Meta-analysis models
We fit random effects (RE) models, firstly without any covariates, and then with a meta-regression on the continuous covariate latitude.
RE meta-analysis (no covariate)
We start by fitting a standard RE model without any covariates. We use \(\mathrm{N}(0, 100^2)\) prior distributions for the treatment effect \(d_\mathrm{Vaccine}\) and study-specific intercepts \(\mu_j\), and a \(\textrm{half-N}(0, 5^2)\) prior distribution for the heterogeneity standard deviation \(\tau\). We can examine the range of parameter values implied by these prior distributions with the summary() method:
summary(normal(scale = 100))
#> A Normal prior distribution: location = 0, scale = 100.
#> 50% of the prior density lies between -67.45 and 67.45.
#> 95% of the prior density lies between -196 and 196.
summary(half_normal(scale = 5))
#> A half-Normal prior distribution: location = 0, scale = 5.
#> 50% of the prior density lies between 0 and 3.37.
#> 95% of the prior density lies between 0 and 9.8.The model is fitted with the nma() function, with a random effects model specified by trt_effects = "random".
bcg_fit_unadj <- nma(bcg_net,
trt_effects = "random",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100),
prior_het = half_normal(scale = 5))Basic parameter summaries are given by the print() method:
bcg_fit_unadj
#> A random effects NMA with a binomial likelihood (logit link).
#> Inference for Stan model: binomial_1par.
#> 4 chains, each with iter=2000; warmup=1000; thin=1;
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#>
#> mean se_mean sd 2.5% 25% 50% 75% 97.5% n_eff Rhat
#> d[Vaccinated] -0.75 0.01 0.22 -1.19 -0.89 -0.75 -0.61 -0.32 1170 1.00
#> lp__ -13453.72 0.15 4.56 -13464.08 -13456.35 -13453.34 -13450.61 -13445.85 886 1.01
#> tau 0.69 0.01 0.20 0.39 0.55 0.66 0.80 1.19 1502 1.00
#>
#> Samples were drawn using NUTS(diag_e) at Thu Feb 24 09:08:24 2022.
#> 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).By default, summaries of the study-specific intercepts \(\mu_j\) and random effects \(\delta_j\) are hidden, but could be examined by changing the pars argument:
# Not run
print(bcg_fit_unadj, pars = c("d", "mu", "delta", "tau"))The prior and posterior distributions can be compared visually using the plot_prior_posterior() function:
plot_prior_posterior(bcg_fit_unadj, prior = c("trt", "het"))
RE meta-regression with covariate latitude
We now fit a RE meta-regression model, adjusting for latitude. We use \(\mathrm{N}(0, 100^2)\) prior distributions for the treatment effect \(d_\mathrm{Vaccine}\), study-specific intercepts \(\mu_j\), and regression coefficient \(\beta\). We use a \(\text{half-N}(0, 5^2)\) prior distribution for the heterogeneity standard deviation \(\tau\). We can examine the range of parameter values implied by these prior distributions with the summary() method:
summary(normal(scale = 100))
#> A Normal prior distribution: location = 0, scale = 100.
#> 50% of the prior density lies between -67.45 and 67.45.
#> 95% of the prior density lies between -196 and 196.
summary(half_normal(scale = 5))
#> A half-Normal prior distribution: location = 0, scale = 5.
#> 50% of the prior density lies between 0 and 3.37.
#> 95% of the prior density lies between 0 and 9.8.Again, the model is fitted with the nma() function. The regression formula ~ .trt:latitude means that the interaction of latitude with treatment will be included; the .trt special variable indicates treatment, and latitude is in the original data set. We increase adapt_delta to 0.99 to remove a small number of divergent transition errors (the default for RE models is set to 0.95).
bcg_fit_lat <- nma(bcg_net,
trt_effects = "random",
regression = ~.trt:latitude,
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100),
prior_reg = normal(scale = 100),
prior_het = half_normal(scale = 5),
adapt_delta = 0.99)#> Note: No treatment classes specified in network, any interactions in `regression` formula will be separate (independent) for each treatment.
#> Use set_*() argument `trt_class` and nma() argument `class_interactions` to change this.Basic parameter summaries are given by the print() method:
bcg_fit_lat
#> A random effects NMA with a binomial likelihood (logit link).
#> Regression model: ~.trt:latitude.
#> Centred covariates at the following overall mean values:
#> latitude
#> 33.46154
#> Inference for Stan model: binomial_1par.
#> 4 chains, each with iter=2000; warmup=1000; thin=1;
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#>
#> mean se_mean sd 2.5% 25% 50% 75%
#> beta[.trtVaccinated:latitude] -0.03 0.00 0.01 -0.05 -0.04 -0.03 -0.03
#> d[Vaccinated] -0.76 0.00 0.12 -1.04 -0.83 -0.76 -0.69
#> lp__ -13457.20 0.23 5.18 -13468.13 -13460.64 -13456.86 -13453.49
#> tau 0.30 0.01 0.19 0.03 0.16 0.27 0.40
#> 97.5% n_eff Rhat
#> beta[.trtVaccinated:latitude] -0.01 1388 1
#> d[Vaccinated] -0.53 1530 1
#> lp__ -13447.86 520 1
#> tau 0.73 625 1
#>
#> Samples were drawn using NUTS(diag_e) at Thu Feb 24 09:08:43 2022.
#> 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).Note that latitude has automatically been centered at 33.46, the mean value for the studies in the network.
By default, summaries of the study-specific intercepts \(\mu_j\) and study-specific relative effects \(\delta_{jk}\) are hidden, but could be examined by changing the pars argument:
# Not run
print(bcg_fit_lat, pars = c("d", "beta", "mu", "delta", "tau"))The prior and posterior distributions can be compared visually using the plot_prior_posterior() function:
plot_prior_posterior(bcg_fit_lat, prior = c("trt", "reg", "het"))
