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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, qgamma

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
#> 1 ACTIVE-W      1 Standard adjusted dose anti-coagulant    3 Anti-coagulant  65 3371 4200
#> 2 ACTIVE-W      1         Low dose aspirin + copidogrel   16  Anti-platelet 106 3335 4180
#> 3 AFASAK 1      2                 Placebo/Standard care    1        Control  19  336  398
#> 4 AFASAK 1      2 Standard adjusted dose anti-coagulant    3 Anti-coagulant   9  335  413
#> 5 AFASAK 1      2                      Low dose aspirin    5  Anti-platelet  16  336  409
#> 6 AFASAK 2      3 Standard adjusted dose anti-coagulant    3 Anti-coagulant  11  170  355
#>   stroke year followup
#> 1   0.15 2006      1.3
#> 2   0.15 2006      1.3
#> 3   0.06 1989      1.2
#> 4   0.06 1989      1.2
#> 5   0.06 1989      1.2
#> 6   0.10 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         Treatment arms                                                                 
#>  ACTIVE-W      2: Standard adjusted dose anti-coagulant | Low dose aspirin + copidogrel       
#>  AFASAK 1      3: Standard adjusted dose anti-coagulant | Low dose aspirin | Placebo/Standa...
#>  AFASAK 2      4: Standard adjusted dose anti-coagulant | Fixed dose warfarin | Fixed dose ...
#>  BAATAF        2: Low adjusted dose anti-coagulant | Placebo/Standard care                    
#>  BAFTA         2: Standard adjusted dose anti-coagulant | Low dose aspirin                    
#>  CAFA          2: Standard adjusted dose anti-coagulant | Placebo/Standard care               
#>  Chinese ATAFS 2: Standard adjusted dose anti-coagulant | Low dose aspirin                    
#>  EAFT          3: Standard adjusted dose anti-coagulant | Medium dose aspirin | Placebo/Sta...
#>  ESPS 2        4: Dipyridamole | Low dose aspirin | Low dose aspirin + dipyridamole | Place...
#>  JAST          2: Low dose aspirin | Placebo/Standard care                                    
#>  ... 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 N(0,1002)\mathrm{N}(0, 100^2) prior distributions for the treatment effects dkd_k and study-specific intercepts μj\mu_j, and a half-N(52)\textrm{half-N}(5^2) prior for the heterogeneity standard deviation τ\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]                                -0.76    0.01 0.83    -2.46    -1.30    -0.73
#> d[Alternate day aspirin]                        -1.02    0.02 1.41    -4.32    -1.77    -0.84
#> d[Dipyridamole]                                  0.60    0.01 0.43    -0.27     0.31     0.60
#> d[Fixed dose warfarin]                           0.93    0.01 0.40     0.12     0.66     0.94
#> d[Fixed dose warfarin + low dose aspirin]        0.48    0.01 0.45    -0.43     0.20     0.49
#> d[Fixed dose warfarin + medium dose aspirin]     0.89    0.01 0.32     0.24     0.68     0.89
#> d[High dose aspirin]                             0.52    0.01 0.76    -1.00     0.02     0.52
#> d[Indobufen]                                     0.25    0.01 0.45    -0.66    -0.05     0.25
#> d[Low adjusted dose anti-coagulant]             -0.29    0.01 0.38    -1.07    -0.55    -0.27
#> d[Low dose aspirin]                              0.62    0.00 0.22     0.18     0.48     0.62
#> d[Low dose aspirin + copidogrel]                 0.51    0.01 0.35    -0.23     0.31     0.51
#> d[Low dose aspirin + dipyridamole]               0.27    0.01 0.48    -0.68    -0.04     0.28
#> d[Medium dose aspirin]                           0.39    0.00 0.20    -0.01     0.26     0.39
#> d[Placebo/Standard care]                         0.76    0.00 0.20     0.37     0.63     0.76
#> d[Triflusal]                                     0.66    0.01 0.63    -0.51     0.23     0.65
#> d[Ximelagatran]                                 -0.08    0.00 0.26    -0.60    -0.24    -0.08
#> lp__                                         -4771.72    0.24 7.27 -4786.34 -4776.46 -4771.43
#> tau                                              0.29    0.00 0.13     0.04     0.19     0.28
#>                                                   75%    97.5% n_eff Rhat
#> d[Acenocoumarol]                                -0.20     0.82  4248    1
#> d[Alternate day aspirin]                        -0.04     1.19  3805    1
#> d[Dipyridamole]                                  0.88     1.48  3151    1
#> d[Fixed dose warfarin]                           1.20     1.73  3758    1
#> d[Fixed dose warfarin + low dose aspirin]        0.77     1.36  2680    1
#> d[Fixed dose warfarin + medium dose aspirin]     1.10     1.53  3988    1
#> d[High dose aspirin]                             1.02     2.03  5605    1
#> d[Indobufen]                                     0.53     1.13  4481    1
#> d[Low adjusted dose anti-coagulant]             -0.04     0.44  3606    1
#> d[Low dose aspirin]                              0.76     1.04  2585    1
#> d[Low dose aspirin + copidogrel]                 0.71     1.22  3944    1
#> d[Low dose aspirin + dipyridamole]               0.60     1.19  3472    1
#> d[Medium dose aspirin]                           0.51     0.77  2720    1
#> d[Placebo/Standard care]                         0.89     1.16  1760    1
#> d[Triflusal]                                     1.09     1.92  3801    1
#> d[Ximelagatran]                                  0.08     0.47  3739    1
#> lp__                                         -4766.62 -4758.17   927    1
#> tau                                              0.37     0.58   720    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Sep 24 09:04:43 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 μj\mu_j and study-specific relative effects δjk\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 -0.63 -0.37     1788
#> d[Acenocoumarol]                             -1.52 0.87 -3.30 -2.08 -1.49 -0.94  0.10     3926
#> d[Alternate day aspirin]                     -1.78 1.41 -5.07 -2.52 -1.60 -0.81  0.45     5177
#> d[Dipyridamole]                              -0.16 0.41 -0.97 -0.43 -0.17  0.10  0.65     4954
#> d[Fixed dose warfarin]                        0.17 0.44 -0.71 -0.12  0.17  0.47  1.04     3646
#> d[Fixed dose warfarin + low dose aspirin]    -0.28 0.41 -1.10 -0.53 -0.27 -0.02  0.52     4639
#> d[Fixed dose warfarin + medium dose aspirin]  0.13 0.37 -0.63 -0.11  0.14  0.38  0.85     3006
#> d[High dose aspirin]                         -0.24 0.75 -1.71 -0.73 -0.24  0.25  1.28     6910
#> d[Indobufen]                                 -0.51 0.50 -1.53 -0.82 -0.51 -0.19  0.48     3476
#> d[Low adjusted dose anti-coagulant]          -1.05 0.36 -1.77 -1.28 -1.04 -0.81 -0.36     6029
#> d[Low dose aspirin]                          -0.14 0.21 -0.55 -0.28 -0.14  0.00  0.28     5535
#> d[Low dose aspirin + copidogrel]             -0.25 0.40 -1.09 -0.49 -0.24  0.00  0.57     3151
#> d[Low dose aspirin + dipyridamole]           -0.48 0.45 -1.40 -0.78 -0.48 -0.18  0.36     5148
#> d[Medium dose aspirin]                       -0.37 0.23 -0.83 -0.52 -0.38 -0.23  0.07     3677
#> d[Triflusal]                                 -0.10 0.67 -1.40 -0.54 -0.11  0.35  1.26     3441
#> d[Ximelagatran]                              -0.84 0.33 -1.47 -1.05 -0.84 -0.63 -0.18     2811
#>                                              Tail_ESS Rhat
#> d[Standard adjusted dose anti-coagulant]         2331    1
#> d[Acenocoumarol]                                 2787    1
#> d[Alternate day aspirin]                         2749    1
#> d[Dipyridamole]                                  2529    1
#> d[Fixed dose warfarin]                           2999    1
#> d[Fixed dose warfarin + low dose aspirin]        3025    1
#> d[Fixed dose warfarin + medium dose aspirin]     2710    1
#> d[High dose aspirin]                             3552    1
#> d[Indobufen]                                     3278    1
#> d[Low adjusted dose anti-coagulant]              3213    1
#> d[Low dose aspirin]                              3386    1
#> d[Low dose aspirin + copidogrel]                 2622    1
#> d[Low dose aspirin + dipyridamole]               3326    1
#> d[Medium dose aspirin]                           3367    1
#> d[Triflusal]                                     2635    1
#> d[Ximelagatran]                                  2983    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
#> rank[Standard adjusted dose anti-coagulant]      5.29 1.40  3.00   4   5   6     8     2826
#> rank[Acenocoumarol]                              3.12 3.15  1.00   1   2   3    13     3536
#> rank[Alternate day aspirin]                      3.74 4.32  1.00   1   2   5    17     5797
#> rank[Dipyridamole]                              11.28 3.69  4.00   9  11  14    17     4235
#> rank[Fixed dose warfarin]                       14.10 3.02  6.00  12  15  16    17     4320
#> rank[Fixed dose warfarin + low dose aspirin]    10.12 3.88  3.00   7  10  13    17     3877
#> rank[Fixed dose warfarin + medium dose aspirin] 14.02 2.74  7.00  12  15  16    17     2932
#> rank[High dose aspirin]                         10.32 5.23  1.98   6  11  16    17     6152
#> rank[Indobufen]                                  8.02 3.99  2.00   5   8  11    16     3954
#> rank[Low adjusted dose anti-coagulant]           3.75 2.15  1.00   2   3   5     9     3925
#> rank[Low dose aspirin]                          11.72 2.24  7.00  10  12  13    16     4489
#> rank[Low dose aspirin + copidogrel]             10.46 3.44  4.00   8  10  13    17     3756
#> rank[Low dose aspirin + dipyridamole]            8.26 3.97  2.00   5   8  11    16     4447
#> rank[Medium dose aspirin]                        9.11 2.16  5.00   8   9  10    14     4408
#> rank[Placebo/Standard care]                     13.38 1.83 10.00  12  14  15    17     3392
#> rank[Triflusal]                                 11.47 4.68  3.00   8  12  16    17     4017
#> rank[Ximelagatran]                               4.84 2.26  2.00   3   4   6    10     3628
#>                                                 Tail_ESS Rhat
#> rank[Standard adjusted dose anti-coagulant]         3727    1
#> rank[Acenocoumarol]                                 2838    1
#> rank[Alternate day aspirin]                         4044    1
#> rank[Dipyridamole]                                    NA    1
#> rank[Fixed dose warfarin]                             NA    1
#> rank[Fixed dose warfarin + low dose aspirin]        3211    1
#> rank[Fixed dose warfarin + medium dose aspirin]       NA    1
#> rank[High dose aspirin]                               NA    1
#> rank[Indobufen]                                     3210    1
#> rank[Low adjusted dose anti-coagulant]              3487    1
#> rank[Low dose aspirin]                              3812    1
#> rank[Low dose aspirin + copidogrel]                 3112    1
#> rank[Low dose aspirin + dipyridamole]               3490    1
#> rank[Medium dose aspirin]                           3363    1
#> rank[Placebo/Standard care]                         3334    1
#> rank[Triflusal]                                       NA    1
#> rank[Ximelagatran]                                  3156    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      0.08      0.20      0.28
#> d[Acenocoumarol]                                  0.37      0.28      0.10      0.05      0.04
#> d[Alternate day aspirin]                          0.47      0.17      0.07      0.04      0.03
#> d[Dipyridamole]                                   0.00      0.00      0.01      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      0.04      0.05
#> d[Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00      0.00      0.00
#> d[High dose aspirin]                              0.03      0.06      0.06      0.06      0.04
#> d[Indobufen]                                      0.01      0.05      0.07      0.07      0.08
#> d[Low adjusted dose anti-coagulant]               0.08      0.24      0.25      0.16      0.09
#> d[Low dose aspirin]                               0.00      0.00      0.00      0.00      0.00
#> d[Low dose aspirin + copidogrel]                  0.00      0.01      0.01      0.02      0.03
#> d[Low dose aspirin + dipyridamole]                0.01      0.04      0.08      0.07      0.08
#> d[Medium dose aspirin]                            0.00      0.00      0.00      0.01      0.03
#> d[Placebo/Standard care]                          0.00      0.00      0.00      0.00      0.00
#> d[Triflusal]                                      0.00      0.02      0.05      0.04      0.04
#> d[Ximelagatran]                                   0.02      0.09      0.19      0.22      0.17
#>                                              p_rank[6] p_rank[7] p_rank[8] p_rank[9]
#> d[Standard adjusted dose anti-coagulant]          0.23      0.13      0.05      0.01
#> d[Acenocoumarol]                                  0.03      0.03      0.02      0.02
#> d[Alternate day aspirin]                          0.03      0.03      0.02      0.02
#> d[Dipyridamole]                                   0.04      0.06      0.07      0.09
#> d[Fixed dose warfarin]                            0.01      0.02      0.02      0.03
#> d[Fixed dose warfarin + low dose aspirin]         0.06      0.08      0.09      0.10
#> d[Fixed dose warfarin + medium dose aspirin]      0.01      0.01      0.02      0.04
#> d[High dose aspirin]                              0.04      0.05      0.06      0.05
#> d[Indobufen]                                      0.11      0.10      0.09      0.08
#> d[Low adjusted dose anti-coagulant]               0.06      0.05      0.03      0.02
#> d[Low dose aspirin]                               0.01      0.02      0.05      0.08
#> d[Low dose aspirin + copidogrel]                  0.05      0.08      0.10      0.10
#> d[Low dose aspirin + dipyridamole]                0.09      0.09      0.09      0.08
#> d[Medium dose aspirin]                            0.07      0.11      0.18      0.19
#> d[Placebo/Standard care]                          0.00      0.00      0.01      0.02
#> d[Triflusal]                                      0.05      0.06      0.06      0.05
#> d[Ximelagatran]                                   0.12      0.08      0.05      0.03
#>                                              p_rank[10] p_rank[11] p_rank[12] p_rank[13]
#> d[Standard adjusted dose anti-coagulant]           0.00       0.00       0.00       0.00
#> d[Acenocoumarol]                                   0.01       0.01       0.01       0.01
#> d[Alternate day aspirin]                           0.01       0.01       0.01       0.02
#> d[Dipyridamole]                                    0.09       0.09       0.08       0.09
#> d[Fixed dose warfarin]                             0.04       0.05       0.06       0.07
#> d[Fixed dose warfarin + low dose aspirin]          0.09       0.09       0.08       0.07
#> d[Fixed dose warfarin + medium dose aspirin]       0.05       0.06       0.07       0.08
#> d[High dose aspirin]                               0.05       0.05       0.04       0.05
#> d[Indobufen]                                       0.07       0.06       0.04       0.04
#> d[Low adjusted dose anti-coagulant]                0.01       0.01       0.00       0.00
#> d[Low dose aspirin]                                0.13       0.15       0.18       0.16
#> d[Low dose aspirin + copidogrel]                   0.11       0.10       0.09       0.09
#> d[Low dose aspirin + dipyridamole]                 0.07       0.06       0.06       0.05
#> d[Medium dose aspirin]                             0.16       0.11       0.07       0.03
#> d[Placebo/Standard care]                           0.04       0.09       0.14       0.19
#> d[Triflusal]                                       0.05       0.05       0.06       0.05
#> d[Ximelagatran]                                    0.02       0.01       0.01       0.00
#>                                              p_rank[14] p_rank[15] p_rank[16] p_rank[17]
#> d[Standard adjusted dose anti-coagulant]           0.00       0.00       0.00       0.00
#> d[Acenocoumarol]                                   0.01       0.00       0.01       0.00
#> d[Alternate day aspirin]                           0.01       0.01       0.02       0.03
#> d[Dipyridamole]                                    0.09       0.09       0.08       0.07
#> d[Fixed dose warfarin]                             0.09       0.14       0.20       0.24
#> d[Fixed dose warfarin + low dose aspirin]          0.07       0.06       0.06       0.04
#> d[Fixed dose warfarin + medium dose aspirin]       0.11       0.16       0.22       0.17
#> d[High dose aspirin]                               0.05       0.06       0.09       0.17
#> d[Indobufen]                                       0.04       0.03       0.03       0.02
#> d[Low adjusted dose anti-coagulant]                0.00       0.00       0.00       0.00
#> d[Low dose aspirin]                                0.11       0.07       0.03       0.01
#> d[Low dose aspirin + copidogrel]                   0.07       0.07       0.05       0.03
#> d[Low dose aspirin + dipyridamole]                 0.04       0.04       0.03       0.02
#> d[Medium dose aspirin]                             0.02       0.01       0.00       0.00
#> d[Placebo/Standard care]                           0.22       0.18       0.09       0.03
#> d[Triflusal]                                       0.06       0.08       0.10       0.19
#> d[Ximelagatran]                                    0.00       0.00       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      0.01      0.09      0.29      0.58
#> d[Acenocoumarol]                                  0.37      0.65      0.75      0.80      0.84
#> d[Alternate day aspirin]                          0.47      0.64      0.71      0.75      0.78
#> d[Dipyridamole]                                   0.00      0.01      0.02      0.04      0.07
#> d[Fixed dose warfarin]                            0.00      0.00      0.00      0.01      0.01
#> d[Fixed dose warfarin + low dose aspirin]         0.00      0.02      0.05      0.09      0.13
#> d[Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00      0.00      0.01
#> d[High dose aspirin]                              0.03      0.08      0.14      0.20      0.24
#> d[Indobufen]                                      0.01      0.06      0.14      0.21      0.30
#> d[Low adjusted dose anti-coagulant]               0.08      0.32      0.57      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      0.05      0.08
#> d[Low dose aspirin + dipyridamole]                0.01      0.05      0.13      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      0.07      0.10      0.14
#> d[Ximelagatran]                                   0.02      0.11      0.30      0.52      0.68
#>                                              p_rank[6] p_rank[7] p_rank[8] p_rank[9]
#> d[Standard adjusted dose anti-coagulant]          0.81      0.94      0.99      1.00
#> d[Acenocoumarol]                                  0.87      0.90      0.92      0.94
#> d[Alternate day aspirin]                          0.81      0.83      0.86      0.88
#> d[Dipyridamole]                                   0.11      0.17      0.24      0.33
#> d[Fixed dose warfarin]                            0.03      0.04      0.07      0.10
#> d[Fixed dose warfarin + low dose aspirin]         0.19      0.26      0.35      0.45
#> d[Fixed dose warfarin + medium dose aspirin]      0.01      0.03      0.05      0.08
#> d[High dose aspirin]                              0.28      0.33      0.39      0.44
#> d[Indobufen]                                      0.40      0.50      0.59      0.67
#> d[Low adjusted dose anti-coagulant]               0.89      0.93      0.96      0.98
#> d[Low dose aspirin]                               0.01      0.03      0.08      0.16
#> d[Low dose aspirin + copidogrel]                  0.12      0.20      0.30      0.40
#> d[Low dose aspirin + dipyridamole]                0.37      0.46      0.55      0.64
#> d[Medium dose aspirin]                            0.11      0.22      0.40      0.59
#> d[Placebo/Standard care]                          0.00      0.00      0.01      0.02
#> d[Triflusal]                                      0.19      0.24      0.31      0.36
#> d[Ximelagatran]                                   0.80      0.88      0.93      0.96
#>                                              p_rank[10] p_rank[11] p_rank[12] p_rank[13]
#> d[Standard adjusted dose anti-coagulant]           1.00       1.00       1.00       1.00
#> d[Acenocoumarol]                                   0.95       0.96       0.97       0.98
#> d[Alternate day aspirin]                           0.89       0.91       0.92       0.94
#> d[Dipyridamole]                                    0.42       0.50       0.59       0.67
#> d[Fixed dose warfarin]                             0.14       0.19       0.25       0.32
#> d[Fixed dose warfarin + low dose aspirin]          0.54       0.62       0.70       0.77
#> d[Fixed dose warfarin + medium dose aspirin]       0.13       0.19       0.25       0.33
#> d[High dose aspirin]                               0.50       0.55       0.59       0.63
#> d[Indobufen]                                       0.73       0.79       0.84       0.88
#> d[Low adjusted dose anti-coagulant]                0.99       1.00       1.00       1.00
#> d[Low dose aspirin]                                0.29       0.44       0.62       0.78
#> d[Low dose aspirin + copidogrel]                   0.51       0.61       0.70       0.78
#> d[Low dose aspirin + dipyridamole]                 0.71       0.77       0.83       0.88
#> d[Medium dose aspirin]                             0.75       0.87       0.94       0.97
#> d[Placebo/Standard care]                           0.07       0.16       0.30       0.49
#> d[Triflusal]                                       0.41       0.46       0.52       0.57
#> d[Ximelagatran]                                    0.98       0.99       0.99       1.00
#>                                              p_rank[14] p_rank[15] p_rank[16] p_rank[17]
#> d[Standard adjusted dose anti-coagulant]           1.00       1.00       1.00          1
#> d[Acenocoumarol]                                   0.98       0.99       1.00          1
#> d[Alternate day aspirin]                           0.95       0.96       0.97          1
#> d[Dipyridamole]                                    0.76       0.85       0.93          1
#> d[Fixed dose warfarin]                             0.42       0.56       0.76          1
#> d[Fixed dose warfarin + low dose aspirin]          0.84       0.90       0.96          1
#> d[Fixed dose warfarin + medium dose aspirin]       0.45       0.61       0.83          1
#> d[High dose aspirin]                               0.69       0.75       0.83          1
#> d[Indobufen]                                       0.92       0.95       0.98          1
#> d[Low adjusted dose anti-coagulant]                1.00       1.00       1.00          1
#> d[Low dose aspirin]                                0.89       0.96       0.99          1
#> d[Low dose aspirin + copidogrel]                   0.86       0.92       0.97          1
#> d[Low dose aspirin + dipyridamole]                 0.92       0.96       0.98          1
#> d[Medium dose aspirin]                             0.99       1.00       1.00          1
#> d[Placebo/Standard care]                           0.71       0.89       0.97          1
#> d[Triflusal]                                       0.63       0.70       0.81          1
#> d[Ximelagatran]                                    1.00       1.00       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 N(0,1002)\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
#> https://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[.trtclassControl:stroke]                    0.70    0.01 0.45    -0.18     0.40     0.69
#> beta[.trtclassAnti-platelet:stroke]              0.94    0.01 0.43     0.11     0.67     0.93
#> beta[.trtclassMixed:stroke]                      3.96    0.03 2.12    -0.14     2.53     3.93
#> d[Acenocoumarol]                                 0.38    0.02 1.02    -1.67    -0.27     0.41
#> d[Alternate day aspirin]                        -0.93    0.03 1.46    -4.45    -1.71    -0.74
#> d[Dipyridamole]                                  0.57    0.01 0.41    -0.26     0.31     0.57
#> d[Fixed dose warfarin]                           0.64    0.01 0.38    -0.12     0.38     0.63
#> d[Fixed dose warfarin + low dose aspirin]        1.47    0.01 0.74     0.02     1.00     1.46
#> d[Fixed dose warfarin + medium dose aspirin]     1.00    0.00 0.30     0.42     0.81     1.00
#> d[High dose aspirin]                             0.42    0.01 0.74    -1.06    -0.06     0.43
#> d[Indobufen]                                    -0.42    0.01 0.49    -1.40    -0.73    -0.41
#> d[Low adjusted dose anti-coagulant]             -0.43    0.01 0.37    -1.17    -0.68    -0.42
#> d[Low dose aspirin]                              0.71    0.00 0.20     0.30     0.59     0.72
#> d[Low dose aspirin + copidogrel]                 0.65    0.01 0.30     0.06     0.48     0.66
#> d[Low dose aspirin + dipyridamole]               0.25    0.01 0.42    -0.60    -0.04     0.25
#> d[Medium dose aspirin]                           0.35    0.00 0.17     0.00     0.23     0.35
#> d[Placebo/Standard care]                         0.79    0.00 0.19     0.43     0.67     0.79
#> d[Triflusal]                                     0.92    0.01 0.60    -0.28     0.53     0.91
#> d[Ximelagatran]                                 -0.08    0.00 0.22    -0.51    -0.21    -0.08
#> lp__                                         -4771.33    0.21 7.21 -4785.67 -4776.02 -4771.02
#> tau                                              0.19    0.01 0.13     0.01     0.08     0.17
#>                                                   75%    97.5% n_eff Rhat
#> beta[.trtclassControl:stroke]                    0.99     1.60  4544 1.00
#> beta[.trtclassAnti-platelet:stroke]              1.21     1.79  4066 1.00
#> beta[.trtclassMixed:stroke]                      5.38     8.24  4887 1.00
#> d[Acenocoumarol]                                 1.08     2.36  3927 1.00
#> d[Alternate day aspirin]                         0.06     1.38  1921 1.00
#> d[Dipyridamole]                                  0.84     1.36  5132 1.00
#> d[Fixed dose warfarin]                           0.89     1.42  4316 1.00
#> d[Fixed dose warfarin + low dose aspirin]        1.94     2.94  5436 1.00
#> d[Fixed dose warfarin + medium dose aspirin]     1.19     1.60  5160 1.00
#> d[High dose aspirin]                             0.93     1.82  5967 1.00
#> d[Indobufen]                                    -0.09     0.55  4083 1.00
#> d[Low adjusted dose anti-coagulant]             -0.18     0.29  4316 1.00
#> d[Low dose aspirin]                              0.84     1.11  4992 1.00
#> d[Low dose aspirin + copidogrel]                 0.83     1.23  3037 1.00
#> d[Low dose aspirin + dipyridamole]               0.53     1.06  5102 1.00
#> d[Medium dose aspirin]                           0.46     0.69  4904 1.00
#> d[Placebo/Standard care]                         0.91     1.16  5042 1.00
#> d[Triflusal]                                     1.31     2.10  4716 1.00
#> d[Ximelagatran]                                  0.05     0.38  2878 1.00
#> lp__                                         -4766.36 -4758.07  1138 1.00
#> tau                                              0.26     0.50   330 1.02
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Sep 24 09:04:55 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).

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 μj\mu_j and study-specific relative effects δjk\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%
#> d[stroke = 0: Standard adjusted dose anti-coagulant]     -0.58 0.24 -1.07 -0.74 -0.58 -0.42
#> d[stroke = 0: Acenocoumarol]                             -1.37 0.84 -3.14 -1.90 -1.34 -0.83
#> d[stroke = 0: Alternate day aspirin]                     -1.79 1.44 -5.28 -2.55 -1.58 -0.82
#> d[stroke = 0: Dipyridamole]                              -0.29 0.45 -1.18 -0.59 -0.29  0.00
#> d[stroke = 0: Fixed dose warfarin]                        0.05 0.44 -0.78 -0.24  0.04  0.35
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]    -0.29 0.34 -0.96 -0.50 -0.28 -0.07
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin] -0.75 0.64 -2.02 -1.17 -0.75 -0.33
#> d[stroke = 0: High dose aspirin]                         -0.44 0.78 -1.96 -0.94 -0.43  0.07
#> d[stroke = 0: Indobufen]                                 -1.28 0.58 -2.45 -1.63 -1.27 -0.91
#> d[stroke = 0: Low adjusted dose anti-coagulant]          -1.01 0.33 -1.68 -1.23 -1.00 -0.78
#> d[stroke = 0: Low dose aspirin]                          -0.15 0.23 -0.60 -0.29 -0.15  0.00
#> d[stroke = 0: Low dose aspirin + copidogrel]             -0.21 0.36 -0.94 -0.42 -0.20  0.01
#> d[stroke = 0: Low dose aspirin + dipyridamole]           -0.61 0.46 -1.52 -0.92 -0.61 -0.31
#> d[stroke = 0: Medium dose aspirin]                       -0.51 0.27 -1.05 -0.69 -0.51 -0.34
#> d[stroke = 0: Triflusal]                                  0.06 0.63 -1.18 -0.35  0.05  0.47
#> d[stroke = 0: Ximelagatran]                              -0.66 0.32 -1.28 -0.86 -0.66 -0.46
#>                                                          97.5% Bulk_ESS Tail_ESS Rhat
#> d[stroke = 0: Standard adjusted dose anti-coagulant]     -0.10     4961     2951    1
#> d[stroke = 0: Acenocoumarol]                              0.19     4136     2471    1
#> d[stroke = 0: Alternate day aspirin]                      0.49     2761     1489    1
#> d[stroke = 0: Dipyridamole]                               0.57     5029     2923    1
#> d[stroke = 0: Fixed dose warfarin]                        0.94     5014     3242    1
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]     0.35     5277     2501    1
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]  0.50     4419     2761    1
#> d[stroke = 0: High dose aspirin]                          1.05     5582     3104    1
#> d[stroke = 0: Indobufen]                                 -0.13     4243     2945    1
#> d[stroke = 0: Low adjusted dose anti-coagulant]          -0.37     4771     2960    1
#> d[stroke = 0: Low dose aspirin]                           0.30     4967     2971    1
#> d[stroke = 0: Low dose aspirin + copidogrel]              0.50     3991     1975    1
#> d[stroke = 0: Low dose aspirin + dipyridamole]            0.27     5123     2683    1
#> d[stroke = 0: Medium dose aspirin]                        0.01     4909     2949    1
#> d[stroke = 0: Triflusal]                                  1.32     5343     2752    1
#> d[stroke = 0: Ximelagatran]                               0.00     3474     2568    1
#> 
#> ------------------------------------------------------------- Study: stroke = 1 ---- 
#> 
#> Covariate values:
#>  stroke
#>       1
#> 
#>                                                           mean   sd  2.5%   25%   50%   75%
#> d[stroke = 1: Standard adjusted dose anti-coagulant]     -1.28 0.36 -2.00 -1.52 -1.27 -1.05
#> d[stroke = 1: Acenocoumarol]                              1.89 2.30 -2.63  0.33  1.88  3.39
#> d[stroke = 1: Alternate day aspirin]                     -1.56 1.49 -5.06 -2.36 -1.36 -0.54
#> d[stroke = 1: Dipyridamole]                              -0.06 0.38 -0.82 -0.31 -0.05  0.20
#> d[stroke = 1: Fixed dose warfarin]                       -0.65 0.53 -1.70 -1.01 -0.65 -0.31
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]     2.97 2.17 -1.23  1.54  2.93  4.37
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]  2.51 1.64 -0.73  1.41  2.47  3.57
#> d[stroke = 1: High dose aspirin]                         -0.20 0.72 -1.64 -0.66 -0.21  0.28
#> d[stroke = 1: Indobufen]                                 -1.04 0.53 -2.09 -1.38 -1.04 -0.69
#> d[stroke = 1: Low adjusted dose anti-coagulant]          -1.71 0.52 -2.76 -2.07 -1.70 -1.36
#> d[stroke = 1: Low dose aspirin]                           0.09 0.29 -0.50 -0.09  0.10  0.28
#> d[stroke = 1: Low dose aspirin + copidogrel]              0.03 0.39 -0.76 -0.20  0.03  0.27
#> d[stroke = 1: Low dose aspirin + dipyridamole]           -0.38 0.41 -1.18 -0.65 -0.36 -0.10
#> d[stroke = 1: Medium dose aspirin]                       -0.28 0.25 -0.80 -0.42 -0.27 -0.12
#> d[stroke = 1: Triflusal]                                  0.30 0.66 -1.05 -0.12  0.29  0.73
#> d[stroke = 1: Ximelagatran]                              -1.36 0.42 -2.18 -1.62 -1.35 -1.09
#>                                                          97.5% Bulk_ESS Tail_ESS Rhat
#> d[stroke = 1: Standard adjusted dose anti-coagulant]     -0.59     4678     2880 1.00
#> d[stroke = 1: Acenocoumarol]                              6.40     4773     2853 1.00
#> d[stroke = 1: Alternate day aspirin]                      0.84     2796     1507 1.00
#> d[stroke = 1: Dipyridamole]                               0.70     5281     2735 1.00
#> d[stroke = 1: Fixed dose warfarin]                        0.42     3967     3029 1.00
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]     7.38     5166     2866 1.00
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]  5.82     5201     3085 1.00
#> d[stroke = 1: High dose aspirin]                          1.22     5824     3250 1.00
#> d[stroke = 1: Indobufen]                                  0.00     4470     2866 1.00
#> d[stroke = 1: Low adjusted dose anti-coagulant]          -0.70     4333     2847 1.00
#> d[stroke = 1: Low dose aspirin]                           0.65     4829     2338 1.00
#> d[stroke = 1: Low dose aspirin + copidogrel]              0.79     3764     2440 1.00
#> d[stroke = 1: Low dose aspirin + dipyridamole]            0.41     5452     3293 1.00
#> d[stroke = 1: Medium dose aspirin]                        0.21     4399     2268 1.01
#> d[stroke = 1: Triflusal]                                  1.58     4603     2921 1.00
#> d[stroke = 1: Ximelagatran]                              -0.55     4235     2734 1.00
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[ , , 2:3] <- sweep(af_4b_beta[ , , 2:3], 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)[1] <- "beta[.trtclassAnti-coagulant:stroke]"

# Summarise
summary(af_4b_beta)
#>                                       mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS
#> beta[.trtclassAnti-coagulant:stroke] -0.70 0.45 -1.60 -0.99 -0.69 -0.40  0.18     4527
#> beta[.trtclassAnti-platelet:stroke]   0.24 0.34 -0.44  0.02  0.24  0.46  0.92     4498
#> beta[.trtclassMixed:stroke]           3.26 2.14 -0.85  1.81  3.24  4.67  7.62     4973
#>                                      Tail_ESS Rhat
#> beta[.trtclassAnti-coagulant:stroke]     2957 1.01
#> beta[.trtclassAnti-platelet:stroke]      2766 1.00
#> beta[.trtclassMixed:stroke]              2927 1.00
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]      7.74 1.89    4   6   8   9    12
#> rank[stroke = 0: Acenocoumarol]                              4.02 3.72    1   1   3   5    15
#> rank[stroke = 0: Alternate day aspirin]                      4.06 4.52    1   1   2   5    17
#> rank[stroke = 0: Dipyridamole]                              11.04 3.75    4   8  11  14    17
#> rank[stroke = 0: Fixed dose warfarin]                       14.07 2.89    7  12  15  16    17
#> rank[stroke = 0: Fixed dose warfarin + low dose aspirin]    11.06 3.66    4   8  11  14    17
#> rank[stroke = 0: Fixed dose warfarin + medium dose aspirin]  7.06 4.42    1   3   6  10    16
#> rank[stroke = 0: High dose aspirin]                          9.63 5.25    1   5  10  15    17
#> rank[stroke = 0: Indobufen]                                  3.65 2.81    1   2   3   4    12
#> rank[stroke = 0: Low adjusted dose anti-coagulant]           4.52 2.46    1   3   4   6    11
#> rank[stroke = 0: Low dose aspirin]                          12.88 1.97    9  12  13  14    16
#> rank[stroke = 0: Low dose aspirin + copidogrel]             12.08 2.92    5  10  12  14    17
#> rank[stroke = 0: Low dose aspirin + dipyridamole]            7.82 3.68    2   5   7  10    16
#> rank[stroke = 0: Medium dose aspirin]                        8.59 2.23    4   7   9  10    13
#> rank[stroke = 0: Placebo/Standard care]                     14.31 1.94   10  13  15  16    17
#> rank[stroke = 0: Triflusal]                                 13.46 3.97    4  11  15  17    17
#> rank[stroke = 0: Ximelagatran]                               7.01 2.64    3   5   7   9    13
#>                                                             Bulk_ESS Tail_ESS Rhat
#> rank[stroke = 0: Standard adjusted dose anti-coagulant]         3854     2898    1
#> rank[stroke = 0: Acenocoumarol]                                 4346     3030    1
#> rank[stroke = 0: Alternate day aspirin]                         4678     3002    1
#> rank[stroke = 0: Dipyridamole]                                  4871       NA    1
#> rank[stroke = 0: Fixed dose warfarin]                           4637       NA    1
#> rank[stroke = 0: Fixed dose warfarin + low dose aspirin]        5615       NA    1
#> rank[stroke = 0: Fixed dose warfarin + medium dose aspirin]     4750     3276    1
#> rank[stroke = 0: High dose aspirin]                             5114       NA    1
#> rank[stroke = 0: Indobufen]                                     3557     3054    1
#> rank[stroke = 0: Low adjusted dose anti-coagulant]              3977     3293    1
#> rank[stroke = 0: Low dose aspirin]                              3921     3059    1
#> rank[stroke = 0: Low dose aspirin + copidogrel]                 3618     2192    1
#> rank[stroke = 0: Low dose aspirin + dipyridamole]               4833     2269    1
#> rank[stroke = 0: Medium dose aspirin]                           4854     3400    1
#> rank[stroke = 0: Placebo/Standard care]                         3626       NA    1
#> rank[stroke = 0: Triflusal]                                     4230       NA    1
#> rank[stroke = 0: Ximelagatran]                                  3215     2498    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]      3.65 1.14    2   3   4   4     6
#> rank[stroke = 1: Acenocoumarol]                             13.28 4.35    1  14  15  16    17
#> rank[stroke = 1: Alternate day aspirin]                      4.53 4.08    1   1   3   7    14
#> rank[stroke = 1: Dipyridamole]                              10.49 2.72    5   8  11  13    15
#> rank[stroke = 1: Fixed dose warfarin]                        7.09 2.80    3   5   6   8    14
#> rank[stroke = 1: Fixed dose warfarin + low dose aspirin]    15.85 2.86    5  16  17  17    17
#> rank[stroke = 1: Fixed dose warfarin + medium dose aspirin] 15.42 1.96    8  15  16  16    17
#> rank[stroke = 1: High dose aspirin]                          9.44 3.93    2   6   9  13    16
#> rank[stroke = 1: Indobufen]                                  4.98 2.21    1   4   5   6    10
#> rank[stroke = 1: Low adjusted dose anti-coagulant]           2.02 1.33    1   1   2   2     6
#> rank[stroke = 1: Low dose aspirin]                          11.85 1.83    8  11  12  13    15
#> rank[stroke = 1: Low dose aspirin + copidogrel]             11.17 2.40    6  10  11  13    15
#> rank[stroke = 1: Low dose aspirin + dipyridamole]            8.19 2.63    3   6   8  10    14
#> rank[stroke = 1: Medium dose aspirin]                        8.60 1.70    6   7   8  10    12
#> rank[stroke = 1: Placebo/Standard care]                     11.11 1.93    7  10  11  12    15
#> rank[stroke = 1: Triflusal]                                 12.15 3.07    5  10  13  14    17
#> rank[stroke = 1: Ximelagatran]                               3.19 1.42    1   2   3   4     6
#>                                                             Bulk_ESS Tail_ESS Rhat
#> rank[stroke = 1: Standard adjusted dose anti-coagulant]         3430     3123    1
#> rank[stroke = 1: Acenocoumarol]                                 4356       NA    1
#> rank[stroke = 1: Alternate day aspirin]                         4322     2597    1
#> rank[stroke = 1: Dipyridamole]                                  4788     3203    1
#> rank[stroke = 1: Fixed dose warfarin]                           3731     3121    1
#> rank[stroke = 1: Fixed dose warfarin + low dose aspirin]        3190       NA    1
#> rank[stroke = 1: Fixed dose warfarin + medium dose aspirin]     3130       NA    1
#> rank[stroke = 1: High dose aspirin]                             4899     3215    1
#> rank[stroke = 1: Indobufen]                                     4053     3112    1
#> rank[stroke = 1: Low adjusted dose anti-coagulant]              3010     2911    1
#> rank[stroke = 1: Low dose aspirin]                              4122     3328    1
#> rank[stroke = 1: Low dose aspirin + copidogrel]                 3662     2420    1
#> rank[stroke = 1: Low dose aspirin + dipyridamole]               4464     2955    1
#> rank[stroke = 1: Medium dose aspirin]                           4235     3679    1
#> rank[stroke = 1: Placebo/Standard care]                         4379     3372    1
#> rank[stroke = 1: Triflusal]                                     4014     3015    1
#> rank[stroke = 1: Ximelagatran]                                  2616     2661    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]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.00      0.00      0.00
#> d[stroke = 0: Acenocoumarol]                                  0.25      0.24      0.14
#> d[stroke = 0: Alternate day aspirin]                          0.44      0.14      0.09
#> d[stroke = 0: Dipyridamole]                                   0.00      0.00      0.02
#> d[stroke = 0: Fixed dose warfarin]                            0.00      0.00      0.00
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.00      0.00      0.01
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.04      0.10      0.12
#> d[stroke = 0: High dose aspirin]                              0.03      0.06      0.07
#> d[stroke = 0: Indobufen]                                      0.17      0.26      0.20
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.05      0.14      0.21
#> d[stroke = 0: Low dose aspirin]                               0.00      0.00      0.00
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.00      0.00      0.01
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.01      0.03      0.07
#> d[stroke = 0: Medium dose aspirin]                            0.00      0.00      0.01
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.00      0.00
#> d[stroke = 0: Triflusal]                                      0.00      0.01      0.01
#> d[stroke = 0: Ximelagatran]                                   0.00      0.01      0.05
#>                                                          p_rank[4] p_rank[5] p_rank[6]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.03      0.08      0.15
#> d[stroke = 0: Acenocoumarol]                                  0.10      0.05      0.03
#> d[stroke = 0: Alternate day aspirin]                          0.06      0.04      0.03
#> d[stroke = 0: Dipyridamole]                                   0.03      0.04      0.05
#> d[stroke = 0: Fixed dose warfarin]                            0.00      0.01      0.01
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.02      0.04      0.05
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.12      0.10      0.07
#> d[stroke = 0: High dose aspirin]                              0.08      0.06      0.05
#> d[stroke = 0: Indobufen]                                      0.12      0.07      0.04
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.19      0.14      0.09
#> d[stroke = 0: Low dose aspirin]                               0.00      0.00      0.00
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.01      0.01      0.02
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.09      0.12      0.11
#> d[stroke = 0: Medium dose aspirin]                            0.02      0.06      0.09
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.00      0.00
#> d[stroke = 0: Triflusal]                                      0.02      0.02      0.03
#> d[stroke = 0: Ximelagatran]                                   0.10      0.15      0.16
#>                                                          p_rank[7] p_rank[8] p_rank[9]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.21      0.19      0.16
#> d[stroke = 0: Acenocoumarol]                                  0.03      0.03      0.02
#> d[stroke = 0: Alternate day aspirin]                          0.02      0.02      0.02
#> d[stroke = 0: Dipyridamole]                                   0.06      0.06      0.07
#> d[stroke = 0: Fixed dose warfarin]                            0.01      0.02      0.03
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.05      0.07      0.07
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.06      0.06      0.05
#> d[stroke = 0: High dose aspirin]                              0.05      0.04      0.05
#> d[stroke = 0: Indobufen]                                      0.03      0.03      0.02
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.05      0.04      0.03
#> d[stroke = 0: Low dose aspirin]                               0.00      0.01      0.03
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.02      0.04      0.06
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.08      0.08      0.08
#> d[stroke = 0: Medium dose aspirin]                            0.14      0.17      0.17
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.00      0.01
#> d[stroke = 0: Triflusal]                                      0.03      0.03      0.04
#> d[stroke = 0: Ximelagatran]                                   0.14      0.12      0.09
#>                                                          p_rank[10] p_rank[11] p_rank[12]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]           0.10       0.05       0.02
#> d[stroke = 0: Acenocoumarol]                                   0.02       0.02       0.01
#> d[stroke = 0: Alternate day aspirin]                           0.02       0.02       0.02
#> d[stroke = 0: Dipyridamole]                                    0.09       0.10       0.10
#> d[stroke = 0: Fixed dose warfarin]                             0.04       0.06       0.07
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]          0.09       0.09       0.10
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]       0.04       0.04       0.04
#> d[stroke = 0: High dose aspirin]                               0.05       0.05       0.05
#> d[stroke = 0: Indobufen]                                       0.02       0.01       0.00
#> d[stroke = 0: Low adjusted dose anti-coagulant]                0.02       0.01       0.01
#> d[stroke = 0: Low dose aspirin]                                0.06       0.12       0.18
#> d[stroke = 0: Low dose aspirin + copidogrel]                   0.10       0.12       0.13
#> d[stroke = 0: Low dose aspirin + dipyridamole]                 0.08       0.07       0.05
#> d[stroke = 0: Medium dose aspirin]                             0.14       0.11       0.05
#> d[stroke = 0: Placebo/Standard care]                           0.02       0.05       0.08
#> d[stroke = 0: Triflusal]                                       0.04       0.05       0.06
#> d[stroke = 0: Ximelagatran]                                    0.07       0.04       0.03
#>                                                          p_rank[13] p_rank[14] p_rank[15]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]           0.01       0.00       0.00
#> d[stroke = 0: Acenocoumarol]                                   0.01       0.01       0.01
#> d[stroke = 0: Alternate day aspirin]                           0.02       0.02       0.01
#> d[stroke = 0: Dipyridamole]                                    0.09       0.07       0.09
#> d[stroke = 0: Fixed dose warfarin]                             0.09       0.11       0.12
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]          0.10       0.09       0.09
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]       0.03       0.04       0.04
#> d[stroke = 0: High dose aspirin]                               0.05       0.05       0.05
#> d[stroke = 0: Indobufen]                                       0.01       0.01       0.00
#> d[stroke = 0: Low adjusted dose anti-coagulant]                0.00       0.00       0.00
#> d[stroke = 0: Low dose aspirin]                                0.19       0.19       0.13
#> d[stroke = 0: Low dose aspirin + copidogrel]                   0.14       0.12       0.11
#> d[stroke = 0: Low dose aspirin + dipyridamole]                 0.04       0.03       0.03
#> d[stroke = 0: Medium dose aspirin]                             0.03       0.01       0.00
#> d[stroke = 0: Placebo/Standard care]                           0.13       0.17       0.23
#> d[stroke = 0: Triflusal]                                       0.06       0.07       0.08
#> d[stroke = 0: Ximelagatran]                                    0.02       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]                                   0.02       0.00
#> d[stroke = 0: Alternate day aspirin]                           0.02       0.03
#> d[stroke = 0: Dipyridamole]                                    0.08       0.06
#> d[stroke = 0: Fixed dose warfarin]                             0.21       0.22
#> 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       0.13
#> 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]                                0.06       0.02
#> d[stroke = 0: Low dose aspirin + copidogrel]                   0.08       0.04
#> 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]                           0.19       0.11
#> d[stroke = 0: Triflusal]                                       0.15       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]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.01      0.12      0.34
#> d[stroke = 1: Acenocoumarol]                                  0.04      0.02      0.01
#> d[stroke = 1: Alternate day aspirin]                          0.37      0.10      0.05
#> d[stroke = 1: Dipyridamole]                                   0.00      0.00      0.00
#> d[stroke = 1: Fixed dose warfarin]                            0.00      0.01      0.02
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.00      0.01      0.01
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00
#> d[stroke = 1: High dose aspirin]                              0.02      0.02      0.03
#> d[stroke = 1: Indobufen]                                      0.04      0.08      0.11
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.43      0.35      0.10
#> d[stroke = 1: Low dose aspirin]                               0.00      0.00      0.00
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.00      0.00      0.00
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.00      0.01      0.01
#> d[stroke = 1: Medium dose aspirin]                            0.00      0.00      0.00
#> d[stroke = 1: Placebo/Standard care]                          0.00      0.00      0.00
#> d[stroke = 1: Triflusal]                                      0.00      0.00      0.01
#> d[stroke = 1: Ximelagatran]                                   0.08      0.26      0.31
#>                                                          p_rank[4] p_rank[5] p_rank[6]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.34      0.14      0.03
#> d[stroke = 1: Acenocoumarol]                                  0.01      0.01      0.02
#> d[stroke = 1: Alternate day aspirin]                          0.06      0.07      0.09
#> d[stroke = 1: Dipyridamole]                                   0.00      0.02      0.04
#> d[stroke = 1: Fixed dose warfarin]                            0.07      0.18      0.25
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.00      0.01      0.01
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00
#> d[stroke = 1: High dose aspirin]                              0.03      0.06      0.09
#> d[stroke = 1: Indobufen]                                      0.18      0.27      0.15
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.06      0.04      0.02
#> d[stroke = 1: Low dose aspirin]                               0.00      0.00      0.00
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.00      0.01      0.02
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.03      0.07      0.14
#> d[stroke = 1: Medium dose aspirin]                            0.00      0.01      0.06
#> d[stroke = 1: Placebo/Standard care]                          0.00      0.00      0.01
#> d[stroke = 1: Triflusal]                                      0.01      0.01      0.03
#> d[stroke = 1: Ximelagatran]                                   0.20      0.09      0.03
#>                                                          p_rank[7] p_rank[8] p_rank[9]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.01      0.00      0.00
#> d[stroke = 1: Acenocoumarol]                                  0.02      0.02      0.02
#> d[stroke = 1: Alternate day aspirin]                          0.05      0.03      0.03
#> d[stroke = 1: Dipyridamole]                                   0.08      0.10      0.11
#> d[stroke = 1: Fixed dose warfarin]                            0.15      0.08      0.06
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.01      0.01      0.01
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.00      0.01      0.01
#> d[stroke = 1: High dose aspirin]                              0.11      0.08      0.07
#> d[stroke = 1: Indobufen]                                      0.08      0.04      0.02
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.01      0.00      0.00
#> d[stroke = 1: Low dose aspirin]                               0.01      0.03      0.06
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.04      0.07      0.10
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.18      0.15      0.12
#> d[stroke = 1: Medium dose aspirin]                            0.18      0.26      0.22
#> d[stroke = 1: Placebo/Standard care]                          0.02      0.06      0.12
#> d[stroke = 1: Triflusal]                                      0.04      0.05      0.05
#> d[stroke = 1: Ximelagatran]                                   0.01      0.01      0.00
#>                                                          p_rank[10] p_rank[11] p_rank[12]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]           0.00       0.00       0.00
#> d[stroke = 1: Acenocoumarol]                                   0.01       0.01       0.01
#> d[stroke = 1: Alternate day aspirin]                           0.02       0.02       0.02
#> d[stroke = 1: Dipyridamole]                                    0.12       0.13       0.12
#> d[stroke = 1: Fixed dose warfarin]                             0.04       0.04       0.03
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]          0.01       0.01       0.01
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.01       0.00       0.01
#> d[stroke = 1: High dose aspirin]                               0.06       0.06       0.06
#> d[stroke = 1: Indobufen]                                       0.01       0.01       0.00
#> d[stroke = 1: Low adjusted dose anti-coagulant]                0.00       0.00       0.00
#> d[stroke = 1: Low dose aspirin]                                0.13       0.18       0.23
#> d[stroke = 1: Low dose aspirin + copidogrel]                   0.12       0.15       0.16
#> d[stroke = 1: Low dose aspirin + dipyridamole]                 0.09       0.07       0.05
#> d[stroke = 1: Medium dose aspirin]                             0.14       0.07       0.03
#> d[stroke = 1: Placebo/Standard care]                           0.18       0.19       0.18
#> d[stroke = 1: Triflusal]                                       0.06       0.06       0.08
#> d[stroke = 1: Ximelagatran]                                    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       0.44
#> d[stroke = 1: Alternate day aspirin]                           0.03       0.03       0.01
#> d[stroke = 1: Dipyridamole]                                    0.12       0.09       0.03
#> d[stroke = 1: Fixed dose warfarin]                             0.03       0.02       0.01
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]          0.01       0.01       0.05
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.01       0.02       0.27
#> d[stroke = 1: High dose aspirin]                               0.09       0.13       0.04
#> d[stroke = 1: Indobufen]                                       0.01       0.00       0.00
#> d[stroke = 1: Low adjusted dose anti-coagulant]                0.00       0.00       0.00
#> d[stroke = 1: Low dose aspirin]                                0.21       0.11       0.03
#> d[stroke = 1: Low dose aspirin + copidogrel]                   0.17       0.10       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]                                       0.12       0.34       0.06
#> 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]                                   0.21       0.07
#> d[stroke = 1: Alternate day aspirin]                           0.00       0.00
#> d[stroke = 1: Dipyridamole]                                    0.02       0.01
#> d[stroke = 1: Fixed dose warfarin]                             0.01       0.00
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]          0.19       0.67
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.50       0.16
#> d[stroke = 1: High dose aspirin]                               0.01       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       0.00
#> 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]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.00      0.00      0.01
#> d[stroke = 0: Acenocoumarol]                                  0.25      0.49      0.63
#> d[stroke = 0: Alternate day aspirin]                          0.44      0.58      0.66
#> d[stroke = 0: Dipyridamole]                                   0.00      0.00      0.02
#> d[stroke = 0: Fixed dose warfarin]                            0.00      0.00      0.00
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.00      0.01      0.02
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.04      0.14      0.26
#> d[stroke = 0: High dose aspirin]                              0.03      0.09      0.16
#> d[stroke = 0: Indobufen]                                      0.17      0.43      0.64
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.05      0.19      0.40
#> d[stroke = 0: Low dose aspirin]                               0.00      0.00      0.00
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.00      0.00      0.01
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.01      0.04      0.11
#> d[stroke = 0: Medium dose aspirin]                            0.00      0.00      0.01
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.00      0.00
#> d[stroke = 0: Triflusal]                                      0.00      0.01      0.02
#> d[stroke = 0: Ximelagatran]                                   0.00      0.01      0.06
#>                                                          p_rank[4] p_rank[5] p_rank[6]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.03      0.11      0.26
#> d[stroke = 0: Acenocoumarol]                                  0.72      0.78      0.81
#> d[stroke = 0: Alternate day aspirin]                          0.73      0.77      0.80
#> d[stroke = 0: Dipyridamole]                                   0.05      0.09      0.14
#> d[stroke = 0: Fixed dose warfarin]                            0.00      0.01      0.02
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.04      0.09      0.14
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.37      0.48      0.55
#> d[stroke = 0: High dose aspirin]                              0.24      0.31      0.35
#> d[stroke = 0: Indobufen]                                      0.76      0.83      0.87
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.59      0.74      0.83
#> d[stroke = 0: Low dose aspirin]                               0.00      0.00      0.00
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.02      0.03      0.05
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.21      0.32      0.43
#> d[stroke = 0: Medium dose aspirin]                            0.03      0.08      0.18
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.00      0.00
#> d[stroke = 0: Triflusal]                                      0.04      0.06      0.09
#> d[stroke = 0: Ximelagatran]                                   0.16      0.31      0.48
#>                                                          p_rank[7] p_rank[8] p_rank[9]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]          0.48      0.66      0.82
#> d[stroke = 0: Acenocoumarol]                                  0.84      0.87      0.89
#> d[stroke = 0: Alternate day aspirin]                          0.82      0.84      0.86
#> d[stroke = 0: Dipyridamole]                                   0.20      0.26      0.33
#> d[stroke = 0: Fixed dose warfarin]                            0.03      0.05      0.09
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]         0.19      0.26      0.33
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]      0.61      0.67      0.72
#> d[stroke = 0: High dose aspirin]                              0.40      0.44      0.49
#> d[stroke = 0: Indobufen]                                      0.90      0.93      0.95
#> d[stroke = 0: Low adjusted dose anti-coagulant]               0.88      0.92      0.95
#> d[stroke = 0: Low dose aspirin]                               0.01      0.02      0.05
#> d[stroke = 0: Low dose aspirin + copidogrel]                  0.07      0.11      0.17
#> d[stroke = 0: Low dose aspirin + dipyridamole]                0.51      0.59      0.67
#> d[stroke = 0: Medium dose aspirin]                            0.32      0.49      0.66
#> d[stroke = 0: Placebo/Standard care]                          0.00      0.01      0.02
#> d[stroke = 0: Triflusal]                                      0.11      0.14      0.18
#> d[stroke = 0: Ximelagatran]                                   0.62      0.74      0.83
#>                                                          p_rank[10] p_rank[11] p_rank[12]
#> d[stroke = 0: Standard adjusted dose anti-coagulant]           0.92       0.97       0.99
#> d[stroke = 0: Acenocoumarol]                                   0.91       0.93       0.94
#> d[stroke = 0: Alternate day aspirin]                           0.87       0.89       0.90
#> d[stroke = 0: Dipyridamole]                                    0.42       0.52       0.62
#> d[stroke = 0: Fixed dose warfarin]                             0.13       0.19       0.26
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]          0.42       0.51       0.61
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]       0.76       0.80       0.84
#> d[stroke = 0: High dose aspirin]                               0.54       0.59       0.64
#> d[stroke = 0: Indobufen]                                       0.96       0.97       0.98
#> d[stroke = 0: Low adjusted dose anti-coagulant]                0.97       0.98       0.99
#> d[stroke = 0: Low dose aspirin]                                0.11       0.23       0.41
#> d[stroke = 0: Low dose aspirin + copidogrel]                   0.27       0.39       0.52
#> d[stroke = 0: Low dose aspirin + dipyridamole]                 0.75       0.82       0.87
#> d[stroke = 0: Medium dose aspirin]                             0.80       0.91       0.96
#> d[stroke = 0: Placebo/Standard care]                           0.04       0.09       0.17
#> d[stroke = 0: Triflusal]                                       0.22       0.27       0.33
#> d[stroke = 0: Ximelagatran]                                    0.90       0.94       0.96
#>                                                          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]                                   0.96       0.97       0.98
#> d[stroke = 0: Alternate day aspirin]                           0.92       0.94       0.95
#> d[stroke = 0: Dipyridamole]                                    0.70       0.78       0.86
#> d[stroke = 0: Fixed dose warfarin]                             0.34       0.45       0.57
#> d[stroke = 0: Fixed dose warfarin + low dose aspirin]          0.71       0.80       0.88
#> d[stroke = 0: Fixed dose warfarin + medium dose aspirin]       0.88       0.91       0.95
#> d[stroke = 0: High dose aspirin]                               0.69       0.74       0.79
#> d[stroke = 0: Indobufen]                                       0.98       0.99       0.99
#> d[stroke = 0: Low adjusted dose anti-coagulant]                1.00       1.00       1.00
#> d[stroke = 0: Low dose aspirin]                                0.60       0.79       0.92
#> d[stroke = 0: Low dose aspirin + copidogrel]                   0.66       0.78       0.89
#> d[stroke = 0: Low dose aspirin + dipyridamole]                 0.92       0.95       0.97
#> d[stroke = 0: Medium dose aspirin]                             0.99       1.00       1.00
#> d[stroke = 0: Placebo/Standard care]                           0.30       0.47       0.70
#> d[stroke = 0: Triflusal]                                       0.38       0.46       0.54
#> d[stroke = 0: Ximelagatran]                                    0.98       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]                             0.78          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]                               0.87          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]                   0.96          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]                           0.89          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]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.01      0.13      0.47
#> d[stroke = 1: Acenocoumarol]                                  0.04      0.06      0.08
#> d[stroke = 1: Alternate day aspirin]                          0.37      0.47      0.52
#> d[stroke = 1: Dipyridamole]                                   0.00      0.00      0.00
#> d[stroke = 1: Fixed dose warfarin]                            0.00      0.02      0.04
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.00      0.01      0.02
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.00      0.00      0.00
#> d[stroke = 1: High dose aspirin]                              0.02      0.04      0.07
#> d[stroke = 1: Indobufen]                                      0.04      0.12      0.23
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.43      0.78      0.88
#> d[stroke = 1: Low dose aspirin]                               0.00      0.00      0.00
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.00      0.00      0.00
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.00      0.01      0.03
#> d[stroke = 1: Medium dose aspirin]                            0.00      0.00      0.00
#> d[stroke = 1: Placebo/Standard care]                          0.00      0.00      0.00
#> d[stroke = 1: Triflusal]                                      0.00      0.01      0.01
#> d[stroke = 1: Ximelagatran]                                   0.08      0.34      0.65
#>                                                          p_rank[4] p_rank[5] p_rank[6]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          0.81      0.95      0.98
#> d[stroke = 1: Acenocoumarol]                                  0.09      0.10      0.12
#> d[stroke = 1: Alternate day aspirin]                          0.58      0.66      0.75
#> d[stroke = 1: Dipyridamole]                                   0.01      0.03      0.07
#> d[stroke = 1: Fixed dose warfarin]                            0.11      0.29      0.53
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.02      0.03      0.03
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.01      0.01      0.01
#> d[stroke = 1: High dose aspirin]                              0.10      0.17      0.26
#> d[stroke = 1: Indobufen]                                      0.40      0.67      0.82
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.94      0.97      0.99
#> d[stroke = 1: Low dose aspirin]                               0.00      0.00      0.01
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.01      0.02      0.04
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.05      0.12      0.27
#> d[stroke = 1: Medium dose aspirin]                            0.00      0.02      0.08
#> d[stroke = 1: Placebo/Standard care]                          0.00      0.00      0.01
#> d[stroke = 1: Triflusal]                                      0.02      0.03      0.06
#> d[stroke = 1: Ximelagatran]                                   0.85      0.94      0.98
#>                                                          p_rank[7] p_rank[8] p_rank[9]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]          1.00      1.00      1.00
#> d[stroke = 1: Acenocoumarol]                                  0.14      0.16      0.18
#> d[stroke = 1: Alternate day aspirin]                          0.80      0.83      0.86
#> d[stroke = 1: Dipyridamole]                                   0.15      0.26      0.37
#> d[stroke = 1: Fixed dose warfarin]                            0.69      0.77      0.83
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]         0.04      0.05      0.06
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]      0.02      0.03      0.04
#> d[stroke = 1: High dose aspirin]                              0.36      0.45      0.52
#> d[stroke = 1: Indobufen]                                      0.90      0.94      0.96
#> d[stroke = 1: Low adjusted dose anti-coagulant]               0.99      1.00      1.00
#> d[stroke = 1: Low dose aspirin]                               0.01      0.04      0.10
#> d[stroke = 1: Low dose aspirin + copidogrel]                  0.07      0.14      0.24
#> d[stroke = 1: Low dose aspirin + dipyridamole]                0.45      0.60      0.72
#> d[stroke = 1: Medium dose aspirin]                            0.26      0.52      0.73
#> d[stroke = 1: Placebo/Standard care]                          0.03      0.08      0.21
#> d[stroke = 1: Triflusal]                                      0.10      0.15      0.21
#> d[stroke = 1: Ximelagatran]                                   0.99      0.99      1.00
#>                                                          p_rank[10] p_rank[11] p_rank[12]
#> d[stroke = 1: Standard adjusted dose anti-coagulant]           1.00       1.00       1.00
#> d[stroke = 1: Acenocoumarol]                                   0.19       0.20       0.21
#> d[stroke = 1: Alternate day aspirin]                           0.88       0.90       0.92
#> d[stroke = 1: Dipyridamole]                                    0.49       0.62       0.73
#> d[stroke = 1: Fixed dose warfarin]                             0.87       0.90       0.93
#> d[stroke = 1: Fixed dose warfarin + low dose aspirin]          0.07       0.07       0.08
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.04       0.05       0.05
#> d[stroke = 1: High dose aspirin]                               0.58       0.64       0.70
#> d[stroke = 1: Indobufen]                                       0.98       0.98       0.99
#> d[stroke = 1: Low adjusted dose anti-coagulant]                1.00       1.00       1.00
#> d[stroke = 1: Low dose aspirin]                                0.22       0.40       0.62
#> d[stroke = 1: Low dose aspirin + copidogrel]                   0.36       0.51       0.67
#> d[stroke = 1: Low dose aspirin + dipyridamole]                 0.81       0.88       0.93
#> d[stroke = 1: Medium dose aspirin]                             0.87       0.95       0.98
#> d[stroke = 1: Placebo/Standard care]                           0.39       0.58       0.76
#> d[stroke = 1: Triflusal]                                       0.26       0.33       0.40
#> d[stroke = 1: Ximelagatran]                                    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]                                   0.24       0.28       0.72
#> 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       0.15
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.06       0.07       0.35
#> d[stroke = 1: High dose aspirin]                               0.79       0.92       0.96
#> d[stroke = 1: Indobufen]                                       0.99       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]                   0.85       0.95       0.98
#> d[stroke = 1: Low dose aspirin + dipyridamole]                 0.97       0.99       1.00
#> 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]                                       0.53       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]          0.33          1
#> d[stroke = 1: Fixed dose warfarin + medium dose aspirin]       0.84          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]                                1.00          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: 60.4 (on 61 data points)
#>                pD: 48.9
#>               DIC: 109.4
(af_dic_4b <- dic(af_fit_4b))
#> Residual deviance: 58.3 (on 61 data points)
#>                pD: 48.3
#>               DIC: 106.6

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, N. J., A. J. Sutton, D. Morris, A. E. Ades, and N. 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.