Meta-analysis models
We fit fixed effect (FE) and random effects (RE) models, with a
meta-regression on the binary covariate prevention.
Fixed effect meta-regression
We start by fitting a FE model. We use \(\mathrm{N}(0, 100^2)\) prior distributions
for the treatment effect \(d_\mathrm{Statin}\), study-specific
intercepts \(\mu_j\), and regression
coefficient \(\beta\). 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.The model is fitted with the nma() function, with a
fixed effect model specified by trt_effects = "fixed". The
regression formula ~ .trt:prevention means
that interaction of primary/secondary prevention with treatment will be
included; the .trt special variable indicates treatment,
and prevention is in the original data set.
statin_fit_FE <- nma(statin_net,
trt_effects = "fixed",
regression = ~.trt:prevention,
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100),
prior_reg = normal(scale = 100))
#> 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:
statin_fit_FE
#> A fixed effects NMA with a binomial likelihood (logit link).
#> Regression model: ~.trt:prevention.
#> 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[.trtStatin:preventionSecondary] -0.21 0.00 0.11 -0.43 -0.28 -0.21 -0.13
#> d[Statin] -0.10 0.00 0.10 -0.30 -0.17 -0.11 -0.04
#> lp__ -7246.71 0.08 3.38 -7254.15 -7248.75 -7246.35 -7244.23
#> 97.5% n_eff Rhat
#> beta[.trtStatin:preventionSecondary] 0.01 2256 1
#> d[Statin] 0.09 2222 1
#> lp__ -7241.21 1688 1
#>
#> Samples were drawn using NUTS(diag_e) at Mon Apr 29 16:42:30 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 \(\mu_j\) 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:
Random effects meta-regression
We now fit a RE model. We use \(\mathrm{N}(0, 100^2)\) prior distributions
for the treatment effect \(d_\mathrm{Statin}\), study-specific
intercepts \(\mu_j\), and regression
coefficient \(\beta\). We use 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.Again, the model is fitted with the nma() function, now
with trt_effects = "random". 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).
statin_fit_RE <- nma(statin_net,
trt_effects = "random",
regression = ~.trt:prevention,
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:
statin_fit_RE
#> A random effects NMA with a binomial likelihood (logit link).
#> Regression model: ~.trt:prevention.
#> 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[.trtStatin:preventionSecondary] -0.31 0.01 0.27 -0.90 -0.45 -0.29 -0.15
#> d[Statin] -0.06 0.01 0.21 -0.49 -0.18 -0.07 0.04
#> lp__ -7255.73 0.16 5.28 -7266.93 -7259.14 -7255.40 -7252.16
#> tau 0.25 0.01 0.20 0.01 0.10 0.20 0.35
#> 97.5% n_eff Rhat
#> beta[.trtStatin:preventionSecondary] 0.19 1221 1
#> d[Statin] 0.38 1290 1
#> lp__ -7246.13 1062 1
#> tau 0.78 779 1
#>
#> Samples were drawn using NUTS(diag_e) at Mon Apr 29 16:42:45 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:
