Setting up the network
We begin by setting up the network. We have arm-level count data
@@ -389,37 +391,37 @@
Setting up the network
(
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
+
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)
+
plot(smknet, weight_edges = TRUE, weight_nodes = TRUE)
@@ -431,180 +433,178 @@
Random effects NMA
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.
+
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))
+
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% n_eff
-#> d[Group counselling] 1.11 0.01 0.44 0.27 0.81 1.09 1.40 2.01 1998
-#> d[Individual counselling] 0.84 0.01 0.25 0.39 0.67 0.83 0.99 1.36 1206
-#> d[Self-help] 0.50 0.01 0.40 -0.27 0.24 0.48 0.75 1.31 1916
-#> lp__ -5919.72 0.20 6.50 -5933.29 -5923.82 -5919.41 -5915.14 -5907.86 1012
-#> tau 0.84 0.01 0.19 0.54 0.70 0.82 0.95 1.28 918
-#> Rhat
-#> d[Group counselling] 1
-#> d[Individual counselling] 1
-#> d[Self-help] 1
-#> lp__ 1
-#> tau 1
-#>
-#> Samples were drawn using NUTS(diag_e) at Tue Jan 9 17:49:33 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).
+
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% n_eff
+#> d[Group counselling] 1.11 0.01 0.43 0.32 0.81 1.09 1.38 2.00 1681
+#> d[Individual counselling] 0.84 0.01 0.23 0.41 0.68 0.83 0.99 1.33 1238
+#> d[Self-help] 0.49 0.01 0.40 -0.29 0.24 0.49 0.74 1.28 1572
+#> lp__ -5767.91 0.21 6.43 -5781.12 -5772.07 -5767.63 -5763.34 -5756.32 902
+#> tau 0.84 0.01 0.19 0.54 0.70 0.81 0.95 1.30 1204
+#> Rhat
+#> d[Group counselling] 1
+#> d[Individual counselling] 1
+#> d[Self-help] 1
+#> lp__ 1
+#> tau 1
+#>
+#> Samples were drawn using NUTS(diag_e) at Wed Mar 6 12:04:00 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:
-
# Not run
-print(smkfit, pars = c("d", "tau", "mu", "delta"))
+
# Not run
+print(smkfit, pars = c("d", "tau", "mu", "delta"))
The prior and posterior distributions can be compared visually using
the plot_prior_posterior() function:
-
plot_prior_posterior(smkfit)
-
+
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")
-
+
plot_prior_posterior(smkfit, prior = "het")
+
Model fit can be checked using the dic() function
-
(dic_consistency <- dic(smkfit))
-#> Residual deviance: 54 (on 50 data points)
-#> pD: 43.7
-#> DIC: 97.7
+
(dic_consistency <- dic(smkfit))
+#> Residual deviance: 53.8 (on 50 data points)
+#> pD: 43.6
+#> DIC: 97.5
and the residual deviance contributions examined with the
corresponding plot() method
-
-
+
+
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$r == 0, ]
-#> studyn trtn trtc r n
-#> 13 6 1 No intervention 0 33
-#> 31 15 1 No intervention 0 20
+
smoking[smoking$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.
+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
+
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% 50%
-#> d[Group counselling vs. No intervention] 1.15 0.02 0.81 -0.35 0.62 1.11
-#> d[Individual counselling vs. No intervention] 0.92 0.01 0.27 0.41 0.75 0.91
-#> d[Self-help vs. No intervention] 0.33 0.01 0.59 -0.83 -0.04 0.33
-#> d[Individual counselling vs. Group counselling] -0.26 0.01 0.60 -1.46 -0.65 -0.26
-#> d[Self-help vs. Group counselling] -0.61 0.01 0.71 -2.02 -1.05 -0.61
-#> d[Self-help vs. Individual counselling] 0.15 0.02 1.07 -2.02 -0.52 0.15
-#> lp__ -5933.38 0.20 6.27 -5946.38 -5937.54 -5933.05
-#> tau 0.94 0.01 0.23 0.59 0.78 0.91
-#> 75% 97.5% n_eff Rhat
-#> d[Group counselling vs. No intervention] 1.64 2.93 2470 1.00
-#> d[Individual counselling vs. No intervention] 1.09 1.51 1094 1.00
-#> d[Self-help vs. No intervention] 0.70 1.55 2000 1.00
-#> d[Individual counselling vs. Group counselling] 0.13 0.93 2775 1.00
-#> d[Self-help vs. Group counselling] -0.16 0.81 2530 1.00
-#> d[Self-help vs. Individual counselling] 0.82 2.27 3983 1.00
-#> lp__ -5928.99 -5921.69 1025 1.01
-#> tau 1.07 1.47 1203 1.00
-#>
-#> Samples were drawn using NUTS(diag_e) at Tue Jan 9 17:49:53 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).
+
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% 50%
+#> d[Group counselling vs. No intervention] 1.11 0.02 0.78 -0.33 0.57 1.08
+#> d[Individual counselling vs. No intervention] 0.89 0.01 0.27 0.37 0.71 0.89
+#> d[Self-help vs. No intervention] 0.35 0.01 0.58 -0.80 0.00 0.34
+#> d[Individual counselling vs. Group counselling] -0.28 0.01 0.61 -1.45 -0.68 -0.29
+#> d[Self-help vs. Group counselling] -0.60 0.02 0.69 -1.94 -1.05 -0.62
+#> d[Self-help vs. Individual counselling] 0.17 0.02 1.06 -1.97 -0.49 0.16
+#> lp__ -5765.20 0.20 6.39 -5778.70 -5769.44 -5764.84
+#> tau 0.93 0.01 0.22 0.58 0.77 0.90
+#> 75% 97.5% n_eff Rhat
+#> d[Group counselling vs. No intervention] 1.60 2.74 2282 1
+#> d[Individual counselling vs. No intervention] 1.07 1.43 961 1
+#> d[Self-help vs. No intervention] 0.72 1.50 1765 1
+#> d[Individual counselling vs. Group counselling] 0.12 0.95 2376 1
+#> d[Self-help vs. Group counselling] -0.16 0.77 1801 1
+#> d[Self-help vs. Individual counselling] 0.83 2.24 3429 1
+#> lp__ -5760.67 -5753.72 982 1
+#> tau 1.04 1.44 984 1
+#>
+#> Samples were drawn using NUTS(diag_e) at Wed Mar 6 12:04:49 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 (on 50 data points)
-#> pD: 43.7
-#> DIC: 97.7
-(dic_ume <- dic(smkfit_ume))
-#> Residual deviance: 53.5 (on 50 data points)
-#> pD: 45
-#> DIC: 98.5
+
dic_consistency
+#> Residual deviance: 53.8 (on 50 data points)
+#> pD: 43.6
+#> DIC: 97.5
+(dic_ume <- dic(smkfit_ume))
+#> Residual deviance: 53.5 (on 50 data points)
+#> pD: 44.7
+#> DIC: 98.2
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)
-
+
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.
+
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
+
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
@@ -614,104 +614,104 @@
Node-splitting
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 1.09 0.44 0.28 0.79 1.07 1.37 1.97 1793 2275 1
-#> d_dir 1.06 0.75 -0.36 0.55 1.03 1.54 2.63 3096 2875 1
-#> d_ind 1.12 0.55 -0.01 0.76 1.12 1.48 2.19 1517 1709 1
-#> omega -0.05 0.91 -1.81 -0.66 -0.08 0.54 1.82 2295 2580 1
-#> tau 0.87 0.20 0.56 0.73 0.84 0.98 1.32 1342 1858 1
-#> tau_consistency 0.84 0.19 0.55 0.71 0.82 0.94 1.29 1092 1786 1
-#>
-#> Residual deviance: 53.9 (on 50 data points)
-#> pD: 44.1
-#> DIC: 98
-#>
-#> 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 0.84 0.24 0.38 0.68 0.83 0.99 1.36 1176 1770 1.00
-#> d_dir 0.89 0.26 0.40 0.72 0.88 1.05 1.42 1218 1860 1.00
-#> d_ind 0.56 0.68 -0.74 0.12 0.55 0.98 1.99 1165 1674 1.00
-#> omega 0.33 0.70 -1.08 -0.10 0.35 0.79 1.65 1163 1632 1.00
-#> tau 0.86 0.19 0.55 0.72 0.83 0.97 1.28 1118 2046 1.01
-#> tau_consistency 0.84 0.19 0.55 0.71 0.82 0.94 1.29 1092 1786 1.00
-#>
-#> Residual deviance: 54.2 (on 50 data points)
-#> pD: 44.1
-#> DIC: 98.3
-#>
-#> Bayesian p-value: 0.6
-#>
-#> -------------------------------------- Node-split Self-help vs. No intervention ----
-#>
-#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
-#> d_net 0.48 0.40 -0.28 0.21 0.48 0.73 1.28 1698 1876 1
-#> d_dir 0.33 0.54 -0.71 -0.01 0.33 0.68 1.41 3339 2529 1
-#> d_ind 0.70 0.65 -0.58 0.28 0.69 1.11 2.03 2165 2405 1
-#> omega -0.36 0.83 -2.09 -0.88 -0.35 0.17 1.26 2351 2589 1
-#> tau 0.87 0.19 0.56 0.73 0.84 0.98 1.31 1311 2303 1
-#> tau_consistency 0.84 0.19 0.55 0.71 0.82 0.94 1.29 1092 1786 1
-#>
-#> Residual deviance: 53.7 (on 50 data points)
-#> pD: 44.1
-#> DIC: 97.8
-#>
-#> 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 0.41 -1.06 -0.52 -0.24 0.03 0.55 2718 2818 1
-#> d_dir -0.10 0.51 -1.09 -0.43 -0.10 0.23 0.90 4213 3127 1
-#> d_ind -0.53 0.61 -1.76 -0.92 -0.51 -0.14 0.67 1866 2406 1
-#> omega 0.43 0.68 -0.88 -0.02 0.42 0.86 1.78 1934 1802 1
-#> tau 0.87 0.20 0.56 0.73 0.84 0.98 1.35 1341 1985 1
-#> tau_consistency 0.84 0.19 0.55 0.71 0.82 0.94 1.29 1092 1786 1
-#>
-#> Residual deviance: 54 (on 50 data points)
-#> pD: 44.4
-#> DIC: 98.5
-#>
-#> Bayesian p-value: 0.52
-#>
-#> ------------------------------------ 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 -1.58 -0.93 -0.61 -0.30 0.36 2737 2551 1
-#> d_dir -0.63 0.66 -1.98 -1.05 -0.63 -0.21 0.68 3657 3014 1
-#> d_ind -0.62 0.67 -2.05 -1.04 -0.61 -0.18 0.70 2132 2344 1
-#> omega -0.02 0.88 -1.75 -0.58 -0.02 0.56 1.72 2421 2701 1
-#> tau 0.86 0.19 0.57 0.72 0.84 0.97 1.29 1013 1808 1
-#> tau_consistency 0.84 0.19 0.55 0.71 0.82 0.94 1.29 1092 1786 1
-#>
-#> Residual deviance: 54 (on 50 data points)
-#> pD: 44.2
-#> DIC: 98.2
-#>
-#> Bayesian p-value: 0.99
-#>
-#> ------------------------------- 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 0.40 -1.15 -0.62 -0.37 -0.11 0.45 2188 2043 1.00
-#> d_dir 0.06 0.65 -1.24 -0.36 0.06 0.50 1.35 3068 2587 1.00
-#> d_ind -0.61 0.51 -1.62 -0.94 -0.61 -0.28 0.38 1883 2496 1.00
-#> omega 0.68 0.80 -0.91 0.15 0.67 1.20 2.27 2164 2423 1.00
-#> tau 0.86 0.19 0.56 0.72 0.83 0.96 1.30 1066 1156 1.01
-#> tau_consistency 0.84 0.19 0.55 0.71 0.82 0.94 1.29 1092 1786 1.00
-#>
-#> Residual deviance: 53.8 (on 50 data points)
-#> pD: 44.1
-#> DIC: 97.9
-#>
-#> Bayesian p-value: 0.38
+
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 0.43 0.27 0.82 1.09 1.38 1.98 2360 2515 1.00
+#> d_dir 1.05 0.74 -0.33 0.55 1.01 1.51 2.61 3575 3129 1.00
+#> d_ind 1.13 0.54 0.10 0.78 1.12 1.48 2.22 1793 2010 1.00
+#> omega -0.08 0.91 -1.81 -0.68 -0.11 0.49 1.77 2659 2417 1.00
+#> tau 0.86 0.20 0.56 0.72 0.83 0.97 1.34 1129 1958 1.01
+#> tau_consistency 0.84 0.18 0.55 0.71 0.82 0.94 1.26 1380 2233 1.00
+#>
+#> Residual deviance: 54.3 (on 50 data points)
+#> pD: 44.2
+#> DIC: 98.5
+#>
+#> Bayesian p-value: 0.9
+#>
+#> ------------------------- 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 0.38 0.68 0.84 1.00 1.35 1430 2196 1
+#> d_dir 0.88 0.25 0.41 0.71 0.87 1.04 1.37 2196 2641 1
+#> d_ind 0.56 0.65 -0.68 0.13 0.54 0.98 1.89 1590 1992 1
+#> omega 0.32 0.67 -1.00 -0.13 0.32 0.78 1.60 1654 2031 1
+#> tau 0.85 0.19 0.55 0.72 0.83 0.96 1.29 1322 1893 1
+#> tau_consistency 0.84 0.18 0.55 0.71 0.82 0.94 1.26 1380 2233 1
+#>
+#> Residual deviance: 54.4 (on 50 data points)
+#> pD: 44.3
+#> DIC: 98.7
+#>
+#> Bayesian p-value: 0.63
+#>
+#> -------------------------------------- Node-split Self-help vs. No intervention ----
+#>
+#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
+#> d_net 0.51 0.40 -0.27 0.24 0.50 0.77 1.33 2229 2107 1
+#> d_dir 0.34 0.55 -0.73 -0.01 0.34 0.68 1.46 3389 2939 1
+#> d_ind 0.70 0.63 -0.52 0.30 0.69 1.09 1.98 2487 2488 1
+#> omega -0.36 0.82 -2.00 -0.88 -0.37 0.17 1.27 2531 2682 1
+#> tau 0.88 0.20 0.56 0.73 0.85 0.99 1.33 1412 1750 1
+#> tau_consistency 0.84 0.18 0.55 0.71 0.82 0.94 1.26 1380 2233 1
+#>
+#> Residual deviance: 53.5 (on 50 data points)
+#> pD: 44
+#> DIC: 97.5
+#>
+#> Bayesian p-value: 0.65
+#>
+#> ----------------------- 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 0.40 -1.05 -0.51 -0.26 0.01 0.53 3147 2866 1
+#> d_dir -0.10 0.48 -1.05 -0.40 -0.10 0.22 0.82 3613 3179 1
+#> d_ind -0.55 0.61 -1.78 -0.94 -0.54 -0.16 0.62 1543 1646 1
+#> omega 0.45 0.66 -0.85 0.02 0.45 0.87 1.79 1692 1832 1
+#> tau 0.87 0.20 0.55 0.73 0.84 0.98 1.33 1215 1784 1
+#> tau_consistency 0.84 0.18 0.55 0.71 0.82 0.94 1.26 1380 2233 1
+#>
+#> Residual deviance: 53.7 (on 50 data points)
+#> pD: 44.1
+#> DIC: 97.8
+#>
+#> Bayesian p-value: 0.47
+#>
+#> ------------------------------------ Node-split Self-help vs. Group counselling ----
+#>
+#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
+#> d_net -0.59 0.48 -1.54 -0.89 -0.59 -0.29 0.35 3241 2871 1
+#> d_dir -0.60 0.64 -1.85 -1.03 -0.62 -0.18 0.66 4527 3486 1
+#> d_ind -0.62 0.69 -2.02 -1.06 -0.59 -0.17 0.70 2189 2282 1
+#> omega 0.01 0.90 -1.68 -0.59 -0.01 0.57 1.82 2583 2545 1
+#> tau 0.88 0.20 0.58 0.74 0.85 0.99 1.33 1305 2176 1
+#> tau_consistency 0.84 0.18 0.55 0.71 0.82 0.94 1.26 1380 2233 1
+#>
+#> Residual deviance: 54 (on 50 data points)
+#> pD: 44.4
+#> DIC: 98.4
+#>
+#> 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 -0.34 0.41 -1.17 -0.60 -0.33 -0.08 0.47 2744 2564 1
+#> d_dir 0.06 0.66 -1.28 -0.36 0.06 0.49 1.35 3032 2784 1
+#> d_ind -0.61 0.53 -1.66 -0.96 -0.60 -0.27 0.42 1647 2275 1
+#> omega 0.67 0.82 -0.93 0.14 0.66 1.19 2.31 2064 2246 1
+#> tau 0.85 0.19 0.55 0.72 0.82 0.96 1.30 1018 1891 1
+#> tau_consistency 0.84 0.18 0.55 0.71 0.82 0.94 1.26 1380 2233 1
+#>
+#> Residual deviance: 54 (on 50 data points)
+#> pD: 44.1
+#> DIC: 98.1
+#>
+#> 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
@@ -724,59 +724,59 @@
Node-splitting
information for the Individual counselling vs. No intervention
comparison, so the network (consistency) estimate is very similar to the
direct estimate for this comparison.
-
plot(smk_nodesplit) +
- ggplot2::theme(legend.position = "bottom", legend.direct = "horizontal")
-
+
plot(smk_nodesplit) +
+ ggplot2::theme(legend.position = "bottom", legend.direction = "horizontal")
+
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% Bulk_ESS
-#> d[Group counselling vs. No intervention] 1.11 0.44 0.27 0.81 1.09 1.40 2.01 2031
-#> d[Individual counselling vs. No intervention] 0.84 0.25 0.39 0.67 0.83 0.99 1.36 1221
-#> d[Self-help vs. No intervention] 0.50 0.40 -0.27 0.24 0.48 0.75 1.31 1930
-#> d[Individual counselling vs. Group counselling] -0.27 0.42 -1.13 -0.54 -0.27 0.01 0.54 2773
-#> d[Self-help vs. Group counselling] -0.61 0.50 -1.58 -0.92 -0.61 -0.29 0.37 2816
-#> d[Self-help vs. Individual counselling] -0.34 0.41 -1.15 -0.61 -0.35 -0.08 0.47 2229
-#> Tail_ESS Rhat
-#> d[Group counselling vs. No intervention] 2307 1
-#> d[Individual counselling vs. No intervention] 1875 1
-#> d[Self-help vs. No intervention] 2362 1
-#> d[Individual counselling vs. Group counselling] 2754 1
-#> d[Self-help vs. Group counselling] 2787 1
-#> d[Self-help vs. Individual counselling] 2425 1
-plot(smk_releff, ref_line = 0)
-
+
(smk_releff <- relative_effects(smkfit, all_contrasts = TRUE))
+#> mean sd 2.5% 25% 50% 75% 97.5% Bulk_ESS
+#> d[Group counselling vs. No intervention] 1.11 0.43 0.32 0.81 1.09 1.38 2.00 1750
+#> d[Individual counselling vs. No intervention] 0.84 0.23 0.41 0.68 0.83 0.99 1.33 1257
+#> d[Self-help vs. No intervention] 0.49 0.40 -0.29 0.24 0.49 0.74 1.28 1639
+#> d[Individual counselling vs. Group counselling] -0.26 0.41 -1.07 -0.53 -0.26 0.01 0.52 2784
+#> d[Self-help vs. Group counselling] -0.61 0.48 -1.58 -0.92 -0.60 -0.29 0.32 2982
+#> d[Self-help vs. Individual counselling] -0.35 0.41 -1.19 -0.61 -0.34 -0.08 0.45 2219
+#> Tail_ESS Rhat
+#> d[Group counselling vs. No intervention] 2675 1
+#> d[Individual counselling vs. No intervention] 1773 1
+#> d[Self-help vs. No intervention] 2303 1
+#> d[Individual counselling vs. Group counselling] 2698 1
+#> d[Self-help vs. Group counselling] 2767 1
+#> d[Self-help vs. Individual counselling] 2406 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] 3.90 0.31 3 4 4 4 4 2465 NA 1
-#> rank[Group counselling] 1.36 0.63 1 1 1 2 3 2834 2832 1
-#> rank[Individual counselling] 1.94 0.63 1 2 2 2 3 2409 2647 1
-#> rank[Self-help] 2.80 0.69 1 3 3 3 4 2581 NA 1
-plot(smk_ranks)
+
(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] 3.90 0.30 3 4 4 4 4 2443 NA 1
+#> rank[Group counselling] 1.35 0.60 1 1 1 2 3 3345 3426 1
+#> rank[Individual counselling] 1.93 0.63 1 2 2 2 3 2611 3205 1
+#> rank[Self-help] 2.82 0.66 1 3 3 3 4 2414 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 0.10 0.9
-#> d[Group counselling] 0.71 0.22 0.07 0.0
-#> d[Individual counselling] 0.23 0.60 0.17 0.0
-#> d[Self-help] 0.06 0.18 0.67 0.1
-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 0.1 1
-#> d[Group counselling] 0.71 0.93 1.0 1
-#> d[Individual counselling] 0.23 0.83 1.0 1
-#> d[Self-help] 0.06 0.24 0.9 1
-plot(smk_cumrankprobs)
-
+
(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 0.10 0.9
+#> d[Group counselling] 0.72 0.22 0.06 0.0
+#> d[Individual counselling] 0.23 0.60 0.16 0.0
+#> d[Self-help] 0.05 0.17 0.68 0.1
+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 0.1 1
+#> d[Group counselling] 0.72 0.94 1.0 1
+#> d[Individual counselling] 0.23 0.84 1.0 1
+#> d[Self-help] 0.05 0.22 0.9 1
+plot(smk_cumrankprobs)
+
