Example: Parkinson’s disease

library(multinma)
options(mc.cores = parallel::detectCores())

This vignette describes the analysis of data on the mean off-time reduction in patients given dopamine agonists as adjunct therapy in Parkinson’s disease, in a network of 7 trials of 4 active drugs plus placebo (Dias et al. 2011). The data are available in this package as parkinsons:

head(parkinsons)
#>   studyn trtn     y    se   n  diff se_diff
#> 1      1    1 -1.22 0.504  54    NA   0.504
#> 2      1    3 -1.53 0.439  95 -0.31   0.668
#> 3      2    1 -0.70 0.282 172    NA   0.282
#> 4      2    2 -2.40 0.258 173 -1.70   0.382
#> 5      3    1 -0.30 0.505  76    NA   0.505
#> 6      3    2 -2.60 0.510  71 -2.30   0.718

We consider analysing these data in three separate ways:

  1. Using arm-based data (means y and corresponding standard errors se);
  2. Using contrast-based data (mean differences diff and corresponding standard errors se_diff);
  3. A combination of the two, where some studies contribute arm-based data, and other contribute contrast-based data.

Note: In this case, with Normal likelihoods for both arms and contrasts, we will see that the three analyses give identical results. In general, unless the arm-based likelihood is Normal, results from a model using a contrast-based likelihood will not exactly match those from a model using an arm-based likelihood, since the contrast-based Normal likelihood is only an approximation. Similarity of results depends on the suitability of the Normal approximation, which may not always be appropriate - e.g. with a small number of events or small sample size for a binary outcome. The use of an arm-based likelihood (sometimes called an “exact” likelihood) is therefore preferable where possible in general.

Analysis of arm-based data

We begin with an analysis of the arm-based data - means and standard errors.

Setting up the network

We have arm-level continuous data giving the mean off-time reduction (y) and standard error (se) in each arm. We use the function set_agd_arm() to set up the network.

arm_net <- set_agd_arm(parkinsons, 
                      study = studyn,
                      trt = trtn,
                      y = y, 
                      se = se,
                      sample_size = n)
arm_net
#> A network with 7 AgD studies (arm-based).
#> 
#> ------------------------------------------------------- AgD studies (arm-based) ---- 
#>  Study Treatment arms
#>  1     2: 1 | 3      
#>  2     2: 1 | 2      
#>  3     3: 4 | 1 | 2  
#>  4     2: 4 | 3      
#>  5     2: 4 | 3      
#>  6     2: 4 | 5      
#>  7     2: 4 | 5      
#> 
#>  Outcome type: continuous
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 5
#> Total number of studies: 7
#> Reference treatment is: 4
#> Network is connected

We let treatment 4 be set by default as the network reference treatment, since this results in considerably improved sampling efficiency over choosing treatment 1 as the network reference. The sample_size argument is optional, but enables the nodes to be weighted by sample size in the network plot.

Plot the network structure.

plot(arm_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.

arm_fit_FE <- nma(arm_net, 
                  trt_effects = "fixed",
                  prior_intercept = normal(scale = 100),
                  prior_trt = normal(scale = 10))
#> Note: Setting "4" as the network reference treatment.

Basic parameter summaries are given by the print() method:

arm_fit_FE
#> A fixed effects NMA with a normal likelihood (identity link).
#> Inference for Stan model: normal.
#> 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 Rhat
#> d[1]   0.52    0.01 0.49  -0.44   0.19   0.51   0.84   1.49  1455    1
#> d[2]  -1.29    0.01 0.53  -2.36  -1.63  -1.30  -0.93  -0.24  1447    1
#> d[3]   0.05    0.01 0.32  -0.59  -0.17   0.05   0.26   0.68  2084    1
#> d[5]  -0.30    0.00 0.21  -0.70  -0.45  -0.31  -0.17   0.11  3020    1
#> lp__ -58.31    0.06 2.45 -64.18 -59.72 -57.98 -56.53 -54.61  1593    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan  9 17:57:15 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:

# Not run
print(arm_fit_FE, pars = c("d", "mu"))

The prior and posterior distributions can be compared visually using the plot_prior_posterior() function:

plot_prior_posterior(arm_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

arm_fit_RE <- nma(arm_net, 
                  seed = 379394727,
                  trt_effects = "random",
                  prior_intercept = normal(scale = 100),
                  prior_trt = normal(scale = 100),
                  prior_het = half_normal(scale = 5),
                  adapt_delta = 0.99)
#> Note: Setting "4" as the network reference treatment.
#> Warning: There were 2 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems

We do see a small number of divergent transition errors, which cannot simply be removed by increasing the value of the adapt_delta argument (by default set to 0.95 for RE models). To diagnose, we use the pairs() method to investigate where in the posterior distribution these divergences are happening (indicated by red crosses):

pairs(arm_fit_RE, pars = c("mu[4]", "d[3]", "delta[4: 3]", "tau"))

The divergent transitions occur in the upper tail of the heterogeneity standard deviation. In this case, with only a small number of studies, there is not very much information to estimate the heterogeneity standard deviation and the prior distribution may be too heavy-tailed. We could consider a more informative prior distribution for the heterogeneity variance to aid estimation.

Basic parameter summaries are given by the print() method:

arm_fit_RE
#> A random effects NMA with a normal likelihood (identity link).
#> Inference for Stan model: normal.
#> 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 Rhat
#> d[1]   0.52    0.02 0.62  -0.63   0.14   0.51   0.90   1.78  1522    1
#> d[2]  -1.32    0.02 0.68  -2.68  -1.74  -1.32  -0.91  -0.01  1633    1
#> d[3]   0.03    0.01 0.48  -0.89  -0.24   0.04   0.31   0.94  2222    1
#> d[5]  -0.31    0.01 0.43  -1.15  -0.50  -0.31  -0.10   0.49  1629    1
#> lp__ -76.05    0.10 3.59 -84.03 -78.19 -75.72 -73.52 -70.04  1358    1
#> tau    0.38    0.01 0.38   0.01   0.12   0.27   0.51   1.37   723    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan  9 17:57:26 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(arm_fit_RE, pars = c("d", "mu", "delta"))

The prior and posterior distributions can be compared visually using the plot_prior_posterior() function:

plot_prior_posterior(arm_fit_RE)

Model comparison

Model fit can be checked using the dic() function:

(arm_dic_FE <- dic(arm_fit_FE))
#> Residual deviance: 13.5 (on 15 data points)
#>                pD: 11.2
#>               DIC: 24.7
(arm_dic_RE <- dic(arm_fit_RE))
#> Residual deviance: 13.5 (on 15 data points)
#>                pD: 12.3
#>               DIC: 25.8

Both models fit the data well, having posterior mean residual deviance close to the number of data points. The DIC is similar between models, so we choose the FE model based on parsimony.

We can also examine the residual deviance contributions with the corresponding plot() method.

plot(arm_dic_FE)

plot(arm_dic_RE)

Further results

For comparison with Dias et al. (2011), we can produce relative effects against placebo using the relative_effects() function with trt_ref = 1:

(arm_releff_FE <- relative_effects(arm_fit_FE, trt_ref = 1))
#>       mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d[4] -0.52 0.49 -1.49 -0.84 -0.51 -0.19  0.44     1462     2068    1
#> d[2] -1.81 0.33 -2.48 -2.03 -1.81 -1.58 -1.16     5387     3162    1
#> d[3] -0.47 0.50 -1.44 -0.82 -0.47 -0.13  0.50     2184     2601    1
#> d[5] -0.82 0.53 -1.87 -1.17 -0.82 -0.47  0.21     1571     2072    1
plot(arm_releff_FE, ref_line = 0)

(arm_releff_RE <- relative_effects(arm_fit_RE, trt_ref = 1))
#>       mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d[4] -0.52 0.62 -1.78 -0.90 -0.51 -0.14  0.63     1551     1917    1
#> d[2] -1.85 0.50 -2.86 -2.12 -1.84 -1.55 -0.91     4153     2853    1
#> d[3] -0.49 0.62 -1.69 -0.87 -0.49 -0.11  0.68     3010     2412    1
#> d[5] -0.83 0.76 -2.38 -1.25 -0.81 -0.38  0.58     1704     1902    1
plot(arm_releff_RE, ref_line = 0)

Following Dias et al. (2011), we produce absolute predictions of the mean off-time reduction on each treatment assuming a Normal distribution for the outcomes on treatment 1 (placebo) with mean \(-0.73\) and precision \(21\). We use the predict() method, where the baseline argument takes a distr() distribution object with which we specify the corresponding Normal distribution, and we specify trt_ref = 1 to indicate that the baseline distribution corresponds to treatment 1. (Strictly speaking, type = "response" is unnecessary here, since the identity link function was used.)

arm_pred_FE <- predict(arm_fit_FE, 
                       baseline = distr(qnorm, mean = -0.73, sd = 21^-0.5),
                       type = "response",
                       trt_ref = 1)
arm_pred_FE
#>          mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[4] -1.24 0.53 -2.29 -1.60 -1.25 -0.88 -0.20     1570     2220    1
#> pred[1] -0.73 0.22 -1.16 -0.88 -0.73 -0.58 -0.29     3910     3936    1
#> pred[2] -2.53 0.40 -3.31 -2.78 -2.53 -2.27 -1.75     4820     3588    1
#> pred[3] -1.20 0.54 -2.25 -1.56 -1.20 -0.81 -0.15     2260     2867    1
#> pred[5] -1.55 0.57 -2.69 -1.92 -1.55 -1.18 -0.45     1664     2609    1
plot(arm_pred_FE)

arm_pred_RE <- predict(arm_fit_RE, 
                       baseline = distr(qnorm, mean = -0.73, sd = 21^-0.5),
                       type = "response",
                       trt_ref = 1)
arm_pred_RE
#>          mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[4] -1.25 0.65 -2.60 -1.66 -1.24 -0.82 -0.06     1653     1893    1
#> pred[1] -0.73 0.22 -1.16 -0.87 -0.73 -0.58 -0.30     3976     3867    1
#> pred[2] -2.58 0.55 -3.64 -2.89 -2.57 -2.25 -1.56     4242     2675    1
#> pred[3] -1.22 0.66 -2.50 -1.64 -1.21 -0.80  0.03     3146     2589    1
#> pred[5] -1.56 0.79 -3.12 -2.00 -1.53 -1.09 -0.14     1760     1765    1
plot(arm_pred_RE)

If the baseline argument is omitted, predictions of mean off-time reduction will be produced for every study in the network based on their estimated baseline response \(\mu_j\):

arm_pred_FE_studies <- predict(arm_fit_FE, type = "response")
arm_pred_FE_studies
#> ---------------------------------------------------------------------- Study: 1 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[1: 4] -1.64 0.46 -2.59 -1.94 -1.64 -1.35 -0.76     1940     2553    1
#> pred[1: 1] -1.13 0.43 -1.97 -1.42 -1.12 -0.84 -0.29     3545     2759    1
#> pred[1: 2] -2.93 0.51 -3.95 -3.29 -2.93 -2.59 -1.96     3165     2850    1
#> pred[1: 3] -1.60 0.40 -2.40 -1.86 -1.59 -1.33 -0.80     3429     3084    1
#> pred[1: 5] -1.95 0.50 -2.96 -2.28 -1.95 -1.61 -0.98     1982     2436    1
#> 
#> ---------------------------------------------------------------------- Study: 2 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[2: 4] -1.16 0.52 -2.17 -1.50 -1.15 -0.82 -0.13     1447     2010    1
#> pred[2: 1] -0.64 0.26 -1.14 -0.81 -0.64 -0.47 -0.13     4796     3864    1
#> pred[2: 2] -2.45 0.25 -2.92 -2.62 -2.44 -2.28 -1.97     4864     3265    1
#> pred[2: 3] -1.11 0.54 -2.18 -1.47 -1.11 -0.76 -0.06     2060     2552    1
#> pred[2: 5] -1.46 0.56 -2.57 -1.82 -1.46 -1.10 -0.37     1557     2098    1
#> 
#> ---------------------------------------------------------------------- Study: 3 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[3: 4] -1.11 0.42 -1.95 -1.39 -1.10 -0.83 -0.29     1714     2578    1
#> pred[3: 1] -0.59 0.38 -1.33 -0.85 -0.60 -0.34  0.13     4890     3255    1
#> pred[3: 2] -2.40 0.40 -3.19 -2.67 -2.39 -2.13 -1.62     4261     2897    1
#> pred[3: 3] -1.06 0.48 -2.01 -1.38 -1.06 -0.75 -0.12     2691     3161    1
#> pred[3: 5] -1.41 0.47 -2.33 -1.72 -1.42 -1.09 -0.49     1803     2763    1
#> 
#> ---------------------------------------------------------------------- Study: 4 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[4: 4] -0.40 0.30 -0.99 -0.60 -0.39 -0.20  0.18     2265     2762    1
#> pred[4: 1]  0.12 0.51 -0.87 -0.24  0.11  0.48  1.09     2127     2799    1
#> pred[4: 2] -1.69 0.56 -2.77 -2.06 -1.69 -1.31 -0.59     1962     2477    1
#> pred[4: 3] -0.35 0.24 -0.83 -0.52 -0.36 -0.18  0.12     4953     3214    1
#> pred[4: 5] -0.70 0.36 -1.40 -0.94 -0.70 -0.46 -0.01     2477     2779    1
#> 
#> ---------------------------------------------------------------------- Study: 5 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[5: 4] -0.56 0.35 -1.25 -0.79 -0.54 -0.33  0.12     2514     2543    1
#> pred[5: 1] -0.04 0.55 -1.15 -0.42 -0.03  0.34  1.02     2143     2665    1
#> pred[5: 2] -1.85 0.60 -3.01 -2.23 -1.84 -1.46 -0.69     1966     2181    1
#> pred[5: 3] -0.51 0.29 -1.08 -0.71 -0.51 -0.32  0.06     4987     2803    1
#> pred[5: 5] -0.86 0.40 -1.67 -1.13 -0.86 -0.59 -0.07     2686     2693    1
#> 
#> ---------------------------------------------------------------------- Study: 6 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[6: 4] -2.19 0.17 -2.53 -2.31 -2.19 -2.08 -1.86     2909     2629    1
#> pred[6: 1] -1.68 0.51 -2.71 -2.02 -1.68 -1.34 -0.65     1584     2059    1
#> pred[6: 2] -3.48 0.56 -4.58 -3.84 -3.49 -3.11 -2.37     1566     2208    1
#> pred[6: 3] -2.15 0.37 -2.87 -2.39 -2.15 -1.90 -1.43     2340     2707    1
#> pred[6: 5] -2.50 0.17 -2.82 -2.61 -2.50 -2.39 -2.17     5579     3339    1
#> 
#> ---------------------------------------------------------------------- Study: 7 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[7: 4] -1.79 0.18 -2.13 -1.91 -1.79 -1.67 -1.45     3591     3141    1
#> pred[7: 1] -1.28 0.52 -2.27 -1.63 -1.28 -0.93 -0.21     1579     2161    1
#> pred[7: 2] -3.08 0.56 -4.17 -3.47 -3.09 -2.71 -1.95     1546     2169    1
#> pred[7: 3] -1.75 0.37 -2.48 -2.00 -1.75 -1.50 -1.05     2259     2571    1
#> pred[7: 5] -2.10 0.21 -2.49 -2.24 -2.10 -1.96 -1.69     4670     3172    1
plot(arm_pred_FE_studies)

We can also produce treatment rankings, rank probabilities, and cumulative rank probabilities.

(arm_ranks <- posterior_ranks(arm_fit_FE))
#>         mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> rank[4] 3.51 0.72    2   3   3   4     5     2091       NA    1
#> rank[1] 4.62 0.79    2   5   5   5     5     1999       NA    1
#> rank[2] 1.06 0.31    1   1   1   1     2     2051     2153    1
#> rank[3] 3.55 0.92    2   3   4   4     5     2948       NA    1
#> rank[5] 2.26 0.67    1   2   2   2     4     2493     2702    1
plot(arm_ranks)

(arm_rankprobs <- posterior_rank_probs(arm_fit_FE))
#>      p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5]
#> d[4]      0.00      0.04      0.49      0.38      0.09
#> d[1]      0.00      0.04      0.07      0.12      0.77
#> d[2]      0.95      0.04      0.01      0.00      0.00
#> d[3]      0.00      0.16      0.26      0.44      0.14
#> d[5]      0.05      0.72      0.17      0.05      0.01
plot(arm_rankprobs)

(arm_cumrankprobs <- posterior_rank_probs(arm_fit_FE, cumulative = TRUE))
#>      p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5]
#> d[4]      0.00      0.05      0.53      0.91         1
#> d[1]      0.00      0.04      0.11      0.23         1
#> d[2]      0.95      0.99      1.00      1.00         1
#> d[3]      0.00      0.16      0.42      0.86         1
#> d[5]      0.05      0.76      0.94      0.99         1
plot(arm_cumrankprobs)

Analysis of contrast-based data

We now perform an analysis using the contrast-based data (mean differences and standard errors).

Setting up the network

With contrast-level data giving the mean difference in off-time reduction (diff) and standard error (se_diff), we use the function set_agd_contrast() to set up the network.

contr_net <- set_agd_contrast(parkinsons, 
                              study = studyn,
                              trt = trtn,
                              y = diff, 
                              se = se_diff,
                              sample_size = n)
contr_net
#> A network with 7 AgD studies (contrast-based).
#> 
#> -------------------------------------------------- AgD studies (contrast-based) ---- 
#>  Study Treatment arms
#>  1     2: 1 | 3      
#>  2     2: 1 | 2      
#>  3     3: 4 | 1 | 2  
#>  4     2: 4 | 3      
#>  5     2: 4 | 3      
#>  6     2: 4 | 5      
#>  7     2: 4 | 5      
#> 
#>  Outcome type: continuous
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 5
#> Total number of studies: 7
#> Reference treatment is: 4
#> Network is connected

The sample_size argument is optional, but enables the nodes to be weighted by sample size in the network plot.

Plot the network structure.

plot(contr_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 \(d_k\). 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.

contr_fit_FE <- nma(contr_net, 
                    trt_effects = "fixed",
                    prior_trt = normal(scale = 100))
#> Note: Setting "4" as the network reference treatment.

Basic parameter summaries are given by the print() method:

contr_fit_FE
#> A fixed effects NMA with a normal likelihood (identity link).
#> Inference for Stan model: normal.
#> 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 Rhat
#> d[1]   0.52    0.01 0.48  -0.40   0.19   0.51   0.85   1.50  2071    1
#> d[2]  -1.29    0.01 0.52  -2.34  -1.64  -1.30  -0.94  -0.27  2191    1
#> d[3]   0.04    0.01 0.33  -0.59  -0.19   0.04   0.27   0.69  2688    1
#> d[5]  -0.30    0.00 0.21  -0.71  -0.44  -0.31  -0.16   0.12  3076    1
#> lp__ -25.27    0.03 1.41 -28.91 -25.98 -24.95 -24.22 -23.49  1720    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan  9 17:57:46 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at 
#> convergence, Rhat=1).

The prior and posterior distributions can be compared visually using the plot_prior_posterior() function:

plot_prior_posterior(contr_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 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

contr_fit_RE <- nma(contr_net, 
                    seed = 1150676438,
                    trt_effects = "random",
                    prior_trt = normal(scale = 100),
                    prior_het = half_normal(scale = 5),
                    adapt_delta = 0.99)
#> Note: Setting "4" as the network reference treatment.
#> Warning: There were 1 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems

We do see a small number of divergent transition errors, which cannot simply be removed by increasing the value of the adapt_delta argument (by default set to 0.95 for RE models). To diagnose, we use the pairs() method to investigate where in the posterior distribution these divergences are happening (indicated by red crosses):

pairs(contr_fit_RE, pars = c("d[3]", "delta[4: 4 vs. 3]", "tau"))

The divergent transitions occur in the upper tail of the heterogeneity standard deviation. In this case, with only a small number of studies, there is not very much information to estimate the heterogeneity standard deviation and the prior distribution may be too heavy-tailed. We could consider a more informative prior distribution for the heterogeneity variance to aid estimation.

Basic parameter summaries are given by the print() method:

contr_fit_RE
#> A random effects NMA with a normal likelihood (identity link).
#> Inference for Stan model: normal.
#> 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 Rhat
#> d[1]   0.53    0.01 0.62  -0.64   0.15   0.52   0.90   1.77  2374    1
#> d[2]  -1.31    0.01 0.70  -2.64  -1.73  -1.31  -0.91   0.08  2315    1
#> d[3]   0.03    0.01 0.48  -0.89  -0.24   0.03   0.30   0.92  2405    1
#> d[5]  -0.30    0.01 0.46  -1.27  -0.51  -0.31  -0.10   0.58  1453    1
#> lp__ -32.87    0.09 2.87 -39.20 -34.57 -32.61 -30.85 -27.99  1067    1
#> tau    0.40    0.02 0.43   0.01   0.12   0.28   0.51   1.51   789    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan  9 17:57:55 2024.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at 
#> convergence, Rhat=1).

By default, summaries of the study-specific relative effects \(\delta_{jk}\) are hidden, but could be examined by changing the pars argument:

# Not run
print(contr_fit_RE, pars = c("d", "delta"))

The prior and posterior distributions can be compared visually using the plot_prior_posterior() function:

plot_prior_posterior(contr_fit_RE)

Model comparison

Model fit can be checked using the dic() function:

(contr_dic_FE <- dic(contr_fit_FE))
#> Residual deviance: 6.3 (on 8 data points)
#>                pD: 4.1
#>               DIC: 10.4
(contr_dic_RE <- dic(contr_fit_RE))
#> Residual deviance: 6.5 (on 8 data points)
#>                pD: 5.3
#>               DIC: 11.8

Both models fit the data well, having posterior mean residual deviance close to the number of data points. The DIC is similar between models, so we choose the FE model based on parsimony.

We can also examine the residual deviance contributions with the corresponding plot() method.

plot(contr_dic_FE)

plot(contr_dic_RE)

Further results

For comparison with Dias et al. (2011), we can produce relative effects against placebo using the relative_effects() function with trt_ref = 1:

(contr_releff_FE <- relative_effects(contr_fit_FE, trt_ref = 1))
#>       mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d[4] -0.52 0.48 -1.50 -0.85 -0.51 -0.19  0.40     2079     1902    1
#> d[2] -1.81 0.33 -2.47 -2.03 -1.81 -1.59 -1.18     5161     3033    1
#> d[3] -0.48 0.49 -1.43 -0.81 -0.46 -0.16  0.49     2944     2987    1
#> d[5] -0.82 0.52 -1.85 -1.18 -0.81 -0.46  0.19     2210     2363    1
plot(contr_releff_FE, ref_line = 0)

(contr_releff_RE <- relative_effects(contr_fit_RE, trt_ref = 1))
#>       mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d[4] -0.53 0.62 -1.77 -0.90 -0.52 -0.15  0.64     2734     1906    1
#> d[2] -1.84 0.51 -2.83 -2.14 -1.84 -1.55 -0.81     3720     2532    1
#> d[3] -0.50 0.63 -1.76 -0.86 -0.50 -0.12  0.72     3618     2669    1
#> d[5] -0.83 0.74 -2.34 -1.26 -0.84 -0.39  0.56     2394     1698    1
plot(contr_releff_RE, ref_line = 0)

Following Dias et al. (2011), we produce absolute predictions of the mean off-time reduction on each treatment assuming a Normal distribution for the outcomes on treatment 1 (placebo) with mean \(-0.73\) and precision \(21\). We use the predict() method, where the baseline argument takes a distr() distribution object with which we specify the corresponding Normal distribution, and we specify trt_ref = 1 to indicate that the baseline distribution corresponds to treatment 1. (Strictly speaking, type = "response" is unnecessary here, since the identity link function was used.)

contr_pred_FE <- predict(contr_fit_FE, 
                       baseline = distr(qnorm, mean = -0.73, sd = 21^-0.5),
                       type = "response",
                       trt_ref = 1)
contr_pred_FE
#>          mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[4] -1.25 0.53 -2.32 -1.60 -1.24 -0.89 -0.22     2373     2538    1
#> pred[1] -0.73 0.22 -1.16 -0.88 -0.73 -0.58 -0.30     3988     3629    1
#> pred[2] -2.54 0.39 -3.34 -2.80 -2.54 -2.28 -1.77     4817     3183    1
#> pred[3] -1.21 0.54 -2.27 -1.56 -1.20 -0.85 -0.15     3205     3270    1
#> pred[5] -1.55 0.57 -2.66 -1.93 -1.55 -1.17 -0.44     2493     2437    1
plot(contr_pred_FE)

contr_pred_RE <- predict(contr_fit_RE, 
                       baseline = distr(qnorm, mean = -0.73, sd = 21^-0.5),
                       type = "response",
                       trt_ref = 1)
contr_pred_RE
#>          mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[4] -1.26 0.66 -2.54 -1.67 -1.24 -0.86 -0.01     2879     2089    1
#> pred[1] -0.73 0.22 -1.16 -0.88 -0.73 -0.59 -0.31     3715     3592    1
#> pred[2] -2.57 0.56 -3.66 -2.91 -2.58 -2.24 -1.46     3722     2467    1
#> pred[3] -1.23 0.67 -2.57 -1.63 -1.24 -0.81  0.05     3715     2838    1
#> pred[5] -1.57 0.78 -3.12 -2.02 -1.56 -1.09 -0.09     2359     1834    1
plot(contr_pred_RE)

If the baseline argument is omitted an error will be raised, as there are no study baselines estimated in this network.

# Not run
predict(contr_fit_FE, type = "response")

We can also produce treatment rankings, rank probabilities, and cumulative rank probabilities.

(contr_ranks <- posterior_ranks(contr_fit_FE))
#>         mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> rank[4] 3.51 0.73    2   3   3   4     5     2462       NA    1
#> rank[1] 4.64 0.77    2   5   5   5     5     2330       NA    1
#> rank[2] 1.05 0.28    1   1   1   1     2     2564     2522    1
#> rank[3] 3.51 0.92    2   3   4   4     5     3676       NA    1
#> rank[5] 2.29 0.67    1   2   2   2     4     2598     2496    1
plot(contr_ranks)

(contr_rankprobs <- posterior_rank_probs(contr_fit_FE))
#>      p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5]
#> d[4]      0.00      0.05      0.48      0.38      0.09
#> d[1]      0.00      0.04      0.06      0.12      0.78
#> d[2]      0.96      0.03      0.01      0.00      0.00
#> d[3]      0.00      0.16      0.27      0.44      0.12
#> d[5]      0.04      0.72      0.18      0.06      0.01
plot(contr_rankprobs)

(contr_cumrankprobs <- posterior_rank_probs(contr_fit_FE, cumulative = TRUE))
#>      p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5]
#> d[4]      0.00      0.05      0.53      0.91         1
#> d[1]      0.00      0.04      0.10      0.22         1
#> d[2]      0.96      0.99      1.00      1.00         1
#> d[3]      0.00      0.17      0.44      0.88         1
#> d[5]      0.04      0.76      0.93      0.99         1
plot(contr_cumrankprobs)

Analysis of mixed arm-based and contrast-based data

We now perform an analysis where some studies contribute arm-based data, and other contribute contrast-based data. Replicating Dias et al. (2011), we consider arm-based data from studies 1-3, and contrast-based data from studies 4-7.

studies <- parkinsons$studyn
(parkinsons_arm <- parkinsons[studies %in% 1:3, ])
#>   studyn trtn     y    se   n  diff se_diff
#> 1      1    1 -1.22 0.504  54    NA   0.504
#> 2      1    3 -1.53 0.439  95 -0.31   0.668
#> 3      2    1 -0.70 0.282 172    NA   0.282
#> 4      2    2 -2.40 0.258 173 -1.70   0.382
#> 5      3    1 -0.30 0.505  76    NA   0.505
#> 6      3    2 -2.60 0.510  71 -2.30   0.718
#> 7      3    4 -1.20 0.478  81 -0.90   0.695
(parkinsons_contr <- parkinsons[studies %in% 4:7, ])
#>    studyn trtn     y    se   n  diff se_diff
#> 8       4    3 -0.24 0.265 128    NA   0.265
#> 9       4    4 -0.59 0.354  72 -0.35   0.442
#> 10      5    3 -0.73 0.335  80    NA   0.335
#> 11      5    4 -0.18 0.442  46  0.55   0.555
#> 12      6    4 -2.20 0.197 137    NA   0.197
#> 13      6    5 -2.50 0.190 131 -0.30   0.274
#> 14      7    4 -1.80 0.200 154    NA   0.200
#> 15      7    5 -2.10 0.250 143 -0.30   0.320

Setting up the network

We use the functions set_agd_arm() and set_agd_contrast() to set up the respective data sources within the network, and then combine together with combine_network().

mix_arm_net <- set_agd_arm(parkinsons_arm, 
                           study = studyn,
                           trt = trtn,
                           y = y, 
                           se = se,
                           sample_size = n)

mix_contr_net <- set_agd_contrast(parkinsons_contr, 
                                  study = studyn,
                                  trt = trtn,
                                  y = diff, 
                                  se = se_diff,
                                  sample_size = n)

mix_net <- combine_network(mix_arm_net, mix_contr_net)
mix_net
#> A network with 3 AgD studies (arm-based), and 4 AgD studies (contrast-based).
#> 
#> ------------------------------------------------------- AgD studies (arm-based) ---- 
#>  Study Treatment arms
#>  1     2: 1 | 3      
#>  2     2: 1 | 2      
#>  3     3: 4 | 1 | 2  
#> 
#>  Outcome type: continuous
#> -------------------------------------------------- AgD studies (contrast-based) ---- 
#>  Study Treatment arms
#>  4     2: 4 | 3      
#>  5     2: 4 | 3      
#>  6     2: 4 | 5      
#>  7     2: 4 | 5      
#> 
#>  Outcome type: continuous
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 5
#> Total number of studies: 7
#> Reference treatment is: 4
#> Network is connected

The sample_size argument is optional, but enables the nodes to be weighted by sample size in the network plot.

Plot the network structure.

plot(mix_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.

mix_fit_FE <- nma(mix_net, 
                  trt_effects = "fixed",
                  prior_intercept = normal(scale = 100),
                  prior_trt = normal(scale = 100))
#> Note: Setting "4" as the network reference treatment.

Basic parameter summaries are given by the print() method:

mix_fit_FE
#> A fixed effects NMA with a normal likelihood (identity link).
#> Inference for Stan model: normal.
#> 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 Rhat
#> d[1]   0.54    0.01 0.48  -0.43   0.21   0.56   0.87   1.46  1400    1
#> d[2]  -1.28    0.01 0.52  -2.29  -1.64  -1.27  -0.93  -0.31  1453    1
#> d[3]   0.05    0.01 0.33  -0.58  -0.17   0.05   0.27   0.72  2193    1
#> d[5]  -0.31    0.00 0.21  -0.71  -0.44  -0.31  -0.17   0.11  3102    1
#> lp__ -43.33    0.05 1.88 -47.85 -44.38 -42.97 -41.97 -40.62  1734    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan  9 17:58:10 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:

# Not run
print(mix_fit_FE, pars = c("d", "mu"))

The prior and posterior distributions can be compared visually using the plot_prior_posterior() function:

plot_prior_posterior(mix_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

mix_fit_RE <- nma(mix_net, 
                  seed = 437219664,
                  trt_effects = "random",
                  prior_intercept = normal(scale = 100),
                  prior_trt = normal(scale = 100),
                  prior_het = half_normal(scale = 5),
                  adapt_delta = 0.99)
#> Note: Setting "4" as the network reference treatment.
#> Warning: There were 2 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems

We do see a small number of divergent transition errors, which cannot simply be removed by increasing the value of the adapt_delta argument (by default set to 0.95 for RE models). To diagnose, we use the pairs() method to investigate where in the posterior distribution these divergences are happening (indicated by red crosses):

pairs(mix_fit_RE, pars = c("d[3]", "delta[4: 4 vs. 3]", "tau"))

The divergent transitions occur in the upper tail of the heterogeneity standard deviation. In this case, with only a small number of studies, there is not very much information to estimate the heterogeneity standard deviation and the prior distribution may be too heavy-tailed. We could consider a more informative prior distribution for the heterogeneity variance to aid estimation.

Basic parameter summaries are given by the print() method:

mix_fit_RE
#> A random effects NMA with a normal likelihood (identity link).
#> Inference for Stan model: normal.
#> 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 Rhat
#> d[1]   0.52    0.02 0.62  -0.68   0.14   0.52   0.90   1.73  1630 1.00
#> d[2]  -1.32    0.02 0.69  -2.70  -1.74  -1.31  -0.90   0.01  1869 1.00
#> d[3]   0.01    0.01 0.46  -0.92  -0.26   0.02   0.28   0.88  2335 1.00
#> d[5]  -0.28    0.01 0.41  -1.07  -0.49  -0.29  -0.08   0.55  2213 1.00
#> lp__ -51.85    0.11 3.31 -59.22 -53.81 -51.51 -49.50 -46.37   938 1.00
#> tau    0.39    0.01 0.37   0.01   0.13   0.28   0.52   1.35   601 1.01
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan  9 17:58:21 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(mix_fit_RE, pars = c("d", "mu", "delta"))

The prior and posterior distributions can be compared visually using the plot_prior_posterior() function:

plot_prior_posterior(mix_fit_RE)

Model comparison

Model fit can be checked using the dic() function:

(mix_dic_FE <- dic(mix_fit_FE))
#> Residual deviance: 9.3 (on 11 data points)
#>                pD: 7
#>               DIC: 16.4
(mix_dic_RE <- dic(mix_fit_RE))
#> Residual deviance: 9.5 (on 11 data points)
#>                pD: 8.3
#>               DIC: 17.8

Both models fit the data well, having posterior mean residual deviance close to the number of data points. The DIC is similar between models, so we choose the FE model based on parsimony.

We can also examine the residual deviance contributions with the corresponding plot() method.

plot(mix_dic_FE)

plot(mix_dic_RE)

Further results

For comparison with Dias et al. (2011), we can produce relative effects against placebo using the relative_effects() function with trt_ref = 1:

(mix_releff_FE <- relative_effects(mix_fit_FE, trt_ref = 1))
#>       mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d[4] -0.54 0.48 -1.46 -0.87 -0.56 -0.21  0.43     1439     2275    1
#> d[2] -1.82 0.33 -2.46 -2.04 -1.82 -1.59 -1.14     5602     3215    1
#> d[3] -0.49 0.48 -1.43 -0.82 -0.49 -0.16  0.46     2255     2644    1
#> d[5] -0.84 0.52 -1.86 -1.21 -0.85 -0.49  0.17     1559     2448    1
plot(mix_releff_FE, ref_line = 0)

(mix_releff_RE <- relative_effects(mix_fit_RE, trt_ref = 1))
#>       mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> d[4] -0.52 0.62 -1.73 -0.90 -0.52 -0.14  0.68     1690     1682    1
#> d[2] -1.84 0.50 -2.86 -2.12 -1.83 -1.55 -0.86     4089     2312    1
#> d[3] -0.51 0.62 -1.72 -0.89 -0.50 -0.12  0.70     2941     2065    1
#> d[5] -0.80 0.76 -2.31 -1.24 -0.79 -0.38  0.70     1791     1643    1
plot(mix_releff_RE, ref_line = 0)

Following Dias et al. (2011), we produce absolute predictions of the mean off-time reduction on each treatment assuming a Normal distribution for the outcomes on treatment 1 (placebo) with mean \(-0.73\) and precision \(21\). We use the predict() method, where the baseline argument takes a distr() distribution object with which we specify the corresponding Normal distribution, and we specify trt_ref = 1 to indicate that the baseline distribution corresponds to treatment 1. (Strictly speaking, type = "response" is unnecessary here, since the identity link function was used.)

mix_pred_FE <- predict(mix_fit_FE, 
                       baseline = distr(qnorm, mean = -0.73, sd = 21^-0.5),
                       type = "response",
                       trt_ref = 1)
mix_pred_FE
#>          mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[4] -1.26 0.54 -2.28 -1.63 -1.27 -0.89 -0.21     1564     2770    1
#> pred[1] -0.72 0.22 -1.14 -0.87 -0.72 -0.58 -0.29     3556     3638    1
#> pred[2] -2.54 0.40 -3.33 -2.81 -2.53 -2.27 -1.77     5100     3406    1
#> pred[3] -1.21 0.54 -2.28 -1.58 -1.21 -0.85 -0.13     2366     3038    1
#> pred[5] -1.57 0.57 -2.69 -1.95 -1.58 -1.17 -0.47     1656     2573    1
plot(mix_pred_FE)

mix_pred_RE <- predict(mix_fit_RE, 
                       baseline = distr(qnorm, mean = -0.73, sd = 21^-0.5),
                       type = "response",
                       trt_ref = 1)
mix_pred_RE
#>          mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[4] -1.25 0.65 -2.51 -1.67 -1.24 -0.84  0.05     1799     1935    1
#> pred[1] -0.73 0.22 -1.17 -0.88 -0.73 -0.58 -0.31     4051     3817    1
#> pred[2] -2.57 0.54 -3.62 -2.89 -2.57 -2.24 -1.53     4082     2741    1
#> pred[3] -1.24 0.66 -2.52 -1.64 -1.24 -0.82  0.05     3016     2476    1
#> pred[5] -1.53 0.79 -3.07 -2.01 -1.53 -1.06  0.04     1830     1872    1
plot(mix_pred_RE)

If the baseline argument is omitted, predictions of mean off-time reduction will be produced for every arm-based study in the network based on their estimated baseline response \(\mu_j\):

mix_pred_FE_studies <- predict(mix_fit_FE, type = "response")
mix_pred_FE_studies
#> ---------------------------------------------------------------------- Study: 1 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[1: 4] -1.66 0.46 -2.57 -1.97 -1.66 -1.35 -0.73     1773     2308    1
#> pred[1: 1] -1.12 0.43 -1.96 -1.42 -1.11 -0.82 -0.31     3431     2745    1
#> pred[1: 2] -2.94 0.51 -3.94 -3.28 -2.92 -2.60 -1.95     3163     2927    1
#> pred[1: 3] -1.61 0.39 -2.33 -1.87 -1.62 -1.34 -0.85     3321     2745    1
#> pred[1: 5] -1.96 0.50 -2.94 -2.31 -1.97 -1.63 -1.00     1872     2319    1
#> 
#> ---------------------------------------------------------------------- Study: 2 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[2: 4] -1.17 0.51 -2.16 -1.53 -1.18 -0.83 -0.17     1429     2350    1
#> pred[2: 1] -0.64 0.26 -1.15 -0.81 -0.64 -0.46 -0.11     5027     3218    1
#> pred[2: 2] -2.45 0.24 -2.92 -2.62 -2.45 -2.28 -1.99     4600     3784    1
#> pred[2: 3] -1.12 0.52 -2.12 -1.48 -1.13 -0.77 -0.10     2135     2645    1
#> pred[2: 5] -1.48 0.55 -2.51 -1.85 -1.49 -1.11 -0.41     1540     2193    1
#> 
#> ---------------------------------------------------------------------- Study: 3 ---- 
#> 
#>             mean   sd  2.5%   25%   50%   75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[3: 4] -1.12 0.42 -1.95 -1.40 -1.11 -0.84 -0.28     1776     2446    1
#> pred[3: 1] -0.58 0.36 -1.29 -0.82 -0.58 -0.34  0.12     4552     3294    1
#> pred[3: 2] -2.40 0.38 -3.17 -2.65 -2.40 -2.14 -1.65     4155     3226    1
#> pred[3: 3] -1.07 0.47 -2.00 -1.38 -1.07 -0.75 -0.13     2816     2808    1
#> pred[3: 5] -1.42 0.47 -2.36 -1.73 -1.42 -1.12 -0.52     2004     2545    1
plot(mix_pred_FE_studies)

We can also produce treatment rankings, rank probabilities, and cumulative rank probabilities.

(mix_ranks <- posterior_ranks(mix_fit_FE))
#>         mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> rank[4] 3.48 0.72    2   3   3   4     5     2118       NA    1
#> rank[1] 4.65 0.76    2   5   5   5     5     1863       NA    1
#> rank[2] 1.05 0.27    1   1   1   1     2     2313     2316    1
#> rank[3] 3.55 0.91    2   3   4   4     5     3138       NA    1
#> rank[5] 2.27 0.64    1   2   2   2     4     2398     2801    1
plot(mix_ranks)

(mix_rankprobs <- posterior_rank_probs(mix_fit_FE))
#>      p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5]
#> d[4]      0.00      0.05      0.50      0.37      0.08
#> d[1]      0.00      0.04      0.06      0.11      0.79
#> d[2]      0.96      0.03      0.01      0.00      0.00
#> d[3]      0.00      0.16      0.26      0.46      0.12
#> d[5]      0.04      0.73      0.17      0.05      0.01
plot(mix_rankprobs)

(mix_cumrankprobs <- posterior_rank_probs(mix_fit_FE, cumulative = TRUE))
#>      p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5]
#> d[4]      0.00      0.05      0.55      0.92         1
#> d[1]      0.00      0.04      0.10      0.21         1
#> d[2]      0.96      0.99      1.00      1.00         1
#> d[3]      0.00      0.16      0.41      0.88         1
#> d[5]      0.04      0.77      0.94      0.99         1
plot(mix_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.