library(multinma)
options(mc.cores = parallel::detectCores())#> For execution on a local, multicore CPU with excess RAM we recommend calling
#> options(mc.cores = parallel::detectCores())
#>
#> Attaching package: 'multinma'
#> The following objects are masked from 'package:stats':
#>
#> dgamma, pgamma, qgammaThis vignette describes the analysis of data on the number of new
cases of diabetes in 22 trials of 6 antihypertensive drugs (Elliott and Meyer 2007;
Dias et al. 2011). The data are available
in this package as diabetes:
head(diabetes)
#> studyn studyc trtn trtc r n time
#> 1 1 MRC-E 1 Diuretic 43 1081 5.8
#> 2 1 MRC-E 2 Placebo 34 2213 5.8
#> 3 1 MRC-E 3 Beta Blocker 37 1102 5.8
#> 4 2 EWPH 1 Diuretic 29 416 4.7
#> 5 2 EWPH 2 Placebo 20 424 4.7
#> 6 3 SHEP 1 Diuretic 140 1631 3.0Setting up the network
We begin by setting up the network. We have arm-level count data
giving the number of new cases of diabetes (r) out of the
total (n) in each arm, so we use the function
set_agd_arm(). For computational efficiency, we let “Beta
Blocker” be set as the network reference treatment by default. Elliott and Meyer (2007) and Dias
et al. (2011) use “Diuretic” as the
reference, but it is a simple matter to transform the results after
fitting the NMA model.1
db_net <- set_agd_arm(diabetes,
study = studyc,
trt = trtc,
r = r,
n = n)
db_net
#> A network with 22 AgD studies (arm-based).
#>
#> ------------------------------------------------------- AgD studies (arm-based) ----
#> Study Treatment arms
#> AASK 3: Beta Blocker | ACE Inhibitor | CCB
#> ALLHAT 3: ACE Inhibitor | CCB | Diuretic
#> ALPINE 2: ARB | Diuretic
#> ANBP-2 2: ACE Inhibitor | Diuretic
#> ASCOT 2: Beta Blocker | CCB
#> CAPPP 2: Beta Blocker | ACE Inhibitor
#> CHARM 2: ARB | Placebo
#> DREAM 2: ACE Inhibitor | Placebo
#> EWPH 2: Diuretic | Placebo
#> FEVER 2: CCB | Placebo
#> ... plus 12 more studies
#>
#> Outcome type: count
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 6
#> Total number of studies: 22
#> Reference treatment is: Beta Blocker
#> Network is connectedWe also have details of length of follow-up in years in each trial
(time), which we will use as an offset with a cloglog link
function to model the data as rates. We do not have to specify this in
the function set_agd_arm(): any additional columns in the
data (e.g. offsets or covariates, here the column time)
will automatically be made available in the network.
Plot the network structure.
plot(db_net, weight_edges = TRUE, weight_nodes = TRUE)
Meta-analysis models
We fit both fixed effect (FE) and random effects (RE) models.
Fixed effect meta-analysis
First, we fit a fixed effect model using the nma()
function with trt_effects = "fixed". We use \mathrm{N}(0, 100^2) prior distributions for
the treatment effects d_k and
study-specific intercepts \mu_j. We can
examine the range of parameter values implied by these prior
distributions with the summary() method:
summary(normal(scale = 100))
#> A Normal prior distribution: location = 0, scale = 100.
#> 50% of the prior density lies between -67.45 and 67.45.
#> 95% of the prior density lies between -196 and 196.The model is fitted using the nma() function. We specify
that a cloglog link will be used with link = "cloglog" (the
Binomial likelihood is the default for these data), and specify the log
follow-up time offset using the regression formula
regression = ~offset(log(time)).
db_fit_FE <- nma(db_net,
trt_effects = "fixed",
link = "cloglog",
regression = ~offset(log(time)),
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100))
#> Note: No treatment classes specified in network, any interactions in `regression` formula will be separate (independent) for each treatment.
#> Use set_*() argument `trt_class` and nma() argument `class_interactions` to change this.
#> Note: Setting "Beta Blocker" as the network reference treatment.Basic parameter summaries are given by the print()
method:
db_fit_FE
#> A fixed effects NMA with a binomial likelihood (cloglog link).
#> Regression model: ~offset(log(time)).
#> Inference for Stan model: binomial_1par.
#> 4 chains, each with iter=2000; warmup=1000; thin=1;
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#>
#> mean se_mean sd 2.5% 25% 50% 75% 97.5%
#> d[ACE Inhibitor] -0.30 0.00 0.05 -0.39 -0.34 -0.30 -0.27 -0.21
#> d[ARB] -0.40 0.00 0.05 -0.49 -0.43 -0.40 -0.37 -0.31
#> d[CCB] -0.20 0.00 0.03 -0.26 -0.22 -0.20 -0.18 -0.14
#> d[Diuretic] 0.05 0.00 0.06 -0.05 0.02 0.06 0.09 0.16
#> d[Placebo] -0.19 0.00 0.05 -0.29 -0.23 -0.19 -0.16 -0.09
#> lp__ -37970.30 0.09 3.64 -37978.50 -37972.56 -37970.06 -37967.69 -37963.89
#> n_eff Rhat
#> d[ACE Inhibitor] 1657 1
#> d[ARB] 2483 1
#> d[CCB] 1915 1
#> d[Diuretic] 1868 1
#> d[Placebo] 1482 1
#> lp__ 1824 1
#>
#> Samples were drawn using NUTS(diag_e) at Thu Nov 5 17:19:52 2026.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).By default, summaries of the study-specific intercepts \mu_j are hidden, but could be examined by
changing the pars argument:
The prior and posterior distributions can be compared visually using
the plot_prior_posterior() function:
plot_prior_posterior(db_fit_FE)
Random effects meta-analysis
We now fit a random effects model using the nma()
function with trt_effects = "random". Again, we use \mathrm{N}(0, 100^2) prior distributions for
the treatment effects d_k and
study-specific intercepts \mu_j, and we
additionally use a \textrm{half-N}(5^2)
prior for the heterogeneity standard deviation \tau. We can examine the range of parameter
values implied by these prior distributions with the
summary() method:
summary(normal(scale = 100))
#> A Normal prior distribution: location = 0, scale = 100.
#> 50% of the prior density lies between -67.45 and 67.45.
#> 95% of the prior density lies between -196 and 196.
summary(half_normal(scale = 5))
#> A half-Normal prior distribution: 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 RE model
db_fit_RE <- nma(db_net,
trt_effects = "random",
link = "cloglog",
regression = ~offset(log(time)),
prior_intercept = normal(scale = 10),
prior_trt = normal(scale = 10),
prior_het = half_normal(scale = 5),
init_r = 0.5)
#> Note: No treatment classes specified in network, any interactions in `regression` formula will be separate (independent) for each treatment.
#> Use set_*() argument `trt_class` and nma() argument `class_interactions` to change this.
#> Note: Setting "Beta Blocker" as the network reference treatment.Basic parameter summaries are given by the print()
method:
db_fit_RE
#> A random effects NMA with a binomial likelihood (cloglog link).
#> Regression model: ~offset(log(time)).
#> Inference for Stan model: binomial_1par.
#> 4 chains, each with iter=2000; warmup=1000; thin=1;
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#>
#> mean se_mean sd 2.5% 25% 50% 75% 97.5%
#> d[ACE Inhibitor] -0.33 0.00 0.08 -0.49 -0.38 -0.33 -0.28 -0.19
#> d[ARB] -0.40 0.00 0.10 -0.60 -0.46 -0.40 -0.34 -0.22
#> d[CCB] -0.17 0.00 0.06 -0.30 -0.21 -0.17 -0.13 -0.04
#> d[Diuretic] 0.07 0.00 0.09 -0.10 0.01 0.07 0.13 0.24
#> d[Placebo] -0.22 0.00 0.08 -0.39 -0.27 -0.21 -0.16 -0.06
#> lp__ -37980.54 0.22 6.96 -37994.94 -37985.04 -37980.25 -37975.70 -37968.10
#> tau 0.13 0.00 0.04 0.05 0.10 0.13 0.16 0.23
#> n_eff Rhat
#> d[ACE Inhibitor] 1875 1
#> d[ARB] 2142 1
#> d[CCB] 2161 1
#> d[Diuretic] 2100 1
#> d[Placebo] 1759 1
#> lp__ 992 1
#> tau 999 1
#>
#> Samples were drawn using NUTS(diag_e) at Thu Nov 5 17:19:59 2026.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).By default, summaries of the study-specific intercepts \mu_j and study-specific relative effects
\delta_{jk} are hidden, but could be
examined by changing the pars argument:
The prior and posterior distributions can be compared visually using
the plot_prior_posterior() function:
plot_prior_posterior(db_fit_RE, prior = c("trt", "het"))
Model comparison
Model fit can be checked using the dic() function:
(dic_FE <- dic(db_fit_FE))
#> Residual deviance: 78.2 (on 48 data points)
#> pD: 27
#> DIC: 105.2
(dic_RE <- dic(db_fit_RE))
#> Residual deviance: 53.3 (on 48 data points)
#> pD: 37.9
#> DIC: 91.2The FE model is a very poor fit to the data, with a residual deviance much higher than the number of data points. The RE model fits the data better, and has a much lower DIC; we prefer the RE model.
We can also examine the residual deviance contributions with the
corresponding plot() method.
plot(dic_FE)
plot(dic_RE)
Further results
For comparison with Elliott and Meyer (2007) and Dias
et al. (2011), we can produce relative
effects against “Diuretic” using the relative_effects()
function with trt_ref = "Diuretic":
(db_releff_FE <- relative_effects(db_fit_FE, trt_ref = "Diuretic"))
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d[Beta Blocker] -0.05 0.06 -0.16 -0.09 -0.06 -0.02 0.05 1895 2372 1
#> d[ACE Inhibitor] -0.36 0.05 -0.46 -0.39 -0.36 -0.32 -0.26 5041 3299 1
#> d[ARB] -0.45 0.06 -0.58 -0.50 -0.45 -0.41 -0.33 4056 3578 1
#> d[CCB] -0.25 0.05 -0.35 -0.29 -0.25 -0.22 -0.15 3358 3133 1
#> d[Placebo] -0.25 0.06 -0.36 -0.28 -0.25 -0.21 -0.14 4578 3624 1
plot(db_releff_FE, ref_line = 0)
(db_releff_RE <- relative_effects(db_fit_RE, trt_ref = "Diuretic"))
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d[Beta Blocker] -0.07 0.09 -0.24 -0.13 -0.07 -0.01 0.10 2120 2478 1
#> d[ACE Inhibitor] -0.40 0.09 -0.58 -0.46 -0.40 -0.34 -0.24 4445 2588 1
#> d[ARB] -0.47 0.11 -0.70 -0.54 -0.47 -0.40 -0.26 3452 2448 1
#> d[CCB] -0.24 0.08 -0.41 -0.30 -0.24 -0.18 -0.07 4137 3318 1
#> d[Placebo] -0.29 0.09 -0.48 -0.34 -0.29 -0.22 -0.12 4330 2852 1
plot(db_releff_RE, ref_line = 0)
Dias et al. (2011) produce absolute predictions of the
probability of developing diabetes after three years, assuming a Normal
distribution on the baseline cloglog probability of developing diabetes
on diuretic treatment with mean -4.2
and precision 1.11. We can replicate
these results using the predict() method. We specify a data
frame of newdata, containing the time
offset(s) at which to produce predictions (here only 3 years). The
baseline argument takes a distr() distribution
object with which we specify the corresponding Normal distribution on
the baseline cloglog probability, and we set
baseline_trt = "Diuretic" to indicate that the baseline
distribution corresponds to “Diuretic” rather than the network reference
“Beta Blocker”. We set type = "response" to produce
predicted event probabilities (type = "link" would produce
predicted cloglog probabilities).
db_pred_FE <- predict(db_fit_FE,
newdata = data.frame(time = 3),
baseline = distr(qnorm, mean = -4.2, sd = 1.11^-0.5),
baseline_trt = "Diuretic",
type = "response")
db_pred_FE
#> ------------------------------------------------------------------ Study: New 1 ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[New 1: Beta Blocker] 0.06 0.06 0.01 0.02 0.04 0.08 0.22 4072 3918 1
#> pred[New 1: ACE Inhibitor] 0.05 0.05 0.01 0.02 0.03 0.06 0.17 4089 3919 1
#> pred[New 1: ARB] 0.04 0.04 0.00 0.02 0.03 0.05 0.16 4034 3878 1
#> pred[New 1: CCB] 0.05 0.05 0.01 0.02 0.03 0.06 0.19 4072 3919 1
#> pred[New 1: Diuretic] 0.06 0.06 0.01 0.02 0.04 0.08 0.23 4064 3878 1
#> pred[New 1: Placebo] 0.05 0.05 0.01 0.02 0.03 0.06 0.19 4065 3879 1
plot(db_pred_FE)
db_pred_RE <- predict(db_fit_RE,
newdata = data.frame(time = 3),
baseline = distr(qnorm, mean = -4.2, sd = 1.11^-0.5),
baseline_trt = "Diuretic",
type = "response")
db_pred_RE
#> ------------------------------------------------------------------ Study: New 1 ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[New 1: Beta Blocker] 0.06 0.07 0.01 0.02 0.04 0.08 0.24 4384 3681 1
#> pred[New 1: ACE Inhibitor] 0.05 0.05 0.00 0.02 0.03 0.06 0.18 4394 3930 1
#> pred[New 1: ARB] 0.04 0.05 0.00 0.02 0.03 0.05 0.17 4365 3829 1
#> pred[New 1: CCB] 0.05 0.06 0.01 0.02 0.04 0.07 0.21 4384 3678 1
#> pred[New 1: Diuretic] 0.07 0.07 0.01 0.02 0.04 0.08 0.26 4365 3649 1
#> pred[New 1: Placebo] 0.05 0.06 0.01 0.02 0.03 0.06 0.20 4386 3780 1
plot(db_pred_RE)
If the baseline and newdata arguments are
omitted, predicted probabilities will be produced for every study in the
network based on their follow-up times and estimated baseline cloglog
probabilities \mu_j:
db_pred_RE_studies <- predict(db_fit_RE, type = "response")
db_pred_RE_studies
#> ------------------------------------------------------------------- Study: AASK ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[AASK: Beta Blocker] 0.17 0.02 0.14 0.16 0.17 0.18 0.20 5901 3262 1
#> pred[AASK: ACE Inhibitor] 0.12 0.01 0.10 0.12 0.12 0.13 0.15 4607 3013 1
#> pred[AASK: ARB] 0.12 0.01 0.09 0.11 0.12 0.13 0.15 4192 2887 1
#> pred[AASK: CCB] 0.15 0.01 0.12 0.13 0.14 0.15 0.18 5481 3140 1
#> pred[AASK: Diuretic] 0.18 0.02 0.14 0.17 0.18 0.19 0.22 4467 3192 1
#> pred[AASK: Placebo] 0.14 0.02 0.11 0.13 0.14 0.15 0.17 4327 3266 1
#>
#> ----------------------------------------------------------------- Study: ALLHAT ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[ALLHAT: Beta Blocker] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 2928 2613 1
#> pred[ALLHAT: ACE Inhibitor] 0.03 0.00 0.02 0.03 0.03 0.03 0.04 4071 2634 1
#> pred[ALLHAT: ARB] 0.03 0.00 0.02 0.03 0.03 0.03 0.04 3752 2530 1
#> pred[ALLHAT: CCB] 0.04 0.00 0.03 0.03 0.04 0.04 0.05 3929 2804 1
#> pred[ALLHAT: Diuretic] 0.05 0.01 0.04 0.04 0.05 0.05 0.06 4497 2952 1
#> pred[ALLHAT: Placebo] 0.03 0.00 0.03 0.03 0.03 0.04 0.04 3797 2493 1
#>
#> ----------------------------------------------------------------- Study: ALPINE ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[ALPINE: Beta Blocker] 0.03 0.01 0.01 0.02 0.03 0.03 0.05 6163 3107 1
#> pred[ALPINE: ACE Inhibitor] 0.02 0.01 0.01 0.01 0.02 0.02 0.03 6487 3135 1
#> pred[ALPINE: ARB] 0.02 0.01 0.01 0.01 0.02 0.02 0.03 6392 3212 1
#> pred[ALPINE: CCB] 0.02 0.01 0.01 0.02 0.02 0.03 0.04 6788 3082 1
#> pred[ALPINE: Diuretic] 0.03 0.01 0.01 0.02 0.03 0.03 0.05 6901 3139 1
#> pred[ALPINE: Placebo] 0.02 0.01 0.01 0.02 0.02 0.03 0.04 6632 3043 1
#>
#> ----------------------------------------------------------------- Study: ANBP-2 ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[ANBP-2: Beta Blocker] 0.07 0.01 0.05 0.06 0.07 0.07 0.09 3029 2447 1
#> pred[ANBP-2: ACE Inhibitor] 0.05 0.01 0.04 0.04 0.05 0.05 0.06 4437 3017 1
#> pred[ANBP-2: ARB] 0.05 0.01 0.03 0.04 0.05 0.05 0.06 3839 2578 1
#> pred[ANBP-2: CCB] 0.06 0.01 0.04 0.05 0.06 0.06 0.07 4298 2909 1
#> pred[ANBP-2: Diuretic] 0.07 0.01 0.06 0.07 0.07 0.08 0.09 4971 3042 1
#> pred[ANBP-2: Placebo] 0.05 0.01 0.04 0.05 0.05 0.06 0.07 4431 2871 1
#>
#> ------------------------------------------------------------------ Study: ASCOT ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[ASCOT: Beta Blocker] 0.11 0.00 0.10 0.11 0.11 0.11 0.12 6494 3085 1
#> pred[ASCOT: ACE Inhibitor] 0.08 0.01 0.07 0.08 0.08 0.09 0.10 2291 2461 1
#> pred[ASCOT: ARB] 0.08 0.01 0.06 0.07 0.08 0.08 0.09 2507 2322 1
#> pred[ASCOT: CCB] 0.10 0.01 0.08 0.09 0.10 0.10 0.11 2558 2923 1
#> pred[ASCOT: Diuretic] 0.12 0.01 0.10 0.11 0.12 0.13 0.14 2431 2940 1
#> pred[ASCOT: Placebo] 0.09 0.01 0.08 0.09 0.09 0.10 0.11 2132 2776 1
#>
#> ------------------------------------------------------------------ Study: CAPPP ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[CAPPP: Beta Blocker] 0.07 0.00 0.07 0.07 0.07 0.08 0.08 5783 3079 1.01
#> pred[CAPPP: ACE Inhibitor] 0.05 0.00 0.05 0.05 0.05 0.06 0.06 2212 2006 1.00
#> pred[CAPPP: ARB] 0.05 0.01 0.04 0.05 0.05 0.05 0.06 2474 2354 1.00
#> pred[CAPPP: CCB] 0.06 0.00 0.05 0.06 0.06 0.07 0.07 3106 3123 1.00
#> pred[CAPPP: Diuretic] 0.08 0.01 0.07 0.08 0.08 0.08 0.10 2640 2634 1.00
#> pred[CAPPP: Placebo] 0.06 0.01 0.05 0.06 0.06 0.06 0.07 2111 2730 1.00
#>
#> ------------------------------------------------------------------ Study: CHARM ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[CHARM: Beta Blocker] 0.09 0.01 0.07 0.08 0.09 0.10 0.12 2860 2468 1
#> pred[CHARM: ACE Inhibitor] 0.07 0.01 0.05 0.06 0.07 0.07 0.09 3556 2378 1
#> pred[CHARM: ARB] 0.06 0.01 0.05 0.06 0.06 0.07 0.08 4652 2996 1
#> pred[CHARM: CCB] 0.08 0.01 0.06 0.07 0.08 0.08 0.10 3803 2830 1
#> pred[CHARM: Diuretic] 0.10 0.02 0.07 0.09 0.10 0.10 0.13 3789 2497 1
#> pred[CHARM: Placebo] 0.07 0.01 0.06 0.07 0.07 0.08 0.09 4305 2506 1
#>
#> ------------------------------------------------------------------ Study: DREAM ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[DREAM: Beta Blocker] 0.23 0.03 0.18 0.21 0.23 0.24 0.29 3016 2285 1
#> pred[DREAM: ACE Inhibitor] 0.17 0.02 0.13 0.16 0.17 0.18 0.21 4980 2874 1
#> pred[DREAM: ARB] 0.16 0.02 0.12 0.15 0.16 0.17 0.21 4220 2383 1
#> pred[DREAM: CCB] 0.20 0.03 0.15 0.18 0.19 0.21 0.25 4383 2866 1
#> pred[DREAM: Diuretic] 0.24 0.03 0.19 0.22 0.24 0.26 0.31 4958 2492 1
#> pred[DREAM: Placebo] 0.19 0.02 0.15 0.17 0.19 0.20 0.23 5096 2976 1
#>
#> ------------------------------------------------------------------- Study: EWPH ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[EWPH: Beta Blocker] 0.06 0.01 0.04 0.05 0.06 0.07 0.09 3960 2586 1
#> pred[EWPH: ACE Inhibitor] 0.05 0.01 0.03 0.04 0.04 0.05 0.06 5446 3085 1
#> pred[EWPH: ARB] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 5099 2476 1
#> pred[EWPH: CCB] 0.05 0.01 0.04 0.05 0.05 0.06 0.08 5268 2576 1
#> pred[EWPH: Diuretic] 0.07 0.01 0.05 0.06 0.07 0.07 0.09 5430 3076 1
#> pred[EWPH: Placebo] 0.05 0.01 0.03 0.04 0.05 0.06 0.07 5815 2613 1
#>
#> ------------------------------------------------------------------ Study: FEVER ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[FEVER: Beta Blocker] 0.04 0.01 0.03 0.04 0.04 0.04 0.05 2955 2712 1
#> pred[FEVER: ACE Inhibitor] 0.03 0.00 0.02 0.03 0.03 0.03 0.04 4989 2866 1
#> pred[FEVER: ARB] 0.03 0.00 0.02 0.03 0.03 0.03 0.04 4534 2710 1
#> pred[FEVER: CCB] 0.04 0.00 0.03 0.03 0.03 0.04 0.05 4676 3035 1
#> pred[FEVER: Diuretic] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 4621 2864 1
#> pred[FEVER: Placebo] 0.03 0.00 0.03 0.03 0.03 0.04 0.04 5193 3183 1
#>
#> ----------------------------------------------------------------- Study: HAPPHY ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[HAPPHY: Beta Blocker] 0.02 0 0.02 0.02 0.02 0.03 0.03 6053 2917 1
#> pred[HAPPHY: ACE Inhibitor] 0.02 0 0.01 0.02 0.02 0.02 0.02 4802 3209 1
#> pred[HAPPHY: ARB] 0.02 0 0.01 0.02 0.02 0.02 0.02 4102 2806 1
#> pred[HAPPHY: CCB] 0.02 0 0.02 0.02 0.02 0.02 0.03 5170 3236 1
#> pred[HAPPHY: Diuretic] 0.03 0 0.02 0.02 0.03 0.03 0.03 4284 3215 1
#> pred[HAPPHY: Placebo] 0.02 0 0.02 0.02 0.02 0.02 0.03 4507 2599 1
#>
#> ------------------------------------------------------------------- Study: HOPE ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[HOPE: Beta Blocker] 0.06 0.01 0.04 0.05 0.06 0.06 0.08 3180 2550 1
#> pred[HOPE: ACE Inhibitor] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 4869 2828 1
#> pred[HOPE: ARB] 0.04 0.01 0.03 0.04 0.04 0.04 0.05 4281 2769 1
#> pred[HOPE: CCB] 0.05 0.01 0.04 0.04 0.05 0.05 0.07 4548 2830 1
#> pred[HOPE: Diuretic] 0.06 0.01 0.05 0.06 0.06 0.07 0.08 4582 2978 1
#> pred[HOPE: Placebo] 0.05 0.01 0.04 0.04 0.05 0.05 0.06 5631 3059 1
#>
#> ---------------------------------------------------------------- Study: INSIGHT ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[INSIGHT: Beta Blocker] 0.07 0.01 0.05 0.06 0.06 0.07 0.09 2860 2638 1
#> pred[INSIGHT: ACE Inhibitor] 0.05 0.01 0.03 0.04 0.05 0.05 0.06 3756 2716 1
#> pred[INSIGHT: ARB] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 3836 2376 1
#> pred[INSIGHT: CCB] 0.06 0.01 0.04 0.05 0.05 0.06 0.07 4002 2676 1
#> pred[INSIGHT: Diuretic] 0.07 0.01 0.05 0.06 0.07 0.08 0.09 4344 2947 1
#> pred[INSIGHT: Placebo] 0.05 0.01 0.04 0.05 0.05 0.06 0.07 4062 2693 1
#>
#> ----------------------------------------------------------------- Study: INVEST ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[INVEST: Beta Blocker] 0.08 0.00 0.08 0.08 0.08 0.08 0.09 7301 2876 1
#> pred[INVEST: ACE Inhibitor] 0.06 0.00 0.05 0.06 0.06 0.06 0.07 2417 2558 1
#> pred[INVEST: ARB] 0.06 0.01 0.05 0.05 0.06 0.06 0.07 2495 2453 1
#> pred[INVEST: CCB] 0.07 0.00 0.06 0.07 0.07 0.07 0.08 2698 2785 1
#> pred[INVEST: Diuretic] 0.09 0.01 0.07 0.08 0.09 0.09 0.11 2458 2573 1
#> pred[INVEST: Placebo] 0.07 0.01 0.06 0.06 0.07 0.07 0.08 2177 2812 1
#>
#> ------------------------------------------------------------------- Study: LIFE ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[LIFE: Beta Blocker] 0.08 0.00 0.07 0.08 0.08 0.08 0.09 7399 3174 1
#> pred[LIFE: ACE Inhibitor] 0.06 0.01 0.05 0.06 0.06 0.06 0.07 2681 2387 1
#> pred[LIFE: ARB] 0.06 0.01 0.05 0.05 0.06 0.06 0.07 2532 2242 1
#> pred[LIFE: CCB] 0.07 0.01 0.06 0.07 0.07 0.07 0.08 3479 2916 1
#> pred[LIFE: Diuretic] 0.09 0.01 0.07 0.08 0.09 0.09 0.10 2901 2432 1
#> pred[LIFE: Placebo] 0.07 0.01 0.05 0.06 0.07 0.07 0.08 2411 2599 1
#>
#> ------------------------------------------------------------------ Study: MRC-E ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[MRC-E: Beta Blocker] 0.03 0 0.02 0.03 0.03 0.03 0.04 4193 3016 1
#> pred[MRC-E: ACE Inhibitor] 0.02 0 0.02 0.02 0.02 0.02 0.03 5088 3141 1
#> pred[MRC-E: ARB] 0.02 0 0.01 0.02 0.02 0.02 0.03 5211 2894 1
#> pred[MRC-E: CCB] 0.03 0 0.02 0.02 0.02 0.03 0.03 5146 3267 1
#> pred[MRC-E: Diuretic] 0.03 0 0.02 0.03 0.03 0.03 0.04 4272 3362 1
#> pred[MRC-E: Placebo] 0.02 0 0.02 0.02 0.02 0.03 0.03 4904 3238 1
#>
#> ----------------------------------------------------------------- Study: NORDIL ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[NORDIL: Beta Blocker] 0.05 0.00 0.04 0.05 0.05 0.05 0.06 6985 3046 1
#> pred[NORDIL: ACE Inhibitor] 0.04 0.00 0.03 0.03 0.04 0.04 0.04 2902 2488 1
#> pred[NORDIL: ARB] 0.03 0.00 0.03 0.03 0.03 0.04 0.04 2926 2654 1
#> pred[NORDIL: CCB] 0.04 0.00 0.04 0.04 0.04 0.04 0.05 3285 2874 1
#> pred[NORDIL: Diuretic] 0.05 0.01 0.04 0.05 0.05 0.06 0.06 2823 2667 1
#> pred[NORDIL: Placebo] 0.04 0.00 0.03 0.04 0.04 0.04 0.05 2613 2971 1
#>
#> ------------------------------------------------------------------ Study: PEACE ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[PEACE: Beta Blocker] 0.14 0.02 0.11 0.13 0.14 0.15 0.18 2855 2335 1
#> pred[PEACE: ACE Inhibitor] 0.10 0.01 0.08 0.09 0.10 0.11 0.13 4699 2481 1
#> pred[PEACE: ARB] 0.09 0.01 0.07 0.09 0.09 0.10 0.13 4311 2738 1
#> pred[PEACE: CCB] 0.12 0.02 0.09 0.11 0.12 0.13 0.15 4017 2231 1
#> pred[PEACE: Diuretic] 0.15 0.02 0.11 0.13 0.15 0.16 0.19 3954 2423 1
#> pred[PEACE: Placebo] 0.11 0.01 0.09 0.10 0.11 0.12 0.14 5177 2685 1
#>
#> ------------------------------------------------------------------ Study: SCOPE ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[SCOPE: Beta Blocker] 0.06 0.01 0.05 0.06 0.06 0.07 0.09 3151 2729 1
#> pred[SCOPE: ACE Inhibitor] 0.05 0.01 0.03 0.04 0.05 0.05 0.06 4625 3115 1
#> pred[SCOPE: ARB] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 4936 3072 1
#> pred[SCOPE: CCB] 0.05 0.01 0.04 0.05 0.05 0.06 0.07 4254 2781 1
#> pred[SCOPE: Diuretic] 0.07 0.01 0.05 0.06 0.07 0.08 0.09 4086 3162 1
#> pred[SCOPE: Placebo] 0.05 0.01 0.04 0.05 0.05 0.06 0.07 4957 2783 1
#>
#> ------------------------------------------------------------------- Study: SHEP ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[SHEP: Beta Blocker] 0.09 0.01 0.06 0.08 0.09 0.09 0.11 2980 2717 1
#> pred[SHEP: ACE Inhibitor] 0.06 0.01 0.05 0.06 0.06 0.07 0.08 4707 3090 1
#> pred[SHEP: ARB] 0.06 0.01 0.04 0.05 0.06 0.06 0.08 4486 3032 1
#> pred[SHEP: CCB] 0.07 0.01 0.05 0.07 0.07 0.08 0.10 4647 3027 1
#> pred[SHEP: Diuretic] 0.09 0.01 0.07 0.08 0.09 0.10 0.12 5141 3014 1
#> pred[SHEP: Placebo] 0.07 0.01 0.05 0.06 0.07 0.08 0.09 4734 2849 1
#>
#> ----------------------------------------------------------------- Study: STOP-2 ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[STOP-2: Beta Blocker] 0.05 0.00 0.05 0.05 0.05 0.06 0.06 4797 2959 1
#> pred[STOP-2: ACE Inhibitor] 0.04 0.00 0.03 0.04 0.04 0.04 0.05 2906 2629 1
#> pred[STOP-2: ARB] 0.04 0.00 0.03 0.03 0.04 0.04 0.04 2810 2496 1
#> pred[STOP-2: CCB] 0.05 0.00 0.04 0.04 0.05 0.05 0.05 3670 2662 1
#> pred[STOP-2: Diuretic] 0.06 0.01 0.05 0.05 0.06 0.06 0.07 3365 2682 1
#> pred[STOP-2: Placebo] 0.04 0.00 0.03 0.04 0.04 0.05 0.05 2711 2868 1
#>
#> ------------------------------------------------------------------ Study: VALUE ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[VALUE: Beta Blocker] 0.20 0.02 0.15 0.18 0.19 0.21 0.25 2933 2458 1
#> pred[VALUE: ACE Inhibitor] 0.15 0.02 0.11 0.13 0.14 0.16 0.19 3770 2582 1
#> pred[VALUE: ARB] 0.14 0.02 0.10 0.13 0.14 0.15 0.17 4152 2795 1
#> pred[VALUE: CCB] 0.17 0.02 0.13 0.16 0.17 0.18 0.21 4056 2601 1
#> pred[VALUE: Diuretic] 0.21 0.03 0.16 0.19 0.21 0.22 0.27 3866 2659 1
#> pred[VALUE: Placebo] 0.16 0.02 0.12 0.15 0.16 0.17 0.21 3909 2445 1
plot(db_pred_RE_studies)
We can also produce treatment rankings, rank probabilities, and cumulative rank probabilities.
(db_ranks <- posterior_ranks(db_fit_RE))
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> rank[Beta Blocker] 5.20 0.42 5 5 5 5 6 2226 NA 1
#> rank[ACE Inhibitor] 1.84 0.53 1 2 2 2 3 3407 3760 1
#> rank[ARB] 1.27 0.53 1 1 1 1 3 3321 3017 1
#> rank[CCB] 3.70 0.52 3 3 4 4 4 3958 3154 1
#> rank[Diuretic] 5.79 0.42 5 6 6 6 6 2438 NA 1
#> rank[Placebo] 3.20 0.59 2 3 3 4 4 4049 3313 1
plot(db_ranks)
(db_rankprobs <- posterior_rank_probs(db_fit_RE))
#> p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
#> d[Beta Blocker] 0.00 0.00 0.00 0.01 0.79 0.2
#> d[ACE Inhibitor] 0.22 0.71 0.06 0.00 0.00 0.0
#> d[ARB] 0.77 0.20 0.03 0.00 0.00 0.0
#> d[CCB] 0.00 0.02 0.27 0.71 0.01 0.0
#> d[Diuretic] 0.00 0.00 0.00 0.00 0.20 0.8
#> d[Placebo] 0.01 0.07 0.65 0.27 0.00 0.0
plot(db_rankprobs)
(db_cumrankprobs <- posterior_rank_probs(db_fit_RE, cumulative = TRUE))
#> p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
#> d[Beta Blocker] 0.00 0.00 0.00 0.01 0.8 1
#> d[ACE Inhibitor] 0.22 0.94 1.00 1.00 1.0 1
#> d[ARB] 0.77 0.97 1.00 1.00 1.0 1
#> d[CCB] 0.00 0.02 0.28 0.99 1.0 1
#> d[Diuretic] 0.00 0.00 0.00 0.00 0.2 1
#> d[Placebo] 0.01 0.08 0.72 1.00 1.0 1
plot(db_cumrankprobs)
References
Dias, S., N. J. Welton, A. J. Sutton, and A. E. Ades. 2011.
NICE DSU Technical Support Document 2: A Generalised
Linear Modelling Framework for Pair-Wise and Network Meta-Analysis of
Randomised Controlled Trials. National Institute for Health
and Care Excellence. https://sheffield.ac.uk/nice-dsu.
Elliott, W. J., and P. M. Meyer. 2007. “Incident Diabetes in
Clinical Trials of Antihypertensive Drugs: A Network
Meta-Analysis.” The Lancet 369 (9557): 201–7. https://doi.org/10.1016/s0140-6736(07)60108-1.