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 -0.33 -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] 0.06 0.00 0.06 -0.05 0.02 0.06 0.09 0.17
#> d[Placebo] -0.19 0.00 0.05 -0.28 -0.22 -0.19 -0.16 -0.10
#> lp__ -37970.22 0.09 3.65 -37978.52 -37972.31 -37969.85 -37967.71 -37963.98
#> n_eff Rhat
#> d[ACE Inhibitor] 1467 1
#> d[ARB] 2027 1
#> d[CCB] 1987 1
#> d[Diuretic] 1971 1
#> d[Placebo] 1391 1
#> lp__ 1836 1
#>
#> Samples were drawn using NUTS(diag_e) at Wed Mar 6 13:20:12 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).By default, summaries of the study-specific intercepts \(\mu_j\) are hidden, but could be examined
by changing the pars argument:
The prior and posterior distributions can be compared visually using
the plot_prior_posterior() function:
plot_prior_posterior(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: 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 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 -0.18
#> d[ARB] -0.40 0.00 0.09 -0.59 -0.46 -0.40 -0.34 -0.22
#> d[CCB] -0.17 0.00 0.07 -0.30 -0.21 -0.17 -0.13 -0.03
#> d[Diuretic] 0.08 0.00 0.09 -0.09 0.02 0.07 0.13 0.26
#> d[Placebo] -0.21 0.00 0.08 -0.38 -0.27 -0.21 -0.16 -0.05
#> lp__ -37980.98 0.24 6.94 -37995.42 -37985.45 -37980.56 -37976.11 -37968.62
#> tau 0.13 0.00 0.04 0.05 0.10 0.12 0.15 0.23
#> n_eff Rhat
#> d[ACE Inhibitor] 1968 1
#> d[ARB] 2466 1
#> d[CCB] 2100 1
#> d[Diuretic] 2123 1
#> d[Placebo] 1960 1
#> lp__ 864 1
#> tau 868 1
#>
#> Samples were drawn using NUTS(diag_e) at Wed Mar 6 13:20:30 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).By default, summaries of the study-specific intercepts \(\mu_j\) 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 (on 48 data points)
#> pD: 26.8
#> DIC: 104.9
(dic_RE <- dic(db_fit_RE))
#> Residual deviance: 53.7 (on 48 data points)
#> pD: 38
#> DIC: 91.7The 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] -0.06 0.06 -0.17 -0.09 -0.06 -0.02 0.05 1999 2665 1
#> d[ACE Inhibitor] -0.36 0.05 -0.46 -0.40 -0.36 -0.32 -0.25 4642 3685 1
#> d[ARB] -0.45 0.06 -0.58 -0.50 -0.45 -0.41 -0.33 3598 3191 1
#> d[CCB] -0.25 0.05 -0.36 -0.29 -0.25 -0.22 -0.15 3395 3128 1
#> d[Placebo] -0.25 0.06 -0.36 -0.29 -0.25 -0.21 -0.13 4229 3411 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] -0.08 0.09 -0.26 -0.13 -0.07 -0.02 0.09 2200 2029 1
#> d[ACE Inhibitor] -0.40 0.09 -0.58 -0.46 -0.40 -0.34 -0.24 4657 2429 1
#> d[ARB] -0.48 0.11 -0.71 -0.54 -0.47 -0.40 -0.26 4095 2468 1
#> d[CCB] -0.24 0.09 -0.42 -0.30 -0.24 -0.19 -0.07 3997 2421 1
#> d[Placebo] -0.29 0.09 -0.47 -0.35 -0.29 -0.23 -0.12 4709 3187 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
trt_ref = "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),
trt_ref = "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 0.25 3654 3954 1
#> pred[New 1: ACE Inhibitor] 0.05 0.05 0.00 0.02 0.03 0.06 0.19 3641 3839 1
#> pred[New 1: ARB] 0.04 0.05 0.00 0.01 0.03 0.05 0.17 3632 3955 1
#> pred[New 1: CCB] 0.05 0.05 0.01 0.02 0.03 0.06 0.21 3637 3858 1
#> pred[New 1: Diuretic] 0.06 0.07 0.01 0.02 0.04 0.08 0.26 3635 3878 1
#> pred[New 1: Placebo] 0.05 0.05 0.01 0.02 0.03 0.06 0.21 3621 3853 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),
trt_ref = "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 0.25 4200 3916 1
#> pred[New 1: ACE Inhibitor] 0.05 0.05 0.00 0.02 0.03 0.06 0.19 4222 4037 1
#> pred[New 1: ARB] 0.04 0.05 0.00 0.01 0.03 0.05 0.17 4229 3728 1
#> pred[New 1: CCB] 0.05 0.06 0.01 0.02 0.03 0.07 0.22 4207 3972 1
#> pred[New 1: Diuretic] 0.07 0.07 0.01 0.02 0.04 0.08 0.26 4203 3734 1
#> pred[New 1: Placebo] 0.05 0.06 0.01 0.02 0.03 0.06 0.21 4215 3878 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 6000 2912 1
#> pred[AASK: ACE Inhibitor] 0.12 0.01 0.10 0.12 0.12 0.13 0.15 3981 2863 1
#> pred[AASK: ARB] 0.12 0.01 0.09 0.11 0.12 0.13 0.15 4409 2990 1
#> pred[AASK: CCB] 0.15 0.02 0.12 0.13 0.14 0.15 0.18 4584 2999 1
#> pred[AASK: Diuretic] 0.18 0.02 0.14 0.17 0.18 0.19 0.23 4188 2454 1
#> pred[AASK: Placebo] 0.14 0.02 0.11 0.13 0.14 0.15 0.17 3858 2977 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 3085 2436 1
#> pred[ALLHAT: ACE Inhibitor] 0.03 0.00 0.02 0.03 0.03 0.03 0.04 4148 2790 1
#> pred[ALLHAT: ARB] 0.03 0.00 0.02 0.03 0.03 0.03 0.04 4063 2336 1
#> pred[ALLHAT: CCB] 0.04 0.00 0.03 0.03 0.04 0.04 0.05 4302 2994 1
#> pred[ALLHAT: Diuretic] 0.05 0.01 0.04 0.04 0.05 0.05 0.06 4384 2682 1
#> pred[ALLHAT: Placebo] 0.03 0.00 0.03 0.03 0.03 0.04 0.04 4047 3064 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 6827 3367 1
#> pred[ALPINE: ACE Inhibitor] 0.02 0.01 0.01 0.01 0.02 0.02 0.03 6979 3225 1
#> pred[ALPINE: ARB] 0.02 0.01 0.01 0.01 0.02 0.02 0.03 7134 3354 1
#> pred[ALPINE: CCB] 0.02 0.01 0.01 0.02 0.02 0.03 0.04 7141 3344 1
#> pred[ALPINE: Diuretic] 0.03 0.01 0.01 0.02 0.03 0.03 0.05 7266 3175 1
#> pred[ALPINE: Placebo] 0.02 0.01 0.01 0.02 0.02 0.03 0.04 7030 3299 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 2726 2440 1
#> pred[ANBP-2: ACE Inhibitor] 0.05 0.01 0.04 0.04 0.05 0.05 0.06 4808 2976 1
#> pred[ANBP-2: ARB] 0.05 0.01 0.03 0.04 0.05 0.05 0.06 4198 2659 1
#> pred[ANBP-2: CCB] 0.06 0.01 0.04 0.05 0.06 0.06 0.08 4095 2617 1
#> pred[ANBP-2: Diuretic] 0.07 0.01 0.06 0.07 0.07 0.08 0.09 4522 2708 1
#> pred[ANBP-2: Placebo] 0.05 0.01 0.04 0.05 0.05 0.06 0.07 4864 3036 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 5891 2765 1
#> pred[ASCOT: ACE Inhibitor] 0.08 0.01 0.07 0.08 0.08 0.09 0.10 2290 2199 1
#> pred[ASCOT: ARB] 0.08 0.01 0.06 0.07 0.08 0.08 0.09 2758 2750 1
#> pred[ASCOT: CCB] 0.10 0.01 0.08 0.09 0.10 0.10 0.11 2377 2272 1
#> pred[ASCOT: Diuretic] 0.12 0.01 0.10 0.11 0.12 0.13 0.14 2351 1910 1
#> pred[ASCOT: Placebo] 0.09 0.01 0.08 0.09 0.09 0.10 0.11 2268 2874 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 6540 3060 1
#> pred[CAPPP: ACE Inhibitor] 0.05 0.00 0.05 0.05 0.05 0.06 0.06 2474 2952 1
#> pred[CAPPP: ARB] 0.05 0.01 0.04 0.05 0.05 0.05 0.06 3055 2892 1
#> pred[CAPPP: CCB] 0.06 0.00 0.05 0.06 0.06 0.07 0.07 3045 2485 1
#> pred[CAPPP: Diuretic] 0.08 0.01 0.07 0.08 0.08 0.09 0.10 3003 2452 1
#> pred[CAPPP: Placebo] 0.06 0.01 0.05 0.06 0.06 0.06 0.07 2363 2864 1
#>
#> ------------------------------------------------------------------ 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 3129 2472 1
#> pred[CHARM: ACE Inhibitor] 0.07 0.01 0.05 0.06 0.07 0.07 0.09 4532 2908 1
#> pred[CHARM: ARB] 0.06 0.01 0.05 0.06 0.06 0.07 0.08 5145 3148 1
#> pred[CHARM: CCB] 0.08 0.01 0.06 0.07 0.08 0.08 0.10 4126 2861 1
#> pred[CHARM: Diuretic] 0.10 0.01 0.07 0.09 0.10 0.11 0.13 4241 3029 1
#> pred[CHARM: Placebo] 0.07 0.01 0.06 0.07 0.07 0.08 0.09 4737 2899 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 3036 2890 1
#> pred[DREAM: ACE Inhibitor] 0.17 0.02 0.13 0.16 0.17 0.18 0.21 4900 2703 1
#> pred[DREAM: ARB] 0.16 0.02 0.12 0.14 0.16 0.17 0.21 4190 2570 1
#> pred[DREAM: CCB] 0.20 0.03 0.15 0.18 0.19 0.21 0.25 4035 3014 1
#> pred[DREAM: Diuretic] 0.24 0.03 0.19 0.22 0.24 0.26 0.31 4474 2821 1
#> pred[DREAM: Placebo] 0.19 0.02 0.15 0.17 0.19 0.20 0.23 5207 3137 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 4194 2895 1
#> pred[EWPH: ACE Inhibitor] 0.05 0.01 0.03 0.04 0.04 0.05 0.06 5485 3172 1
#> pred[EWPH: ARB] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 4680 2778 1
#> pred[EWPH: CCB] 0.05 0.01 0.03 0.05 0.05 0.06 0.08 5229 2877 1
#> pred[EWPH: Diuretic] 0.07 0.01 0.05 0.06 0.07 0.07 0.09 5726 2972 1
#> pred[EWPH: Placebo] 0.05 0.01 0.03 0.04 0.05 0.06 0.07 5234 2984 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 3057 2724 1
#> pred[FEVER: ACE Inhibitor] 0.03 0.00 0.02 0.03 0.03 0.03 0.04 4289 2677 1
#> pred[FEVER: ARB] 0.03 0.00 0.02 0.03 0.03 0.03 0.04 4285 2964 1
#> pred[FEVER: CCB] 0.04 0.00 0.03 0.03 0.03 0.04 0.05 4216 3067 1
#> pred[FEVER: Diuretic] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 4240 2568 1
#> pred[FEVER: Placebo] 0.03 0.00 0.03 0.03 0.03 0.04 0.04 4160 2752 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 5902 3242 1
#> pred[HAPPHY: ACE Inhibitor] 0.02 0 0.01 0.02 0.02 0.02 0.02 4853 3490 1
#> pred[HAPPHY: ARB] 0.02 0 0.01 0.02 0.02 0.02 0.02 4609 2664 1
#> pred[HAPPHY: CCB] 0.02 0 0.02 0.02 0.02 0.02 0.03 4693 3075 1
#> pred[HAPPHY: Diuretic] 0.03 0 0.02 0.02 0.03 0.03 0.03 3895 2937 1
#> pred[HAPPHY: Placebo] 0.02 0 0.02 0.02 0.02 0.02 0.03 4538 3255 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 3086 2802 1
#> pred[HOPE: ACE Inhibitor] 0.04 0.01 0.03 0.04 0.04 0.05 0.05 4869 2950 1
#> pred[HOPE: ARB] 0.04 0.01 0.03 0.04 0.04 0.04 0.05 4486 2924 1
#> pred[HOPE: CCB] 0.05 0.01 0.04 0.04 0.05 0.05 0.07 4182 2898 1
#> pred[HOPE: Diuretic] 0.06 0.01 0.05 0.06 0.06 0.07 0.08 4790 3110 1
#> pred[HOPE: Placebo] 0.05 0.01 0.04 0.04 0.05 0.05 0.06 4709 3045 1
#>
#> ---------------------------------------------------------------- Study: INSIGHT ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[INSIGHT: Beta Blocker] 0.06 0.01 0.05 0.06 0.06 0.07 0.08 3414 2939 1
#> pred[INSIGHT: ACE Inhibitor] 0.05 0.01 0.03 0.04 0.05 0.05 0.06 4478 3202 1
#> pred[INSIGHT: ARB] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 4217 2789 1
#> pred[INSIGHT: CCB] 0.06 0.01 0.04 0.05 0.05 0.06 0.07 4433 2926 1
#> pred[INSIGHT: Diuretic] 0.07 0.01 0.05 0.06 0.07 0.07 0.09 5312 3128 1
#> pred[INSIGHT: Placebo] 0.05 0.01 0.04 0.05 0.05 0.06 0.07 4569 2615 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 6765 3019 1
#> pred[INVEST: ACE Inhibitor] 0.06 0.01 0.05 0.06 0.06 0.06 0.07 2401 2467 1
#> pred[INVEST: ARB] 0.06 0.01 0.05 0.05 0.06 0.06 0.07 2722 2568 1
#> pred[INVEST: CCB] 0.07 0.00 0.06 0.07 0.07 0.07 0.08 2549 2378 1
#> pred[INVEST: Diuretic] 0.09 0.01 0.07 0.08 0.09 0.09 0.11 2529 2185 1
#> pred[INVEST: Placebo] 0.07 0.01 0.06 0.06 0.07 0.07 0.08 2280 2582 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 6442 2844 1
#> pred[LIFE: ACE Inhibitor] 0.06 0.01 0.05 0.06 0.06 0.06 0.07 2743 2454 1
#> pred[LIFE: ARB] 0.06 0.01 0.05 0.05 0.06 0.06 0.07 2894 2639 1
#> pred[LIFE: CCB] 0.07 0.01 0.06 0.07 0.07 0.07 0.08 3133 2826 1
#> pred[LIFE: Diuretic] 0.09 0.01 0.07 0.08 0.09 0.09 0.11 2935 2514 1
#> pred[LIFE: Placebo] 0.07 0.01 0.05 0.06 0.07 0.07 0.08 2585 2513 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 4456 3100 1
#> pred[MRC-E: ACE Inhibitor] 0.02 0 0.02 0.02 0.02 0.02 0.03 5918 3337 1
#> pred[MRC-E: ARB] 0.02 0 0.01 0.02 0.02 0.02 0.03 5187 3030 1
#> pred[MRC-E: CCB] 0.03 0 0.02 0.02 0.02 0.03 0.03 4801 3081 1
#> pred[MRC-E: Diuretic] 0.03 0 0.02 0.03 0.03 0.03 0.04 4628 2846 1
#> pred[MRC-E: Placebo] 0.02 0 0.02 0.02 0.02 0.03 0.03 5680 3482 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 6926 3009 1
#> pred[NORDIL: ACE Inhibitor] 0.04 0.00 0.03 0.03 0.04 0.04 0.04 3097 2871 1
#> pred[NORDIL: ARB] 0.03 0.00 0.03 0.03 0.03 0.04 0.04 3462 2849 1
#> pred[NORDIL: CCB] 0.04 0.00 0.04 0.04 0.04 0.04 0.05 3228 2844 1
#> pred[NORDIL: Diuretic] 0.05 0.01 0.04 0.05 0.05 0.06 0.07 3174 2946 1
#> pred[NORDIL: Placebo] 0.04 0.00 0.03 0.04 0.04 0.04 0.05 2960 3023 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 0.10 0.13 0.14 0.15 0.18 2958 2546 1
#> pred[PEACE: ACE Inhibitor] 0.10 0.01 0.08 0.09 0.10 0.11 0.13 4648 2915 1
#> pred[PEACE: ARB] 0.09 0.01 0.07 0.09 0.09 0.10 0.13 4234 2674 1
#> pred[PEACE: CCB] 0.12 0.02 0.09 0.11 0.12 0.13 0.15 3948 2628 1
#> pred[PEACE: Diuretic] 0.15 0.02 0.11 0.13 0.15 0.16 0.19 4243 2680 1
#> pred[PEACE: Placebo] 0.11 0.01 0.09 0.10 0.11 0.12 0.14 4892 2564 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 0.08 3368 3083 1
#> pred[SCOPE: ACE Inhibitor] 0.05 0.01 0.03 0.04 0.05 0.05 0.06 5312 2937 1
#> pred[SCOPE: ARB] 0.04 0.01 0.03 0.04 0.04 0.05 0.06 4673 2796 1
#> pred[SCOPE: CCB] 0.05 0.01 0.04 0.05 0.05 0.06 0.07 4582 2770 1
#> pred[SCOPE: Diuretic] 0.07 0.01 0.05 0.06 0.07 0.08 0.09 4755 3025 1
#> pred[SCOPE: Placebo] 0.05 0.01 0.04 0.05 0.05 0.06 0.07 4956 2953 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 2645 2824 1
#> pred[SHEP: ACE Inhibitor] 0.06 0.01 0.05 0.06 0.06 0.07 0.08 4355 2988 1
#> pred[SHEP: ARB] 0.06 0.01 0.04 0.05 0.06 0.06 0.08 4073 3152 1
#> pred[SHEP: CCB] 0.07 0.01 0.05 0.07 0.07 0.08 0.10 3744 2687 1
#> pred[SHEP: Diuretic] 0.09 0.01 0.07 0.08 0.09 0.10 0.12 4797 2766 1
#> pred[SHEP: Placebo] 0.07 0.01 0.05 0.06 0.07 0.08 0.09 4488 3254 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 0.04 0.05 0.05 0.06 0.06 4141 3184 1
#> pred[STOP-2: ACE Inhibitor] 0.04 0.00 0.03 0.04 0.04 0.04 0.05 3060 2756 1
#> pred[STOP-2: ARB] 0.04 0.00 0.03 0.03 0.04 0.04 0.04 3461 2693 1
#> pred[STOP-2: CCB] 0.05 0.00 0.04 0.04 0.05 0.05 0.05 4088 3146 1
#> pred[STOP-2: Diuretic] 0.06 0.01 0.05 0.05 0.06 0.06 0.07 3553 2951 1
#> pred[STOP-2: Placebo] 0.04 0.00 0.03 0.04 0.04 0.05 0.05 3009 2711 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 0.20 0.21 0.25 3241 2613 1
#> pred[VALUE: ACE Inhibitor] 0.15 0.02 0.11 0.13 0.14 0.16 0.19 4155 2717 1
#> pred[VALUE: ARB] 0.14 0.02 0.11 0.13 0.14 0.15 0.17 4978 2264 1
#> pred[VALUE: CCB] 0.17 0.02 0.13 0.16 0.17 0.18 0.21 4598 2600 1
#> pred[VALUE: Diuretic] 0.21 0.03 0.16 0.19 0.21 0.22 0.27 4174 2121 1
#> pred[VALUE: Placebo] 0.16 0.02 0.12 0.15 0.16 0.17 0.21 4162 2620 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] 5.17 0.42 5 5 5 5 6 2621 NA 1
#> rank[ACE Inhibitor] 1.84 0.54 1 2 2 2 3 3784 3286 1
#> rank[ARB] 1.26 0.50 1 1 1 1 3 3376 3435 1
#> rank[CCB] 3.70 0.53 3 3 4 4 4 3683 2483 1
#> rank[Diuretic] 5.81 0.40 5 6 6 6 6 2908 NA 1
#> rank[Placebo] 3.22 0.58 2 3 3 4 4 4021 2653 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 0.80 0.18
#> d[ACE Inhibitor] 0.23 0.70 0.06 0.01 0.00 0.00
#> d[ARB] 0.76 0.21 0.02 0.00 0.00 0.00
#> d[CCB] 0.00 0.02 0.27 0.70 0.01 0.00
#> d[Diuretic] 0.00 0.00 0.00 0.00 0.18 0.81
#> d[Placebo] 0.00 0.07 0.64 0.28 0.00 0.00
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 0.82 1
#> d[ACE Inhibitor] 0.23 0.94 0.99 1.00 1.00 1
#> d[ARB] 0.76 0.97 1.00 1.00 1.00 1
#> d[CCB] 0.00 0.02 0.29 0.99 1.00 1
#> d[Diuretic] 0.00 0.00 0.00 0.00 0.18 1
#> d[Placebo] 0.00 0.07 0.72 1.00 1.00 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://www.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.