# Example: Thrombolytic treatments

library(multinma)
options(mc.cores = parallel::detectCores())

This vignette describes the analysis of 50 trials of 8 thrombolytic drugs (streptokinase, SK; alteplase, t-PA; accelerated alteplase, Acc t-PA; streptokinase plus alteplase, SK+tPA; reteplase, r-PA; tenocteplase, TNK; urokinase, UK; anistreptilase, ASPAC) plus per-cutaneous transluminal coronary angioplasty (PTCA) . The number of deaths in 30 or 35 days following acute myocardial infarction are recorded. The data are available in this package as thrombolytics:

#>   studyn trtn      trtc    r     n
#> 1      1    1        SK 1472 20251
#> 2      1    3  Acc t-PA  652 10396
#> 3      1    4 SK + t-PA  723 10374
#> 4      2    1        SK    9   130
#> 5      2    2      t-PA    6   123
#> 6      3    1        SK    5    63

## Setting up the network

We begin by setting up the network. 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(). By default, SK is set as the network reference treatment.

thrombo_net <- set_agd_arm(thrombolytics,
study = studyn,
trt = trtc,
r = r,
n = n)
thrombo_net
#> A network with 50 AgD studies (arm-based).
#>
#> ------------------------------------------------------- AgD studies (arm-based) ----
#>  Study Treatment arms
#>  1     3: SK | Acc t-PA | SK + t-PA
#>  2     2: SK | t-PA
#>  3     2: SK | t-PA
#>  4     2: SK | t-PA
#>  5     2: SK | t-PA
#>  6     3: SK | ASPAC | t-PA
#>  7     2: SK | t-PA
#>  8     2: SK | t-PA
#>  9     2: SK | t-PA
#>  10    2: SK | SK + t-PA
#>  ... plus 40 more studies
#>
#>  Outcome type: count
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 9
#> Total number of studies: 50
#> Reference treatment is: SK
#> Network is connected

Plot the network structure.

plot(thrombo_net, weight_edges = TRUE, weight_nodes = TRUE)

## Fixed effects NMA

Following TSD 4 , we fit a fixed effects NMA model, using the nma() function with trt_effects = "fixed". We use $$\mathrm{N}(0, 100^2)$$ prior distributions for the treatment effects $$d_k$$ and study-specific intercepts $$\mu_j$$. 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 using the nma() function. By default, this will use a Binomial likelihood and a logit link function, auto-detected from the data.

thrombo_fit <- nma(thrombo_net,
trt_effects = "fixed",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100))
#> Note: Setting "SK" as the network reference treatment.

Basic parameter summaries are given by the print() method:

thrombo_fit
#> A fixed 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[Acc t-PA]      -0.18    0.00 0.04     -0.26     -0.21     -0.18     -0.15     -0.09  2542    1
#> d[ASPAC]          0.02    0.00 0.04     -0.05     -0.01      0.02      0.04      0.09  5131    1
#> d[PTCA]          -0.48    0.00 0.10     -0.68     -0.55     -0.48     -0.41     -0.28  4674    1
#> d[r-PA]          -0.13    0.00 0.06     -0.24     -0.17     -0.13     -0.08     -0.01  3382    1
#> d[SK + t-PA]     -0.05    0.00 0.05     -0.14     -0.08     -0.05     -0.02      0.04  5819    1
#> d[t-PA]           0.00    0.00 0.03     -0.06     -0.02      0.00      0.02      0.06  4583    1
#> d[TNK]           -0.17    0.00 0.08     -0.33     -0.22     -0.17     -0.12     -0.02  3598    1
#> d[UK]            -0.20    0.00 0.22     -0.64     -0.36     -0.20     -0.05      0.24  5560    1
#> lp__         -43042.84    0.14 5.46 -43054.57 -43046.34 -43042.39 -43038.88 -43033.42  1508    1
#>
#> Samples were drawn using NUTS(diag_e) at Thu Feb 24 08:53:12 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 $$\mu_j$$ are hidden, but could be examined by changing the pars argument:

# Not run
print(thrombo_fit, pars = c("d", "mu"))

The prior and posterior distributions can be compared visually using the plot_prior_posterior() function:

plot_prior_posterior(thrombo_fit, prior = "trt")

Model fit can be checked using the dic() function

(dic_consistency <- dic(thrombo_fit))
#> Residual deviance: 105.9 (on 102 data points)
#>                pD: 58.7
#>               DIC: 164.5

and the residual deviance contributions examined with the corresponding plot() method.

plot(dic_consistency)

There are a number of points which are not very well fit by the model, having posterior mean residual deviance contributions greater than 1.

## Checking for inconsistency

Note: The results of the inconsistency models here are slightly different to those of Dias et al. (2010, 2011), although the overall conclusions are the same. This is due to the presence of multi-arm trials and a different ordering of treatments, meaning that inconsistency is parameterised differently within the multi-arm trials. The same results as Dias et al. are obtained if the network is instead set up with trtn as the treatment variable.

### Unrelated mean effects model

We first fit an unrelated mean effects (UME) model to assess the consistency assumption. Again, we use the function nma(), but now with the argument consistency = "ume".

thrombo_fit_ume <- nma(thrombo_net,
consistency = "ume",
trt_effects = "fixed",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100))
#> Note: Setting "SK" as the network reference treatment.
thrombo_fit_ume
#> A fixed effects NMA with a binomial likelihood (logit link).
#> An inconsistency model ('ume') was fitted.
#> 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%
#> d[Acc t-PA vs. SK]        -0.16    0.00 0.05     -0.25     -0.19     -0.16     -0.13     -0.06
#> d[ASPAC vs. SK]            0.01    0.00 0.04     -0.07     -0.02      0.01      0.03      0.08
#> d[PTCA vs. SK]            -0.67    0.00 0.19     -1.04     -0.79     -0.66     -0.54     -0.31
#> d[r-PA vs. SK]            -0.06    0.00 0.09     -0.23     -0.12     -0.06      0.00      0.11
#> d[SK + t-PA vs. SK]       -0.04    0.00 0.05     -0.14     -0.07     -0.04     -0.01      0.05
#> d[t-PA vs. SK]             0.00    0.00 0.03     -0.06     -0.02      0.00      0.02      0.06
#> d[UK vs. SK]              -0.36    0.01 0.52     -1.40     -0.71     -0.35     -0.01      0.64
#> d[ASPAC vs. Acc t-PA]      1.42    0.01 0.42      0.64      1.13      1.39      1.70      2.30
#> d[PTCA vs. Acc t-PA]      -0.22    0.00 0.12     -0.45     -0.29     -0.21     -0.14      0.01
#> d[r-PA vs. Acc t-PA]       0.02    0.00 0.07     -0.11     -0.03      0.02      0.06      0.15
#> d[TNK vs. Acc t-PA]        0.01    0.00 0.06     -0.12     -0.04      0.01      0.05      0.13
#> d[UK vs. Acc t-PA]         0.14    0.01 0.36     -0.56     -0.09      0.14      0.39      0.86
#> d[t-PA vs. ASPAC]          0.28    0.01 0.36     -0.40      0.04      0.28      0.52      1.01
#> d[t-PA vs. PTCA]           0.55    0.01 0.41     -0.25      0.27      0.54      0.82      1.37
#> d[UK vs. t-PA]            -0.30    0.00 0.35     -0.99     -0.53     -0.30     -0.06      0.36
#> lp__                  -43039.85    0.16 5.86 -43052.32 -43043.64 -43039.41 -43035.75 -43029.59
#>                       n_eff Rhat
#> d[Acc t-PA vs. SK]     7030    1
#> d[ASPAC vs. SK]        5416    1
#> d[PTCA vs. SK]         5734    1
#> d[r-PA vs. SK]         5622    1
#> d[SK + t-PA vs. SK]    7308    1
#> d[t-PA vs. SK]         4718    1
#> d[UK vs. SK]           6793    1
#> d[ASPAC vs. Acc t-PA]  3769    1
#> d[PTCA vs. Acc t-PA]   4742    1
#> d[r-PA vs. Acc t-PA]   5732    1
#> d[TNK vs. Acc t-PA]    5402    1
#> d[UK vs. Acc t-PA]     4234    1
#> d[t-PA vs. ASPAC]      4804    1
#> d[t-PA vs. PTCA]       4297    1
#> d[UK vs. t-PA]         4959    1
#> lp__                   1276    1
#>
#> Samples were drawn using NUTS(diag_e) at Thu Feb 24 08:53:30 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).

Comparing the model fit statistics

dic_consistency
#> Residual deviance: 105.9 (on 102 data points)
#>                pD: 58.7
#>               DIC: 164.5
(dic_ume <- dic(thrombo_fit_ume))
#> Residual deviance: 99.9 (on 102 data points)
#>                pD: 66.1
#>               DIC: 166

Whilst the UME model fits the data better, having a lower residual deviance, the additional parameters in the UME model mean that the DIC is very similar between both models. However, it is also important to examine the individual contributions to model fit of each data point under the two models (a so-called “dev-dev” plot). Passing two nma_dic objects produced by the dic() function to the plot() method produces this dev-dev plot:

plot(dic_consistency, dic_ume, show_uncertainty = FALSE)

The four points lying in the lower right corner of the plot have much lower posterior mean residual deviance under the UME model, indicating that these data are potentially inconsistent. These points correspond to trials 44 and 45, the only two trials comparing Acc t-PA to ASPAC. The ASPAC vs. Acc t-PA estimates are very different under the consistency model and inconsistency (UME) model, suggesting that these two trials may be systematically different from the others in the network.

### Node-splitting

Another method for assessing inconsistency is node-splitting . Whereas the UME model assesses inconsistency globally, node-splitting assesses inconsistency locally for each potentially inconsistent comparison (those with both direct and indirect evidence) in turn.

Node-splitting can be performed using the nma() function with the argument consistency = "nodesplit". By default, all possible comparisons will be split (as determined by the get_nodesplits() function). Alternatively, a specific comparison or comparisons to split can be provided to the nodesplit argument.

thrombo_nodesplit <- nma(thrombo_net,
consistency = "nodesplit",
trt_effects = "fixed",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100))
#> Fitting model 1 of 15, node-split: Acc t-PA vs. SK
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 2 of 15, node-split: ASPAC vs. SK
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 3 of 15, node-split: PTCA vs. SK
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 4 of 15, node-split: r-PA vs. SK
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 5 of 15, node-split: t-PA vs. SK
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 6 of 15, node-split: UK vs. SK
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 7 of 15, node-split: ASPAC vs. Acc t-PA
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 8 of 15, node-split: PTCA vs. Acc t-PA
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 9 of 15, node-split: r-PA vs. Acc t-PA
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 10 of 15, node-split: SK + t-PA vs. Acc t-PA
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 11 of 15, node-split: UK vs. Acc t-PA
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 12 of 15, node-split: t-PA vs. ASPAC
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 13 of 15, node-split: t-PA vs. PTCA
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 14 of 15, node-split: UK vs. t-PA
#> Note: Setting "SK" as the network reference treatment.
#> Fitting model 15 of 15, consistency model
#> Note: Setting "SK" as the network reference treatment.

The summary() method summarises the node-splitting results, displaying the direct and indirect estimates $$d_\mathrm{dir}$$ and $$d_\mathrm{ind}$$ from each node-split model, the network estimate $$d_\mathrm{net}$$ from the consistency model, the inconsistency factor $$\omega = d_\mathrm{dir} - d_\mathrm{ind}$$, and a Bayesian $$p$$-value for inconsistency on each comparison. The DIC model fit statistics are also provided. (If a random effects model was fitted, the heterogeneity standard deviation $$\tau$$ under each node-split model and under the consistency model would also be displayed.)

summary(thrombo_nodesplit)
#> Node-splitting models fitted for 14 comparisons.
#>
#> ---------------------------------------------------- Node-split Acc t-PA vs. SK ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.18 0.04 -0.26 -0.21 -0.18 -0.15 -0.09     3027     3074    1
#> d_dir -0.16 0.05 -0.26 -0.19 -0.16 -0.12 -0.06     3674     3666    1
#> d_ind -0.25 0.09 -0.43 -0.31 -0.25 -0.19 -0.08      628      872    1
#> omega  0.09 0.10 -0.10  0.03  0.09  0.16  0.29      783     1322    1
#>
#> Residual deviance: 106.4 (on 102 data points)
#>                pD: 59.9
#>               DIC: 166.2
#>
#> Bayesian p-value: 0.34
#>
#> ------------------------------------------------------- Node-split ASPAC vs. SK ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net  0.02 0.04 -0.05 -0.01  0.02  0.04  0.09     4251     2997    1
#> d_dir  0.01 0.04 -0.06 -0.02  0.01  0.03  0.08     4942     3205    1
#> d_ind  0.42 0.25 -0.05  0.25  0.42  0.58  0.90     2419     2634    1
#> omega -0.41 0.25 -0.90 -0.57 -0.41 -0.24  0.07     2477     2546    1
#>
#> Residual deviance: 104 (on 102 data points)
#>                pD: 59.5
#>               DIC: 163.5
#>
#> Bayesian p-value: 0.1
#>
#> -------------------------------------------------------- Node-split PTCA vs. SK ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.47 0.10 -0.68 -0.54 -0.47 -0.41 -0.28     4268     3553    1
#> d_dir -0.67 0.18 -1.03 -0.79 -0.66 -0.54 -0.31     5455     3188    1
#> d_ind -0.39 0.12 -0.63 -0.47 -0.39 -0.31 -0.16     3497     3622    1
#> omega -0.27 0.22 -0.70 -0.42 -0.27 -0.12  0.15     4296     2879    1
#>
#> Residual deviance: 105.8 (on 102 data points)
#>                pD: 60
#>               DIC: 165.8
#>
#> Bayesian p-value: 0.21
#>
#> -------------------------------------------------------- Node-split r-PA vs. SK ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.12 0.06 -0.24 -0.16 -0.12 -0.08 -0.01     3930     3299    1
#> d_dir -0.06 0.09 -0.24 -0.12 -0.06  0.00  0.12     5806     3041    1
#> d_ind -0.18 0.08 -0.33 -0.23 -0.18 -0.12 -0.02     2253     2821    1
#> omega  0.12 0.12 -0.12  0.03  0.11  0.20  0.36     3105     3035    1
#>
#> Residual deviance: 105.8 (on 102 data points)
#>                pD: 59.6
#>               DIC: 165.4
#>
#> Bayesian p-value: 0.34
#>
#> -------------------------------------------------------- Node-split t-PA vs. SK ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net  0.00 0.03 -0.06 -0.02  0.00  0.02  0.06     4064     3292    1
#> d_dir  0.00 0.03 -0.06 -0.02  0.00  0.02  0.06     3791     3049    1
#> d_ind  0.19 0.24 -0.28  0.03  0.18  0.35  0.66     1329     2033    1
#> omega -0.19 0.24 -0.66 -0.35 -0.19 -0.03  0.28     1327     2104    1
#>
#> Residual deviance: 105.9 (on 102 data points)
#>                pD: 59.3
#>               DIC: 165.2
#>
#> Bayesian p-value: 0.43
#>
#> ---------------------------------------------------------- Node-split UK vs. SK ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.20 0.22 -0.64 -0.35 -0.20 -0.05  0.22     4756     3141    1
#> d_dir -0.37 0.53 -1.40 -0.72 -0.36  0.00  0.64     5392     3525    1
#> d_ind -0.17 0.25 -0.67 -0.33 -0.17  0.00  0.32     4975     3183    1
#> omega -0.20 0.59 -1.37 -0.59 -0.19  0.20  0.91     5078     3155    1
#>
#> Residual deviance: 106.7 (on 102 data points)
#>                pD: 59.6
#>               DIC: 166.3
#>
#> Bayesian p-value: 0.75
#>
#> ------------------------------------------------- Node-split ASPAC vs. Acc t-PA ----
#>
#>       mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net 0.19 0.06 0.09 0.16 0.19 0.23  0.30     3684     3237    1
#> d_dir 1.40 0.41 0.65 1.12 1.39 1.67  2.22     3880     2694    1
#> d_ind 0.16 0.06 0.05 0.13 0.16 0.20  0.28     3147     3289    1
#> omega 1.24 0.41 0.48 0.95 1.23 1.50  2.07     3722     2793    1
#>
#> Residual deviance: 96.6 (on 102 data points)
#>                pD: 59.5
#>               DIC: 156.2
#>
#> Bayesian p-value: <0.01
#>
#> -------------------------------------------------- Node-split PTCA vs. Acc t-PA ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.30 0.10 -0.49 -0.36 -0.29 -0.23 -0.11     5088     3571    1
#> d_dir -0.22 0.12 -0.45 -0.30 -0.22 -0.14  0.02     4828     3575    1
#> d_ind -0.48 0.18 -0.83 -0.59 -0.47 -0.36 -0.14     3283     3303    1
#> omega  0.26 0.21 -0.14  0.12  0.26  0.40  0.68     3079     2996    1
#>
#> Residual deviance: 105.6 (on 102 data points)
#>                pD: 59.9
#>               DIC: 165.5
#>
#> Bayesian p-value: 0.2
#>
#> -------------------------------------------------- Node-split r-PA vs. Acc t-PA ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net  0.05 0.06 -0.06  0.02  0.05  0.09  0.16     6103     3503    1
#> d_dir  0.02 0.06 -0.11 -0.02  0.02  0.06  0.15     4904     3584    1
#> d_ind  0.13 0.10 -0.06  0.07  0.13  0.20  0.33     1815     2288    1
#> omega -0.11 0.12 -0.34 -0.19 -0.11 -0.03  0.12     1997     2520    1
#>
#> Residual deviance: 106.1 (on 102 data points)
#>                pD: 59.8
#>               DIC: 165.9
#>
#> Bayesian p-value: 0.33
#>
#> --------------------------------------------- Node-split SK + t-PA vs. Acc t-PA ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net  0.13 0.05  0.02  0.09  0.13  0.16  0.24     5767     3285    1
#> d_dir  0.13 0.05  0.02  0.09  0.13  0.16  0.23     3644     3302    1
#> d_ind  0.65 0.69 -0.67  0.17  0.62  1.09  2.05     3220     2165    1
#> omega -0.52 0.69 -1.90 -0.97 -0.50 -0.05  0.80     3230     2275    1
#>
#> Residual deviance: 106.6 (on 102 data points)
#>                pD: 59.9
#>               DIC: 166.5
#>
#> Bayesian p-value: 0.46
#>
#> ---------------------------------------------------- Node-split UK vs. Acc t-PA ----
#>
#>        mean   sd  2.5%   25%   50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.03 0.22 -0.45 -0.17 -0.03 0.13  0.40     4986     3319    1
#> d_dir  0.15 0.36 -0.53 -0.10  0.14 0.38  0.88     5251     3610    1
#> d_ind -0.13 0.29 -0.72 -0.33 -0.13 0.07  0.41     3801     3153    1
#> omega  0.28 0.46 -0.63 -0.02  0.28 0.57  1.21     4017     3268    1
#>
#> Residual deviance: 107.3 (on 102 data points)
#>                pD: 60.4
#>               DIC: 167.7
#>
#> Bayesian p-value: 0.54
#>
#> ----------------------------------------------------- Node-split t-PA vs. ASPAC ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.01 0.04 -0.09 -0.04 -0.01  0.01  0.06     5769     3144    1
#> d_dir -0.02 0.04 -0.10 -0.05 -0.02  0.00  0.05     4796     3179    1
#> d_ind  0.03 0.06 -0.10 -0.02  0.03  0.07  0.15     3245     3277    1
#> omega -0.05 0.06 -0.18 -0.09 -0.05 -0.01  0.07     3024     2956    1
#>
#> Residual deviance: 106.6 (on 102 data points)
#>                pD: 60.1
#>               DIC: 166.7
#>
#> Bayesian p-value: 0.44
#>
#> ------------------------------------------------------ Node-split t-PA vs. PTCA ----
#>
#>       mean   sd  2.5%   25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net 0.48 0.10  0.27  0.41 0.47 0.55  0.68     4551     3284    1
#> d_dir 0.55 0.41 -0.24  0.27 0.53 0.83  1.38     3974     2943    1
#> d_ind 0.48 0.11  0.27  0.40 0.48 0.55  0.69     3665     3519    1
#> omega 0.07 0.43 -0.74 -0.23 0.05 0.36  0.94     3671     2889    1
#>
#> Residual deviance: 107 (on 102 data points)
#>                pD: 59.8
#>               DIC: 166.9
#>
#> Bayesian p-value: 0.9
#>
#> -------------------------------------------------------- Node-split UK vs. t-PA ----
#>
#>        mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.21 0.22 -0.65 -0.35 -0.20 -0.06  0.21     4817     2922    1
#> d_dir -0.29 0.35 -0.97 -0.52 -0.29 -0.06  0.38     5166     3579    1
#> d_ind -0.15 0.29 -0.70 -0.35 -0.14  0.06  0.42     3980     3164    1
#> omega -0.15 0.45 -1.05 -0.45 -0.14  0.17  0.73     4118     3285    1
#>
#> Residual deviance: 106.8 (on 102 data points)
#>                pD: 59.8
#>               DIC: 166.6
#>
#> Bayesian p-value: 0.74

Node-splitting the ASPAC vs. Acc t-PA comparison results the lowest DIC, and this is lower than the consistency model. The posterior distribution for the inconsistency factor $$\omega$$ for this comparison lies far from 0 and the Bayesian $$p$$-value for inconsistency is small (< 0.01), meaning that there is substantial disagreement between the direct and indirect evidence on this comparison.

We can visually compare the direct, indirect, and network estimates using the plot() method.

plot(thrombo_nodesplit)

We can also plot the posterior distributions of the inconsistency factors $$\omega$$, again using the plot() method. Here, we specify a “halfeye” plot of the posterior density with median and credible intervals, and customise the plot layout with standard ggplot2 functions.

plot(thrombo_nodesplit, pars = "omega", stat = "halfeye", ref_line = 0) +
ggplot2::aes(y = comparison) +
ggplot2::facet_null()

Notice again that the posterior distribution of the inconsistency factor for the ASPAC vs. Acc t-PA comparison lies far from 0, indicating substantial inconsistency between the direct and indirect evidence on this comparison.

## Further results

Relative effects for all pairwise contrasts between treatments can be produced using the relative_effects() function, with all_contrasts = TRUE.

(thrombo_releff <- relative_effects(thrombo_fit, all_contrasts = TRUE))
#>                            mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d[Acc t-PA vs. SK]        -0.18 0.04 -0.26 -0.21 -0.18 -0.15 -0.09     2585     3063    1
#> d[ASPAC vs. SK]            0.02 0.04 -0.05 -0.01  0.02  0.04  0.09     5217     3016    1
#> d[PTCA vs. SK]            -0.48 0.10 -0.68 -0.55 -0.48 -0.41 -0.28     4500     3420    1
#> d[r-PA vs. SK]            -0.13 0.06 -0.24 -0.17 -0.13 -0.08 -0.01     3408     3245    1
#> d[SK + t-PA vs. SK]       -0.05 0.05 -0.14 -0.08 -0.05 -0.02  0.04     5919     3153    1
#> d[t-PA vs. SK]             0.00 0.03 -0.06 -0.02  0.00  0.02  0.06     4689     3245    1
#> d[TNK vs. SK]             -0.17 0.08 -0.33 -0.22 -0.17 -0.12 -0.02     3681     3302    1
#> d[UK vs. SK]              -0.20 0.22 -0.64 -0.36 -0.20 -0.05  0.24     5604     3198    1
#> d[ASPAC vs. Acc t-PA]      0.19 0.06  0.08  0.15  0.19  0.23  0.31     3132     2915    1
#> d[PTCA vs. Acc t-PA]      -0.30 0.10 -0.50 -0.37 -0.30 -0.23 -0.10     5913     3133    1
#> d[r-PA vs. Acc t-PA]       0.05 0.05 -0.06  0.02  0.05  0.09  0.16     5305     3189    1
#> d[SK + t-PA vs. Acc t-PA]  0.13 0.05  0.02  0.09  0.13  0.16  0.23     5049     3566    1
#> d[t-PA vs. Acc t-PA]       0.18 0.05  0.08  0.14  0.18  0.21  0.29     3029     3191    1
#> d[TNK vs. Acc t-PA]        0.01 0.06 -0.12 -0.04  0.01  0.05  0.13     5126     3001    1
#> d[UK vs. Acc t-PA]        -0.02 0.22 -0.47 -0.18 -0.02  0.13  0.42     5877     3164    1
#> d[PTCA vs. ASPAC]         -0.49 0.11 -0.71 -0.57 -0.49 -0.42 -0.29     4477     3337    1
#> d[r-PA vs. ASPAC]         -0.14 0.07 -0.28 -0.19 -0.14 -0.09  0.00     3856     3076    1
#> d[SK + t-PA vs. ASPAC]    -0.07 0.06 -0.19 -0.11 -0.07 -0.02  0.05     5588     2774    1
#> d[t-PA vs. ASPAC]         -0.01 0.04 -0.09 -0.04 -0.01  0.01  0.06     6596     3708    1
#> d[TNK vs. ASPAC]          -0.19 0.09 -0.36 -0.24 -0.19 -0.13 -0.02     3825     3160    1
#> d[UK vs. ASPAC]           -0.22 0.23 -0.66 -0.37 -0.22 -0.06  0.22     5603     2924    1
#> d[r-PA vs. PTCA]           0.35 0.11  0.14  0.28  0.35  0.43  0.57     5963     3205    1
#> d[SK + t-PA vs. PTCA]      0.43 0.11  0.20  0.35  0.43  0.50  0.65     5784     3314    1
#> d[t-PA vs. PTCA]           0.48 0.11  0.27  0.41  0.48  0.55  0.69     4495     3546    1
#> d[TNK vs. PTCA]            0.31 0.12  0.07  0.23  0.31  0.39  0.54     6189     3110    1
#> d[UK vs. PTCA]             0.28 0.24 -0.20  0.11  0.27  0.44  0.76     5851     2984    1
#> d[SK + t-PA vs. r-PA]      0.08 0.07 -0.06  0.03  0.07  0.12  0.21     5389     3131    1
#> d[t-PA vs. r-PA]           0.13 0.07  0.00  0.08  0.13  0.17  0.26     3747     3296    1
#> d[TNK vs. r-PA]           -0.05 0.09 -0.22 -0.10 -0.05  0.01  0.13     6801     3085    1
#> d[UK vs. r-PA]            -0.08 0.23 -0.52 -0.23 -0.07  0.08  0.37     5526     3238    1
#> d[t-PA vs. SK + t-PA]      0.05 0.06 -0.06  0.01  0.05  0.09  0.16     5568     3179    1
#> d[TNK vs. SK + t-PA]      -0.12 0.09 -0.29 -0.18 -0.12 -0.06  0.05     5677     2970    1
#> d[UK vs. SK + t-PA]       -0.15 0.23 -0.59 -0.31 -0.15  0.00  0.30     5833     3339    1
#> d[TNK vs. t-PA]           -0.17 0.08 -0.34 -0.23 -0.17 -0.12 -0.01     4000     3586    1
#> d[UK vs. t-PA]            -0.20 0.22 -0.64 -0.36 -0.20 -0.05  0.23     5653     3014    1
#> d[UK vs. TNK]             -0.03 0.24 -0.49 -0.19 -0.03  0.14  0.42     5717     3401    1
plot(thrombo_releff, ref_line = 0)

Treatment rankings, rank probabilities, and cumulative rank probabilities.

(thrombo_ranks <- posterior_ranks(thrombo_fit))
#>                 mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> rank[SK]        7.44 0.97    6   7   7   8     9     3798       NA    1
#> rank[Acc t-PA]  3.18 0.79    2   3   3   4     5     4218     3854    1
#> rank[ASPAC]     7.96 1.14    5   7   8   9     9     4643       NA    1
#> rank[PTCA]      1.14 0.35    1   1   1   1     2     3984     4030    1
#> rank[r-PA]      4.36 1.15    2   4   4   5     7     4654     3204    1
#> rank[SK + t-PA] 6.00 1.26    4   5   6   6     9     5114       NA    1
#> rank[t-PA]      7.46 1.10    5   7   8   8     9     4771       NA    1
#> rank[TNK]       3.51 1.30    2   2   3   4     6     5383     3245    1
#> rank[UK]        3.95 2.72    1   2   3   6     9     5546       NA    1
plot(thrombo_ranks)

(thrombo_rankprobs <- posterior_rank_probs(thrombo_fit))
#>              p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6] p_rank[7] p_rank[8]
#> d[SK]             0.00      0.00      0.00      0.00      0.02      0.14      0.38      0.31
#> d[Acc t-PA]       0.00      0.20      0.47      0.29      0.04      0.00      0.00      0.00
#> d[ASPAC]          0.00      0.00      0.00      0.01      0.02      0.10      0.18      0.26
#> d[PTCA]           0.86      0.13      0.00      0.00      0.00      0.00      0.00      0.00
#> d[r-PA]           0.00      0.06      0.15      0.32      0.37      0.07      0.01      0.01
#> d[SK + t-PA]      0.00      0.00      0.01      0.06      0.25      0.45      0.10      0.06
#> d[t-PA]           0.00      0.00      0.00      0.00      0.04      0.16      0.29      0.32
#> d[TNK]            0.00      0.25      0.30      0.23      0.16      0.03      0.01      0.00
#> d[UK]             0.13      0.36      0.07      0.09      0.10      0.06      0.02      0.02
#>              p_rank[9]
#> d[SK]             0.15
#> d[Acc t-PA]       0.00
#> d[ASPAC]          0.43
#> d[PTCA]           0.00
#> d[r-PA]           0.00
#> d[SK + t-PA]      0.06
#> d[t-PA]           0.19
#> d[TNK]            0.01
#> d[UK]             0.15
plot(thrombo_rankprobs)

(thrombo_cumrankprobs <- posterior_rank_probs(thrombo_fit, cumulative = TRUE))
#>              p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6] p_rank[7] p_rank[8]
#> d[SK]             0.00      0.00      0.00      0.00      0.02      0.16      0.54      0.85
#> d[Acc t-PA]       0.00      0.20      0.67      0.96      1.00      1.00      1.00      1.00
#> d[ASPAC]          0.00      0.00      0.00      0.01      0.03      0.12      0.31      0.57
#> d[PTCA]           0.86      1.00      1.00      1.00      1.00      1.00      1.00      1.00
#> d[r-PA]           0.00      0.06      0.20      0.52      0.90      0.97      0.98      1.00
#> d[SK + t-PA]      0.00      0.00      0.01      0.08      0.32      0.77      0.87      0.94
#> d[t-PA]           0.00      0.00      0.00      0.00      0.04      0.20      0.49      0.81
#> d[TNK]            0.00      0.25      0.55      0.78      0.95      0.98      0.99      0.99
#> d[UK]             0.13      0.49      0.56      0.65      0.74      0.80      0.82      0.85
#>              p_rank[9]
#> d[SK]                1
#> d[Acc t-PA]          1
#> d[ASPAC]             1
#> d[PTCA]              1
#> d[r-PA]              1
#> d[SK + t-PA]         1
#> d[t-PA]              1
#> d[TNK]               1
#> d[UK]                1
plot(thrombo_cumrankprobs)

## References

Boland, A., Y. Dundar, A. Bagust, A. Haycox, R. Hill, R. Mujica Mota, T. Walley, and R. Dickson. 2003. “Early Thrombolysis for the Treatment of Acute Myocardial Infarction: A Systematic Review and Economic Evaluation.” Health Technology Assessment 7 (15). https://doi.org/10.3310/hta7150.
Dias, S., N. J. Welton, D. M. Caldwell, and A. E. Ades. 2010. “Checking Consistency in Mixed Treatment Comparison Meta-Analysis.” Statistics in Medicine 29 (7-8): 932–44. https://doi.org/10.1002/sim.3767.
Dias, S., N. J. Welton, A. J. Sutton, D. M. Caldwell, G. Lu, and A. E. Ades. 2011. NICE DSU Technical Support Document 4: Inconsistency in Networks of Evidence Based on Randomised Controlled Trials.” National Institute for Health and Care Excellence. https://nicedsu.sites.sheffield.ac.uk.
Lu, G. B., and A. E. Ades. 2006. “Assessing Evidence Inconsistency in Mixed Treatment Comparisons.” Journal of the American Statistical Association 101 (474): 447–59. https://doi.org/10.1198/016214505000001302.