Example: Plaque psoriasis HTA report

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

This vignette describes the analysis of treatments for moderate-to-severe plaque psoriasis from an HTA report (Woolacott et al. 2006), replicating the analysis in NICE Technical Support Document 2 (Dias et al. 2011). The data are available in this package as hta_psoriasis:

head(hta_psoriasis)
#>   studyn   studyc year trtn             trtc sample_size PASI50 PASI75 PASI90
#> 1      1  Elewski 2004    1  Supportive care         193     12      5      1
#> 2      1  Elewski 2004    2 Etanercept 25 mg         196     59     46     21
#> 3      1  Elewski 2004    3 Etanercept 50 mg         194     54     56     40
#> 4      2 Gottlieb 2003    1  Supportive care          55      5      1      0
#> 5      2 Gottlieb 2003    2 Etanercept 25 mg          57     23     11      6
#> 6      3  Lebwohl 2003    1  Supportive care         122     13      5      1

Outcomes are ordered multinomial success/failure to achieve 50%, 75%, or 90% reduction in symptoms on the Psoriasis Area and Severity Index (PASI) scale. Some studies report ordered outcomes at all three cutpoints, others only one or two:

dplyr::filter(hta_psoriasis, studyc %in% c("Elewski", "Gordon", "ACD2058g", "Altmeyer"))
#>   studyn   studyc year trtn             trtc sample_size PASI50 PASI75 PASI90
#> 1      1  Elewski 2004    1  Supportive care         193     12      5      1
#> 2      1  Elewski 2004    2 Etanercept 25 mg         196     59     46     21
#> 3      1  Elewski 2004    3 Etanercept 50 mg         194     54     56     40
#> 4      5   Gordon 2003    1  Supportive care         187     18      8     NA
#> 5      5   Gordon 2003    4       Efalizumab         369    118     98     NA
#> 6      6 ACD2058g 2004    1  Supportive care         170     25     NA     NA
#> 7      6 ACD2058g 2004    4       Efalizumab         162     99     NA     NA
#> 8     10 Altmeyer 1994    1  Supportive care          51     NA      1     NA
#> 9     10 Altmeyer 1994    6         Fumaderm          49     NA     12     NA

Here, the outcome counts are given as “exclusive” counts. That is, for a study reporting all outcomes (e.g. Elewski), the counts represent the categories 50 < PASI < 75, 75 < PASI < 90, and 90 < PASI < 100, and the corresponding columns are named by the lower end of the interval. Missing values are used where studies only report a subset of the outcomes. For a study reporting only two outcomes, say PASI50 and PASI75 as in Gordon, the counts represent the categories 50 < PASI < 75 and 75 < PASI < 100. For a study reporting only one outcome, say PASI70 as in Altmeyer, the count represents 70 < PASI < 100. We also need the count for the lowest category (i.e. no higher outcomes achieved), which is equal to the sample size minus the counts in the other observed categories.

Setting up the network

We begin by setting up the network. We have arm-level ordered multinomial count data, so we use the function set_agd_arm(). The function multi() helps us to specify the ordered outcomes correctly.

pso_net <- set_agd_arm(hta_psoriasis, 
                       study = paste(studyc, year), 
                       trt = trtc, 
                       r = multi(r0 = sample_size - rowSums(cbind(PASI50, PASI75, PASI90), na.rm = TRUE), 
                                 PASI50, PASI75, PASI90,
                                 inclusive = FALSE, 
                                 type = "ordered"))
pso_net
#> A network with 16 AgD studies (arm-based).
#> 
#> ------------------------------------------------------- AgD studies (arm-based) ---- 
#>  Study         Treatment arms                                          
#>  ACD2058g 2004 2: Supportive care | Efalizumab                         
#>  ACD2600g 2004 2: Supportive care | Efalizumab                         
#>  Altmeyer 1994 2: Supportive care | Fumaderm                           
#>  Chaudari 2001 2: Supportive care | Infliximab                         
#>  Elewski 2004  3: Supportive care | Etanercept 25 mg | Etanercept 50 mg
#>  Ellis 1991    3: Supportive care | Ciclosporin | Ciclosporin          
#>  Gordon 2003   2: Supportive care | Efalizumab                         
#>  Gottlieb 2003 2: Supportive care | Etanercept 25 mg                   
#>  Gottlieb 2004 3: Supportive care | Infliximab | Infliximab            
#>  Guenther 1991 2: Supportive care | Ciclosporin                        
#>  ... plus 6 more studies
#> 
#>  Outcome type: ordered (4 categories)
#> ------------------------------------------------------------------------------------
#> Total number of treatments: 8
#> Total number of studies: 16
#> Reference treatment is: Supportive care
#> Network is connected

Plot the network structure.

plot(pso_net, weight_edges = TRUE, weight_nodes = TRUE) + 
  # Nudge the legend over
  ggplot2::theme(legend.box.spacing = ggplot2::unit(0.75, "in"),
                 plot.margin = ggplot2::margin(0.1, 0, 0.1, 0.75, "in"))

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", using a probit link function link = "probit". We use \(\mathrm{N}(0, 10^2)\) prior distributions for the treatment effects \(d_k\), and \(\mathrm{N}(0, 100^2)\) prior distributions for the 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 = 10))
#> A Normal prior distribution: location = 0, scale = 10.
#> 50% of the prior density lies between -6.74 and 6.74.
#> 95% of the prior density lies between -19.6 and 19.6.
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.

We also need to specify prior distributions for the latent cutpoints \(c_\textrm{PASI75}\) and \(c_\textrm{PASI90}\) on the underlying scale - here the PASI standardised mean difference due to the probit link (the cutpoint \(c_\textrm{PASI50}=0\)). To make these easier to reason about, we actually specify priors on the differences between adjacent cutpoints, e.g. \(c_\textrm{PASI90} - c_\textrm{PASI75}\) and \(c_\textrm{PASI75} - c_\textrm{PASI50}\). These can be given any positive-valued prior distribution, and Stan will automatically impose the necessary ordering constraints behind the scenes. We choose to give these implicit flat priors flat().

The model is fitted using the nma() function.

pso_fit_FE <- nma(pso_net, 
                  trt_effects = "fixed",
                  link = "probit",
                  prior_intercept = normal(scale = 100),
                  prior_trt = normal(scale = 10),
                  prior_aux = flat())
#> Note: Setting "Supportive care" as the network reference treatment.

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

pso_fit_FE
#> A fixed effects NMA with a ordered likelihood (probit link).
#> Inference for Stan model: ordered_multinomial.
#> 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[Ciclosporin]          1.93    0.01 0.34     1.31     1.68     1.91     2.15     2.63  1320    1
#> d[Efalizumab]           1.19    0.00 0.06     1.08     1.15     1.19     1.23     1.30  1911    1
#> d[Etanercept 25 mg]     1.51    0.00 0.09     1.33     1.44     1.51     1.57     1.69  2287    1
#> d[Etanercept 50 mg]     1.92    0.00 0.10     1.73     1.85     1.92     1.98     2.11  2514    1
#> d[Fumaderm]             1.48    0.01 0.49     0.61     1.14     1.45     1.79     2.51  2348    1
#> d[Infliximab]           2.34    0.01 0.28     1.83     2.16     2.33     2.52     2.91  2842    1
#> d[Methotrexate]         1.62    0.01 0.44     0.77     1.32     1.62     1.92     2.51  1540    1
#> lp__                -3516.17    0.09 3.58 -3524.40 -3518.36 -3515.86 -3513.56 -3510.21  1454    1
#> cc[PASI50]              0.00     NaN 0.00     0.00     0.00     0.00     0.00     0.00   NaN  NaN
#> cc[PASI75]              0.76    0.00 0.03     0.70     0.73     0.75     0.78     0.82  5385    1
#> cc[PASI90]              1.56    0.00 0.05     1.46     1.53     1.56     1.60     1.67  6851    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan  9 18:07:36 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).

Note: the treatment effects are the opposite sign to those in TSD 2 (Dias et al. 2011). This is because we parameterise the linear predictor as \(\mu_j + d_k + c_m\), rather than \(\mu_j + d_k - c_m\). The interpretation here thus follows that of a standard binomial probit (or logit) regression; SMDs (or log ORs) greater than zero mean that the treatment increases the probability of an event compared to the comparator (and less than zero mean a reduction in probability). Here higher outcomes are positive, and all of the active treatments are estimated to increase the response (i.e. a greater reduction) on the PASI scale compared to the network reference (supportive care).

By default, summaries of the study-specific intercepts \(\mu_j\) are hidden, but could be examined by changing the pars argument:

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

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

plot_prior_posterior(pso_fit_FE)

Focusing specifically on the cutpoints we see that these are highly identified by the data, which is why the implicit flat priors work for these parameters.

plot_prior_posterior(pso_fit_FE, prior = "aux")

Random effects meta-analysis

We now fit a random effects model using the nma() function with trt_effects = "random". Again, we use \(\mathrm{N}(0, 10^2)\) prior distributions for the treatment effects \(d_k\), \(\mathrm{N}(0, 100^2)\) prior distributions for the study-specific intercepts \(\mu_j\), implicit flat prior distributions for the latent cutpoints, and we additionally use a \(\textrm{half-N}(2.5^2)\) 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 = 10))
#> A Normal prior distribution: location = 0, scale = 10.
#> 50% of the prior density lies between -6.74 and 6.74.
#> 95% of the prior density lies between -19.6 and 19.6.
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 = 2.5))
#> A half-Normal prior distribution: location = 0, scale = 2.5.
#> 50% of the prior density lies between 0 and 1.69.
#> 95% of the prior density lies between 0 and 4.9.

Fitting the RE model

pso_fit_RE <- nma(pso_net, 
                  trt_effects = "random",
                  link = "probit",
                  prior_intercept = normal(scale = 100),
                  prior_trt = normal(scale = 10),
                  prior_aux = flat(),
                  prior_het = half_normal(scale = 2.5),
                  adapt_delta = 0.99)
#> Note: Setting "Supportive care" as the network reference treatment.

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

pso_fit_RE
#> A random effects NMA with a ordered likelihood (probit link).
#> Inference for Stan model: ordered_multinomial.
#> 4 chains, each with iter=5000; warmup=2500; thin=1; 
#> post-warmup draws per chain=2500, total post-warmup draws=10000.
#> 
#>                         mean se_mean   sd     2.5%      25%      50%      75%    97.5% n_eff Rhat
#> d[Ciclosporin]          2.02    0.01 0.42     1.29     1.74     1.99     2.27     2.95  2437    1
#> d[Efalizumab]           1.19    0.00 0.18     0.80     1.10     1.19     1.27     1.56  3907    1
#> d[Etanercept 25 mg]     1.53    0.00 0.24     1.04     1.40     1.52     1.65     2.03  5361    1
#> d[Etanercept 50 mg]     1.92    0.00 0.28     1.35     1.78     1.92     2.06     2.49  4927    1
#> d[Fumaderm]             1.48    0.01 0.61     0.34     1.09     1.46     1.85     2.78  8575    1
#> d[Infliximab]           2.32    0.00 0.38     1.55     2.08     2.31     2.55     3.08  6738    1
#> d[Methotrexate]         1.71    0.01 0.63     0.57     1.31     1.68     2.06     3.09  3771    1
#> lp__                -3523.36    0.20 6.74 -3537.03 -3527.97 -3523.18 -3518.57 -3510.96  1170    1
#> tau                     0.31    0.01 0.22     0.02     0.15     0.27     0.43     0.84   860    1
#> cc[PASI50]              0.00     NaN 0.00     0.00     0.00     0.00     0.00     0.00   NaN  NaN
#> cc[PASI75]              0.76    0.00 0.03     0.70     0.73     0.76     0.78     0.82 18439    1
#> cc[PASI90]              1.56    0.00 0.05     1.46     1.53     1.56     1.60     1.67 21740    1
#> 
#> Samples were drawn using NUTS(diag_e) at Tue Jan  9 18:08:43 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(pso_fit_RE, pars = c("d", "cc", "mu", "delta"))

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

plot_prior_posterior(pso_fit_RE, prior = c("trt", "aux", "het"))

Model comparison

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

(dic_FE <- dic(pso_fit_FE))
#> Residual deviance: 75.1 (on 58 data points)
#>                pD: 25.6
#>               DIC: 100.6
(dic_RE <- dic(pso_fit_RE))
#> Residual deviance: 62.7 (on 58 data points)
#>                pD: 33.4
#>               DIC: 96.1

The random effects model has a lower DIC and the residual deviance is closer to the number of data points, so is preferred in this case.

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

plot(dic_FE)

plot(dic_RE)

Most data points are fit well, with posterior mean residual deviances close to the degrees of freedom. The Meffert 1997 study has a substantially higher residual deviance contribution, which could be investigated further to see why this study appears to be an outlier.

Further results

Predicted probabilities of response

Dias et al. (2011) produce absolute predictions of probability of achieving responses at each PASI cutoff, assuming a Normal distribution for the baseline probit probability of PASI50 response on supportive care with mean \(-1.097\) and precision \(123\). We can replicate these results using the predict() method. The baseline argument takes a distr() distribution object, with which we specify the corresponding Normal distribution. We set type = "response" to produce predicted probabilities (type = "link" would produce predicted probit probabilities).

pred_FE <- predict(pso_fit_FE, 
                   baseline = distr(qnorm, mean = -1.097, sd = 123^-0.5), 
                   type = "response")
pred_FE
#>                                mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[Supportive care, PASI50]  0.14 0.02 0.10 0.12 0.14 0.15  0.18     3631     3432    1
#> pred[Supportive care, PASI75]  0.03 0.01 0.02 0.03 0.03 0.04  0.05     3798     3795    1
#> pred[Supportive care, PASI90]  0.00 0.00 0.00 0.00 0.00 0.00  0.01     4176     3581    1
#> pred[Ciclosporin, PASI50]      0.78 0.10 0.57 0.72 0.79 0.85  0.94     1465     1547    1
#> pred[Ciclosporin, PASI75]      0.53 0.13 0.28 0.43 0.52 0.62  0.79     1462     1773    1
#> pred[Ciclosporin, PASI90]      0.24 0.11 0.08 0.16 0.23 0.31  0.50     1481     1647    1
#> pred[Efalizumab, PASI50]       0.54 0.04 0.45 0.51 0.53 0.57  0.62     3061     3193    1
#> pred[Efalizumab, PASI75]       0.25 0.04 0.19 0.23 0.25 0.28  0.33     3153     3293    1
#> pred[Efalizumab, PASI90]       0.07 0.02 0.04 0.06 0.07 0.08  0.11     3409     3327    1
#> pred[Etanercept 25 mg, PASI50] 0.66 0.05 0.56 0.63 0.66 0.69  0.75     2707     3322    1
#> pred[Etanercept 25 mg, PASI75] 0.37 0.05 0.27 0.33 0.37 0.40  0.46     2738     3788    1
#> pred[Etanercept 25 mg, PASI90] 0.13 0.03 0.08 0.11 0.12 0.14  0.19     2906     3155    1
#> pred[Etanercept 50 mg, PASI50] 0.79 0.04 0.71 0.77 0.79 0.82  0.86     2779     3495    1
#> pred[Etanercept 50 mg, PASI75] 0.53 0.05 0.42 0.49 0.53 0.56  0.63     2769     3678    1
#> pred[Etanercept 50 mg, PASI90] 0.23 0.04 0.16 0.20 0.23 0.26  0.32     2892     3481    1
#> pred[Fumaderm, PASI50]         0.63 0.17 0.31 0.52 0.64 0.76  0.92     2793     2222    1
#> pred[Fumaderm, PASI75]         0.37 0.17 0.10 0.24 0.35 0.48  0.76     2781     2275    1
#> pred[Fumaderm, PASI90]         0.15 0.12 0.02 0.06 0.11 0.20  0.45     2763     2260    1
#> pred[Infliximab, PASI50]       0.88 0.06 0.75 0.85 0.89 0.93  0.97     2927     2764    1
#> pred[Infliximab, PASI75]       0.68 0.10 0.47 0.62 0.68 0.75  0.86     2890     2912    1
#> pred[Infliximab, PASI90]       0.38 0.11 0.19 0.30 0.37 0.45  0.61     2904     2750    1
#> pred[Methotrexate, PASI50]     0.68 0.15 0.37 0.59 0.70 0.80  0.93     1654     1948    1
#> pred[Methotrexate, PASI75]     0.42 0.16 0.13 0.30 0.41 0.53  0.75     1642     1918    1
#> pred[Methotrexate, PASI90]     0.17 0.11 0.03 0.09 0.15 0.23  0.45     1658     1987    1
plot(pred_FE)

pred_RE <- predict(pso_fit_RE, 
                   baseline = distr(qnorm, mean = -1.097, sd = 123^-0.5), 
                   type = "response")
pred_RE
#>                                mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[Supportive care, PASI50]  0.14 0.02 0.10 0.12 0.14 0.15  0.18     9886     9993    1
#> pred[Supportive care, PASI75]  0.03 0.01 0.02 0.03 0.03 0.04  0.05    10471     9755    1
#> pred[Supportive care, PASI90]  0.00 0.00 0.00 0.00 0.00 0.00  0.01    11189     9998    1
#> pred[Ciclosporin, PASI50]      0.80 0.10 0.57 0.74 0.81 0.88  0.97     2891     2681    1
#> pred[Ciclosporin, PASI75]      0.56 0.15 0.28 0.45 0.55 0.66  0.87     2857     2603    1
#> pred[Ciclosporin, PASI90]      0.28 0.14 0.08 0.18 0.25 0.35  0.62     2896     2773    1
#> pred[Efalizumab, PASI50]       0.53 0.08 0.37 0.49 0.54 0.58  0.69     5018     3190    1
#> pred[Efalizumab, PASI75]       0.26 0.06 0.14 0.22 0.25 0.29  0.40     5080     3220    1
#> pred[Efalizumab, PASI90]       0.07 0.03 0.03 0.06 0.07 0.09  0.14     5195     3201    1
#> pred[Etanercept 25 mg, PASI50] 0.66 0.09 0.46 0.61 0.67 0.72  0.83     5965     4297    1
#> pred[Etanercept 25 mg, PASI75] 0.38 0.09 0.20 0.32 0.37 0.43  0.58     5967     4145    1
#> pred[Etanercept 25 mg, PASI90] 0.14 0.06 0.05 0.10 0.13 0.16  0.27     6030     4166    1
#> pred[Etanercept 50 mg, PASI50] 0.79 0.08 0.59 0.75 0.80 0.84  0.92     5795     3901    1
#> pred[Etanercept 50 mg, PASI75] 0.53 0.11 0.30 0.47 0.53 0.59  0.74     5828     4022    1
#> pred[Etanercept 50 mg, PASI90] 0.24 0.09 0.09 0.19 0.23 0.28  0.44     5865     4122    1
#> pred[Fumaderm, PASI50]         0.63 0.19 0.22 0.50 0.64 0.78  0.95     8804     6212    1
#> pred[Fumaderm, PASI75]         0.37 0.20 0.06 0.22 0.35 0.51  0.82     8801     6264    1
#> pred[Fumaderm, PASI90]         0.16 0.14 0.01 0.06 0.11 0.21  0.55     8856     6381    1
#> pred[Infliximab, PASI50]       0.87 0.08 0.67 0.83 0.89 0.93  0.98     7199     5104    1
#> pred[Infliximab, PASI75]       0.67 0.13 0.38 0.59 0.68 0.76  0.89     7201     5220    1
#> pred[Infliximab, PASI90]       0.37 0.14 0.13 0.28 0.37 0.46  0.67     7243     5377    1
#> pred[Methotrexate, PASI50]     0.70 0.18 0.29 0.58 0.72 0.83  0.98     4456     2702    1
#> pred[Methotrexate, PASI75]     0.45 0.21 0.10 0.29 0.43 0.59  0.89     4447     2722    1
#> pred[Methotrexate, PASI90]     0.21 0.17 0.02 0.09 0.16 0.28  0.67     4419     2758    1
plot(pred_RE)

If instead of information on the baseline PASI 50 response probit probability we have PASI 50 event counts, we can use these to construct a Beta distribution for the baseline probability of PASI 50 response. For example, if 56 out of 408 individuals achieved PASI 50 response on supportive care in the target population of interest, the appropriate Beta distribution for the response probability would be \(\textrm{Beta}(56, 408-56)\). We can specify this Beta distribution for the baseline response using the baseline_type = "reponse" argument (the default is "link", used above for the baseline probit probability).

pred_FE_beta <- predict(pso_fit_FE, 
                        baseline = distr(qbeta, 56, 408-56),
                        baseline_type = "response",
                        type = "response")
pred_FE_beta
#>                                mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[Supportive care, PASI50]  0.14 0.02 0.10 0.12 0.14 0.15  0.17     3940     4002    1
#> pred[Supportive care, PASI75]  0.03 0.01 0.02 0.03 0.03 0.04  0.05     4150     3816    1
#> pred[Supportive care, PASI90]  0.00 0.00 0.00 0.00 0.00 0.00  0.01     4410     3897    1
#> pred[Ciclosporin, PASI50]      0.78 0.10 0.58 0.72 0.79 0.85  0.94     1410     1921    1
#> pred[Ciclosporin, PASI75]      0.53 0.13 0.29 0.43 0.52 0.62  0.79     1412     1768    1
#> pred[Ciclosporin, PASI90]      0.24 0.11 0.08 0.16 0.23 0.30  0.50     1427     1631    1
#> pred[Efalizumab, PASI50]       0.54 0.04 0.46 0.51 0.54 0.56  0.61     3267     3490    1
#> pred[Efalizumab, PASI75]       0.25 0.03 0.19 0.23 0.25 0.27  0.32     3330     3666    1
#> pred[Efalizumab, PASI90]       0.07 0.01 0.04 0.06 0.07 0.08  0.10     3498     3813    1
#> pred[Etanercept 25 mg, PASI50] 0.66 0.04 0.57 0.63 0.66 0.69  0.74     2871     2833    1
#> pred[Etanercept 25 mg, PASI75] 0.37 0.05 0.28 0.33 0.37 0.40  0.46     2982     3380    1
#> pred[Etanercept 25 mg, PASI90] 0.13 0.03 0.08 0.11 0.12 0.14  0.18     3225     3574    1
#> pred[Etanercept 50 mg, PASI50] 0.79 0.04 0.72 0.77 0.79 0.82  0.86     2867     3036    1
#> pred[Etanercept 50 mg, PASI75] 0.53 0.05 0.43 0.49 0.52 0.56  0.62     3050     3340    1
#> pred[Etanercept 50 mg, PASI90] 0.23 0.04 0.16 0.20 0.23 0.25  0.31     3249     3249    1
#> pred[Fumaderm, PASI50]         0.63 0.16 0.32 0.52 0.64 0.76  0.92     2796     2015    1
#> pred[Fumaderm, PASI75]         0.37 0.17 0.11 0.24 0.34 0.48  0.75     2834     2037    1
#> pred[Fumaderm, PASI90]         0.15 0.12 0.02 0.06 0.11 0.20  0.45     2801     1995    1
#> pred[Infliximab, PASI50]       0.88 0.05 0.76 0.85 0.89 0.92  0.97     2960     2277    1
#> pred[Infliximab, PASI75]       0.68 0.10 0.48 0.62 0.69 0.75  0.87     2921     2330    1
#> pred[Infliximab, PASI90]       0.38 0.11 0.19 0.30 0.37 0.45  0.61     2932     2409    1
#> pred[Methotrexate, PASI50]     0.68 0.15 0.37 0.59 0.70 0.79  0.92     1582     1997    1
#> pred[Methotrexate, PASI75]     0.42 0.16 0.14 0.29 0.41 0.53  0.74     1577     2074    1
#> pred[Methotrexate, PASI90]     0.17 0.11 0.03 0.09 0.15 0.23  0.44     1590     2137    1
plot(pred_FE_beta)

pred_RE_beta <- predict(pso_fit_RE, 
                        baseline = distr(qbeta, 56, 408-56),
                        baseline_type = "response",
                        type = "response")
pred_RE_beta
#>                                mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
#> pred[Supportive care, PASI50]  0.14 0.02 0.11 0.13 0.14 0.15  0.17     9843     9606    1
#> pred[Supportive care, PASI75]  0.03 0.01 0.02 0.03 0.03 0.04  0.05    10410     9613    1
#> pred[Supportive care, PASI90]  0.00 0.00 0.00 0.00 0.00 0.00  0.01    12274     9519    1
#> pred[Ciclosporin, PASI50]      0.80 0.10 0.57 0.74 0.81 0.88  0.97     3035     2838    1
#> pred[Ciclosporin, PASI75]      0.56 0.15 0.28 0.45 0.55 0.66  0.87     3003     2786    1
#> pred[Ciclosporin, PASI90]      0.28 0.14 0.08 0.18 0.25 0.35  0.62     3060     2825    1
#> pred[Efalizumab, PASI50]       0.53 0.07 0.38 0.49 0.54 0.58  0.68     4688     3388    1
#> pred[Efalizumab, PASI75]       0.26 0.06 0.14 0.22 0.25 0.29  0.39     4765     3360    1
#> pred[Efalizumab, PASI90]       0.07 0.03 0.03 0.06 0.07 0.09  0.14     4943     3392    1
#> pred[Etanercept 25 mg, PASI50] 0.66 0.09 0.47 0.61 0.67 0.71  0.83     5847     4057    1
#> pred[Etanercept 25 mg, PASI75] 0.38 0.09 0.20 0.32 0.37 0.43  0.58     5850     4035    1
#> pred[Etanercept 25 mg, PASI90] 0.14 0.06 0.05 0.10 0.13 0.16  0.27     5991     4071    1
#> pred[Etanercept 50 mg, PASI50] 0.79 0.08 0.59 0.75 0.80 0.84  0.92     5774     3766    1
#> pred[Etanercept 50 mg, PASI75] 0.53 0.11 0.30 0.47 0.53 0.59  0.74     5801     3891    1
#> pred[Etanercept 50 mg, PASI90] 0.24 0.09 0.09 0.19 0.23 0.28  0.44     5909     4035    1
#> pred[Fumaderm, PASI50]         0.63 0.19 0.22 0.50 0.64 0.78  0.95     8902     6259    1
#> pred[Fumaderm, PASI75]         0.37 0.20 0.06 0.22 0.35 0.50  0.82     8891     6286    1
#> pred[Fumaderm, PASI90]         0.16 0.14 0.01 0.06 0.11 0.21  0.55     8960     6369    1
#> pred[Infliximab, PASI50]       0.87 0.08 0.67 0.84 0.89 0.93  0.98     7029     5043    1
#> pred[Infliximab, PASI75]       0.67 0.13 0.38 0.59 0.68 0.76  0.89     7031     5042    1
#> pred[Infliximab, PASI90]       0.37 0.14 0.13 0.28 0.36 0.46  0.67     7095     5071    1
#> pred[Methotrexate, PASI50]     0.70 0.18 0.30 0.58 0.72 0.84  0.98     4471     2855    1
#> pred[Methotrexate, PASI75]     0.45 0.21 0.10 0.29 0.43 0.59  0.89     4472     2878    1
#> pred[Methotrexate, PASI90]     0.21 0.17 0.02 0.09 0.16 0.28  0.67     4484     2720    1
plot(pred_RE_beta)

(Notice that these results are equivalent to those calculated above using the Normal distribution for the baseline probit probability, since these event counts correspond to the same probit probability.)

We can modify the plots using standard ggplot2 functions. For example, to plot the cutpoints together with a colour coding (instead of split into facets):

library(ggplot2)
plot(pred_RE, position = position_dodge(width = 0.75)) +
  facet_null() +
  aes(colour = Category) +
  scale_colour_brewer(palette = "Blues")

If the baseline argument is omitted, predicted probabilities will be produced for every study in the network based on their estimated baseline probit probability \(\mu_j\).

Ranks and rank probabilities

Treatment rankings, rank probabilities, and cumulative rank probabilities can also be produced. We set lower_better = FALSE since higher outcome categories are better (the outcomes are positive).

(pso_ranks <- posterior_ranks(pso_fit_RE, lower_better = FALSE))
#>                        mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
#> rank[Supportive care]  7.99 0.11    8   8   8   8     8     6794       NA    1
#> rank[Ciclosporin]      2.77 1.27    1   2   3   4     5     6291     6805    1
#> rank[Efalizumab]       6.35 0.81    4   6   7   7     7     5929       NA    1
#> rank[Etanercept 25 mg] 4.92 1.07    3   4   5   6     7     6353     5633    1
#> rank[Etanercept 50 mg] 3.05 1.22    1   2   3   4     6     4814     5452    1
#> rank[Fumaderm]         4.91 1.93    1   3   5   7     7     8323     6866    1
#> rank[Infliximab]       1.77 1.17    1   1   1   2     5     3843     4744    1
#> rank[Methotrexate]     4.24 1.86    1   3   4   6     7     5385     5667    1
plot(pso_ranks)

(pso_rankprobs <- posterior_rank_probs(pso_fit_RE, lower_better = FALSE))
#>                     p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6] p_rank[7] p_rank[8]
#> d[Supportive care]       0.00      0.00      0.00      0.00      0.00      0.00      0.01      0.99
#> d[Ciclosporin]           0.17      0.29      0.27      0.17      0.07      0.02      0.00      0.00
#> d[Efalizumab]            0.00      0.00      0.01      0.03      0.10      0.36      0.52      0.00
#> d[Etanercept 25 mg]      0.00      0.01      0.08      0.21      0.39      0.26      0.05      0.00
#> d[Etanercept 50 mg]      0.08      0.30      0.27      0.24      0.09      0.02      0.01      0.00
#> d[Fumaderm]              0.07      0.09      0.10      0.11      0.16      0.19      0.27      0.01
#> d[Infliximab]            0.60      0.18      0.12      0.06      0.02      0.01      0.00      0.00
#> d[Methotrexate]          0.09      0.12      0.15      0.18      0.17      0.15      0.14      0.00
plot(pso_rankprobs)

(pso_cumrankprobs <- posterior_rank_probs(pso_fit_RE, lower_better = FALSE, cumulative = TRUE))
#>                     p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6] p_rank[7] p_rank[8]
#> d[Supportive care]       0.00      0.00      0.00      0.00      0.00      0.00      0.01         1
#> d[Ciclosporin]           0.17      0.46      0.73      0.90      0.98      1.00      1.00         1
#> d[Efalizumab]            0.00      0.00      0.01      0.03      0.13      0.48      1.00         1
#> d[Etanercept 25 mg]      0.00      0.02      0.10      0.31      0.70      0.95      1.00         1
#> d[Etanercept 50 mg]      0.08      0.38      0.65      0.89      0.97      0.99      1.00         1
#> d[Fumaderm]              0.07      0.15      0.25      0.37      0.53      0.72      0.99         1
#> d[Infliximab]            0.60      0.78      0.90      0.96      0.99      1.00      1.00         1
#> d[Methotrexate]          0.09      0.21      0.36      0.54      0.71      0.85      1.00         1
plot(pso_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.
Woolacott, N., N. Hawkins, A. Mason, A. Kainth, Z. Khadjesari, Y. Bravo Vergel, K. Misso, et al. 2006. “Etanercept and Efalizumab for the Treatment of Psoriasis: A Systematic Review.” Health Technology Assessment 10 (46). https://doi.org/10.3310/hta10460.