\tau\), introducing an additional level of uncertainty into the estimate. I would expect that the estimate of the overall treatment effect \(\Delta\) will have a wider 95% CI (credible interval in this context) than the 95% CI (confidence interval) for \(\delta_0\) in the mixed effects model. This added measure of uncertainty is a strength of the Bayesian approach.

data { intN; // number of observations int K; // number of studies real y[N]; // vector of continuous outcomes int kk[N]; // study for individual int x[N]; // treatment arm for individual } parameters { vector[K] beta; // study-specific intercept vector[K] delta; // study effects real sigma[K]; // sd of outcome dist - study specific real Delta; // average treatment effect real tau; // variation of treatment effects } transformed parameters{ vector[N] yhat; for (i in 1:N) yhat[i] = beta[kk[i]] + x[i] * delta[kk[i]]; } model { // priors sigma ~ normal(0, 2.5); beta ~ normal(0, 10); tau ~ normal(0, 2.5); Delta ~ normal(0, 10); delta ~ normal(Delta, tau); // outcome model for (i in 1:N) y[i] ~ normal(yhat[i], sigma[kk[i]]); }

### Generate the posterior distributions

With the model in place, we transform the data into a `list`

so that Stan can make sense of it:

N <- nrow(d.obs) ## number of observations K <- dc[, length(unique(study))] ## number of studies y <- d.obs$y ## vector of continuous outcomes kk <- d.obs$study ## study for individual x <- d.obs$x ## treatment arm for individual ddata <- list(N = N, K = K, y = y, kk = kk, x = x)

And then we compile the Stan code:

rt <- stanc("model.stan") sm <- stan_model(stanc_ret = rt, verbose=FALSE)

Finally, we can sample data from the posterior distribution:

fit <- sampling(sm, data=ddata, seed = 3327, iter = 10000, warmup = 2500, control=list(adapt_delta=0.9))

### Check the diagonstic plots

Before looking at any of the output, it is imperative to convince ourselves that the MCMC process was a stable one. The *trace* plot is the most basic way to assess this. Here, I am only showing these plots for \(\Delta\) and \(\tau\), but the plots for the other parameters looked similar, which is to say everything looks good:

pname <- c("Delta", "tau") stan_trace(object = fit, pars = pname)

### Look at the results

It is possible to look inspect the distribution of any or all parameters. In this case, I am particularly interested in the treatment effects at the study level, and overall. That is, the focus here is on \(\Delta\), \(\delta_k\), and \(\tau\).

pname <- c("delta", "Delta","tau") print(fit, pars=pname, probs = c(0.05, 0.5, 0.95)) ## Inference for Stan model: model. ## 4 chains, each with iter=10000; warmup=2500; thin=1; ## post-warmup draws per chain=7500, total post-warmup draws=30000. ## ## mean se_mean sd 5% 50% 95% n_eff Rhat ## delta[1] 6.39 0.01 1.13 4.51 6.41 8.22 29562 1 ## delta[2] -0.78 0.01 1.62 -3.45 -0.78 1.85 28188 1 ## delta[3] -0.14 0.01 1.39 -2.37 -0.16 2.18 28909 1 ## delta[4] 3.08 0.00 0.59 2.09 3.08 4.05 34277 1 ## delta[5] -0.16 0.01 1.01 -1.77 -0.18 1.52 27491 1 ## delta[6] 3.87 0.00 0.86 2.47 3.87 5.27 35079 1 ## delta[7] 4.04 0.01 1.11 2.21 4.03 5.87 32913 1 ## delta[8] 5.23 0.01 1.29 3.12 5.23 7.36 33503 1 ## delta[9] 1.79 0.01 1.25 -0.27 1.78 3.82 30709 1 ## delta[10] 1.38 0.01 1.12 -0.46 1.38 3.21 30522 1 ## delta[11] 4.47 0.01 1.25 2.43 4.47 6.54 34573 1 ## delta[12] 0.79 0.01 1.45 -1.60 0.80 3.16 33422 1 ## Delta 2.48 0.00 0.89 1.01 2.50 3.89 31970 1 ## tau 2.72 0.00 0.71 1.72 2.64 4.01 24118 1 ## ## Samples were drawn using NUTS(diag_e) at Sat Jun 27 15:47:15 2020. ## For each parameter, n_eff is a crude measure of effective sample size, ## and Rhat is the potential scale reduction factor on split chains (at ## convergence, Rhat=1).

The forest plot is quite similar to the one based on the mixed effects model, though as predicted, the 95% CI is considerably wider:

As a comparison, here is the plot from the mixed effects model estimated using the `nlme`

package in the previous post. The bootstrapped estimates of uncertainty at the study level are quite close to the Bayesian measure of uncertainty; the difference really lies in the uncertainty around the global estimate.

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