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) (Boland et al. 2003; Lu and Ades 2006; Dias et al. 2011, 2010). The number of deaths in 30 or 35 days following acute myocardial infarction are recorded. The data are available in this package as thrombolytics:

head(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 (Dias et al. 2011), 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     -0.10  2797    1
#> d[ASPAC]          0.02    0.00 0.04     -0.05     -0.01      0.02      0.04      0.09  4743    1
#> d[PTCA]          -0.48    0.00 0.10     -0.67     -0.55     -0.47     -0.41     -0.28  4449    1
#> d[r-PA]          -0.12    0.00 0.06     -0.24     -0.16     -0.12     -0.08     -0.01  4056    1
#> d[SK + t-PA]     -0.05    0.00 0.05     -0.14     -0.08     -0.05     -0.02      0.04  5889    1
#> d[t-PA]           0.00    0.00 0.03     -0.05     -0.02      0.00      0.02      0.06  4672    1
#> d[TNK]           -0.17    0.00 0.08     -0.33     -0.22     -0.17     -0.12     -0.02  3843    1
#> d[UK]            -0.20    0.00 0.22     -0.64     -0.35     -0.21     -0.05      0.23  5138    1
#> lp__         -43043.04    0.14 5.44 -43054.58 -43046.51 -43042.70 -43039.21 -43033.41  1490    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue May 23 11:29:30 2023.
#> 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: 106.2 (on 102 data points)
#>                pD: 59.1
#>               DIC: 165.3

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 (Dias et al. 2011) 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     -0.12     -0.06
#> d[ASPAC vs. SK]            0.00    0.00 0.04     -0.07     -0.02      0.00      0.03      0.08
#> d[PTCA vs. SK]            -0.66    0.00 0.19     -1.04     -0.79     -0.66     -0.54     -0.29
#> 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     -0.13     -0.07     -0.04     -0.01      0.05
#> d[t-PA vs. SK]             0.00    0.00 0.03     -0.06     -0.03     -0.01      0.02      0.05
#> d[UK vs. SK]              -0.37    0.01 0.51     -1.35     -0.70     -0.37     -0.03      0.66
#> d[ASPAC vs. Acc t-PA]      1.40    0.01 0.41      0.63      1.11      1.38      1.66      2.24
#> d[PTCA vs. Acc t-PA]      -0.22    0.00 0.12     -0.46     -0.30     -0.22     -0.14      0.02
#> d[r-PA vs. Acc t-PA]       0.02    0.00 0.06     -0.11     -0.02      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     -0.54     -0.10      0.13      0.38      0.84
#> d[t-PA vs. ASPAC]          0.30    0.00 0.36     -0.40      0.05      0.30      0.54      1.04
#> d[t-PA vs. PTCA]           0.54    0.01 0.42     -0.26      0.26      0.54      0.80      1.39
#> d[UK vs. t-PA]            -0.30    0.00 0.34     -0.98     -0.52     -0.30     -0.07      0.39
#> lp__                  -43039.66    0.16 5.75 -43052.14 -43043.36 -43039.36 -43035.64 -43029.28
#>                       n_eff Rhat
#> d[Acc t-PA vs. SK]     5645    1
#> d[ASPAC vs. SK]        4345    1
#> d[PTCA vs. SK]         4442    1
#> d[r-PA vs. SK]         5581    1
#> d[SK + t-PA vs. SK]    5521    1
#> d[t-PA vs. SK]         4137    1
#> d[UK vs. SK]           5257    1
#> d[ASPAC vs. Acc t-PA]  3443    1
#> d[PTCA vs. Acc t-PA]   4459    1
#> d[r-PA vs. Acc t-PA]   5587    1
#> d[TNK vs. Acc t-PA]    5692    1
#> d[UK vs. Acc t-PA]     4370    1
#> d[t-PA vs. ASPAC]      5376    1
#> d[t-PA vs. PTCA]       4331    1
#> d[UK vs. t-PA]         5231    1
#> lp__                   1363    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue May 23 11:29:51 2023.
#> 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: 106.2 (on 102 data points)
#>                pD: 59.1
#>               DIC: 165.3
(dic_ume <- dic(thrombo_fit_ume))
#> Residual deviance: 99.5 (on 102 data points)
#>                pD: 65.7
#>               DIC: 165.2

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 (Dias et al. 2011, 2010). 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     2521     3141    1
#> d_dir -0.16 0.05 -0.25 -0.19 -0.16 -0.12 -0.06     4551     3852    1
#> d_ind -0.25 0.09 -0.42 -0.31 -0.25 -0.19 -0.07      630     1212    1
#> omega  0.09 0.10 -0.11  0.02  0.09  0.16  0.29      772     1396    1
#> 
#> Residual deviance: 106.3 (on 102 data points)
#>                pD: 59.8
#>               DIC: 166.1
#> 
#> Bayesian p-value: 0.38
#> 
#> ------------------------------------------------------- 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 -0.06 -0.01  0.02  0.04  0.09     5295     3456    1
#> d_dir  0.01 0.04 -0.07 -0.02  0.01  0.03  0.08     4535     3493    1
#> d_ind  0.42 0.25 -0.06  0.25  0.41  0.58  0.92     2746     2609    1
#> omega -0.41 0.25 -0.92 -0.58 -0.41 -0.24  0.08     2793     2726    1
#> 
#> Residual deviance: 104.8 (on 102 data points)
#>                pD: 60.2
#>               DIC: 165.1
#> 
#> 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 -0.67 -0.54 -0.47 -0.40 -0.27     3926     3359    1
#> d_dir -0.66 0.18 -1.03 -0.79 -0.66 -0.54 -0.31     4503     3800    1
#> d_ind -0.39 0.12 -0.62 -0.47 -0.39 -0.31 -0.16     3489     3105    1
#> omega -0.27 0.22 -0.70 -0.42 -0.27 -0.12  0.18     3825     3197    1
#> 
#> Residual deviance: 105.6 (on 102 data points)
#>                pD: 59.9
#>               DIC: 165.5
#> 
#> Bayesian p-value: 0.22
#> 
#> -------------------------------------------------------- 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  0.00     3802     3353    1
#> d_dir -0.06 0.09 -0.24 -0.12 -0.06  0.00  0.11     5975     3549    1
#> d_ind -0.18 0.08 -0.33 -0.23 -0.18 -0.12 -0.02     2087     2702    1
#> omega  0.11 0.12 -0.12  0.03  0.11  0.20  0.35     2868     2988    1
#> 
#> Residual deviance: 106.3 (on 102 data points)
#>                pD: 60
#>               DIC: 166.3
#> 
#> 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     4941     3656    1
#> d_dir  0.00 0.03 -0.06 -0.02  0.00  0.02  0.06     3754     3571    1
#> d_ind  0.17 0.23 -0.26  0.02  0.18  0.34  0.60     1252     2003    1
#> omega -0.17 0.23 -0.60 -0.34 -0.18 -0.01  0.27     1273     2201    1
#> 
#> Residual deviance: 105.9 (on 102 data points)
#>                pD: 59.4
#>               DIC: 165.3
#> 
#> Bayesian p-value: 0.46
#> 
#> ---------------------------------------------------------- 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 -0.63 -0.34 -0.20 -0.05  0.23     4501     3364    1
#> d_dir -0.36 0.53 -1.43 -0.71 -0.37  0.00  0.65     5612     2999    1
#> d_ind -0.17 0.25 -0.65 -0.34 -0.17 -0.01  0.33     3854     3133    1
#> omega -0.19 0.59 -1.39 -0.59 -0.18  0.21  0.93     5006     2822    1
#> 
#> Residual deviance: 107.1 (on 102 data points)
#>                pD: 60
#>               DIC: 167.2
#> 
#> 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 0.08 0.15 0.19 0.23  0.30     3229     3158    1
#> d_dir 1.40 0.41 0.63 1.11 1.39 1.66  2.25     3683     2856    1
#> d_ind 0.16 0.06 0.05 0.12 0.16 0.20  0.28     2851     3010    1
#> omega 1.23 0.41 0.48 0.95 1.22 1.50  2.09     3525     2658    1
#> 
#> Residual deviance: 96.8 (on 102 data points)
#>                pD: 59.7
#>               DIC: 156.5
#> 
#> 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 -0.37 -0.30 -0.23 -0.10     5422     3485    1
#> d_dir -0.22 0.12 -0.44 -0.30 -0.22 -0.14  0.02     4788     3685    1
#> d_ind -0.47 0.17 -0.82 -0.59 -0.48 -0.36 -0.14     3208     3305    1
#> omega  0.26 0.21 -0.14  0.12  0.26  0.40  0.66     3408     3077    1
#> 
#> Residual deviance: 105.1 (on 102 data points)
#>                pD: 59.4
#>               DIC: 164.5
#> 
#> Bayesian p-value: 0.22
#> 
#> -------------------------------------------------- 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     5889     3168    1
#> d_dir  0.02 0.07 -0.11 -0.03  0.02  0.06  0.15     5343     3910    1
#> d_ind  0.13 0.10 -0.07  0.06  0.13  0.20  0.34     1867     2746    1
#> omega -0.11 0.12 -0.35 -0.20 -0.12 -0.03  0.12     1943     2521    1
#> 
#> Residual deviance: 106.1 (on 102 data points)
#>                pD: 59.9
#>               DIC: 166
#> 
#> Bayesian p-value: 0.36
#> 
#> --------------------------------------------- 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  0.17  0.23     5277     3247    1
#> d_dir  0.13 0.05  0.02  0.09  0.13  0.16  0.23     3551     3485    1
#> d_ind  0.63 0.69 -0.71  0.16  0.61  1.07  2.03     2928     2351    1
#> omega -0.50 0.69 -1.91 -0.95 -0.49 -0.04  0.84     2939     2464    1
#> 
#> Residual deviance: 106.6 (on 102 data points)
#>                pD: 59.9
#>               DIC: 166.5
#> 
#> Bayesian p-value: 0.47
#> 
#> ---------------------------------------------------- Node-split UK vs. Acc t-PA ---- 
#> 
#>        mean   sd  2.5%   25%   50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.02 0.22 -0.47 -0.17 -0.02 0.13  0.41     4277     3064    1
#> d_dir  0.13 0.35 -0.57 -0.11  0.13 0.37  0.82     4832     3421    1
#> d_ind -0.14 0.28 -0.70 -0.32 -0.13 0.06  0.42     4221     3376    1
#> omega  0.27 0.46 -0.61 -0.05  0.26 0.57  1.19     3940     2953    1
#> 
#> Residual deviance: 106.5 (on 102 data points)
#>                pD: 59.7
#>               DIC: 166.2
#> 
#> Bayesian p-value: 0.56
#> 
#> ----------------------------------------------------- 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     7086     3248    1
#> d_dir -0.02 0.04 -0.10 -0.05 -0.02  0.00  0.05     4822     3243    1
#> d_ind  0.02 0.06 -0.10 -0.02  0.03  0.07  0.15     2950     3120    1
#> omega -0.05 0.06 -0.17 -0.09 -0.05 -0.01  0.07     3041     3089    1
#> 
#> Residual deviance: 106.5 (on 102 data points)
#>                pD: 60
#>               DIC: 166.5
#> 
#> 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 0.11  0.27  0.40 0.48 0.55  0.69     4207     3416    1
#> d_dir 0.55 0.43 -0.27  0.26 0.55 0.82  1.40     4256     3356    1
#> d_ind 0.47 0.11  0.26  0.40 0.47 0.55  0.68     3440     3047    1
#> omega 0.08 0.44 -0.77 -0.22 0.07 0.37  0.96     3842     3243    1
#> 
#> Residual deviance: 107.1 (on 102 data points)
#>                pD: 59.9
#>               DIC: 167.1
#> 
#> Bayesian p-value: 0.86
#> 
#> -------------------------------------------------------- Node-split UK vs. t-PA ---- 
#> 
#>        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  0.23     4599     3160    1
#> d_dir -0.30 0.34 -0.97 -0.52 -0.30 -0.06  0.36     4567     3797    1
#> d_ind -0.14 0.29 -0.72 -0.34 -0.14  0.05  0.42     3970     3038    1
#> omega -0.16 0.44 -1.07 -0.44 -0.15  0.14  0.68     4252     3498    1
#> 
#> Residual deviance: 107.1 (on 102 data points)
#>                pD: 60
#>               DIC: 167.1
#> 
#> 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 -0.10     2787     2902    1
#> d[ASPAC vs. SK]            0.02 0.04 -0.05 -0.01  0.02  0.04  0.09     4882     3323    1
#> d[PTCA vs. SK]            -0.48 0.10 -0.67 -0.55 -0.47 -0.41 -0.28     4458     3124    1
#> d[r-PA vs. SK]            -0.12 0.06 -0.24 -0.16 -0.12 -0.08 -0.01     4104     3362    1
#> d[SK + t-PA vs. SK]       -0.05 0.05 -0.14 -0.08 -0.05 -0.02  0.04     5956     2726    1
#> d[t-PA vs. SK]             0.00 0.03 -0.05 -0.02  0.00  0.02  0.06     4715     3221    1
#> d[TNK vs. SK]             -0.17 0.08 -0.33 -0.22 -0.17 -0.12 -0.02     3912     2912    1
#> d[UK vs. SK]              -0.20 0.22 -0.64 -0.35 -0.21 -0.05  0.23     5190     3606    1
#> d[ASPAC vs. Acc t-PA]      0.19 0.06  0.08  0.16  0.19  0.23  0.30     3265     3187    1
#> d[PTCA vs. Acc t-PA]      -0.30 0.10 -0.48 -0.36 -0.30 -0.23 -0.11     5685     3538    1
#> d[r-PA vs. Acc t-PA]       0.05 0.06 -0.06  0.02  0.05  0.09  0.16     5474     3708    1
#> d[SK + t-PA vs. Acc t-PA]  0.13 0.05  0.02  0.09  0.13  0.17  0.23     5195     3166    1
#> d[t-PA vs. Acc t-PA]       0.18 0.05  0.08  0.15  0.18  0.22  0.28     3277     3466    1
#> d[TNK vs. Acc t-PA]        0.01 0.06 -0.12 -0.04  0.01  0.05  0.13     5682     3836    1
#> d[UK vs. Acc t-PA]        -0.03 0.22 -0.46 -0.18 -0.03  0.13  0.41     5249     3612    1
#> d[PTCA vs. ASPAC]         -0.49 0.11 -0.69 -0.57 -0.49 -0.42 -0.29     4510     3429    1
#> d[r-PA vs. ASPAC]         -0.14 0.07 -0.28 -0.19 -0.14 -0.09  0.00     4158     3396    1
#> d[SK + t-PA vs. ASPAC]    -0.06 0.06 -0.18 -0.10 -0.06 -0.02  0.05     5529     3127    1
#> d[t-PA vs. ASPAC]         -0.01 0.04 -0.08 -0.04 -0.01  0.01  0.06     6265     3193    1
#> d[TNK vs. ASPAC]          -0.19 0.09 -0.35 -0.25 -0.19 -0.13 -0.02     4030     3025    1
#> d[UK vs. ASPAC]           -0.22 0.23 -0.66 -0.37 -0.22 -0.07  0.23     5290     3385    1
#> d[r-PA vs. PTCA]           0.35 0.11  0.14  0.28  0.35  0.42  0.56     5839     3792    1
#> d[SK + t-PA vs. PTCA]      0.43 0.11  0.22  0.36  0.43  0.50  0.63     5556     3400    1
#> d[t-PA vs. PTCA]           0.48 0.10  0.28  0.41  0.48  0.55  0.68     4474     3187    1
#> d[TNK vs. PTCA]            0.30 0.12  0.08  0.22  0.31  0.38  0.53     6395     3267    1
#> d[UK vs. PTCA]             0.27 0.24 -0.20  0.11  0.26  0.44  0.75     5336     3466    1
#> d[SK + t-PA vs. r-PA]      0.08 0.07 -0.06  0.03  0.07  0.12  0.21     5975     3297    1
#> d[t-PA vs. r-PA]           0.13 0.07  0.00  0.08  0.13  0.17  0.26     4018     3209    1
#> d[TNK vs. r-PA]           -0.05 0.09 -0.22 -0.11 -0.05  0.01  0.12     6506     2948    1
#> d[UK vs. r-PA]            -0.08 0.23 -0.52 -0.24 -0.08  0.08  0.38     5161     3397    1
#> d[t-PA vs. SK + t-PA]      0.05 0.06 -0.06  0.01  0.05  0.09  0.16     5763     3209    1
#> d[TNK vs. SK + t-PA]      -0.12 0.08 -0.29 -0.18 -0.12 -0.07  0.04     5837     2839    1
#> d[UK vs. SK + t-PA]       -0.15 0.23 -0.58 -0.31 -0.16  0.00  0.29     5225     3477    1
#> d[TNK vs. t-PA]           -0.17 0.08 -0.34 -0.23 -0.17 -0.12 -0.01     4156     3355    1
#> d[UK vs. t-PA]            -0.21 0.22 -0.63 -0.35 -0.21 -0.05  0.24     5186     3373    1
#> d[UK vs. TNK]             -0.03 0.23 -0.48 -0.19 -0.03  0.13  0.42     5506     3248    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 0.95    6   7   7   8     9     3578       NA    1
#> rank[Acc t-PA]  3.20 0.81    2   3   3   4     5     4327     3751    1
#> rank[ASPAC]     7.98 1.14    5   7   8   9     9     4684       NA    1
#> rank[PTCA]      1.13 0.35    1   1   1   1     2     3764     3831    1
#> rank[r-PA]      4.40 1.16    2   4   4   5     7     4505     3328    1
#> rank[SK + t-PA] 5.98 1.22    4   5   6   6     9     5388       NA    1
#> rank[t-PA]      7.50 1.10    5   7   8   8     9     4778       NA    1
#> rank[TNK]       3.49 1.27    2   3   3   4     6     5358     3134    1
#> rank[UK]        3.88 2.69    1   2   2   5     9     5234       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      0.13      0.40      0.31
#> d[Acc t-PA]       0.00      0.19      0.46      0.30      0.05      0.00      0.00      0.00
#> d[ASPAC]          0.00      0.00      0.00      0.00      0.03      0.10      0.17      0.26
#> d[PTCA]           0.87      0.12      0.00      0.00      0.00      0.00      0.00      0.00
#> d[r-PA]           0.00      0.06      0.15      0.30      0.38      0.08      0.02      0.01
#> d[SK + t-PA]      0.00      0.00      0.01      0.06      0.25      0.46      0.10      0.06
#> d[t-PA]           0.00      0.00      0.00      0.01      0.04      0.14      0.29      0.33
#> d[TNK]            0.00      0.24      0.32      0.24      0.15      0.03      0.01      0.01
#> d[UK]             0.12      0.39      0.07      0.08      0.09      0.06      0.02      0.02
#>              p_rank[9]
#> d[SK]             0.14
#> d[Acc t-PA]       0.00
#> d[ASPAC]          0.44
#> d[PTCA]           0.00
#> d[r-PA]           0.01
#> d[SK + t-PA]      0.06
#> d[t-PA]           0.20
#> 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      0.15      0.54      0.86
#> d[Acc t-PA]       0.00      0.19      0.65      0.95      1.00      1.00      1.00      1.00
#> d[ASPAC]          0.00      0.00      0.00      0.00      0.03      0.13      0.30      0.56
#> d[PTCA]           0.87      1.00      1.00      1.00      1.00      1.00      1.00      1.00
#> d[r-PA]           0.00      0.06      0.20      0.50      0.89      0.97      0.99      0.99
#> d[SK + t-PA]      0.00      0.00      0.01      0.08      0.32      0.79      0.88      0.94
#> d[t-PA]           0.00      0.00      0.00      0.01      0.04      0.18      0.47      0.80
#> d[TNK]            0.00      0.24      0.56      0.80      0.95      0.98      0.99      0.99
#> d[UK]             0.12      0.51      0.57      0.66      0.75      0.81      0.83      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://www.sheffield.ac.uk/nice-dsu.
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.