This vignette describes the analysis of 6 trials comparing
transfusion of granulocytes (white blood cells) to control for
preventing mortality in patients with neutropenia or neutrophil
dysfunction (Stanworth et al. 2005; Turner et al. 2012). The data are
available in this package as `transfusion`

:

```
head(transfusion)
#> studyc trtc r n
#> 1 Bow 1984 Transfusion 5 13
#> 2 Bow 1984 Control 4 11
#> 3 Herzig 1977 Transfusion 1 13
#> 4 Herzig 1977 Control 3 14
#> 5 Higby 1975 Transfusion 2 17
#> 6 Higby 1975 Control 14 19
```

Turner et al. (2012) previously used this dataset to demonstrate the application of informative priors for heterogeneity, an analysis which we recreate here.

We begin by setting up the network - here just a pairwise
meta-analysis. We have arm-level count data giving the number of deaths
(`r`

) out of the total (`n`

) in each arm, so we
use the function `set_agd_arm()`

. We set “Control” as the
reference treatment.

```
tr_net <- set_agd_arm(transfusion,
study = studyc,
trt = trtc,
r = r,
n = n,
trt_ref = "Control")
tr_net
#> A network with 6 AgD studies (arm-based).
#>
#> ------------------------------------------------------- AgD studies (arm-based) ----
#> Study Treatment arms
#> Bow 1984 2: Control | Transfusion
#> Herzig 1977 2: Control | Transfusion
#> Higby 1975 2: Control | Transfusion
#> Scali 1978 2: Control | Transfusion
#> Vogler 1977 2: Control | Transfusion
#> Winston 1982a 2: Control | Transfusion
#>
#> Outcome type: count
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 2
#> Total number of studies: 6
#> Reference treatment is: Control
#> Network is connected
```

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).

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 problems
```

Basic 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

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 heterogeneityvariance\(\tau^2\) is plotted here since the prior was specified on \(\tau^2\).

The posterior distribution for the heterogeneity variance \(\tau^2\) is summarised by

Stanworth, S., E. Massey, C. Hyde, S. J. Brunskill, C. Navarette, G.
Lucas, D. Marks, and U. Paulus. 2005. “Granulocyte Transfusions
for Treating Infections in Patients with Neutropenia or Neutrophil
Dysfunction.” *Cochrane Database of Systematic Reviews*,
no. 3. https://doi.org/10.1002/14651858.CD005339.

Turner, R. M., J. Davey, M. J. Clarke, S. G. Thompson, and J. P. T.
Higgins. 2012. “Predicting the Extent of Heterogeneity in
Meta-Analysis, Using Empirical Data from the Cochrane Database of
Systematic Reviews.” *International Journal of
Epidemiology* 41 (3): 818–27. https://doi.org/10.1093/ije/dys041.