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 Treatments                  
#>  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 | t-PA | ASPAC        
#>  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     -0.09  2705    1
#> d[ASPAC]          0.02    0.00 0.04     -0.06     -0.01      0.02      0.04      0.09  6390    1
#> d[PTCA]          -0.47    0.00 0.10     -0.67     -0.54     -0.47     -0.40     -0.28  3837    1
#> d[r-PA]          -0.12    0.00 0.06     -0.24     -0.16     -0.12     -0.09     -0.01  3021    1
#> d[SK + t-PA]     -0.05    0.00 0.05     -0.14     -0.08     -0.05     -0.02      0.04  5781    1
#> d[t-PA]           0.00    0.00 0.03     -0.06     -0.02      0.00      0.02      0.06  4755    1
#> d[TNK]           -0.17    0.00 0.07     -0.32     -0.22     -0.17     -0.12     -0.03  4145    1
#> d[UK]            -0.20    0.00 0.23     -0.65     -0.36     -0.20     -0.05      0.24  4314    1
#> lp__         -43042.85    0.14 5.40 -43053.87 -43046.36 -43042.61 -43039.08 -43033.16  1519    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan 18 09:33:18 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 (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     -0.13     -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     -1.05     -0.80     -0.67     -0.54     -0.31
#> d[r-PA vs. SK]            -0.06    0.00 0.09     -0.23     -0.12     -0.06      0.00      0.12
#> d[SK + t-PA vs. SK]       -0.04    0.00 0.05     -0.13     -0.08     -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]              -0.37    0.01 0.52     -1.39     -0.70     -0.37     -0.03      0.64
#> d[ASPAC vs. Acc t-PA]      1.39    0.01 0.40      0.64      1.12      1.38      1.66      2.21
#> d[PTCA vs. Acc t-PA]      -0.21    0.00 0.12     -0.45     -0.29     -0.21     -0.13      0.02
#> 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 0.35     -0.53     -0.10      0.15      0.38      0.83
#> d[t-PA vs. ASPAC]          0.29    0.01 0.36     -0.41      0.05      0.28      0.52      0.99
#> d[t-PA vs. PTCA]           0.55    0.01 0.41     -0.22      0.26      0.54      0.81      1.43
#> d[UK vs. t-PA]            -0.30    0.00 0.34     -0.99     -0.53     -0.29     -0.07      0.38
#> lp__                  -43039.70    0.14 5.70 -43051.93 -43043.34 -43039.43 -43035.75 -43029.45
#>                       n_eff Rhat
#> d[Acc t-PA vs. SK]     5609    1
#> d[ASPAC vs. SK]        5146    1
#> d[PTCA vs. SK]         5906    1
#> d[r-PA vs. SK]         6166    1
#> d[SK + t-PA vs. SK]    6016    1
#> d[t-PA vs. SK]         4584    1
#> d[UK vs. SK]           6129    1
#> d[ASPAC vs. Acc t-PA]  3543    1
#> d[PTCA vs. Acc t-PA]   4540    1
#> d[r-PA vs. Acc t-PA]   5625    1
#> d[TNK vs. Acc t-PA]    4849    1
#> d[UK vs. Acc t-PA]     4471    1
#> d[t-PA vs. ASPAC]      4221    1
#> d[t-PA vs. PTCA]       3844    1
#> d[UK vs. t-PA]         5355    1
#> lp__                   1687    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan 18 09:33:38 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: 99.6 (on 102 data points)
#>                pD: 65.8
#>               DIC: 165.4

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     2764     3008    1
#> d_dir -0.16 0.05 -0.25 -0.19 -0.16 -0.12 -0.06     3551     3830    1
#> d_ind -0.24 0.09 -0.42 -0.31 -0.25 -0.18 -0.07      647     1181    1
#> omega  0.09 0.10 -0.13  0.01  0.09  0.16  0.29      754     1563    1
#> 
#> Residual deviance: 106.6 (on 102 data points)
#>                pD: 60.1
#>               DIC: 166.7
#> 
#> Bayesian p-value: 0.41
#> 
#> ------------------------------------------------------- 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     5040     3351    1
#> d_dir  0.01 0.04 -0.06 -0.02  0.01  0.03  0.08     4229     3246    1
#> d_ind  0.42 0.25 -0.06  0.25  0.42  0.59  0.92     2680     3210    1
#> omega -0.41 0.25 -0.91 -0.58 -0.41 -0.24  0.08     2679     3003    1
#> 
#> Residual deviance: 104.4 (on 102 data points)
#>                pD: 59.8
#>               DIC: 164.3
#> 
#> Bayesian p-value: 0.092
#> 
#> -------------------------------------------------------- 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 -0.27     4262     2754    1
#> d_dir -0.66 0.19 -1.03 -0.79 -0.67 -0.54 -0.29     5634     3172    1
#> d_ind -0.39 0.12 -0.63 -0.47 -0.40 -0.31 -0.16     3893     3299    1
#> omega -0.27 0.22 -0.70 -0.41 -0.27 -0.13  0.16     4721     3358    1
#> 
#> Residual deviance: 105.7 (on 102 data points)
#>                pD: 60
#>               DIC: 165.6
#> 
#> 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     3924     2629    1
#> d_dir -0.06 0.09 -0.23 -0.12 -0.06  0.00  0.11     4832     3126    1
#> d_ind -0.18 0.08 -0.33 -0.23 -0.18 -0.12 -0.02     2580     3046    1
#> omega  0.11 0.12 -0.12  0.04  0.12  0.19  0.35     3084     3089    1
#> 
#> Residual deviance: 106.4 (on 102 data points)
#>                pD: 60.1
#>               DIC: 166.5
#> 
#> 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     4747     3756    1
#> d_dir  0.00 0.03 -0.06 -0.02  0.00  0.02  0.06     4078     3569    1
#> d_ind  0.19 0.23 -0.25  0.03  0.19  0.35  0.66     1146     1884    1
#> omega -0.19 0.23 -0.67 -0.35 -0.19 -0.03  0.25     1156     1877    1
#> 
#> Residual deviance: 106.5 (on 102 data points)
#>                pD: 60
#>               DIC: 166.5
#> 
#> Bayesian p-value: 0.42
#> 
#> ---------------------------------------------------------- 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 -0.62 -0.35 -0.20 -0.05  0.22     4727     3192    1
#> d_dir -0.37 0.53 -1.44 -0.72 -0.37 -0.02  0.63     7298     3061    1
#> d_ind -0.16 0.25 -0.65 -0.33 -0.17  0.01  0.31     4335     2784    1
#> omega -0.21 0.59 -1.38 -0.59 -0.20  0.18  0.90     6233     2998    1
#> 
#> Residual deviance: 106.6 (on 102 data points)
#>                pD: 59.5
#>               DIC: 166.1
#> 
#> Bayesian p-value: 0.73
#> 
#> ------------------------------------------------- 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     3329     2914    1
#> d_dir 1.40 0.41 0.64 1.11 1.38 1.67  2.24     4006     2643    1
#> d_ind 0.16 0.06 0.05 0.12 0.16 0.20  0.28     3386     3224    1
#> omega 1.24 0.42 0.46 0.95 1.22 1.52  2.09     3934     2562    1
#> 
#> Residual deviance: 97.8 (on 102 data points)
#>                pD: 60.6
#>               DIC: 158.4
#> 
#> 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 -0.30 -0.23 -0.10     5358     2948    1
#> d_dir -0.21 0.12 -0.45 -0.29 -0.21 -0.13  0.02     4779     3460    1
#> d_ind -0.48 0.18 -0.83 -0.60 -0.48 -0.36 -0.13     3033     3243    1
#> omega  0.27 0.21 -0.14  0.12  0.26  0.41  0.68     3044     2918    1
#> 
#> Residual deviance: 105.2 (on 102 data points)
#>                pD: 59.5
#>               DIC: 164.7
#> 
#> Bayesian p-value: 0.21
#> 
#> -------------------------------------------------- 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 0.05 -0.05  0.02  0.05  0.09  0.16     6185     3557    1
#> d_dir  0.02 0.07 -0.11 -0.02  0.02  0.06  0.15     5014     3573    1
#> d_ind  0.13 0.10 -0.06  0.06  0.13  0.20  0.33     1788     2421    1
#> omega -0.11 0.12 -0.35 -0.19 -0.11 -0.03  0.13     1851     2454    1
#> 
#> Residual deviance: 105.5 (on 102 data points)
#>                pD: 59.2
#>               DIC: 164.7
#> 
#> 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  0.03  0.09  0.13  0.16  0.23     4890     3585    1
#> d_dir  0.13 0.05  0.02  0.09  0.13  0.16  0.23     3022     3122    1
#> d_ind  0.62 0.70 -0.69  0.16  0.60  1.04  2.07     2798     2546    1
#> omega -0.49 0.69 -1.96 -0.92 -0.48 -0.03  0.82     2799     2573    1
#> 
#> Residual deviance: 106.2 (on 102 data points)
#>                pD: 59.5
#>               DIC: 165.8
#> 
#> Bayesian p-value: 0.48
#> 
#> ---------------------------------------------------- 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 -0.44 -0.17 -0.02 0.13  0.39     4776     3153    1
#> d_dir  0.14 0.34 -0.51 -0.09  0.14 0.36  0.85     4920     2945    1
#> d_ind -0.13 0.29 -0.71 -0.32 -0.14 0.07  0.45     4437     3021    1
#> omega  0.27 0.45 -0.61 -0.03  0.27 0.58  1.15     4108     2908    1
#> 
#> Residual deviance: 106.6 (on 102 data points)
#>                pD: 59.8
#>               DIC: 166.4
#> 
#> 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 -0.08 -0.04 -0.01  0.01  0.06     5457     3156    1
#> d_dir -0.02 0.04 -0.10 -0.05 -0.02  0.00  0.05     4907     3444    1
#> d_ind  0.03 0.06 -0.09 -0.02  0.03  0.07  0.15     3416     3238    1
#> omega -0.05 0.06 -0.18 -0.09 -0.05 -0.01  0.07     3416     2876    1
#> 
#> Residual deviance: 106.5 (on 102 data points)
#>                pD: 59.9
#>               DIC: 166.4
#> 
#> Bayesian p-value: 0.41
#> 
#> ------------------------------------------------------ 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 0.48 0.54  0.68     4171     2831    1
#> d_dir 0.54 0.42 -0.26  0.25 0.54 0.82  1.37     4654     3244    1
#> d_ind 0.47 0.11  0.26  0.40 0.47 0.54  0.68     4139     3609    1
#> omega 0.07 0.43 -0.76 -0.23 0.06 0.36  0.91     4204     3193    1
#> 
#> Residual deviance: 107.1 (on 102 data points)
#>                pD: 59.9
#>               DIC: 167
#> 
#> Bayesian p-value: 0.88
#> 
#> -------------------------------------------------------- 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 -0.62 -0.35 -0.20 -0.05  0.22     4777     3033    1
#> d_dir -0.29 0.34 -0.98 -0.53 -0.29 -0.06  0.38     5388     3354    1
#> d_ind -0.14 0.30 -0.72 -0.34 -0.14  0.06  0.44     3836     3307    1
#> omega -0.15 0.46 -1.07 -0.46 -0.16  0.16  0.75     4318     3386    1
#> 
#> Residual deviance: 107 (on 102 data points)
#>                pD: 60
#>               DIC: 167
#> 
#> Bayesian p-value: 0.73

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     2766     3228    1
#> d[ASPAC vs. SK]            0.02 0.04 -0.06 -0.01  0.02  0.04  0.09     6415     3816    1
#> d[PTCA vs. SK]            -0.47 0.10 -0.67 -0.54 -0.47 -0.40 -0.28     3922     3176    1
#> d[r-PA vs. SK]            -0.12 0.06 -0.24 -0.16 -0.12 -0.09 -0.01     3088     3019    1
#> d[SK + t-PA vs. SK]       -0.05 0.05 -0.14 -0.08 -0.05 -0.02  0.04     5906     3396    1
#> d[t-PA vs. SK]             0.00 0.03 -0.06 -0.02  0.00  0.02  0.06     4886     3461    1
#> d[TNK vs. SK]             -0.17 0.07 -0.32 -0.22 -0.17 -0.12 -0.03     4223     3245    1
#> d[UK vs. SK]              -0.20 0.23 -0.65 -0.36 -0.20 -0.05  0.24     4319     3243    1
#> d[ASPAC vs. Acc t-PA]      0.19 0.06  0.08  0.16  0.19  0.23  0.31     4017     3650    1
#> d[PTCA vs. Acc t-PA]      -0.30 0.10 -0.48 -0.36 -0.30 -0.23 -0.10     5298     3778    1
#> d[r-PA vs. Acc t-PA]       0.05 0.05 -0.05  0.02  0.05  0.09  0.16     6180     3518    1
#> d[SK + t-PA vs. Acc t-PA]  0.13 0.05  0.03  0.09  0.13  0.16  0.23     5629     3601    1
#> d[t-PA vs. Acc t-PA]       0.18 0.05  0.08  0.14  0.18  0.21  0.28     3527     3235    1
#> d[TNK vs. Acc t-PA]        0.01 0.06 -0.12 -0.04  0.01  0.05  0.13     5848     3514    1
#> d[UK vs. Acc t-PA]        -0.03 0.23 -0.47 -0.18 -0.03  0.13  0.42     4640     2957    1
#> d[PTCA vs. ASPAC]         -0.49 0.11 -0.70 -0.56 -0.49 -0.42 -0.28     4454     3582    1
#> d[r-PA vs. ASPAC]         -0.14 0.07 -0.28 -0.19 -0.14 -0.09  0.00     4106     3171    1
#> d[SK + t-PA vs. ASPAC]    -0.06 0.06 -0.18 -0.10 -0.06 -0.03  0.05     6049     3717    1
#> d[t-PA vs. ASPAC]         -0.01 0.04 -0.09 -0.04 -0.01  0.01  0.06     7459     3403    1
#> d[TNK vs. ASPAC]          -0.19 0.08 -0.35 -0.25 -0.19 -0.13 -0.02     4691     3398    1
#> d[UK vs. ASPAC]           -0.22 0.23 -0.68 -0.37 -0.22 -0.07  0.23     4497     3482    1
#> d[r-PA vs. PTCA]           0.35 0.11  0.14  0.27  0.35  0.43  0.56     5664     3411    1
#> d[SK + t-PA vs. PTCA]      0.43 0.11  0.22  0.35  0.43  0.50  0.63     5090     3410    1
#> d[t-PA vs. PTCA]           0.48 0.10  0.27  0.40  0.48  0.55  0.68     4002     3636    1
#> d[TNK vs. PTCA]            0.30 0.11  0.07  0.23  0.30  0.39  0.52     6106     3378    1
#> d[UK vs. PTCA]             0.27 0.25 -0.23  0.11  0.27  0.44  0.74     4761     3479    1
#> d[SK + t-PA vs. r-PA]      0.08 0.07 -0.06  0.03  0.08  0.12  0.21     5474     3159    1
#> d[t-PA vs. r-PA]           0.13 0.07  0.00  0.08  0.13  0.17  0.26     3759     3382    1
#> d[TNK vs. r-PA]           -0.05 0.08 -0.21 -0.10 -0.05  0.01  0.11     7765     3097    1
#> d[UK vs. r-PA]            -0.08 0.23 -0.54 -0.23 -0.08  0.08  0.38     4555     3211    1
#> d[t-PA vs. SK + t-PA]      0.05 0.05 -0.05  0.02  0.05  0.09  0.16     5696     3325    1
#> d[TNK vs. SK + t-PA]      -0.12 0.08 -0.29 -0.18 -0.12 -0.07  0.04     6340     3296    1
#> d[UK vs. SK + t-PA]       -0.15 0.23 -0.61 -0.31 -0.16  0.00  0.30     4248     3289    1
#> d[TNK vs. t-PA]           -0.17 0.08 -0.34 -0.23 -0.17 -0.12 -0.02     4386     3483    1
#> d[UK vs. t-PA]            -0.20 0.23 -0.66 -0.36 -0.21 -0.05  0.23     4343     3412    1
#> d[UK vs. TNK]             -0.03 0.24 -0.50 -0.19 -0.04  0.13  0.43     4645     3120    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]        7.45 0.98    6   7   7   8     9     3862       NA    1
#> rank[Acc t-PA]  3.18 0.80    2   3   3   4     5     4536     3459    1
#> rank[ASPAC]     7.98 1.14    5   7   8   9     9     5203       NA    1
#> rank[PTCA]      1.14 0.35    1   1   1   1     2     3960     3863    1
#> rank[r-PA]      4.42 1.14    2   4   5   5     7     4815     2767    1
#> rank[SK + t-PA] 5.97 1.18    4   5   6   6     9     4465     3385    1
#> rank[t-PA]      7.48 1.08    5   7   8   8     9     4713       NA    1
#> rank[TNK]       3.47 1.23    2   3   3   4     6     5158     3479    1
#> rank[UK]        3.93 2.75    1   2   2   5     9     4215       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      0.37      0.31
#> d[Acc t-PA]       0.00      0.21      0.46      0.29      0.04      0.00      0.00      0.00
#> d[ASPAC]          0.00      0.00      0.00      0.00      0.03      0.09      0.18      0.26
#> d[PTCA]           0.87      0.13      0.00      0.00      0.00      0.00      0.00      0.00
#> d[r-PA]           0.00      0.05      0.14      0.31      0.39      0.08      0.01      0.01
#> d[SK + t-PA]      0.00      0.00      0.01      0.07      0.24      0.47      0.10      0.07
#> d[t-PA]           0.00      0.00      0.00      0.00      0.03      0.14      0.30      0.33
#> d[TNK]            0.00      0.24      0.32      0.24      0.15      0.03      0.01      0.00
#> d[UK]             0.13      0.38      0.07      0.08      0.10      0.05      0.02      0.02
#>              p_rank[9]
#> d[SK]             0.16
#> d[Acc t-PA]       0.00
#> d[ASPAC]          0.44
#> d[PTCA]           0.00
#> d[r-PA]           0.01
#> d[SK + t-PA]      0.05
#> d[t-PA]           0.19
#> d[TNK]            0.00
#> d[UK]             0.16
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      0.53      0.84
#> d[Acc t-PA]       0.00      0.21      0.66      0.96      1.00      1.00      1.00      1.00
#> d[ASPAC]          0.00      0.00      0.00      0.00      0.03      0.12      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      0.05      0.19      0.50      0.89      0.97      0.98      0.99
#> d[SK + t-PA]      0.00      0.00      0.01      0.08      0.32      0.79      0.89      0.95
#> d[t-PA]           0.00      0.00      0.00      0.00      0.04      0.18      0.49      0.81
#> d[TNK]            0.00      0.24      0.56      0.80      0.96      0.98      0.99      1.00
#> d[UK]             0.13      0.51      0.57      0.65      0.75      0.80      0.82      0.84
#>              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. http://nicedsu.org.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.