Meta-analysis models
We fit two random effects models, first with a non-informative prior for the heterogeneity, then using the informative prior described by Turner et al. (2012).
Random effects meta-analysis with non-informative heterogeneity prior
We fit a random effects model using the nma() function
with trt_effects = "random". We use \(\mathrm{N}(0, 100^2)\) prior distributions
for the treatment effects \(d_k\) and
study-specific intercepts \(\mu_j\),
and a non-informative \(\textrm{half-N}(5^2)\) prior 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.Fitting the RE model
tr_fit_RE_noninf <- nma(tr_net,
trt_effects = "random",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100),
prior_het = half_normal(scale = 5))#> Warning: There were 1 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problemsBasic parameter summaries are given by the print()
method:
tr_fit_RE_noninf
#> 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[Transfusion] -1.15 0.03 0.95 -3.19 -1.67 -1.12 -0.59 0.70 842 1.01
#> lp__ -134.46 0.10 3.24 -141.75 -136.39 -134.08 -132.10 -129.27 1026 1.00
#> tau 1.82 0.03 1.01 0.54 1.12 1.58 2.27 4.53 1077 1.00
#>
#> Samples were drawn using NUTS(diag_e) at Mon Apr 29 16:41:16 2024.
#> 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 study-specific relative effects
\(\delta_{jk}\) are hidden, but could
be examined by changing the pars argument:
The prior and posterior distributions can be compared visually using
the plot_prior_posterior() function:
The posterior distribution for the heterogeneity variance \(\tau^2\) is summarised by
Random effects meta-analysis with informative heterogeneity prior
Keeping the rest of the model setup the same, we now use an
informative \(\textrm{log-N}(-3.93,
1.51^2)\) prior for the heterogeneity variance \(\tau^2\). We can examine the range of
parameter values implied by this prior distribution with the
summary() method:
summary(log_normal(-3.93, 1.51))
#> A log-Normal prior distribution: location = -3.93, scale = 1.51.
#> 50% of the prior density lies between 0.01 and 0.05.
#> 95% of the prior density lies between 0 and 0.38.Fitting the RE model, we specify the log_normal prior
distribution in the prior_het argument, and set
prior_het_type = "var" to indicate that this prior
distribution is on the variance scale (instead of the standard
deviation, the default).
tr_fit_RE_inf <- nma(tr_net,
trt_effects = "random",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100),
prior_het = log_normal(-3.93, 1.51),
prior_het_type = "var")Basic parameter summaries are given by the print()
method:
tr_fit_RE_inf
#> 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[Transfusion] -0.77 0.01 0.44 -1.71 -1.03 -0.75 -0.48 0.01 1938 1
#> lp__ -141.03 0.07 2.83 -147.30 -142.71 -140.74 -138.97 -136.43 1454 1
#> tau 0.49 0.01 0.36 0.05 0.21 0.43 0.70 1.36 1389 1
#>
#> Samples were drawn using NUTS(diag_e) at Mon Apr 29 16:41:26 2024.
#> 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 study-specific relative effects
\(\delta_{jk}\) are hidden, but could
be examined by changing the pars argument:
The prior and posterior distributions can be compared visually using
the plot_prior_posterior() function:
Note: The heterogeneity variance \(\tau^2\) is plotted here since the prior was specified on \(\tau^2\).
The posterior distribution for the heterogeneity variance \(\tau^2\) is summarised by