Example: Thrombolytic treatments
Source:vignettes/example_thrombolytics.Rmd
example_thrombolytics.Rmd
library(multinma)
options(mc.cores = parallel::detectCores())#> For execution on a local, multicore CPU with excess RAM we recommend calling
#> options(mc.cores = parallel::detectCores())
#>
#> Attaching package: 'multinma'
#> The following objects are masked from 'package:stats':
#>
#> dgamma, pgamma, qgammaThis 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 63Setting 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 connectedPlot 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
#> d[Acc t-PA] -0.18 0.00 0.04 -0.26 -0.21 -0.18 -0.15 -0.09 2760
#> d[ASPAC] 0.02 0.00 0.04 -0.06 -0.01 0.02 0.04 0.09 5923
#> d[PTCA] -0.48 0.00 0.10 -0.67 -0.54 -0.48 -0.41 -0.28 4065
#> d[r-PA] -0.12 0.00 0.06 -0.24 -0.16 -0.12 -0.08 0.00 3931
#> d[SK + t-PA] -0.05 0.00 0.05 -0.14 -0.08 -0.05 -0.02 0.04 5710
#> d[t-PA] 0.00 0.00 0.03 -0.06 -0.02 0.00 0.02 0.06 5374
#> d[TNK] -0.17 0.00 0.08 -0.32 -0.22 -0.17 -0.12 -0.02 4421
#> d[UK] -0.20 0.00 0.22 -0.64 -0.35 -0.20 -0.06 0.23 5063
#> lp__ -43042.80 0.14 5.48 -43054.54 -43046.38 -43042.43 -43039.00 -43033.13 1514
#> Rhat
#> 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
#> lp__ 1
#>
#> Samples were drawn using NUTS(diag_e) at Wed Mar 6 13:34:20 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).By default, summaries of the study-specific intercepts \(\mu_j\) are hidden, but could be examined
by changing the pars argument:
The prior and posterior distributions can be compared visually using
the plot_prior_posterior() function:
plot_prior_posterior(thrombo_fit, prior = "trt")
Model fit can be checked using the dic() function
(dic_consistency <- dic(thrombo_fit))
#> Residual deviance: 105.8 (on 102 data points)
#> pD: 58.6
#> DIC: 164.4and 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
trtnas 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] 0.01 0.00 0.04 -0.07 -0.02 0.01 0.03 0.08
#> d[PTCA vs. SK] -0.67 0.00 0.18 -1.02 -0.79 -0.67 -0.54 -0.32
#> 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 -0.08 -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 -1.41 -0.73 -0.36 -0.01 0.62
#> d[ASPAC vs. Acc t-PA] 1.40 0.01 0.41 0.64 1.12 1.39 1.67 2.27
#> d[PTCA vs. Acc t-PA] -0.22 0.00 0.12 -0.45 -0.30 -0.22 -0.13 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] 0.00 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 -0.54 -0.09 0.13 0.38 0.85
#> d[t-PA vs. ASPAC] 0.30 0.01 0.36 -0.40 0.05 0.30 0.53 1.02
#> d[t-PA vs. PTCA] 0.55 0.01 0.41 -0.24 0.27 0.54 0.81 1.37
#> d[UK vs. t-PA] -0.30 0.00 0.35 -1.00 -0.53 -0.29 -0.06 0.38
#> lp__ -43039.57 0.15 5.79 -43051.95 -43043.42 -43039.20 -43035.40 -43029.49
#> n_eff Rhat
#> d[Acc t-PA vs. SK] 5993 1
#> d[ASPAC vs. SK] 4623 1
#> d[PTCA vs. SK] 5222 1
#> d[r-PA vs. SK] 4753 1
#> d[SK + t-PA vs. SK] 5823 1
#> d[t-PA vs. SK] 4145 1
#> d[UK vs. SK] 5509 1
#> d[ASPAC vs. Acc t-PA] 3506 1
#> d[PTCA vs. Acc t-PA] 4085 1
#> d[r-PA vs. Acc t-PA] 4229 1
#> d[TNK vs. Acc t-PA] 5760 1
#> d[UK vs. Acc t-PA] 4009 1
#> d[t-PA vs. ASPAC] 4722 1
#> d[t-PA vs. PTCA] 3925 1
#> d[UK vs. t-PA] 5219 1
#> lp__ 1530 1
#>
#> Samples were drawn using NUTS(diag_e) at Wed Mar 6 13:34:31 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).Comparing the model fit statistics
dic_consistency
#> Residual deviance: 105.8 (on 102 data points)
#> pD: 58.6
#> DIC: 164.4
(dic_ume <- dic(thrombo_fit_ume))
#> Residual deviance: 99.3 (on 102 data points)
#> pD: 65.6
#> DIC: 164.9Whilst 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 -0.20 -0.18 -0.15 -0.09 2540 3148 1.00
#> d_dir -0.16 0.05 -0.26 -0.19 -0.16 -0.12 -0.06 4404 3826 1.00
#> d_ind -0.24 0.09 -0.42 -0.31 -0.24 -0.18 -0.07 610 1376 1.01
#> omega 0.09 0.10 -0.12 0.02 0.09 0.15 0.29 722 1816 1.01
#>
#> Residual deviance: 106 (on 102 data points)
#> pD: 59.5
#> DIC: 165.5
#>
#> Bayesian p-value: 0.39
#>
#> ------------------------------------------------------- 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 5967 3467 1
#> d_dir 0.01 0.04 -0.07 -0.02 0.01 0.03 0.08 4326 3410 1
#> d_ind 0.42 0.25 -0.06 0.25 0.41 0.58 0.91 2588 2561 1
#> omega -0.41 0.25 -0.91 -0.57 -0.41 -0.24 0.07 2672 2718 1
#>
#> Residual deviance: 104.2 (on 102 data points)
#> pD: 59.6
#> DIC: 163.8
#>
#> 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 -0.48 -0.40 -0.27 4536 3325 1
#> d_dir -0.66 0.19 -1.04 -0.79 -0.66 -0.54 -0.29 5841 3688 1
#> d_ind -0.40 0.12 -0.63 -0.48 -0.40 -0.32 -0.17 3887 3660 1
#> omega -0.27 0.22 -0.72 -0.42 -0.26 -0.12 0.17 4401 3512 1
#>
#> Residual deviance: 105.2 (on 102 data points)
#> pD: 59.5
#> DIC: 164.7
#>
#> 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 -0.13 -0.08 -0.01 3698 3326 1
#> d_dir -0.06 0.09 -0.25 -0.12 -0.06 0.00 0.11 5016 3541 1
#> d_ind -0.18 0.08 -0.33 -0.23 -0.18 -0.12 -0.02 2421 3385 1
#> omega 0.12 0.12 -0.12 0.04 0.12 0.20 0.35 2909 3292 1
#>
#> Residual deviance: 106.2 (on 102 data points)
#> pD: 59.9
#> DIC: 166.2
#>
#> 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 5010 3566 1
#> d_dir 0.00 0.03 -0.06 -0.02 0.00 0.02 0.06 3309 3300 1
#> d_ind 0.19 0.23 -0.25 0.04 0.19 0.35 0.64 1057 1796 1
#> omega -0.19 0.23 -0.65 -0.35 -0.19 -0.03 0.25 1088 2031 1
#>
#> Residual deviance: 105.9 (on 102 data points)
#> pD: 59.4
#> DIC: 165.3
#>
#> Bayesian p-value: 0.41
#>
#> ---------------------------------------------------------- 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 -0.35 -0.20 -0.05 0.24 4712 3165 1
#> d_dir -0.39 0.53 -1.44 -0.74 -0.37 -0.01 0.61 6044 3449 1
#> d_ind -0.17 0.24 -0.63 -0.34 -0.17 -0.01 0.30 3998 3484 1
#> omega -0.21 0.57 -1.40 -0.61 -0.20 0.17 0.88 5322 3506 1
#>
#> Residual deviance: 107.1 (on 102 data points)
#> pD: 60
#> DIC: 167
#>
#> Bayesian p-value: 0.72
#>
#> ------------------------------------------------- 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 3583 3514 1
#> d_dir 1.41 0.41 0.64 1.13 1.40 1.68 2.24 4007 3178 1
#> d_ind 0.16 0.06 0.05 0.13 0.17 0.20 0.28 3550 3595 1
#> omega 1.25 0.41 0.47 0.97 1.24 1.52 2.09 3866 3232 1
#>
#> Residual deviance: 97 (on 102 data points)
#> pD: 59.9
#> DIC: 157
#>
#> 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 -0.11 5432 3387 1
#> d_dir -0.22 0.12 -0.46 -0.30 -0.22 -0.14 0.02 4277 3403 1
#> d_ind -0.47 0.17 -0.81 -0.58 -0.47 -0.35 -0.16 3129 3109 1
#> omega 0.26 0.21 -0.14 0.11 0.25 0.39 0.66 3028 2866 1
#>
#> Residual deviance: 105.1 (on 102 data points)
#> pD: 59.5
#> DIC: 164.6
#>
#> 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 5710 3445 1
#> d_dir 0.02 0.07 -0.11 -0.03 0.02 0.06 0.15 4875 3717 1
#> d_ind 0.13 0.10 -0.06 0.06 0.13 0.20 0.33 1681 2266 1
#> omega -0.11 0.12 -0.34 -0.20 -0.12 -0.04 0.13 1622 2381 1
#>
#> Residual deviance: 106.2 (on 102 data points)
#> pD: 59.9
#> DIC: 166.2
#>
#> Bayesian p-value: 0.34
#>
#> --------------------------------------------- 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 5169 2976 1
#> d_dir 0.13 0.05 0.03 0.09 0.13 0.16 0.23 3402 3584 1
#> d_ind 0.62 0.71 -0.70 0.14 0.61 1.09 2.08 2582 2055 1
#> omega -0.50 0.71 -1.95 -0.96 -0.48 -0.02 0.85 2567 2038 1
#>
#> Residual deviance: 106.9 (on 102 data points)
#> pD: 60.2
#> DIC: 167.1
#>
#> 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 -0.45 -0.18 -0.02 0.13 0.41 4872 3174 1
#> d_dir 0.14 0.36 -0.53 -0.10 0.13 0.38 0.85 4488 3522 1
#> d_ind -0.14 0.29 -0.71 -0.33 -0.14 0.06 0.43 3626 3146 1
#> omega 0.28 0.46 -0.62 -0.02 0.28 0.59 1.17 3587 3075 1
#>
#> Residual deviance: 107.1 (on 102 data points)
#> pD: 60.2
#> DIC: 167.3
#>
#> Bayesian p-value: 0.53
#>
#> ----------------------------------------------------- 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 7274 3513 1
#> d_dir -0.02 0.04 -0.10 -0.05 -0.02 0.00 0.05 4912 3121 1
#> d_ind 0.02 0.06 -0.10 -0.02 0.03 0.07 0.15 3211 3061 1
#> omega -0.05 0.06 -0.17 -0.09 -0.05 -0.01 0.07 3143 2932 1
#>
#> Residual deviance: 106.3 (on 102 data points)
#> pD: 59.8
#> DIC: 166.2
#>
#> Bayesian p-value: 0.43
#>
#> ------------------------------------------------------ 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 0.68 4721 3218 1
#> d_dir 0.54 0.42 -0.24 0.25 0.53 0.82 1.41 4872 2748 1
#> d_ind 0.48 0.11 0.27 0.40 0.48 0.55 0.69 3740 3793 1
#> omega 0.07 0.43 -0.74 -0.23 0.06 0.35 0.96 4372 2788 1
#>
#> Residual deviance: 107 (on 102 data points)
#> pD: 59.8
#> DIC: 166.8
#>
#> 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 -0.20 0.22 -0.62 -0.35 -0.20 -0.06 0.23 4810 3391 1
#> d_dir -0.29 0.34 -0.98 -0.51 -0.28 -0.06 0.38 4912 3610 1
#> d_ind -0.15 0.29 -0.74 -0.34 -0.15 0.05 0.43 4028 3324 1
#> omega -0.14 0.45 -1.04 -0.44 -0.13 0.16 0.75 3904 3125 1
#>
#> Residual deviance: 106.4 (on 102 data points)
#> pD: 59.3
#> DIC: 165.8
#>
#> Bayesian p-value: 0.75Node-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 2760 2716 1
#> d[ASPAC vs. SK] 0.02 0.04 -0.06 -0.01 0.02 0.04 0.09 6008 3632 1
#> d[PTCA vs. SK] -0.48 0.10 -0.67 -0.54 -0.48 -0.41 -0.28 4044 3190 1
#> d[r-PA vs. SK] -0.12 0.06 -0.24 -0.16 -0.12 -0.08 0.00 3986 3208 1
#> d[SK + t-PA vs. SK] -0.05 0.05 -0.14 -0.08 -0.05 -0.02 0.04 5664 2864 1
#> d[t-PA vs. SK] 0.00 0.03 -0.06 -0.02 0.00 0.02 0.06 5268 3386 1
#> d[TNK vs. SK] -0.17 0.08 -0.32 -0.22 -0.17 -0.12 -0.02 4431 3368 1
#> d[UK vs. SK] -0.20 0.22 -0.64 -0.35 -0.20 -0.06 0.23 4991 3232 1
#> d[ASPAC vs. Acc t-PA] 0.19 0.06 0.08 0.15 0.19 0.23 0.30 3907 2972 1
#> d[PTCA vs. Acc t-PA] -0.30 0.10 -0.48 -0.36 -0.30 -0.23 -0.11 4934 3320 1
#> d[r-PA vs. Acc t-PA] 0.06 0.06 -0.06 0.02 0.06 0.09 0.17 5521 2935 1
#> d[SK + t-PA vs. Acc t-PA] 0.13 0.05 0.02 0.09 0.13 0.16 0.24 5728 2914 1
#> d[t-PA vs. Acc t-PA] 0.18 0.05 0.08 0.14 0.18 0.22 0.28 3380 3185 1
#> d[TNK vs. Acc t-PA] 0.01 0.06 -0.12 -0.04 0.01 0.05 0.13 6016 3561 1
#> d[UK vs. Acc t-PA] -0.02 0.22 -0.46 -0.17 -0.02 0.12 0.41 5220 3351 1
#> d[PTCA vs. ASPAC] -0.49 0.11 -0.69 -0.56 -0.49 -0.42 -0.28 4399 3005 1
#> d[r-PA vs. ASPAC] -0.14 0.07 -0.27 -0.19 -0.14 -0.09 0.00 4613 3161 1
#> d[SK + t-PA vs. ASPAC] -0.07 0.06 -0.18 -0.11 -0.07 -0.03 0.05 6138 3240 1
#> d[t-PA vs. ASPAC] -0.01 0.04 -0.09 -0.04 -0.01 0.01 0.06 6668 3250 1
#> d[TNK vs. ASPAC] -0.19 0.09 -0.35 -0.25 -0.19 -0.13 -0.02 5017 3006 1
#> d[UK vs. ASPAC] -0.22 0.22 -0.65 -0.37 -0.21 -0.07 0.21 5097 3432 1
#> d[r-PA vs. PTCA] 0.35 0.11 0.14 0.28 0.35 0.43 0.56 5023 3369 1
#> d[SK + t-PA vs. PTCA] 0.43 0.11 0.21 0.36 0.43 0.50 0.63 5404 2903 1
#> d[t-PA vs. PTCA] 0.48 0.10 0.27 0.41 0.48 0.55 0.68 4166 3207 1
#> d[TNK vs. PTCA] 0.30 0.12 0.06 0.23 0.31 0.38 0.53 5551 3048 1
#> d[UK vs. PTCA] 0.27 0.24 -0.18 0.11 0.27 0.43 0.73 5256 3114 1
#> d[SK + t-PA vs. r-PA] 0.07 0.07 -0.07 0.02 0.07 0.12 0.21 6692 3109 1
#> d[t-PA vs. r-PA] 0.12 0.07 -0.01 0.08 0.12 0.17 0.26 4169 3169 1
#> d[TNK vs. r-PA] -0.05 0.08 -0.21 -0.10 -0.05 0.01 0.12 7662 2726 1
#> d[UK vs. r-PA] -0.08 0.23 -0.53 -0.23 -0.08 0.07 0.37 5197 3605 1
#> d[t-PA vs. SK + t-PA] 0.05 0.05 -0.06 0.01 0.05 0.09 0.16 5596 3253 1
#> d[TNK vs. SK + t-PA] -0.12 0.08 -0.29 -0.18 -0.12 -0.07 0.04 6689 2927 1
#> d[UK vs. SK + t-PA] -0.15 0.22 -0.60 -0.30 -0.15 0.00 0.30 5257 3406 1
#> d[TNK vs. t-PA] -0.17 0.08 -0.34 -0.23 -0.17 -0.12 -0.01 4687 3053 1
#> d[UK vs. t-PA] -0.20 0.22 -0.64 -0.35 -0.20 -0.06 0.23 5081 3356 1
#> d[UK vs. TNK] -0.03 0.23 -0.48 -0.18 -0.03 0.12 0.42 5247 3393 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 4168 NA 1
#> rank[Acc t-PA] 3.18 0.82 2 3 3 4 5 4356 3834 1
#> rank[ASPAC] 7.96 1.14 5 7 8 9 9 4980 NA 1
#> rank[PTCA] 1.13 0.36 1 1 1 1 2 4068 3530 1
#> rank[r-PA] 4.45 1.18 2 4 5 5 7 4874 3259 1
#> rank[SK + t-PA] 5.98 1.25 4 5 6 6 9 5575 NA 1
#> rank[t-PA] 7.48 1.10 5 7 8 8 9 4537 NA 1
#> rank[TNK] 3.48 1.26 2 3 3 4 6 5191 3162 1
#> rank[UK] 3.89 2.69 1 2 2 5 9 4900 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 0.38 0.31
#> d[Acc t-PA] 0.00 0.21 0.46 0.28 0.05 0.00 0.00 0.00
#> d[ASPAC] 0.00 0.00 0.00 0.00 0.03 0.10 0.18 0.27
#> 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 0.13 0.31 0.38 0.09 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 0.14 0.31 0.32
#> d[TNK] 0.00 0.23 0.33 0.24 0.14 0.04 0.01 0.00
#> d[UK] 0.13 0.38 0.06 0.09 0.09 0.05 0.02 0.03
#> p_rank[9]
#> d[SK] 0.16
#> d[Acc t-PA] 0.00
#> d[ASPAC] 0.43
#> 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] 0.14
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 0.67 0.95 1.00 1.00 1.00 1.00
#> d[ASPAC] 0.00 0.00 0.00 0.00 0.03 0.13 0.31 0.57
#> 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 0.88 0.97 0.98 0.99
#> d[SK + t-PA] 0.00 0.00 0.02 0.08 0.33 0.78 0.88 0.94
#> d[t-PA] 0.00 0.00 0.00 0.00 0.04 0.18 0.49 0.80
#> d[TNK] 0.00 0.24 0.56 0.80 0.95 0.98 0.99 0.99
#> d[UK] 0.13 0.51 0.57 0.66 0.75 0.81 0.83 0.86
#> 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)