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 smoking cessation data (Hasselblad
1998), replicating the analysis in NICE Technical Support
Document 4 (Dias et al.
2011). The data are available in this package as
smoking
:
head(smoking)
#> studyn trtn trtc r n
#> 1 1 1 No intervention 9 140
#> 2 1 3 Individual counselling 23 140
#> 3 1 4 Group counselling 10 138
#> 4 2 2 Self-help 11 78
#> 5 2 3 Individual counselling 12 85
#> 6 2 4 Group counselling 29 170
Setting up the network
We begin by setting up the network. We have arm-level count data
giving the number quitting smoking (r
) out of the total
(n
) in each arm, so we use the function
set_agd_arm()
. Treatment “No intervention” is set as the
network reference treatment.
smknet <- set_agd_arm(smoking,
study = studyn,
trt = trtc,
r = r,
n = n,
trt_ref = "No intervention")
smknet
#> A network with 24 AgD studies (arm-based).
#>
#> ------------------------------------------------------- AgD studies (arm-based) ----
#> Study Treatment arms
#> 1 3: No intervention | Group counselling | Individual counselling
#> 2 3: Group counselling | Individual counselling | Self-help
#> 3 2: No intervention | Individual counselling
#> 4 2: No intervention | Individual counselling
#> 5 2: No intervention | Individual counselling
#> 6 2: No intervention | Individual counselling
#> 7 2: No intervention | Individual counselling
#> 8 2: No intervention | Individual counselling
#> 9 2: No intervention | Individual counselling
#> 10 2: No intervention | Self-help
#> ... plus 14 more studies
#>
#> Outcome type: count
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 4
#> Total number of studies: 24
#> Reference treatment is: No intervention
#> Network is connected
Plot the network structure.
plot(smknet, weight_edges = TRUE, weight_nodes = TRUE)
Random effects NMA
Following TSD 4, we fit a random effects NMA model, using the
nma()
function with trt_effects = "random"
. We
use \(\mathrm{N}(0, 100^2)\) prior
distributions for the treatment effects \(d_k\) and study-specific intercepts \(\mu_j\), and a \(\textrm{half-N}(5^2)\) prior distribution
for the between-study 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.
The model is fitted using the nma()
function. By
default, this will use a Binomial likelihood and a logit link function,
auto-detected from the data.
smkfit <- nma(smknet,
trt_effects = "random",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100),
prior_het = normal(scale = 5))
Basic parameter summaries are given by the print()
method:
smkfit
#> 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% 75% 97.5%
#> d[Group counselling] 1.10 0.01 0.44 0.28 0.80 1.08 1.38 1.95
#> d[Individual counselling] 0.84 0.01 0.23 0.41 0.68 0.83 0.99 1.33
#> d[Self-help] 0.48 0.01 0.40 -0.30 0.21 0.48 0.74 1.25
#> lp__ -5768.06 0.20 6.44 -5781.69 -5772.21 -5767.77 -5763.58 -5756.14
#> tau 0.84 0.01 0.18 0.55 0.71 0.81 0.93 1.25
#> n_eff Rhat
#> d[Group counselling] 2243 1
#> d[Individual counselling] 1053 1
#> d[Self-help] 1888 1
#> lp__ 1078 1
#> tau 935 1
#>
#> Samples were drawn using NUTS(diag_e) at Wed Mar 6 13:32:03 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(smkfit)
By default, this displays all model parameters given prior
distributions (in this case \(d_k\),
\(\mu_j\), and \(\tau\)), but this may be changed using the
prior
argument:
plot_prior_posterior(smkfit, prior = "het")
Model fit can be checked using the dic()
function
(dic_consistency <- dic(smkfit))
#> Residual deviance: 54.1 (on 50 data points)
#> pD: 43.8
#> DIC: 97.8
and the residual deviance contributions examined with the
corresponding plot()
method
plot(dic_consistency)
Overall model fit seems to be adequate, with almost all points showing good fit (mean residual deviance contribution of 1). The only two points with higher residual deviance (i.e. worse fit) correspond to the two zero counts in the data:
smoking[smoking$r == 0, ]
#> studyn trtn trtc r n
#> 13 6 1 No intervention 0 33
#> 31 15 1 No intervention 0 20
Checking for inconsistency
Note: The results of the inconsistency models here are slightly different to those of Dias et al. (2010, 2011), although the overall conclusions are the same. This is due to the presence of multi-arm trials and a different ordering of treatments, meaning that inconsistency is parameterised differently within the multi-arm trials. The same results as Dias et al. are obtained if the network is instead set up with
trtn
as the treatment variable.
Unrelated mean effects
We first fit an unrelated mean effects (UME) model (Dias et al. 2011) to
assess the consistency assumption. Again, we use the function
nma()
, but now with the argument
consistency = "ume"
.
smkfit_ume <- nma(smknet,
consistency = "ume",
trt_effects = "random",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100),
prior_het = normal(scale = 5))
smkfit_ume
#> A random effects NMA with a binomial likelihood (logit link).
#> An inconsistency model ('ume') was fitted.
#> Inference for Stan model: binomial_1par.
#> 4 chains, each with iter=2000; warmup=1000; thin=1;
#> post-warmup draws per chain=1000, total post-warmup draws=4000.
#>
#> mean se_mean sd 2.5% 25%
#> d[Group counselling vs. No intervention] 1.13 0.02 0.80 -0.38 0.59
#> d[Individual counselling vs. No intervention] 0.90 0.01 0.27 0.42 0.71
#> d[Self-help vs. No intervention] 0.34 0.01 0.59 -0.80 -0.05
#> d[Individual counselling vs. Group counselling] -0.29 0.01 0.60 -1.49 -0.68
#> d[Self-help vs. Group counselling] -0.63 0.02 0.75 -2.15 -1.09
#> d[Self-help vs. Individual counselling] 0.13 0.02 1.04 -1.95 -0.54
#> lp__ -5765.49 0.23 6.47 -5779.46 -5769.62
#> tau 0.93 0.01 0.22 0.59 0.77
#> 50% 75% 97.5% n_eff Rhat
#> d[Group counselling vs. No intervention] 1.10 1.64 2.79 2012 1
#> d[Individual counselling vs. No intervention] 0.89 1.07 1.46 1279 1
#> d[Self-help vs. No intervention] 0.33 0.72 1.52 2312 1
#> d[Individual counselling vs. Group counselling] -0.30 0.10 0.90 2359 1
#> d[Self-help vs. Group counselling] -0.63 -0.14 0.88 2188 1
#> d[Self-help vs. Individual counselling] 0.12 0.80 2.25 3626 1
#> lp__ -5765.01 -5760.99 -5753.81 764 1
#> tau 0.89 1.05 1.46 787 1
#>
#> Samples were drawn using NUTS(diag_e) at Wed Mar 6 13:32:20 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).
Comparing the model fit statistics
dic_consistency
#> Residual deviance: 54.1 (on 50 data points)
#> pD: 43.8
#> DIC: 97.8
(dic_ume <- dic(smkfit_ume))
#> Residual deviance: 54.1 (on 50 data points)
#> pD: 45.2
#> DIC: 99.4
We see that there is little to choose between the two models.
However, it is also important to examine the individual contributions to
model fit of each data point under the two models (a so-called “dev-dev”
plot). Passing two nma_dic
objects produced by the
dic()
function to the plot()
method produces
this dev-dev plot:
plot(dic_consistency, dic_ume, point_alpha = 0.5, interval_alpha = 0.2)
All points lie roughly on the line of equality, so there is no evidence for inconsistency here.
Node-splitting
Another method for assessing inconsistency is node-splitting (Dias et al. 2011, 2010). Whereas the UME model assesses inconsistency globally, node-splitting assesses inconsistency locally for each potentially inconsistent comparison (those with both direct and indirect evidence) in turn.
Node-splitting can be performed using the nma()
function
with the argument consistency = "nodesplit"
. By default,
all possible comparisons will be split (as determined by the
get_nodesplits()
function). Alternatively, a specific
comparison or comparisons to split can be provided to the
nodesplit
argument.
smk_nodesplit <- nma(smknet,
consistency = "nodesplit",
trt_effects = "random",
prior_intercept = normal(scale = 100),
prior_trt = normal(scale = 100),
prior_het = normal(scale = 5))
#> Fitting model 1 of 7, node-split: Group counselling vs. No intervention
#> Fitting model 2 of 7, node-split: Individual counselling vs. No intervention
#> Fitting model 3 of 7, node-split: Self-help vs. No intervention
#> Fitting model 4 of 7, node-split: Individual counselling vs. Group counselling
#> Fitting model 5 of 7, node-split: Self-help vs. Group counselling
#> Fitting model 6 of 7, node-split: Self-help vs. Individual counselling
#> Fitting model 7 of 7, consistency model
The summary()
method summarises the node-splitting
results, displaying the direct and indirect estimates \(d_\mathrm{dir}\) and \(d_\mathrm{ind}\) from each node-split
model, the network estimate \(d_\mathrm{net}\) from the consistency
model, the inconsistency factor \(\omega =
d_\mathrm{dir} - d_\mathrm{ind}\), and a Bayesian \(p\)-value for inconsistency on each
comparison. Since random effects models are fitted, the heterogeneity
standard deviation \(\tau\) under each
node-split model and under the consistency model is also displayed. The
DIC model fit statistics are also provided.
summary(smk_nodesplit)
#> Node-splitting models fitted for 6 comparisons.
#>
#> ------------------------------ Node-split Group counselling vs. No intervention ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net 1.10 0.44 0.25 0.80 1.10 1.38 2.01 1842 2310 1
#> d_dir 1.08 0.77 -0.34 0.57 1.03 1.55 2.73 3363 2708 1
#> d_ind 1.12 0.55 0.06 0.77 1.12 1.47 2.20 1957 2237 1
#> omega -0.04 0.94 -1.81 -0.66 -0.08 0.53 1.96 2529 2748 1
#> tau 0.88 0.20 0.56 0.74 0.85 1.00 1.34 1151 1940 1
#> tau_consistency 0.84 0.19 0.54 0.71 0.82 0.95 1.29 1080 1667 1
#>
#> Residual deviance: 53.7 (on 50 data points)
#> pD: 43.9
#> DIC: 97.6
#>
#> Bayesian p-value: 0.93
#>
#> ------------------------- Node-split Individual counselling vs. No intervention ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net 0.85 0.24 0.39 0.69 0.84 1.00 1.35 814 1946 1
#> d_dir 0.88 0.25 0.39 0.71 0.88 1.04 1.39 2757 2731 1
#> d_ind 0.57 0.66 -0.71 0.14 0.58 0.98 1.91 2121 2178 1
#> omega 0.31 0.69 -1.07 -0.13 0.32 0.76 1.68 2174 2358 1
#> tau 0.86 0.19 0.55 0.72 0.84 0.98 1.28 1627 2142 1
#> tau_consistency 0.84 0.19 0.54 0.71 0.82 0.95 1.29 1080 1667 1
#>
#> Residual deviance: 54.3 (on 50 data points)
#> pD: 44.3
#> DIC: 98.6
#>
#> Bayesian p-value: 0.64
#>
#> -------------------------------------- Node-split Self-help vs. No intervention ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net 0.49 0.41 -0.32 0.22 0.49 0.75 1.34 1702 2168 1
#> d_dir 0.34 0.55 -0.76 -0.02 0.35 0.70 1.44 3677 2791 1
#> d_ind 0.71 0.65 -0.52 0.29 0.70 1.13 2.04 2128 2465 1
#> omega -0.37 0.85 -2.09 -0.91 -0.35 0.18 1.35 2278 2408 1
#> tau 0.88 0.20 0.56 0.74 0.85 0.99 1.32 1377 2307 1
#> tau_consistency 0.84 0.19 0.54 0.71 0.82 0.95 1.29 1080 1667 1
#>
#> Residual deviance: 53.5 (on 50 data points)
#> pD: 44.1
#> DIC: 97.5
#>
#> Bayesian p-value: 0.64
#>
#> ----------------------- Node-split Individual counselling vs. Group counselling ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.25 0.42 -1.09 -0.52 -0.25 0.02 0.59 2986 2816 1
#> d_dir -0.10 0.48 -1.09 -0.42 -0.10 0.22 0.85 3462 3361 1
#> d_ind -0.53 0.61 -1.79 -0.92 -0.51 -0.13 0.65 1505 1984 1
#> omega 0.42 0.68 -0.93 -0.01 0.42 0.85 1.81 1698 2254 1
#> tau 0.87 0.20 0.56 0.73 0.85 0.99 1.34 1197 2073 1
#> tau_consistency 0.84 0.19 0.54 0.71 0.82 0.95 1.29 1080 1667 1
#>
#> Residual deviance: 53.4 (on 50 data points)
#> pD: 43.9
#> DIC: 97.2
#>
#> Bayesian p-value: 0.51
#>
#> ------------------------------------ Node-split Self-help vs. Group counselling ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.61 0.49 -1.59 -0.93 -0.59 -0.28 0.34 3229 3131 1
#> d_dir -0.62 0.68 -1.98 -1.05 -0.62 -0.17 0.69 3958 2729 1
#> d_ind -0.61 0.68 -1.97 -1.02 -0.61 -0.19 0.72 1837 2170 1
#> omega -0.01 0.87 -1.72 -0.59 0.00 0.58 1.67 2118 2599 1
#> tau 0.87 0.20 0.56 0.73 0.85 0.99 1.34 1135 1759 1
#> tau_consistency 0.84 0.19 0.54 0.71 0.82 0.95 1.29 1080 1667 1
#>
#> Residual deviance: 53.9 (on 50 data points)
#> pD: 44.2
#> DIC: 98.2
#>
#> Bayesian p-value: 1
#>
#> ------------------------------- Node-split Self-help vs. Individual counselling ----
#>
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d_net -0.36 0.42 -1.20 -0.62 -0.36 -0.09 0.47 2304 2681 1
#> d_dir 0.06 0.64 -1.19 -0.36 0.05 0.50 1.30 3381 3282 1
#> d_ind -0.62 0.50 -1.64 -0.95 -0.62 -0.29 0.35 1776 2539 1
#> omega 0.68 0.79 -0.85 0.16 0.67 1.22 2.21 2295 3012 1
#> tau 0.86 0.19 0.55 0.72 0.83 0.97 1.30 1218 1967 1
#> tau_consistency 0.84 0.19 0.54 0.71 0.82 0.95 1.29 1080 1667 1
#>
#> Residual deviance: 53.8 (on 50 data points)
#> pD: 44.2
#> DIC: 98
#>
#> Bayesian p-value: 0.38
The DIC of each inconsistency model is unchanged from the consistency model, no node-splits result in reduced heterogeneity standard deviation \(\tau\) compared to the consistency model, and the Bayesian \(p\)-values are all large. There is no evidence of inconsistency.
We can visually compare the posterior distributions of the direct,
indirect, and network estimates using the plot()
method.
These are all in agreement; the posterior densities of the direct and
indirect estimates overlap. Notice that there is not much indirect
information for the Individual counselling vs. No intervention
comparison, so the network (consistency) estimate is very similar to the
direct estimate for this comparison.
Further results
Pairwise relative effects, for all pairwise contrasts with
all_contrasts = TRUE
.
(smk_releff <- relative_effects(smkfit, all_contrasts = TRUE))
#> mean sd 2.5% 25% 50% 75% 97.5%
#> d[Group counselling vs. No intervention] 1.10 0.44 0.28 0.80 1.08 1.38 1.95
#> d[Individual counselling vs. No intervention] 0.84 0.23 0.41 0.68 0.83 0.99 1.33
#> d[Self-help vs. No intervention] 0.48 0.40 -0.30 0.21 0.48 0.74 1.25
#> d[Individual counselling vs. Group counselling] -0.26 0.41 -1.06 -0.52 -0.26 0.01 0.52
#> d[Self-help vs. Group counselling] -0.62 0.48 -1.59 -0.92 -0.60 -0.31 0.33
#> d[Self-help vs. Individual counselling] -0.36 0.41 -1.18 -0.62 -0.35 -0.10 0.43
#> Bulk_ESS Tail_ESS Rhat
#> d[Group counselling vs. No intervention] 2398 2689 1
#> d[Individual counselling vs. No intervention] 1068 1727 1
#> d[Self-help vs. No intervention] 1946 2294 1
#> d[Individual counselling vs. Group counselling] 3070 2868 1
#> d[Self-help vs. Group counselling] 3028 2505 1
#> d[Self-help vs. Individual counselling] 2213 2320 1
plot(smk_releff, ref_line = 0)
Treatment rankings, rank probabilities, and cumulative rank
probabilities. We set lower_better = FALSE
since a higher
log odds of cessation is better (the outcome is positive).
(smk_ranks <- posterior_ranks(smkfit, lower_better = FALSE))
#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> rank[No intervention] 3.89 0.32 3 4 4 4 4 2574 NA 1
#> rank[Group counselling] 1.36 0.61 1 1 1 2 3 3285 3179 1
#> rank[Individual counselling] 1.93 0.63 1 2 2 2 3 2440 NA 1
#> rank[Self-help] 2.83 0.68 1 3 3 3 4 2536 NA 1
plot(smk_ranks)
(smk_rankprobs <- posterior_rank_probs(smkfit, lower_better = FALSE))
#> p_rank[1] p_rank[2] p_rank[3] p_rank[4]
#> d[No intervention] 0.00 0.00 0.11 0.89
#> d[Group counselling] 0.71 0.23 0.06 0.00
#> d[Individual counselling] 0.24 0.60 0.16 0.00
#> d[Self-help] 0.05 0.17 0.67 0.10
plot(smk_rankprobs)
(smk_cumrankprobs <- posterior_rank_probs(smkfit, lower_better = FALSE, cumulative = TRUE))
#> p_rank[1] p_rank[2] p_rank[3] p_rank[4]
#> d[No intervention] 0.00 0.00 0.11 1
#> d[Group counselling] 0.71 0.94 1.00 1
#> d[Individual counselling] 0.24 0.84 1.00 1
#> d[Self-help] 0.05 0.22 0.90 1
plot(smk_cumrankprobs)