Example: Stroke prevention in atrial fibrillation patients

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

This vignette describes the analysis of 26 trials comparing 17 treatments in 4 classes for the prevention of stroke in patients with atrial fibrillation (Cooper et al. 2009). The data are available in this package as atrial_fibrillation:

head(atrial_fibrillation)
#>     studyc studyn                                  trtc trtn      trt_class   r    n    E stroke
#> 1 ACTIVE-W      1 Standard adjusted dose anti-coagulant    3 Anti-coagulant  65 3371 4200   0.15
#> 2 ACTIVE-W      1         Low dose aspirin + copidogrel   16  Anti-platelet 106 3335 4180   0.15
#> 3 AFASAK 1      2                 Placebo/Standard care    1        Control  19  336  398   0.06
#> 4 AFASAK 1      2 Standard adjusted dose anti-coagulant    3 Anti-coagulant   9  335  413   0.06
#> 5 AFASAK 1      2                      Low dose aspirin    5  Anti-platelet  16  336  409   0.06
#> 6 AFASAK 2      3 Standard adjusted dose anti-coagulant    3 Anti-coagulant  11  170  355   0.10
#>   year followup
#> 1 2006      1.3
#> 2 2006      1.3
#> 3 1989      1.2
#> 4 1989      1.2
#> 5 1989      1.2
#> 6 1998      2.2

Cooper et al. (2009) used this data to demonstrate meta-regression models, which we recreate here.

Setting up the network

Whilst we have data on the patient-years at risk in each study (E), we ignore this here to follow the analysis of Cooper et al. (2009), instead analysing the number of patients with stroke (r) out of the total (n) in each arm. We use the function set_agd_arm() to set up the network, making sure to specify the treatment classes trt_class. We remove the WASPO study from the network as both arms had zero events, and this study therefore contributes no information.

af_net <- set_agd_arm(atrial_fibrillation[atrial_fibrillation$studyc != "WASPO", ], 
                      study = studyc,
                      trt = trtc,
                      r = r, 
                      n = n,
                      trt_class = trt_class)
af_net
#> A network with 25 AgD studies (arm-based).
#> 
#> ------------------------------------------------------- AgD studies (arm-based) ---- 
#>  Study         Treatments                                                                         
#>  ACTIVE-W      2: Standard adjusted dose anti-coagulant | Low dose aspirin + copidogrel           
#>  AFASAK 1      3: Placebo/Standard care | Standard adjusted dose anti-coagulant | Low dose aspirin
#>  AFASAK 2      4: Standard adjusted dose anti-coagulant | Fixed dose warfarin | Medium dose asp...
#>  BAATAF        2: Placebo/Standard care | Low adjusted dose anti-coagulant                        
#>  BAFTA         2: Standard adjusted dose anti-coagulant | Low dose aspirin                        
#>  CAFA          2: Placebo/Standard care | Standard adjusted dose anti-coagulant                   
#>  Chinese ATAFS 2: Standard adjusted dose anti-coagulant | Low dose aspirin                        
#>  EAFT          3: Placebo/Standard care | Standard adjusted dose anti-coagulant | Medium dose a...
#>  ESPS 2        4: Placebo/Standard care | Low dose aspirin | Dipyridamole | Low dose aspirin + ...
#>  JAST          2: Placebo/Standard care | Low dose aspirin                                        
#>  ... plus 15 more studies
#> 
#>  Outcome type: count
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 17, in 4 classes
#> Total number of studies: 25
#> Reference treatment is: Standard adjusted dose anti-coagulant
#> Network is connected

(A better analysis, accounting for differences in the patient-years at risk between studies, can be performed by specifying a rate outcome with r and E in set_agd_arm() above. The following code remains identical.)

Plot the network with the plot() method:

plot(af_net, weight_nodes = TRUE, weight_edges = TRUE, show_trt_class = TRUE) + 
  ggplot2::theme(legend.position = "bottom", legend.box = "vertical")

Meta-analysis models

We fit two (random effects) models:

  1. A standard NMA model without any covariates (model 1 of Cooper et al. (2009));
  2. A meta-regression model adjusting for the proportion of individuals in each study with prior stroke, with shared interaction coefficients by treatment class (model 4b of Cooper et al. (2009)).

NMA with no covariates

We fit a random effects model using the nma() function with trt_effects = "random". We use \(\mathrm{N}(0, 100^2)\) prior distributions for the treatment effects \(d_k\) and study-specific intercepts \(\mu_j\), and a \(\textrm{half-N}(5^2)\) prior for the heterogeneity standard deviation \(\tau\). We can examine the range of parameter values implied by these prior distributions with the summary() method:

summary(normal(scale = 100))
#> A Normal prior distribution: location = 0, scale = 100.
#> 50% of the prior density lies between -67.45 and 67.45.
#> 95% of the prior density lies between -196 and 196.
summary(half_normal(scale = 5))
#> A half-Normal prior distribution: location = 0, scale = 5.
#> 50% of the prior density lies between 0 and 3.37.
#> 95% of the prior density lies between 0 and 9.8.

Fitting the model with the nma() function. We increase the target acceptance rate adapt_delta = 0.99 to minimise divergent transition warnings.

af_fit_1 <- nma(af_net, 
                trt_effects = "random",
                prior_intercept = normal(scale = 100),
                prior_trt = normal(scale = 100),
                prior_het = half_normal(scale = 5),
                adapt_delta = 0.99)
#> Note: Setting "Standard adjusted dose anti-coagulant" as the network reference treatment.

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

af_fit_1
#> A random effects NMA with a binomial likelihood (logit link).
#> Inference for Stan model: binomial_1par.
#> 4 chains, each with iter=2000; warmup=1000; thin=1; 
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#> 
#>                                                  mean se_mean   sd     2.5%      25%      50%
#> d[Acenocoumarol]                                -0.77    0.01 0.81    -2.45    -1.31    -0.75
#> d[Alternate day aspirin]                        -0.99    0.02 1.38    -4.18    -1.77    -0.83
#> d[Dipyridamole]                                  0.58    0.01 0.43    -0.32     0.30     0.59
#> d[Fixed dose warfarin]                           0.93    0.01 0.40     0.13     0.67     0.93
#> d[Fixed dose warfarin + low dose aspirin]        0.48    0.01 0.44    -0.41     0.20     0.48
#> d[Fixed dose warfarin + medium dose aspirin]     0.89    0.01 0.31     0.25     0.69     0.90
#> d[High dose aspirin]                             0.52    0.01 0.76    -1.04     0.02     0.54
#> d[Indobufen]                                     0.24    0.01 0.44    -0.63    -0.05     0.25
#> d[Low adjusted dose anti-coagulant]             -0.29    0.01 0.38    -1.05    -0.54    -0.29
#> d[Low dose aspirin]                              0.62    0.00 0.21     0.18     0.48     0.62
#> d[Low dose aspirin + copidogrel]                 0.52    0.01 0.34    -0.18     0.31     0.52
#> d[Low dose aspirin + dipyridamole]               0.27    0.01 0.46    -0.63    -0.04     0.27
#> d[Medium dose aspirin]                           0.39    0.00 0.20    -0.01     0.28     0.40
#> d[Placebo/Standard care]                         0.76    0.01 0.20     0.37     0.64     0.76
#> d[Triflusal]                                     0.66    0.01 0.61    -0.49     0.26     0.64
#> d[Ximelagatran]                                 -0.09    0.00 0.26    -0.61    -0.25    -0.09
#> lp__                                         -4771.79    0.21 6.98 -4786.49 -4776.32 -4771.46
#> tau                                              0.28    0.00 0.13     0.04     0.19     0.27
#>                                                   75%    97.5% n_eff Rhat
#> d[Acenocoumarol]                                -0.22     0.72  3306    1
#> d[Alternate day aspirin]                        -0.02     1.19  4115    1
#> d[Dipyridamole]                                  0.87     1.41  2423    1
#> d[Fixed dose warfarin]                           1.20     1.72  3174    1
#> d[Fixed dose warfarin + low dose aspirin]        0.75     1.37  2378    1
#> d[Fixed dose warfarin + medium dose aspirin]     1.09     1.50  3095    1
#> d[High dose aspirin]                             1.02     1.96  4029    1
#> d[Indobufen]                                     0.53     1.13  3297    1
#> d[Low adjusted dose anti-coagulant]             -0.04     0.44  3037    1
#> d[Low dose aspirin]                              0.75     1.02  1999    1
#> d[Low dose aspirin + copidogrel]                 0.72     1.21  3687    1
#> d[Low dose aspirin + dipyridamole]               0.57     1.15  2564    1
#> d[Medium dose aspirin]                           0.53     0.75  2473    1
#> d[Placebo/Standard care]                         0.89     1.16  1503    1
#> d[Triflusal]                                     1.05     1.89  3152    1
#> d[Ximelagatran]                                  0.07     0.42  2846    1
#> lp__                                         -4767.06 -4758.85  1121    1
#> tau                                              0.36     0.57   843    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jun 23 16:23:13 2020.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at 
#> convergence, Rhat=1).

By default, summaries of the study-specific intercepts \(\mu_j\) and study-specific relative effects \(\delta_{jk}\) are hidden, but could be examined by changing the pars argument:

# Not run
print(af_fit_1, pars = c("d", "mu", "delta"))

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

plot_prior_posterior(af_fit_1, prior = c("trt", "het"))

We can compute relative effects against placebo/standard care with the relative_effects() function with the trt_ref argument:

(af_1_releff <- relative_effects(af_fit_1, trt_ref = "Placebo/Standard care"))
#>                                               mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS
#> d[Standard adjusted dose anti-coagulant]     -0.76 0.20 -1.16 -0.89 -0.76 -0.64 -0.37     1516
#> d[Acenocoumarol]                             -1.53 0.83 -3.23 -2.08 -1.50 -0.97 -0.01     3399
#> d[Alternate day aspirin]                     -1.76 1.37 -4.94 -2.51 -1.61 -0.80  0.41     5333
#> d[Dipyridamole]                              -0.18 0.41 -1.02 -0.46 -0.17  0.09  0.64     3860
#> d[Fixed dose warfarin]                        0.17 0.44 -0.69 -0.12  0.16  0.45  1.04     3281
#> d[Fixed dose warfarin + low dose aspirin]    -0.28 0.40 -1.09 -0.53 -0.28 -0.04  0.53     3910
#> d[Fixed dose warfarin + medium dose aspirin]  0.13 0.36 -0.59 -0.11  0.13  0.36  0.84     2904
#> d[High dose aspirin]                         -0.24 0.76 -1.78 -0.75 -0.23  0.25  1.23     5014
#> d[Indobufen]                                 -0.52 0.49 -1.48 -0.84 -0.51 -0.20  0.47     2788
#> d[Low adjusted dose anti-coagulant]          -1.05 0.35 -1.76 -1.29 -1.05 -0.83 -0.38     4670
#> d[Low dose aspirin]                          -0.14 0.21 -0.59 -0.28 -0.14  0.00  0.25     4376
#> d[Low dose aspirin + copidogrel]             -0.25 0.39 -1.03 -0.49 -0.25  0.00  0.54     2794
#> d[Low dose aspirin + dipyridamole]           -0.50 0.44 -1.39 -0.78 -0.49 -0.20  0.35     4339
#> d[Medium dose aspirin]                       -0.37 0.22 -0.82 -0.51 -0.36 -0.23  0.06     3080
#> d[Triflusal]                                 -0.10 0.63 -1.31 -0.53 -0.12  0.32  1.17     3146
#> d[Ximelagatran]                              -0.85 0.32 -1.51 -1.06 -0.85 -0.65 -0.21     2391
#>                                              Tail_ESS Rhat
#> d[Standard adjusted dose anti-coagulant]         2269    1
#> d[Acenocoumarol]                                 2824    1
#> d[Alternate day aspirin]                         2419    1
#> d[Dipyridamole]                                  3091    1
#> d[Fixed dose warfarin]                           2922    1
#> d[Fixed dose warfarin + low dose aspirin]        2681    1
#> d[Fixed dose warfarin + medium dose aspirin]     2916    1
#> d[High dose aspirin]                             2975    1
#> d[Indobufen]                                     2552    1
#> d[Low adjusted dose anti-coagulant]              3250    1
#> d[Low dose aspirin]                              2623    1
#> d[Low dose aspirin + copidogrel]                 2581    1
#> d[Low dose aspirin + dipyridamole]               3156    1
#> d[Medium dose aspirin]                           3112    1
#> d[Triflusal]                                     2508    1
#> d[Ximelagatran]                                  2541    1

These estimates can easily be plotted with the plot() method:

plot(af_1_releff, ref_line = 0)

We can also produce treatment rankings, rank probabilities, and cumulative rank probabilities.

(af_1_ranks <- posterior_ranks(af_fit_1))
#>                                                  mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS
#> rank[Standard adjusted dose anti-coagulant]      5.26 1.40    3   4   5   6     8     2175     2661
#> rank[Acenocoumarol]                              3.00 2.99    1   1   2   3    12     3406     2405
#> rank[Alternate day aspirin]                      3.84 4.40    1   1   2   5    16     5614     3709
#> rank[Dipyridamole]                              11.19 3.75    4   8  11  14    17     3445       NA
#> rank[Fixed dose warfarin]                       14.10 3.04    6  12  15  16    17     3499       NA
#> rank[Fixed dose warfarin + low dose aspirin]    10.15 3.82    3   7  10  13    17     3268     3025
#> rank[Fixed dose warfarin + medium dose aspirin] 14.08 2.64    8  13  15  16    17     3100       NA
#> rank[High dose aspirin]                         10.33 5.28    1   6  11  16    17     4680       NA
#> rank[Indobufen]                                  8.00 3.97    2   5   8  11    16     3214     3121
#> rank[Low adjusted dose anti-coagulant]           3.73 2.17    1   2   3   5     9     3506     3207
#> rank[Low dose aspirin]                          11.73 2.27    7  10  12  13    16     3961     3502
#> rank[Low dose aspirin + copidogrel]             10.56 3.39    4   8  10  13    17     3248     2933
#> rank[Low dose aspirin + dipyridamole]            8.12 3.87    2   5   8  11    16     3832     3437
#> rank[Medium dose aspirin]                        9.18 2.14    5   8   9  11    14     3682     3349
#> rank[Placebo/Standard care]                     13.43 1.81   10  12  14  15    17     3692     3733
#> rank[Triflusal]                                 11.53 4.55    3   8  12  16    17     3377       NA
#> rank[Ximelagatran]                               4.77 2.20    2   3   4   6    10     2977     2546
#>                                                 Rhat
#> rank[Standard adjusted dose anti-coagulant]        1
#> rank[Acenocoumarol]                                1
#> rank[Alternate day aspirin]                        1
#> rank[Dipyridamole]                                 1
#> rank[Fixed dose warfarin]                          1
#> rank[Fixed dose warfarin + low dose aspirin]       1
#> rank[Fixed dose warfarin + medium dose aspirin]    1
#> rank[High dose aspirin]                            1
#> rank[Indobufen]                                    1
#> rank[Low adjusted dose anti-coagulant]             1
#> rank[Low dose aspirin]                             1
#> rank[Low dose aspirin + copidogrel]                1
#> rank[Low dose aspirin + dipyridamole]              1
#> rank[Medium dose aspirin]                          1
#> rank[Placebo/Standard care]                        1
#> rank[Triflusal]                                    1
#> rank[Ximelagatran]                                 1
plot(af_1_ranks)

(af_1_rankprobs <- posterior_rank_probs(af_fit_1))
#>                                              p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5]
#> d[Standard adjusted dose anti-coagulant]          0.00      0.01      0.07      0.20      0.29
#> d[Acenocoumarol]                                  0.37      0.29      0.10      0.06      0.04
#> d[Alternate day aspirin]                          0.46      0.17      0.07      0.04      0.03
#> d[Dipyridamole]                                   0.00      0.01      0.02      0.02      0.03
#> d[Fixed dose warfarin]                            0.00      0.00      0.00      0.00      0.01
#> d[Fixed dose warfarin + low dose aspirin]         0.00      0.02      0.03      0.03      0.04
#> d[Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00      0.00      0.00
#> d[High dose aspirin]                              0.03      0.05      0.06      0.06      0.04
#> d[Indobufen]                                      0.01      0.05      0.08      0.08      0.08
#> d[Low adjusted dose anti-coagulant]               0.08      0.24      0.27      0.15      0.09
#> d[Low dose aspirin]                               0.00      0.00      0.00      0.00      0.00
#> d[Low dose aspirin + copidogrel]                  0.00      0.00      0.01      0.02      0.03
#> d[Low dose aspirin + dipyridamole]                0.01      0.04      0.07      0.08      0.08
#> d[Medium dose aspirin]                            0.00      0.00      0.00      0.01      0.02
#> d[Placebo/Standard care]                          0.00      0.00      0.00      0.00      0.00
#> d[Triflusal]                                      0.00      0.01      0.03      0.04      0.04
#> d[Ximelagatran]                                   0.02      0.10      0.19      0.21      0.17
#>                                              p_rank[6] p_rank[7] p_rank[8] p_rank[9] p_rank[10]
#> d[Standard adjusted dose anti-coagulant]          0.23      0.12      0.04      0.01       0.00
#> d[Acenocoumarol]                                  0.03      0.02      0.02      0.01       0.01
#> d[Alternate day aspirin]                          0.03      0.03      0.02      0.02       0.02
#> d[Dipyridamole]                                   0.04      0.07      0.07      0.08       0.09
#> d[Fixed dose warfarin]                            0.01      0.02      0.02      0.03       0.04
#> d[Fixed dose warfarin + low dose aspirin]         0.06      0.08      0.09      0.09       0.09
#> d[Fixed dose warfarin + medium dose aspirin]      0.01      0.01      0.02      0.03       0.04
#> d[High dose aspirin]                              0.05      0.05      0.05      0.05       0.05
#> d[Indobufen]                                      0.09      0.10      0.09      0.08       0.07
#> d[Low adjusted dose anti-coagulant]               0.07      0.05      0.03      0.02       0.01
#> d[Low dose aspirin]                               0.01      0.02      0.05      0.08       0.13
#> d[Low dose aspirin + copidogrel]                  0.05      0.07      0.10      0.11       0.11
#> d[Low dose aspirin + dipyridamole]                0.10      0.11      0.09      0.08       0.07
#> d[Medium dose aspirin]                            0.06      0.12      0.17      0.19       0.17
#> d[Placebo/Standard care]                          0.00      0.00      0.00      0.02       0.04
#> d[Triflusal]                                      0.05      0.06      0.06      0.06       0.05
#> d[Ximelagatran]                                   0.11      0.08      0.05      0.03       0.01
#>                                              p_rank[11] p_rank[12] p_rank[13] p_rank[14] p_rank[15]
#> d[Standard adjusted dose anti-coagulant]           0.00       0.00       0.00       0.00       0.00
#> d[Acenocoumarol]                                   0.01       0.01       0.00       0.00       0.01
#> d[Alternate day aspirin]                           0.01       0.01       0.01       0.01       0.01
#> d[Dipyridamole]                                    0.08       0.08       0.08       0.09       0.09
#> d[Fixed dose warfarin]                             0.05       0.06       0.08       0.09       0.14
#> d[Fixed dose warfarin + low dose aspirin]          0.10       0.08       0.06       0.07       0.05
#> d[Fixed dose warfarin + medium dose aspirin]       0.06       0.07       0.09       0.12       0.17
#> d[High dose aspirin]                               0.05       0.04       0.04       0.05       0.07
#> d[Indobufen]                                       0.06       0.05       0.04       0.03       0.03
#> d[Low adjusted dose anti-coagulant]                0.01       0.00       0.00       0.00       0.00
#> d[Low dose aspirin]                                0.16       0.17       0.15       0.11       0.07
#> d[Low dose aspirin + copidogrel]                   0.09       0.10       0.08       0.08       0.06
#> d[Low dose aspirin + dipyridamole]                 0.06       0.05       0.04       0.04       0.03
#> d[Medium dose aspirin]                             0.12       0.07       0.04       0.02       0.01
#> d[Placebo/Standard care]                           0.08       0.14       0.21       0.22       0.17
#> d[Triflusal]                                       0.05       0.05       0.06       0.06       0.08
#> d[Ximelagatran]                                    0.01       0.00       0.00       0.00       0.00
#>                                              p_rank[16] p_rank[17]
#> d[Standard adjusted dose anti-coagulant]           0.00       0.00
#> d[Acenocoumarol]                                   0.00       0.00
#> d[Alternate day aspirin]                           0.02       0.02
#> d[Dipyridamole]                                    0.09       0.07
#> d[Fixed dose warfarin]                             0.19       0.25
#> d[Fixed dose warfarin + low dose aspirin]          0.06       0.05
#> d[Fixed dose warfarin + medium dose aspirin]       0.21       0.17
#> d[High dose aspirin]                               0.08       0.17
#> d[Indobufen]                                       0.03       0.02
#> d[Low adjusted dose anti-coagulant]                0.00       0.00
#> d[Low dose aspirin]                                0.04       0.01
#> d[Low dose aspirin + copidogrel]                   0.05       0.03
#> d[Low dose aspirin + dipyridamole]                 0.02       0.02
#> d[Medium dose aspirin]                             0.00       0.00
#> d[Placebo/Standard care]                           0.09       0.03
#> d[Triflusal]                                       0.11       0.18
#> d[Ximelagatran]                                    0.00       0.00
plot(af_1_rankprobs)

(af_1_cumrankprobs <- posterior_rank_probs(af_fit_1, cumulative = TRUE))
#>                                              p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5]
#> d[Standard adjusted dose anti-coagulant]          0.00      0.02      0.09      0.29      0.59
#> d[Acenocoumarol]                                  0.37      0.66      0.76      0.81      0.86
#> d[Alternate day aspirin]                          0.46      0.63      0.70      0.74      0.77
#> d[Dipyridamole]                                   0.00      0.01      0.02      0.04      0.08
#> d[Fixed dose warfarin]                            0.00      0.00      0.00      0.01      0.02
#> d[Fixed dose warfarin + low dose aspirin]         0.00      0.02      0.05      0.08      0.12
#> d[Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00      0.00      0.00
#> d[High dose aspirin]                              0.03      0.08      0.15      0.20      0.25
#> d[Indobufen]                                      0.01      0.06      0.14      0.22      0.30
#> d[Low adjusted dose anti-coagulant]               0.08      0.32      0.59      0.73      0.82
#> d[Low dose aspirin]                               0.00      0.00      0.00      0.00      0.00
#> d[Low dose aspirin + copidogrel]                  0.00      0.01      0.02      0.04      0.07
#> d[Low dose aspirin + dipyridamole]                0.01      0.05      0.12      0.20      0.28
#> d[Medium dose aspirin]                            0.00      0.00      0.00      0.01      0.04
#> d[Placebo/Standard care]                          0.00      0.00      0.00      0.00      0.00
#> d[Triflusal]                                      0.00      0.02      0.05      0.09      0.13
#> d[Ximelagatran]                                   0.02      0.12      0.31      0.52      0.69
#>                                              p_rank[6] p_rank[7] p_rank[8] p_rank[9] p_rank[10]
#> d[Standard adjusted dose anti-coagulant]          0.82      0.94      0.99      1.00       1.00
#> d[Acenocoumarol]                                  0.89      0.91      0.93      0.95       0.96
#> d[Alternate day aspirin]                          0.80      0.82      0.85      0.87       0.89
#> d[Dipyridamole]                                   0.12      0.18      0.26      0.34       0.43
#> d[Fixed dose warfarin]                            0.03      0.04      0.07      0.10       0.14
#> d[Fixed dose warfarin + low dose aspirin]         0.18      0.26      0.35      0.44       0.54
#> d[Fixed dose warfarin + medium dose aspirin]      0.01      0.02      0.05      0.08       0.12
#> d[High dose aspirin]                              0.29      0.34      0.39      0.44       0.49
#> d[Indobufen]                                      0.39      0.50      0.59      0.67       0.74
#> d[Low adjusted dose anti-coagulant]               0.89      0.93      0.96      0.98       0.99
#> d[Low dose aspirin]                               0.01      0.03      0.08      0.17       0.29
#> d[Low dose aspirin + copidogrel]                  0.12      0.19      0.29      0.40       0.51
#> d[Low dose aspirin + dipyridamole]                0.38      0.49      0.58      0.66       0.73
#> d[Medium dose aspirin]                            0.09      0.22      0.38      0.57       0.74
#> d[Placebo/Standard care]                          0.00      0.00      0.01      0.02       0.07
#> d[Triflusal]                                      0.17      0.23      0.29      0.36       0.41
#> d[Ximelagatran]                                   0.81      0.89      0.94      0.97       0.98
#>                                              p_rank[11] p_rank[12] p_rank[13] p_rank[14] p_rank[15]
#> d[Standard adjusted dose anti-coagulant]           1.00       1.00       1.00       1.00       1.00
#> d[Acenocoumarol]                                   0.97       0.98       0.98       0.99       0.99
#> d[Alternate day aspirin]                           0.90       0.92       0.93       0.94       0.96
#> d[Dipyridamole]                                    0.51       0.59       0.68       0.76       0.85
#> d[Fixed dose warfarin]                             0.19       0.25       0.33       0.42       0.56
#> d[Fixed dose warfarin + low dose aspirin]          0.63       0.71       0.78       0.84       0.90
#> d[Fixed dose warfarin + medium dose aspirin]       0.17       0.24       0.33       0.45       0.62
#> d[High dose aspirin]                               0.54       0.58       0.63       0.67       0.75
#> d[Indobufen]                                       0.79       0.84       0.89       0.92       0.95
#> d[Low adjusted dose anti-coagulant]                0.99       1.00       1.00       1.00       1.00
#> d[Low dose aspirin]                                0.45       0.62       0.77       0.89       0.96
#> d[Low dose aspirin + copidogrel]                   0.60       0.70       0.77       0.86       0.92
#> d[Low dose aspirin + dipyridamole]                 0.79       0.85       0.89       0.93       0.96
#> d[Medium dose aspirin]                             0.86       0.94       0.97       0.99       1.00
#> d[Placebo/Standard care]                           0.14       0.28       0.49       0.71       0.88
#> d[Triflusal]                                       0.46       0.52       0.58       0.63       0.71
#> d[Ximelagatran]                                    0.99       0.99       1.00       1.00       1.00
#>                                              p_rank[16] p_rank[17]
#> d[Standard adjusted dose anti-coagulant]           1.00          1
#> d[Acenocoumarol]                                   1.00          1
#> d[Alternate day aspirin]                           0.98          1
#> d[Dipyridamole]                                    0.93          1
#> d[Fixed dose warfarin]                             0.75          1
#> d[Fixed dose warfarin + low dose aspirin]          0.95          1
#> d[Fixed dose warfarin + medium dose aspirin]       0.83          1
#> d[High dose aspirin]                               0.83          1
#> d[Indobufen]                                       0.98          1
#> d[Low adjusted dose anti-coagulant]                1.00          1
#> d[Low dose aspirin]                                0.99          1
#> d[Low dose aspirin + copidogrel]                   0.97          1
#> d[Low dose aspirin + dipyridamole]                 0.98          1
#> d[Medium dose aspirin]                             1.00          1
#> d[Placebo/Standard care]                           0.97          1
#> d[Triflusal]                                       0.82          1
#> d[Ximelagatran]                                    1.00          1
plot(af_1_cumrankprobs)

Network meta-regression adjusting for proportion of prior stroke

We now consider a meta-regression model adjusting for the proportion of individuals in each study with prior stroke, with shared interaction coefficients by treatment class. The regression model is specified in the nma() function using a formula in the regression argument. The formula ~ .trt:stroke means that interactions of prior stroke with treatment will be included; the .trt special variable indicates treatment, and stroke is in the original data set. We specify class_interactions = "common" to denote that the interaction parameters are to be common (i.e. shared) between treatments within each class. (Setting class_interactions = "independent" would fit model 2 of Cooper et al. (2009) with separate interactions for each treatment, data permitting.) We use the same prior distributions as above, but additionally require a prior distribution for the regression coefficients prior_reg; we use a \(\mathrm{N}(0, 100^2)\) prior distribution. The QR decomposition can greatly improve the efficiency of sampling for regression models by decorrelating the sampling space; we specify that this should be used with QR = TRUE, and increase the target acceptance rate adapt_delta = 0.99 to minimise divergent transition warnings.

af_fit_4b <- nma(af_net, 
                 trt_effects = "random",
                 regression = ~ .trt:stroke,
                 class_interactions = "common",
                 QR = TRUE,
                 prior_intercept = normal(scale = 100),
                 prior_trt = normal(scale = 100),
                 prior_reg = normal(scale = 100),
                 prior_het = half_normal(scale = 5),
                 adapt_delta = 0.99)
#> Note: Setting "Standard adjusted dose anti-coagulant" as the network reference treatment.
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> http://mc-stan.org/misc/warnings.html#bulk-ess

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

af_fit_4b
#> A random effects NMA with a binomial likelihood (logit link).
#> Regression model: ~.trt:stroke.
#> Centred covariates at the following overall mean values:
#>    stroke 
#> 0.2957377 
#> 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%
#> beta[.trtclassMixed:stroke]                      3.98    0.03 2.05    -0.01     2.62     3.97
#> beta[.trtclassAnti-platelet:stroke]              0.94    0.01 0.40     0.16     0.68     0.93
#> beta[.trtclassControl:stroke]                    0.70    0.01 0.43    -0.15     0.42     0.69
#> d[Acenocoumarol]                                 0.39    0.01 0.96    -1.50    -0.25     0.43
#> d[Alternate day aspirin]                        -0.90    0.03 1.34    -4.14    -1.66    -0.72
#> d[Dipyridamole]                                  0.57    0.01 0.40    -0.24     0.31     0.58
#> d[Fixed dose warfarin]                           0.63    0.01 0.39    -0.13     0.38     0.62
#> d[Fixed dose warfarin + low dose aspirin]        1.48    0.01 0.72     0.05     1.00     1.49
#> d[Fixed dose warfarin + medium dose aspirin]     1.01    0.00 0.30     0.42     0.81     1.01
#> d[High dose aspirin]                             0.43    0.01 0.74    -1.05    -0.06     0.44
#> d[Indobufen]                                    -0.42    0.01 0.48    -1.36    -0.72    -0.41
#> d[Low adjusted dose anti-coagulant]             -0.43    0.01 0.36    -1.16    -0.68    -0.44
#> d[Low dose aspirin]                              0.72    0.00 0.20     0.33     0.59     0.72
#> d[Low dose aspirin + copidogrel]                 0.66    0.00 0.27     0.11     0.50     0.66
#> d[Low dose aspirin + dipyridamole]               0.25    0.01 0.43    -0.63    -0.04     0.26
#> d[Medium dose aspirin]                           0.35    0.00 0.17     0.02     0.23     0.35
#> d[Placebo/Standard care]                         0.79    0.00 0.19     0.42     0.66     0.79
#> d[Triflusal]                                     0.90    0.01 0.57    -0.20     0.52     0.89
#> d[Ximelagatran]                                 -0.09    0.00 0.21    -0.53    -0.21    -0.09
#> lp__                                         -4771.41    0.20 6.93 -4786.35 -4775.66 -4771.20
#> tau                                              0.17    0.01 0.13     0.01     0.07     0.15
#>                                                   75%    97.5% n_eff Rhat
#> beta[.trtclassMixed:stroke]                      5.35     8.08  4458 1.00
#> beta[.trtclassAnti-platelet:stroke]              1.20     1.75  4392 1.00
#> beta[.trtclassControl:stroke]                    0.97     1.56  4272 1.00
#> d[Acenocoumarol]                                 1.04     2.26  4356 1.00
#> d[Alternate day aspirin]                         0.01     1.25  2044 1.00
#> d[Dipyridamole]                                  0.84     1.37  5911 1.00
#> d[Fixed dose warfarin]                           0.88     1.39  4961 1.00
#> d[Fixed dose warfarin + low dose aspirin]        1.96     2.84  4292 1.00
#> d[Fixed dose warfarin + medium dose aspirin]     1.20     1.64  4798 1.00
#> d[High dose aspirin]                             0.93     1.86  5984 1.00
#> d[Indobufen]                                    -0.11     0.52  4122 1.00
#> d[Low adjusted dose anti-coagulant]             -0.18     0.28  4930 1.00
#> d[Low dose aspirin]                              0.85     1.10  4265 1.00
#> d[Low dose aspirin + copidogrel]                 0.82     1.22  3668 1.00
#> d[Low dose aspirin + dipyridamole]               0.54     1.07  5926 1.00
#> d[Medium dose aspirin]                           0.46     0.69  4577 1.00
#> d[Placebo/Standard care]                         0.92     1.15  3823 1.00
#> d[Triflusal]                                     1.29     2.07  4277 1.00
#> d[Ximelagatran]                                  0.04     0.33  3943 1.00
#> lp__                                         -4766.51 -4759.00  1218 1.00
#> tau                                              0.25     0.46   345 1.01
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jun 23 16:24:05 2020.
#> 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).

The estimated treatment effects d[] shown here correspond to relative effects at the reference level of the covariate, here proportion of prior stroke centered at the network mean value 0.296.

By default, summaries of the study-specific intercepts \(\mu_j\) and study-specific relative effects \(\delta_{jk}\) are hidden, but could be examined by changing the pars argument:

# Not run
print(af_fit_4b, pars = c("d", "mu", "delta"))

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

plot_prior_posterior(af_fit_4b, prior = c("reg", "het"))

We can compute relative effects against placebo/standard care with the relative_effects() function with the trt_ref argument, which by default produces relative effects for the observed proportions of prior stroke in each study:

# Not run
(af_4b_releff <- relative_effects(af_fit_4b, trt_ref = "Placebo/Standard care"))
plot(af_4b_releff, ref_line = 0)

We can produce estimated treatment effects for particular covariate values using the newdata argument. For example, treatment effects when no individuals or all individuals have prior stroke are produced by

(af_4b_releff_01 <- relative_effects(af_fit_4b, 
                                     trt_ref = "Placebo/Standard care",
                                     newdata = data.frame(stroke = c(0, 1), 
                                                          label = c("stroke = 0", "stroke = 1")),
                                     study = label))
#> ------------------------------------------------------------- Study: stroke = 0 ---- 
#> 
#> Covariate values:
#>  stroke
#>       0
#> 
#>                                                           mean   sd  2.5%   25%   50%   75% 97.5%
#> d[stroke = 0: Standard adjusted dose anti-coagulant]     -0.58 0.23 -1.04 -0.73 -0.58 -0.43 -0.13
#> d[stroke = 0: Acenocoumarol]                             -1.37 0.79 -3.03 -1.91 -1.33 -0.84  0.12
#> d[stroke = 0: Alternate day aspirin]                     -1.76 1.33 -4.91 -2.49 -1.57 -0.86  0.37
#> d[stroke = 0: Dipyridamole]                              -0.29 0.43 -1.16 -0.57 -0.28 -0.01  0.57
#> d[stroke = 0: Fixed dose warfarin]                        0.05 0.43 -0.76 -0.24  0.04  0.34  0.92
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]    -0.28 0.35 -0.97 -0.50 -0.28 -0.07  0.42
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin] -0.75 0.63 -1.99 -1.17 -0.74 -0.34  0.47
#> d[stroke = 0: High dose aspirin]                         -0.43 0.77 -1.94 -0.93 -0.42  0.09  1.05
#> d[stroke = 0: Indobufen]                                 -1.28 0.57 -2.37 -1.64 -1.27 -0.91 -0.16
#> d[stroke = 0: Low adjusted dose anti-coagulant]          -1.02 0.33 -1.66 -1.24 -1.01 -0.79 -0.38
#> d[stroke = 0: Low dose aspirin]                          -0.14 0.22 -0.58 -0.29 -0.13  0.00  0.29
#> d[stroke = 0: Low dose aspirin + copidogrel]             -0.20 0.36 -0.89 -0.42 -0.20  0.03  0.53
#> d[stroke = 0: Low dose aspirin + dipyridamole]           -0.61 0.47 -1.55 -0.91 -0.60 -0.31  0.29
#> d[stroke = 0: Medium dose aspirin]                       -0.51 0.26 -1.01 -0.68 -0.52 -0.34  0.01
#> d[stroke = 0: Triflusal]                                  0.05 0.61 -1.16 -0.36  0.04  0.44  1.25
#> d[stroke = 0: Ximelagatran]                              -0.67 0.32 -1.30 -0.86 -0.67 -0.47 -0.05
#>                                                          Bulk_ESS Tail_ESS Rhat
#> d[stroke = 0: Standard adjusted dose anti-coagulant]         3431     2792    1
#> d[stroke = 0: Acenocoumarol]                                 4416     2853    1
#> d[stroke = 0: Alternate day aspirin]                         2710     1792    1
#> d[stroke = 0: Dipyridamole]                                  5166     2824    1
#> d[stroke = 0: Fixed dose warfarin]                           4582     2985    1
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]        5042     1950    1
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]     4455     3096    1
#> d[stroke = 0: High dose aspirin]                             5693     2982    1
#> d[stroke = 0: Indobufen]                                     4047     3200    1
#> d[stroke = 0: Low adjusted dose anti-coagulant]              5667     3319    1
#> d[stroke = 0: Low dose aspirin]                              3792     2685    1
#> d[stroke = 0: Low dose aspirin + copidogrel]                 3910     2861    1
#> d[stroke = 0: Low dose aspirin + dipyridamole]               5481     2816    1
#> d[stroke = 0: Medium dose aspirin]                           4137     3246    1
#> d[stroke = 0: Triflusal]                                     4629     2746    1
#> d[stroke = 0: Ximelagatran]                                  3883     2679    1
#> 
#> ------------------------------------------------------------- Study: stroke = 1 ---- 
#> 
#> Covariate values:
#>  stroke
#>       1
#> 
#>                                                           mean   sd  2.5%   25%   50%   75% 97.5%
#> d[stroke = 1: Standard adjusted dose anti-coagulant]     -1.28 0.35 -2.01 -1.51 -1.27 -1.06 -0.61
#> d[stroke = 1: Acenocoumarol]                              1.91 2.20 -2.43  0.45  1.92  3.36  6.17
#> d[stroke = 1: Alternate day aspirin]                     -1.52 1.36 -4.72 -2.31 -1.33 -0.59  0.70
#> d[stroke = 1: Dipyridamole]                              -0.05 0.39 -0.85 -0.31 -0.04  0.22  0.69
#> d[stroke = 1: Fixed dose warfarin]                       -0.65 0.53 -1.68 -1.00 -0.66 -0.30  0.40
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]     3.00 2.11 -1.18  1.61  3.00  4.41  7.10
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]  2.53 1.59 -0.63  1.51  2.52  3.60  5.66
#> d[stroke = 1: High dose aspirin]                         -0.19 0.73 -1.64 -0.68 -0.19  0.31  1.20
#> d[stroke = 1: Indobufen]                                 -1.04 0.53 -2.12 -1.37 -1.03 -0.70 -0.03
#> d[stroke = 1: Low adjusted dose anti-coagulant]          -1.72 0.49 -2.68 -2.05 -1.72 -1.38 -0.75
#> d[stroke = 1: Low dose aspirin]                           0.10 0.29 -0.48 -0.09  0.10  0.29  0.64
#> d[stroke = 1: Low dose aspirin + copidogrel]              0.04 0.37 -0.73 -0.19  0.05  0.28  0.79
#> d[stroke = 1: Low dose aspirin + dipyridamole]           -0.37 0.42 -1.21 -0.65 -0.37 -0.09  0.45
#> d[stroke = 1: Medium dose aspirin]                       -0.27 0.24 -0.77 -0.42 -0.26 -0.12  0.17
#> d[stroke = 1: Triflusal]                                  0.28 0.64 -0.96 -0.15  0.28  0.70  1.55
#> d[stroke = 1: Ximelagatran]                              -1.37 0.40 -2.20 -1.62 -1.37 -1.10 -0.61
#>                                                          Bulk_ESS Tail_ESS Rhat
#> d[stroke = 1: Standard adjusted dose anti-coagulant]         5037     2970    1
#> d[stroke = 1: Acenocoumarol]                                 4606     2927    1
#> d[stroke = 1: Alternate day aspirin]                         2817     1795    1
#> d[stroke = 1: Dipyridamole]                                  5889     3018    1
#> d[stroke = 1: Fixed dose warfarin]                           4740     3031    1
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]        4468     2820    1
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]     4659     2645    1
#> d[stroke = 1: High dose aspirin]                             6231     3101    1
#> d[stroke = 1: Indobufen]                                     4513     2390    1
#> d[stroke = 1: Low adjusted dose anti-coagulant]              4508     3520    1
#> d[stroke = 1: Low dose aspirin]                              4787     2932    1
#> d[stroke = 1: Low dose aspirin + copidogrel]                 4195     2851    1
#> d[stroke = 1: Low dose aspirin + dipyridamole]               6057     2912    1
#> d[stroke = 1: Medium dose aspirin]                           5147     2788    1
#> d[stroke = 1: Triflusal]                                     4507     3100    1
#> d[stroke = 1: Ximelagatran]                                  4423     2952    1
plot(af_4b_releff_01, ref_line = 0)

The estimated class interactions (against the reference “Mixed” class) are very uncertain.

plot(af_fit_4b, pars = "beta", stat = "halfeye", ref_line = 0)

The interactions are more straightforward to interpret if we transform the interaction coefficients (using the consistency equations) so that they are against the control class:

af_4b_beta <- as.array(af_fit_4b, pars = "beta")

# Subtract beta[Control:stroke] from the other class interactions
af_4b_beta[ , , 1:2] <- sweep(af_4b_beta[ , , 1:2], 1:2, 
                              af_4b_beta[ , , "beta[.trtclassControl:stroke]"], FUN = "-")

# Set beta[Anti-coagulant:stroke] = -beta[Control:stroke]
af_4b_beta[ , , "beta[.trtclassControl:stroke]"] <- -af_4b_beta[ , , "beta[.trtclassControl:stroke]"]
names(af_4b_beta)[3] <- "beta[.trtclassAnti-coagulant:stroke]"

# Summarise
summary(af_4b_beta)
#>                                       mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS
#> beta[.trtclassMixed:stroke]           3.28 2.08 -0.80  1.92  3.26  4.68  7.43     4568     3148
#> beta[.trtclassAnti-platelet:stroke]   0.24 0.33 -0.43  0.04  0.24  0.45  0.87     4280     2818
#> beta[.trtclassAnti-coagulant:stroke] -0.70 0.43 -1.56 -0.97 -0.69 -0.42  0.15     4381     3079
#>                                      Rhat
#> beta[.trtclassMixed:stroke]             1
#> beta[.trtclassAnti-platelet:stroke]     1
#> beta[.trtclassAnti-coagulant:stroke]    1
plot(summary(af_4b_beta), stat = "halfeye", ref_line = 0)

There is some evidence that the effect of anti-coagulants increases (compared to control) with prior stroke. There is little evidence the effect of anti-platelets reduces with prior stroke, although the point estimate represents a substantial reduction in effectiveness, and the 95% Credible Interval includes values that correspond to substantial increases in treatment effect. The interaction effect of stroke on mixed treatments is very uncertain, but potentially indicates a substantial reduction in treatment effects with prior stroke.

We can also produce treatment rankings, rank probabilities, and cumulative rank probabilities. By default (without the newdata argument specified), these are produced at the value of stroke for each study in the network in turn. To instead produce rankings for when no individuals or all individuals have prior stroke, we specify the newdata argument.

(af_4b_ranks <- posterior_ranks(af_fit_4b,
                                newdata = data.frame(stroke = c(0, 1), 
                                                     label = c("stroke = 0", "stroke = 1")), 
                                study = label))
#> ------------------------------------------------------------- Study: stroke = 0 ---- 
#> 
#> Covariate values:
#>  stroke
#>       0
#> 
#>                                                              mean   sd 2.5% 25% 50% 75% 97.5%
#> rank[stroke = 0: Standard adjusted dose anti-coagulant]      7.76 1.85    4   6   8   9    11
#> rank[stroke = 0: Acenocoumarol]                              3.93 3.55    1   1   3   5    14
#> rank[stroke = 0: Alternate day aspirin]                      3.99 4.45    1   1   2   5    17
#> rank[stroke = 0: Dipyridamole]                              11.11 3.68    4   8  11  14    17
#> rank[stroke = 0: Fixed dose warfarin]                       14.08 2.92    7  12  15  16    17
#> rank[stroke = 0: Fixed dose warfarin + low dose aspirin]    11.07 3.73    4   8  11  14    17
#> rank[stroke = 0: Fixed dose warfarin + medium dose aspirin]  7.07 4.43    1   3   6  10    16
#> rank[stroke = 0: High dose aspirin]                          9.65 5.29    1   5  10  15    17
#> rank[stroke = 0: Indobufen]                                  3.58 2.67    1   2   3   4    11
#> rank[stroke = 0: Low adjusted dose anti-coagulant]           4.51 2.38    1   3   4   6    11
#> rank[stroke = 0: Low dose aspirin]                          12.94 1.94    9  12  13  14    16
#> rank[stroke = 0: Low dose aspirin + copidogrel]             12.14 2.87    6  10  12  14    17
#> rank[stroke = 0: Low dose aspirin + dipyridamole]            7.90 3.74    2   5   7  11    16
#> rank[stroke = 0: Medium dose aspirin]                        8.64 2.21    4   7   9  10    13
#> rank[stroke = 0: Placebo/Standard care]                     14.32 1.91   10  13  15  16    17
#> rank[stroke = 0: Triflusal]                                 13.39 3.98    4  11  15  17    17
#> rank[stroke = 0: Ximelagatran]                               6.93 2.53    3   5   7   8    13
#>                                                             Bulk_ESS Tail_ESS Rhat
#> rank[stroke = 0: Standard adjusted dose anti-coagulant]         4533     3605    1
#> rank[stroke = 0: Acenocoumarol]                                 4600     3320    1
#> rank[stroke = 0: Alternate day aspirin]                         4887     3539    1
#> rank[stroke = 0: Dipyridamole]                                  5322       NA    1
#> rank[stroke = 0: Fixed dose warfarin]                           4675       NA    1
#> rank[stroke = 0: Fixed dose warfarin + low dose aspirin]        4846       NA    1
#> rank[stroke = 0: Fixed dose warfarin + medium dose aspirin]     4551     3295    1
#> rank[stroke = 0: High dose aspirin]                             5602       NA    1
#> rank[stroke = 0: Indobufen]                                     3728     3190    1
#> rank[stroke = 0: Low adjusted dose anti-coagulant]              4783     3218    1
#> rank[stroke = 0: Low dose aspirin]                              3365     2570    1
#> rank[stroke = 0: Low dose aspirin + copidogrel]                 3836     2740    1
#> rank[stroke = 0: Low dose aspirin + dipyridamole]               5398     3104    1
#> rank[stroke = 0: Medium dose aspirin]                           4555     3064    1
#> rank[stroke = 0: Placebo/Standard care]                         3103       NA    1
#> rank[stroke = 0: Triflusal]                                     4009       NA    1
#> rank[stroke = 0: Ximelagatran]                                  3972     3263    1
#> 
#> ------------------------------------------------------------- Study: stroke = 1 ---- 
#> 
#> Covariate values:
#>  stroke
#>       1
#> 
#>                                                              mean   sd 2.5% 25%  50% 75% 97.5%
#> rank[stroke = 1: Standard adjusted dose anti-coagulant]      3.63 1.08    2   3  4.0   4     6
#> rank[stroke = 1: Acenocoumarol]                             13.34 4.23    1  14 15.0  16    17
#> rank[stroke = 1: Alternate day aspirin]                      4.49 4.00    1   1  3.0   6    14
#> rank[stroke = 1: Dipyridamole]                              10.55 2.68    5   9 11.0  13    15
#> rank[stroke = 1: Fixed dose warfarin]                        7.06 2.71    3   5  6.0   8    14
#> rank[stroke = 1: Fixed dose warfarin + low dose aspirin]    15.86 2.82    6  16 17.0  17    17
#> rank[stroke = 1: Fixed dose warfarin + medium dose aspirin] 15.46 1.83    9  15 16.0  16    17
#> rank[stroke = 1: High dose aspirin]                          9.45 3.98    2   6  9.0  13    16
#> rank[stroke = 1: Indobufen]                                  4.94 2.12    1   4  5.0   6    10
#> rank[stroke = 1: Low adjusted dose anti-coagulant]           1.96 1.24    1   1  2.0   2     5
#> rank[stroke = 1: Low dose aspirin]                          11.89 1.79    8  11 12.0  13    15
#> rank[stroke = 1: Low dose aspirin + copidogrel]             11.21 2.32    6  10 11.0  13    15
#> rank[stroke = 1: Low dose aspirin + dipyridamole]            8.24 2.70    3   6  8.0  10    14
#> rank[stroke = 1: Medium dose aspirin]                        8.64 1.65    6   8  8.5  10    12
#> rank[stroke = 1: Placebo/Standard care]                     11.06 1.94    7  10 11.0  12    15
#> rank[stroke = 1: Triflusal]                                 12.09 3.04    5  10 13.0  14    17
#> rank[stroke = 1: Ximelagatran]                               3.11 1.35    1   2  3.0   4     6
#>                                                             Bulk_ESS Tail_ESS Rhat
#> rank[stroke = 1: Standard adjusted dose anti-coagulant]         3747     3609    1
#> rank[stroke = 1: Acenocoumarol]                                 4140       NA    1
#> rank[stroke = 1: Alternate day aspirin]                         4693     2577    1
#> rank[stroke = 1: Dipyridamole]                                  4975     2997    1
#> rank[stroke = 1: Fixed dose warfarin]                           4461     3371    1
#> rank[stroke = 1: Fixed dose warfarin + low dose aspirin]        3403       NA    1
#> rank[stroke = 1: Fixed dose warfarin + medium dose aspirin]     3276       NA    1
#> rank[stroke = 1: High dose aspirin]                             5685     3104    1
#> rank[stroke = 1: Indobufen]                                     4339     3083    1
#> rank[stroke = 1: Low adjusted dose anti-coagulant]              3237     3221    1
#> rank[stroke = 1: Low dose aspirin]                              3404     3017    1
#> rank[stroke = 1: Low dose aspirin + copidogrel]                 3906     2939    1
#> rank[stroke = 1: Low dose aspirin + dipyridamole]               4837     2811    1
#> rank[stroke = 1: Medium dose aspirin]                           4082     3034    1
#> rank[stroke = 1: Placebo/Standard care]                         4841     3141    1
#> rank[stroke = 1: Triflusal]                                     3329     3078    1
#> rank[stroke = 1: Ximelagatran]                                  3474     3056    1
plot(af_4b_ranks)

(af_4b_rankprobs <- posterior_rank_probs(af_fit_4b,
                                         newdata = data.frame(stroke = c(0, 1), 
                                                              label = c("stroke = 0", "stroke = 1")), 
                                         study = label))
#> ------------------------------------------------------------- Study: stroke = 0 ---- 
#> 
#> Covariate values:
#>  stroke
#>       0
#> 
#>                                                          p_rank[1] p_rank[2] p_rank[3] p_rank[4]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.00      0.00      0.01      0.03
#> d[stroke = 0: Acenocoumarol]                                  0.25      0.24      0.14      0.10
#> d[stroke = 0: Alternate day aspirin]                          0.43      0.15      0.09      0.05
#> d[stroke = 0: Dipyridamole]                                   0.00      0.01      0.01      0.02
#> d[stroke = 0: Fixed dose warfarin]                            0.00      0.00      0.00      0.00
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.00      0.01      0.02      0.03
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.05      0.10      0.11      0.11
#> d[stroke = 0: High dose aspirin]                              0.03      0.06      0.07      0.07
#> d[stroke = 0: Indobufen]                                      0.18      0.25      0.20      0.13
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.04      0.14      0.22      0.20
#> d[stroke = 0: Low dose aspirin]                               0.00      0.00      0.00      0.00
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.00      0.00      0.00      0.01
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.01      0.03      0.06      0.11
#> d[stroke = 0: Medium dose aspirin]                            0.00      0.00      0.01      0.02
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.00      0.00      0.00
#> d[stroke = 0: Triflusal]                                      0.00      0.00      0.01      0.02
#> d[stroke = 0: Ximelagatran]                                   0.00      0.01      0.05      0.10
#>                                                          p_rank[5] p_rank[6] p_rank[7] p_rank[8]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.07      0.15      0.20      0.20
#> d[stroke = 0: Acenocoumarol]                                  0.06      0.04      0.03      0.02
#> d[stroke = 0: Alternate day aspirin]                          0.04      0.03      0.02      0.02
#> d[stroke = 0: Dipyridamole]                                   0.04      0.05      0.06      0.06
#> d[stroke = 0: Fixed dose warfarin]                            0.01      0.01      0.01      0.02
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.04      0.05      0.05      0.06
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.10      0.08      0.06      0.05
#> d[stroke = 0: High dose aspirin]                              0.06      0.06      0.05      0.04
#> d[stroke = 0: Indobufen]                                      0.08      0.05      0.03      0.02
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.14      0.09      0.06      0.04
#> d[stroke = 0: Low dose aspirin]                               0.00      0.00      0.00      0.01
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.01      0.02      0.03      0.04
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.11      0.10      0.08      0.07
#> d[stroke = 0: Medium dose aspirin]                            0.05      0.09      0.13      0.18
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.00      0.00      0.00
#> d[stroke = 0: Triflusal]                                      0.03      0.03      0.03      0.03
#> d[stroke = 0: Ximelagatran]                                   0.15      0.17      0.15      0.13
#>                                                          p_rank[9] p_rank[10] p_rank[11] p_rank[12]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.17       0.10       0.05       0.02
#> d[stroke = 0: Acenocoumarol]                                  0.02       0.02       0.02       0.02
#> d[stroke = 0: Alternate day aspirin]                          0.02       0.01       0.02       0.01
#> d[stroke = 0: Dipyridamole]                                   0.07       0.09       0.10       0.10
#> d[stroke = 0: Fixed dose warfarin]                            0.02       0.04       0.06       0.08
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.06       0.09       0.10       0.09
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.06       0.05       0.05       0.04
#> d[stroke = 0: High dose aspirin]                              0.04       0.05       0.05       0.05
#> d[stroke = 0: Indobufen]                                      0.02       0.01       0.01       0.01
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.03       0.02       0.02       0.01
#> d[stroke = 0: Low dose aspirin]                               0.03       0.06       0.12       0.17
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.06       0.09       0.12       0.13
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.09       0.08       0.07       0.06
#> d[stroke = 0: Medium dose aspirin]                            0.17       0.15       0.10       0.06
#> d[stroke = 0: Placebo/Standard care]                          0.01       0.02       0.04       0.07
#> d[stroke = 0: Triflusal]                                      0.04       0.05       0.05       0.06
#> d[stroke = 0: Ximelagatran]                                   0.09       0.07       0.04       0.02
#>                                                          p_rank[13] p_rank[14] p_rank[15]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]           0.00       0.00       0.00
#> d[stroke = 0: Acenocoumarol]                                   0.02       0.01       0.01
#> d[stroke = 0: Alternate day aspirin]                           0.01       0.01       0.01
#> d[stroke = 0: Dipyridamole]                                    0.09       0.09       0.08
#> d[stroke = 0: Fixed dose warfarin]                             0.08       0.10       0.13
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]          0.10       0.08       0.09
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]       0.03       0.03       0.04
#> d[stroke = 0: High dose aspirin]                               0.05       0.04       0.05
#> d[stroke = 0: Indobufen]                                       0.00       0.00       0.00
#> d[stroke = 0: Low adjusted dose anti-coagulant]                0.00       0.00       0.00
#> d[stroke = 0: Low dose aspirin]                                0.21       0.20       0.12
#> d[stroke = 0: Low dose aspirin + copidogrel]                   0.13       0.12       0.11
#> d[stroke = 0: Low dose aspirin + dipyridamole]                 0.04       0.04       0.02
#> d[stroke = 0: Medium dose aspirin]                             0.02       0.01       0.00
#> d[stroke = 0: Placebo/Standard care]                           0.14       0.18       0.23
#> d[stroke = 0: Triflusal]                                       0.06       0.07       0.08
#> d[stroke = 0: Ximelagatran]                                    0.01       0.01       0.00
#>                                                          p_rank[16] p_rank[17]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]           0.00       0.00
#> d[stroke = 0: Acenocoumarol]                                   0.01       0.00
#> d[stroke = 0: Alternate day aspirin]                           0.02       0.03
#> d[stroke = 0: Dipyridamole]                                    0.07       0.06
#> d[stroke = 0: Fixed dose warfarin]                             0.20       0.23
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]          0.07       0.05
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]       0.03       0.02
#> d[stroke = 0: High dose aspirin]                               0.08       0.14
#> d[stroke = 0: Indobufen]                                       0.00       0.00
#> d[stroke = 0: Low adjusted dose anti-coagulant]                0.00       0.00
#> d[stroke = 0: Low dose aspirin]                                0.07       0.02
#> d[stroke = 0: Low dose aspirin + copidogrel]                   0.08       0.03
#> d[stroke = 0: Low dose aspirin + dipyridamole]                 0.02       0.01
#> d[stroke = 0: Medium dose aspirin]                             0.00       0.00
#> d[stroke = 0: Placebo/Standard care]                           0.20       0.10
#> d[stroke = 0: Triflusal]                                       0.14       0.31
#> d[stroke = 0: Ximelagatran]                                    0.00       0.00
#> 
#> ------------------------------------------------------------- Study: stroke = 1 ---- 
#> 
#> Covariate values:
#>  stroke
#>       1
#> 
#>                                                          p_rank[1] p_rank[2] p_rank[3] p_rank[4]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.01      0.12      0.33      0.35
#> d[stroke = 1: Acenocoumarol]                                  0.03      0.02      0.01      0.01
#> d[stroke = 1: Alternate day aspirin]                          0.36      0.10      0.05      0.07
#> d[stroke = 1: Dipyridamole]                                   0.00      0.00      0.00      0.01
#> d[stroke = 1: Fixed dose warfarin]                            0.01      0.01      0.02      0.06
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.00      0.01      0.01      0.00
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00      0.00
#> d[stroke = 1: High dose aspirin]                              0.02      0.03      0.03      0.03
#> d[stroke = 1: Indobufen]                                      0.03      0.09      0.11      0.18
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.45      0.34      0.10      0.06
#> d[stroke = 1: Low dose aspirin]                               0.00      0.00      0.00      0.00
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.00      0.00      0.00      0.00
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.00      0.01      0.02      0.03
#> d[stroke = 1: Medium dose aspirin]                            0.00      0.00      0.00      0.00
#> d[stroke = 1: Placebo/Standard care]                          0.00      0.00      0.00      0.00
#> d[stroke = 1: Triflusal]                                      0.00      0.00      0.00      0.01
#> d[stroke = 1: Ximelagatran]                                   0.08      0.28      0.31      0.18
#>                                                          p_rank[5] p_rank[6] p_rank[7] p_rank[8]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.14      0.03      0.01      0.00
#> d[stroke = 1: Acenocoumarol]                                  0.02      0.02      0.02      0.02
#> d[stroke = 1: Alternate day aspirin]                          0.09      0.08      0.06      0.04
#> d[stroke = 1: Dipyridamole]                                   0.01      0.04      0.08      0.11
#> d[stroke = 1: Fixed dose warfarin]                            0.18      0.25      0.15      0.09
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.00      0.01      0.01      0.01
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00      0.01
#> d[stroke = 1: High dose aspirin]                              0.07      0.10      0.10      0.07
#> d[stroke = 1: Indobufen]                                      0.26      0.16      0.08      0.04
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.03      0.01      0.00      0.00
#> d[stroke = 1: Low dose aspirin]                               0.00      0.00      0.01      0.03
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.01      0.02      0.04      0.07
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.06      0.14      0.18      0.16
#> d[stroke = 1: Medium dose aspirin]                            0.01      0.06      0.17      0.25
#> d[stroke = 1: Placebo/Standard care]                          0.00      0.01      0.02      0.06
#> d[stroke = 1: Triflusal]                                      0.01      0.03      0.05      0.05
#> d[stroke = 1: Ximelagatran]                                   0.10      0.03      0.01      0.00
#>                                                          p_rank[9] p_rank[10] p_rank[11] p_rank[12]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.00       0.00       0.00       0.00
#> d[stroke = 1: Acenocoumarol]                                  0.01       0.01       0.01       0.01
#> d[stroke = 1: Alternate day aspirin]                          0.03       0.02       0.02       0.02
#> d[stroke = 1: Dipyridamole]                                   0.12       0.12       0.12       0.12
#> d[stroke = 1: Fixed dose warfarin]                            0.06       0.04       0.03       0.03
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.01       0.01       0.01       0.01
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.01       0.00       0.01       0.01
#> d[stroke = 1: High dose aspirin]                              0.07       0.06       0.06       0.06
#> d[stroke = 1: Indobufen]                                      0.02       0.01       0.01       0.01
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.00       0.00       0.00       0.00
#> d[stroke = 1: Low dose aspirin]                               0.06       0.11       0.18       0.23
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.09       0.13       0.15       0.18
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.11       0.09       0.07       0.06
#> d[stroke = 1: Medium dose aspirin]                            0.23       0.15       0.08       0.03
#> d[stroke = 1: Placebo/Standard care]                          0.12       0.19       0.19       0.18
#> d[stroke = 1: Triflusal]                                      0.06       0.06       0.06       0.07
#> d[stroke = 1: Ximelagatran]                                   0.00       0.00       0.00       0.00
#>                                                          p_rank[13] p_rank[14] p_rank[15]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]           0.00       0.00       0.00
#> d[stroke = 1: Acenocoumarol]                                   0.02       0.04       0.46
#> d[stroke = 1: Alternate day aspirin]                           0.02       0.03       0.01
#> d[stroke = 1: Dipyridamole]                                    0.13       0.09       0.03
#> d[stroke = 1: Fixed dose warfarin]                             0.02       0.02       0.01
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]          0.01       0.01       0.04
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.01       0.02       0.26
#> d[stroke = 1: High dose aspirin]                               0.09       0.14       0.04
#> d[stroke = 1: Indobufen]                                       0.00       0.00       0.00
#> d[stroke = 1: Low adjusted dose anti-coagulant]                0.00       0.00       0.00
#> d[stroke = 1: Low dose aspirin]                                0.22       0.11       0.03
#> d[stroke = 1: Low dose aspirin + copidogrel]                   0.16       0.11       0.03
#> d[stroke = 1: Low dose aspirin + dipyridamole]                 0.04       0.02       0.01
#> d[stroke = 1: Medium dose aspirin]                             0.01       0.00       0.00
#> d[stroke = 1: Placebo/Standard care]                           0.13       0.07       0.02
#> d[stroke = 1: Triflusal]                                       0.13       0.33       0.05
#> d[stroke = 1: Ximelagatran]                                    0.00       0.00       0.00
#>                                                          p_rank[16] p_rank[17]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]           0.00       0.00
#> d[stroke = 1: Acenocoumarol]                                   0.20       0.07
#> d[stroke = 1: Alternate day aspirin]                           0.00       0.00
#> d[stroke = 1: Dipyridamole]                                    0.01       0.01
#> d[stroke = 1: Fixed dose warfarin]                             0.00       0.00
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]          0.18       0.68
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.51       0.15
#> d[stroke = 1: High dose aspirin]                               0.02       0.02
#> d[stroke = 1: Indobufen]                                       0.00       0.00
#> d[stroke = 1: Low adjusted dose anti-coagulant]                0.00       0.00
#> d[stroke = 1: Low dose aspirin]                                0.02       0.01
#> d[stroke = 1: Low dose aspirin + copidogrel]                   0.01       0.01
#> d[stroke = 1: Low dose aspirin + dipyridamole]                 0.00       0.00
#> d[stroke = 1: Medium dose aspirin]                             0.00       0.00
#> d[stroke = 1: Placebo/Standard care]                           0.01       0.00
#> d[stroke = 1: Triflusal]                                       0.03       0.05
#> d[stroke = 1: Ximelagatran]                                    0.00       0.00

# Modify the default output with ggplot2 functionality
library(ggplot2)
plot(af_4b_rankprobs) + 
  facet_grid(Treatment~Study, labeller = label_wrap_gen(20)) + 
  theme(strip.text.y = element_text(angle = 0))

(af_4b_cumrankprobs <- posterior_rank_probs(af_fit_4b, cumulative = TRUE,
                                            newdata = data.frame(stroke = c(0, 1), 
                                                                 label = c("stroke = 0", "stroke = 1")), 
                                            study = label))
#> ------------------------------------------------------------- Study: stroke = 0 ---- 
#> 
#> Covariate values:
#>  stroke
#>       0
#> 
#>                                                          p_rank[1] p_rank[2] p_rank[3] p_rank[4]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.00      0.00      0.01      0.03
#> d[stroke = 0: Acenocoumarol]                                  0.25      0.49      0.63      0.72
#> d[stroke = 0: Alternate day aspirin]                          0.43      0.59      0.68      0.73
#> d[stroke = 0: Dipyridamole]                                   0.00      0.01      0.02      0.04
#> d[stroke = 0: Fixed dose warfarin]                            0.00      0.00      0.00      0.00
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.00      0.01      0.02      0.05
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.05      0.14      0.26      0.37
#> d[stroke = 0: High dose aspirin]                              0.03      0.09      0.16      0.24
#> d[stroke = 0: Indobufen]                                      0.18      0.42      0.63      0.76
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.04      0.18      0.40      0.60
#> d[stroke = 0: Low dose aspirin]                               0.00      0.00      0.00      0.00
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.00      0.00      0.00      0.01
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.01      0.04      0.10      0.21
#> d[stroke = 0: Medium dose aspirin]                            0.00      0.00      0.01      0.03
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.00      0.00      0.00
#> d[stroke = 0: Triflusal]                                      0.00      0.00      0.02      0.03
#> d[stroke = 0: Ximelagatran]                                   0.00      0.02      0.06      0.16
#>                                                          p_rank[5] p_rank[6] p_rank[7] p_rank[8]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.10      0.26      0.46      0.66
#> d[stroke = 0: Acenocoumarol]                                  0.79      0.82      0.85      0.87
#> d[stroke = 0: Alternate day aspirin]                          0.77      0.80      0.82      0.85
#> d[stroke = 0: Dipyridamole]                                   0.08      0.13      0.19      0.26
#> d[stroke = 0: Fixed dose warfarin]                            0.01      0.02      0.04      0.06
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.10      0.14      0.20      0.26
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.47      0.55      0.60      0.65
#> d[stroke = 0: High dose aspirin]                              0.30      0.36      0.40      0.45
#> d[stroke = 0: Indobufen]                                      0.84      0.89      0.92      0.94
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.74      0.82      0.89      0.92
#> d[stroke = 0: Low dose aspirin]                               0.00      0.00      0.01      0.02
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.02      0.04      0.07      0.11
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.32      0.42      0.51      0.58
#> d[stroke = 0: Medium dose aspirin]                            0.08      0.17      0.30      0.48
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.00      0.00      0.01
#> d[stroke = 0: Triflusal]                                      0.06      0.09      0.12      0.15
#> d[stroke = 0: Ximelagatran]                                   0.31      0.48      0.62      0.75
#>                                                          p_rank[9] p_rank[10] p_rank[11] p_rank[12]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.83       0.93       0.98       1.00
#> d[stroke = 0: Acenocoumarol]                                  0.90       0.92       0.94       0.95
#> d[stroke = 0: Alternate day aspirin]                          0.86       0.88       0.90       0.91
#> d[stroke = 0: Dipyridamole]                                   0.33       0.41       0.51       0.61
#> d[stroke = 0: Fixed dose warfarin]                            0.08       0.12       0.18       0.26
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.32       0.41       0.51       0.60
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.71       0.76       0.80       0.85
#> d[stroke = 0: High dose aspirin]                              0.49       0.54       0.59       0.64
#> d[stroke = 0: Indobufen]                                      0.95       0.97       0.98       0.98
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.95       0.97       0.99       0.99
#> d[stroke = 0: Low dose aspirin]                               0.05       0.11       0.22       0.39
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.17       0.26       0.38       0.52
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.66       0.74       0.81       0.87
#> d[stroke = 0: Medium dose aspirin]                            0.65       0.80       0.90       0.96
#> d[stroke = 0: Placebo/Standard care]                          0.02       0.04       0.08       0.16
#> d[stroke = 0: Triflusal]                                      0.18       0.23       0.28       0.34
#> d[stroke = 0: Ximelagatran]                                   0.84       0.91       0.95       0.97
#>                                                          p_rank[13] p_rank[14] p_rank[15]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]           1.00       1.00       1.00
#> d[stroke = 0: Acenocoumarol]                                   0.97       0.98       0.99
#> d[stroke = 0: Alternate day aspirin]                           0.92       0.94       0.95
#> d[stroke = 0: Dipyridamole]                                    0.70       0.79       0.87
#> d[stroke = 0: Fixed dose warfarin]                             0.34       0.44       0.57
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]          0.70       0.78       0.88
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]       0.88       0.91       0.95
#> d[stroke = 0: High dose aspirin]                               0.68       0.73       0.78
#> d[stroke = 0: Indobufen]                                       0.99       0.99       1.00
#> d[stroke = 0: Low adjusted dose anti-coagulant]                1.00       1.00       1.00
#> d[stroke = 0: Low dose aspirin]                                0.59       0.79       0.91
#> d[stroke = 0: Low dose aspirin + copidogrel]                   0.65       0.77       0.88
#> d[stroke = 0: Low dose aspirin + dipyridamole]                 0.91       0.95       0.97
#> d[stroke = 0: Medium dose aspirin]                             0.98       1.00       1.00
#> d[stroke = 0: Placebo/Standard care]                           0.29       0.47       0.70
#> d[stroke = 0: Triflusal]                                       0.40       0.47       0.55
#> d[stroke = 0: Ximelagatran]                                    0.99       0.99       1.00
#>                                                          p_rank[16] p_rank[17]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]           1.00          1
#> d[stroke = 0: Acenocoumarol]                                   1.00          1
#> d[stroke = 0: Alternate day aspirin]                           0.97          1
#> d[stroke = 0: Dipyridamole]                                    0.94          1
#> d[stroke = 0: Fixed dose warfarin]                             0.77          1
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]          0.95          1
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]       0.98          1
#> d[stroke = 0: High dose aspirin]                               0.86          1
#> d[stroke = 0: Indobufen]                                       1.00          1
#> d[stroke = 0: Low adjusted dose anti-coagulant]                1.00          1
#> d[stroke = 0: Low dose aspirin]                                0.98          1
#> d[stroke = 0: Low dose aspirin + copidogrel]                   0.97          1
#> d[stroke = 0: Low dose aspirin + dipyridamole]                 0.99          1
#> d[stroke = 0: Medium dose aspirin]                             1.00          1
#> d[stroke = 0: Placebo/Standard care]                           0.90          1
#> d[stroke = 0: Triflusal]                                       0.69          1
#> d[stroke = 0: Ximelagatran]                                    1.00          1
#> 
#> ------------------------------------------------------------- Study: stroke = 1 ---- 
#> 
#> Covariate values:
#>  stroke
#>       1
#> 
#>                                                          p_rank[1] p_rank[2] p_rank[3] p_rank[4]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.01      0.13      0.47      0.82
#> d[stroke = 1: Acenocoumarol]                                  0.03      0.05      0.07      0.08
#> d[stroke = 1: Alternate day aspirin]                          0.36      0.46      0.51      0.58
#> d[stroke = 1: Dipyridamole]                                   0.00      0.00      0.00      0.01
#> d[stroke = 1: Fixed dose warfarin]                            0.01      0.02      0.04      0.10
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.00      0.01      0.01      0.02
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00      0.00
#> d[stroke = 1: High dose aspirin]                              0.02      0.05      0.07      0.10
#> d[stroke = 1: Indobufen]                                      0.03      0.12      0.22      0.40
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.45      0.79      0.89      0.95
#> d[stroke = 1: Low dose aspirin]                               0.00      0.00      0.00      0.00
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.00      0.00      0.00      0.00
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.00      0.01      0.03      0.06
#> d[stroke = 1: Medium dose aspirin]                            0.00      0.00      0.00      0.00
#> d[stroke = 1: Placebo/Standard care]                          0.00      0.00      0.00      0.00
#> d[stroke = 1: Triflusal]                                      0.00      0.00      0.01      0.01
#> d[stroke = 1: Ximelagatran]                                   0.08      0.36      0.67      0.85
#>                                                          p_rank[5] p_rank[6] p_rank[7] p_rank[8]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.96      0.99      1.00      1.00
#> d[stroke = 1: Acenocoumarol]                                  0.09      0.12      0.14      0.16
#> d[stroke = 1: Alternate day aspirin]                          0.67      0.75      0.81      0.85
#> d[stroke = 1: Dipyridamole]                                   0.03      0.06      0.14      0.24
#> d[stroke = 1: Fixed dose warfarin]                            0.28      0.53      0.68      0.77
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.02      0.03      0.04      0.05
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.01      0.01      0.01      0.02
#> d[stroke = 1: High dose aspirin]                              0.17      0.27      0.37      0.45
#> d[stroke = 1: Indobufen]                                      0.67      0.83      0.91      0.95
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.98      1.00      1.00      1.00
#> d[stroke = 1: Low dose aspirin]                               0.00      0.00      0.01      0.04
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.01      0.03      0.06      0.14
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.12      0.26      0.44      0.60
#> d[stroke = 1: Medium dose aspirin]                            0.01      0.07      0.25      0.50
#> d[stroke = 1: Placebo/Standard care]                          0.00      0.01      0.03      0.09
#> d[stroke = 1: Triflusal]                                      0.03      0.06      0.11      0.16
#> d[stroke = 1: Ximelagatran]                                   0.95      0.98      0.99      1.00
#>                                                          p_rank[9] p_rank[10] p_rank[11] p_rank[12]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          1.00       1.00       1.00       1.00
#> d[stroke = 1: Acenocoumarol]                                  0.17       0.19       0.20       0.21
#> d[stroke = 1: Alternate day aspirin]                          0.87       0.89       0.91       0.92
#> d[stroke = 1: Dipyridamole]                                   0.36       0.48       0.60       0.73
#> d[stroke = 1: Fixed dose warfarin]                            0.84       0.88       0.91       0.94
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.06       0.07       0.07       0.08
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.03       0.04       0.04       0.05
#> d[stroke = 1: High dose aspirin]                              0.51       0.57       0.63       0.69
#> d[stroke = 1: Indobufen]                                      0.97       0.98       0.99       0.99
#> d[stroke = 1: Low adjusted dose anti-coagulant]               1.00       1.00       1.00       1.00
#> d[stroke = 1: Low dose aspirin]                               0.09       0.20       0.39       0.62
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.23       0.36       0.50       0.68
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.71       0.80       0.87       0.92
#> d[stroke = 1: Medium dose aspirin]                            0.73       0.87       0.95       0.98
#> d[stroke = 1: Placebo/Standard care]                          0.21       0.40       0.59       0.77
#> d[stroke = 1: Triflusal]                                      0.22       0.27       0.34       0.41
#> d[stroke = 1: Ximelagatran]                                   1.00       1.00       1.00       1.00
#>                                                          p_rank[13] p_rank[14] p_rank[15]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]           1.00       1.00       1.00
#> d[stroke = 1: Acenocoumarol]                                   0.23       0.27       0.73
#> d[stroke = 1: Alternate day aspirin]                           0.95       0.98       0.99
#> d[stroke = 1: Dipyridamole]                                    0.86       0.95       0.98
#> d[stroke = 1: Fixed dose warfarin]                             0.96       0.98       0.99
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]          0.09       0.10       0.14
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.06       0.07       0.34
#> d[stroke = 1: High dose aspirin]                               0.78       0.92       0.96
#> d[stroke = 1: Indobufen]                                       1.00       1.00       1.00
#> d[stroke = 1: Low adjusted dose anti-coagulant]                1.00       1.00       1.00
#> d[stroke = 1: Low dose aspirin]                                0.83       0.95       0.98
#> d[stroke = 1: Low dose aspirin + copidogrel]                   0.84       0.95       0.98
#> d[stroke = 1: Low dose aspirin + dipyridamole]                 0.96       0.99       0.99
#> d[stroke = 1: Medium dose aspirin]                             0.99       1.00       1.00
#> d[stroke = 1: Placebo/Standard care]                           0.90       0.96       0.99
#> d[stroke = 1: Triflusal]                                       0.54       0.87       0.92
#> d[stroke = 1: Ximelagatran]                                    1.00       1.00       1.00
#>                                                          p_rank[16] p_rank[17]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]           1.00          1
#> d[stroke = 1: Acenocoumarol]                                   0.93          1
#> d[stroke = 1: Alternate day aspirin]                           1.00          1
#> d[stroke = 1: Dipyridamole]                                    0.99          1
#> d[stroke = 1: Fixed dose warfarin]                             1.00          1
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]          0.32          1
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.85          1
#> d[stroke = 1: High dose aspirin]                               0.98          1
#> d[stroke = 1: Indobufen]                                       1.00          1
#> d[stroke = 1: Low adjusted dose anti-coagulant]                1.00          1
#> d[stroke = 1: Low dose aspirin]                                0.99          1
#> d[stroke = 1: Low dose aspirin + copidogrel]                   0.99          1
#> d[stroke = 1: Low dose aspirin + dipyridamole]                 1.00          1
#> d[stroke = 1: Medium dose aspirin]                             1.00          1
#> d[stroke = 1: Placebo/Standard care]                           1.00          1
#> d[stroke = 1: Triflusal]                                       0.95          1
#> d[stroke = 1: Ximelagatran]                                    1.00          1

plot(af_4b_cumrankprobs) + 
  facet_grid(Treatment~Study, labeller = label_wrap_gen(20)) + 
  theme(strip.text.y = element_text(angle = 0))

Model fit and comparison

Model fit can be checked using the dic() function:

(af_dic_1 <- dic(af_fit_1))
#> Residual deviance: 60.3 (on 61 data points)
#>                pD: 48.6
#>               DIC: 108.9
(af_dic_4b <- dic(af_fit_4b))
#> Residual deviance: 58.3 (on 61 data points)
#>                pD: 47.9
#>               DIC: 106.2

Both models fit the data well, having posterior mean residual deviance close to the number of data points. The DIC is slightly lower for the meta-regression model, although only by a couple of points (substantial differences are usually considered 3-5 points). The estimated heterogeneity standard deviation is much lower for the meta-regression model, suggesting that adjusting for the proportion of patients with prior stroke is explaining some of the heterogeneity in the data.

We can also examine the residual deviance contributions with the corresponding plot() method.

plot(af_dic_1)

plot(af_dic_4b)

References

Cooper, Nicola J., Alex J. Sutton, Danielle Morris, A. E. Ades, and Nicky J. Welton. 2009. “Addressing Between-Study Heterogeneity and Inconsistency in Mixed Treatment Comparisons: Application to Stroke Prevention Treatments in Individuals with Non-Rheumatic Atrial Fibrillation.” Statistics in Medicine 28 (14): 1861–81. https://doi.org/10.1002/sim.3594.