# Example: Plaque psoriasis ML-NMR

    library(multinma)
    #> For execution on a local, multicore CPU with excess RAM we recommend calling
    #> options(mc.cores = parallel::detectCores())
    #>
    #> Attaching package: 'multinma'
    #> The following objects are masked from 'package:stats':
    #>
    #>     dgamma, pgamma, qgamma
    library(dplyr)      # dplyr and tidyr for data manipulation
    #>
    #> Attaching package: 'dplyr'
    #> The following objects are masked from 'package:stats':
    #>
    #>     filter, lag
    #> The following objects are masked from 'package:base':
    #>
    #>     intersect, setdiff, setequal, union
    library(tidyr)
    library(ggplot2)    # ggplot2 for plotting covariate distributions

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

Simulated individual patient data (IPD) from trials comparing treatments for plaque psoriasis are found in the data set `plaque_psoriasis_ipd`. Aggregate data (AgD) are available on a further set of trials, found in the data set `plaque_psoriasis_agd`. In this vignette, we recreate the multilevel network meta-regression (ML-NMR) analyses performed by Phillippo et al. (2020) and Phillippo et al. (2022; see also Phillippo 2019).

In the first analysis (Phillippo et al. 2020), we consider a network of four studies with a binary outcome (success/failure to achieve a 75% reduction on the psoriasis area and severity index, PASI 75).

In the second analysis (Phillippo et al. 2022), we extend this network with a further five studies and demonstrate how the key assumptions of population adjustment can be assessed in this larger network. We also demonstrate how to produce estimates for three external target populations, and fit a multinomial model to incorporate ordered categorical outcomes (PASI 75, PASI 90, and PASI 100).

# Initial analysis

We start by recreating the analysis presented by Phillippo et al. (2020). We will analyse IPD from three studies, UNCOVER-1, UNCOVER-2, and UNCOVER-3 (Griffiths et al. 2015; Gordon et al. 2016), and AgD from one study, FIXTURE (Langley et al. 2014).

    pso_ipd <- filter(plaque_psoriasis_ipd,
                      studyc %in% c("UNCOVER-1", "UNCOVER-2", "UNCOVER-3"))

    pso_agd <- filter(plaque_psoriasis_agd,
                      studyc == "FIXTURE")

    head(pso_ipd)
    #>      studyc      trtc_long    trtc trtn pasi75 pasi90 pasi100 age  bmi pasi_w0  male bsa weight
    #> 1 UNCOVER-1 Ixekizumab Q2W IXE_Q2W    2      0      0       0  34 32.2    18.2  TRUE  18   98.1
    #> 2 UNCOVER-1 Ixekizumab Q2W IXE_Q2W    2      1      0       0  64 41.9    23.4  TRUE  33  129.6
    #> 3 UNCOVER-1 Ixekizumab Q2W IXE_Q2W    2      1      1       0  42 26.2    12.8  TRUE  33   78.0
    #> 4 UNCOVER-1 Ixekizumab Q2W IXE_Q2W    2      0      0       0  45 52.9    36.0 FALSE  50  139.9
    #> 5 UNCOVER-1 Ixekizumab Q2W IXE_Q2W    2      1      0       0  67 22.9    20.9 FALSE  35   54.2
    #> 6 UNCOVER-1 Ixekizumab Q2W IXE_Q2W    2      1      1       1  57 22.4    18.2  TRUE  29   67.5
    #>   durnpso prevsys   psa
    #> 1     6.7    TRUE  TRUE
    #> 2    14.5   FALSE  TRUE
    #> 3    26.5    TRUE FALSE
    #> 4    25.0    TRUE  TRUE
    #> 5    11.9    TRUE FALSE
    #> 6    15.2    TRUE FALSE
    head(pso_agd)
    #>    studyc          trtc_long    trtc trtn pasi75_r pasi75_n pasi90_r pasi90_n pasi100_r pasi100_n
    #> 1 FIXTURE         Etanercept     ETN    4      142      323       67      323        14       323
    #> 2 FIXTURE            Placebo     PBO    1       16      324        5      324         0       324
    #> 3 FIXTURE Secukinumab 150 mg SEC_150    5      219      327      137      327        47       327
    #> 4 FIXTURE Secukinumab 300 mg SEC_300    6      249      323      175      323        78       323
    #>   sample_size_w0 age_mean age_sd bmi_mean bmi_sd pasi_w0_mean pasi_w0_sd male bsa_mean bsa_sd
    #> 1            326     43.8   13.0     28.7    5.9         23.2        9.8 71.2     33.6   18.0
    #> 2            326     44.1   12.6     27.9    6.1         24.1       10.5 72.7     35.2   19.1
    #> 3            327     45.4   12.9     28.4    5.9         23.7       10.5 72.2     34.5   19.4
    #> 4            327     44.5   13.2     28.4    6.4         23.9        9.9 68.5     34.3   19.2
    #>   weight_mean weight_sd durnpso_mean durnpso_sd prevsys  psa
    #> 1        84.6      20.5         16.4       12.0    65.6 13.5
    #> 2        82.0      20.4         16.6       11.6    62.6 15.0
    #> 3        83.6      20.8         17.3       12.2    64.8 15.0
    #> 4        83.0      21.6         15.8       12.3    63.0 15.3

We consider running a ML-NMR adjusting for five potential effect-modifying covariates: duration of psoriasis `durnpso`, weight `weight`, previous systemic treatment `prevsys`, body surface area `bsa`, and psoriatic arthritis `psa`.

## Setup

### Preparing the data

We need to prepare the data so that it is in an acceptable format to run a ML-NMR model. Firstly, we need to handle the binary covariates `prevsys` and `psa`. In the IPD, these are coded as `TRUE` or `FALSE`, but in the AgD these are coded as percentages (out of 100). We need these to transform both of these sets of variables so that they are numeric and lie in the interval \([0,1]\), so that the variables are compatible across the data sources. Whilst we are here, we also transform body surface area `bsa` (a percentage) to lie in \([0,1]\), since that will make specifying an appropriate marginal distribution easier later, and rescale `weight` and `duration` to aid interpretation of the regression coefficients (in terms of 10 kilos and 10 years respectively). We also add in a `trtclass` variable, indicating which treatments belong to which classes. Finally, we check for missing values in the IPD.

    pso_ipd <- pso_ipd %>%
      mutate(# Variable transformations
             bsa = bsa / 100,
             prevsys = as.numeric(prevsys),
             psa = as.numeric(psa),
             weight = weight / 10,
             durnpso = durnpso / 10,
             # Treatment classes
             trtclass = case_when(trtn == 1 ~ "Placebo",
                                  trtn %in% c(2, 3, 5, 6) ~ "IL blocker",
                                  trtn == 4 ~ "TNFa blocker"),
             # Check complete cases for covariates of interest
             complete = complete.cases(durnpso, prevsys, bsa, weight, psa)
      )

    pso_agd <- pso_agd %>%
      mutate(
        # Variable transformations
        bsa_mean = bsa_mean / 100,
        bsa_sd = bsa_sd / 100,
        prevsys = prevsys / 100,
        psa = psa / 100,
        weight_mean = weight_mean / 10,
        weight_sd = weight_sd / 10,
        durnpso_mean = durnpso_mean / 10,
        durnpso_sd = durnpso_sd / 10,
        # Treatment classes
        trtclass = case_when(trtn == 1 ~ "Placebo",
                                  trtn %in% c(2, 3, 5, 6) ~ "IL blocker",
                                  trtn == 4 ~ "TNFa blocker")
      )

A small number of individuals have missing covariates:

    sum(!pso_ipd$complete)
    #> [1] 4
    mean(!pso_ipd$complete)
    #> [1] 0.001036807

Since the proportion of missing data is so small, we will simply exclude these individuals from the analysis.

    pso_ipd <- filter(pso_ipd, complete)

### Creating the network

Set up the network, setting the IPD with `set_ipd()`, AgD (arm-based) with `set_agd_arm()`, and combining together using `combine_network()`. We specify the binary `pasi75` outcome as `r` in the IPD, and the count outcome `pasi75_r` and denominator `pasi75_n` as `r` and `n` in the AgD. We specify the treatment classes with `trt_class = trtclass`.

    pso_net <- combine_network(
      set_ipd(pso_ipd,
              study = studyc,
              trt = trtc,
              r = pasi75,
              trt_class = trtclass),
      set_agd_arm(pso_agd,
                  study = studyc,
                  trt = trtc,
                  r = pasi75_r,
                  n = pasi75_n,
                  trt_class = trtclass)
    )

    pso_net
    #> A network with 3 IPD studies, and 1 AgD study (arm-based).
    #>
    #> ------------------------------------------------------------------- IPD studies ----
    #>  Study     Treatment arms
    #>  UNCOVER-1 3: IXE_Q2W | IXE_Q4W | PBO
    #>  UNCOVER-2 4: ETN | IXE_Q2W | IXE_Q4W | PBO
    #>  UNCOVER-3 4: ETN | IXE_Q2W | IXE_Q4W | PBO
    #>
    #>  Outcome type: binary
    #> ------------------------------------------------------- AgD studies (arm-based) ----
    #>  Study   Treatment arms
    #>  FIXTURE 4: PBO | ETN | SEC_150 | SEC_300
    #>
    #>  Outcome type: count
    #> ------------------------------------------------------------------------------------
    #> Total number of treatments: 6, in 3 classes
    #> Total number of studies: 4
    #> Reference treatment is: PBO
    #> Network is connected

We can produce a network plot with the `plot()` method:

    plot(pso_net, weight_nodes = TRUE, weight_edges = TRUE, show_trt_class = TRUE) +
      ggplot2::theme(legend.position = "bottom", legend.box = "vertical")

### Numerical integration for ML-NMR

ML-NMR models define the meta-regression model at the individual level, in exactly the same manner as a full-IPD meta-regression. ML-NMR then incorporates the AgD into the model by integrating this individual-level model over the covariate distribution in each AgD study (Phillippo et al. 2020; Phillippo 2019). Using integration, instead of simply “plugging-in” mean covariate values for the AgD studies, avoids aggregation bias when the link function is not the identity function.

This package utilises numerical integration to incorporate the aggregate data - specifically, quasi-Monte Carlo (QMC) integration with a Gaussian copula (Phillippo et al. 2020; Phillippo 2019). QMC integration is a very general and flexible integration approach, which typically requires far fewer integration points than standard (pseudo-random) Monte-Carlo integration to achieve the same numerical accuracy.1 A Gaussian copula allows us to account for correlations between covariates, which may have any specified marginal distributions.

We now set up the numerical integration for the network. The five covariates that we will consider adjusting for are body surface area `bsa`, duration of psoriasis `durnpso`, previous systemic treatment `prevsys`, psoriatic arthritis `psa`, and weight `weight`. We need to choose suitable marginal distributions for these covariates to draw the integration points from. `prevsys` and `psa` are binary covariates, so these are given a Bernoulli distribution. `bsa` is a percentage, so we choose a logit-Normal distribution. We choose Gamma distributions for `durnpso` and `weight` to account for skewness. These choices seem to match well the marginal distributions observed in the IPD:

    # Get mean and sd of covariates in each study
    ipd_summary <- pso_ipd %>%
      group_by(studyc) %>%
      summarise_at(vars(weight, durnpso, bsa), list(mean = mean, sd = sd, min = min, max = max)) %>%
      pivot_longer(weight_mean:bsa_max, names_sep = "_", names_to = c("covariate", ".value")) %>%
      # Assign distributions
      mutate(dist = recode(covariate,
                           bsa = "dlogitnorm",
                           durnpso = "dgamma",
                           weight = "dgamma")) %>%
      # Compute density curves
      group_by(studyc, covariate) %>%
      mutate(value = if_else(dist == "dlogitnorm",
                             list(seq(0, 1, length.out = 101)),
                             list(seq(min*0.8, max*1.2, length.out = 101)))) %>%
      unnest(cols = value) %>%
      mutate(dens = eval(call(first(dist), x = value, mean = first(mean), sd = first(sd))))

    # Plot histograms and assumed densities
    pso_ipd %>%
      pivot_longer(c(weight, durnpso, bsa), names_to = "covariate", values_to = "value") %>%
    ggplot(aes(x = value)) +
      geom_histogram(aes(y = after_stat(density)),
                     binwidth = function(x) diff(range(x)) / nclass.Sturges(x),
                     boundary = 0,
                     fill = "grey50") +
      geom_line(aes(y = dens), data = ipd_summary,
                colour = "darkred", linewidth = 0.5) +
      facet_wrap(~studyc + covariate, scales = "free", ncol = 3) +
      theme_multinma()
    #> Warning: Removed 1 row containing missing values or values outside the scale range (`geom_line()`).

We add integration points to the AgD studies in the network using the `add_integration()` function. Marginal distributions for each covariate are specified using the `distr()` function, which takes a cumulative distribution function corresponding to the chosen marginal distribution, and arguments to that distribution as column names in the aggregate data. Since we do not know the correlations between covariates in the AgD studies, we impute these with the weighted mean of the correlations in the IPD studies (the default option).

    pso_net <- add_integration(pso_net,
      durnpso = distr(qgamma, mean = durnpso_mean, sd = durnpso_sd),
      prevsys = distr(qbern, prob = prevsys),
      bsa = distr(qlogitnorm, mean = bsa_mean, sd = bsa_sd),
      weight = distr(qgamma, mean = weight_mean, sd = weight_sd),
      psa = distr(qbern, prob = psa),
      n_int = 64
    )
    #> Using weighted average correlation matrix computed from IPD studies.

> Note: This package provides several convenience functions for specifying these distributions, including `qgamma()` which allows for a parameterisation of the Gamma distribution in terms of mean and standard deviation, `qbern()` which provides the Bernoulli distribution, and `qlogitnorm()` which provides the logit-Normal distribution allowing for a parameterisation in terms of mean and standard deviation.

## ML-NMR models

We fit both fixed effect (FE) and random effects (RE) ML-NMR models.

### Fixed effect ML-NMR

First, we fit a FE ML-NMR model using the function `nma()`. Following (Phillippo et al. 2020) we specify weakly-informative \(N(0, 10^2)\) priors on each parameter. The range of parameter values implied by these prior distributions can be checked using 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.

The regression model is specified with `regression = ~(durnpso + prevsys + bsa + weight + psa)*.trt`, which will include the main (prognostic) effects of each covariate as well as interactions with treatment. We use a probit link function (`link = "probit"`), and specify that the two-parameter Binomial approximation for the aggregate-level likelihood should be used (`likelihood = "bernoulli2"`, where “bernoulli” refers to the individual-level likelihood, and “2” denotes the two-parameter adjustment to the aggregate-level likelihood) (Phillippo et al. 2020). We utilise the shared effect modifier assumption to help identify the model, setting treatment-covariate interactions to be equal within each class (`class_interactions = "common"`). We narrow the possible range for random initial values with `init_r = 0.1` (the default is `init_r = 2`), since probit models in particular are often hard to initialise. Using the QR decomposition (`QR = TRUE`) greatly improves sampling efficiency here, as is often the case for regression models.

    pso_fit_FE <- nma(pso_net,
                      trt_effects = "fixed",
                      link = "probit",
                      likelihood = "bernoulli2",
                      regression = ~(durnpso + prevsys + bsa + weight + psa)*.trt,
                      class_interactions = "common",
                      prior_intercept = normal(scale = 10),
                      prior_trt = normal(scale = 10),
                      prior_reg = normal(scale = 10),
                      init_r = 0.1,
                      QR = TRUE)
    #> Note: Setting "PBO" as the network reference treatment.

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

    print(pso_fit_FE)
    #> A fixed effects ML-NMR with a bernoulli2 likelihood (probit link).
    #> Regression model: ~(durnpso + prevsys + bsa + weight + psa) * .trt.
    #> Centred covariates at the following overall mean values:
    #>   durnpso   prevsys       bsa    weight       psa
    #> 1.8134535 0.6450416 0.2909089 8.9369318 0.2147914
    #> Inference for Stan model: binomial_2par.
    #> 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%
    #> beta[durnpso]                           0.04    0.00 0.06    -0.07     0.00     0.04     0.08
    #> beta[prevsys]                          -0.14    0.00 0.16    -0.45    -0.25    -0.14    -0.03
    #> beta[bsa]                              -0.07    0.01 0.44    -0.96    -0.37    -0.05     0.24
    #> beta[weight]                            0.04    0.00 0.03    -0.02     0.02     0.04     0.06
    #> beta[psa]                              -0.07    0.00 0.17    -0.42    -0.18    -0.07     0.04
    #> beta[durnpso:.trtclassTNFa blocker]    -0.03    0.00 0.07    -0.17    -0.08    -0.03     0.02
    #> beta[durnpso:.trtclassIL blocker]      -0.01    0.00 0.07    -0.14    -0.06    -0.01     0.03
    #> beta[prevsys:.trtclassTNFa blocker]     0.19    0.00 0.19    -0.18     0.07     0.19     0.32
    #> beta[prevsys:.trtclassIL blocker]       0.07    0.00 0.17    -0.28    -0.05     0.07     0.19
    #> beta[bsa:.trtclassTNFa blocker]         0.06    0.01 0.51    -0.94    -0.30     0.05     0.40
    #> beta[bsa:.trtclassIL blocker]           0.29    0.01 0.48    -0.61    -0.05     0.29     0.62
    #> beta[weight:.trtclassTNFa blocker]     -0.17    0.00 0.04    -0.24    -0.19    -0.17    -0.14
    #> beta[weight:.trtclassIL blocker]       -0.10    0.00 0.03    -0.16    -0.12    -0.10    -0.08
    #> beta[psa:.trtclassTNFa blocker]        -0.06    0.00 0.21    -0.46    -0.19    -0.06     0.08
    #> beta[psa:.trtclassIL blocker]           0.00    0.00 0.19    -0.35    -0.12     0.00     0.13
    #> d[ETN]                                  1.55    0.00 0.08     1.40     1.50     1.55     1.60
    #> d[IXE_Q2W]                              2.95    0.00 0.09     2.78     2.89     2.95     3.01
    #> d[IXE_Q4W]                              2.54    0.00 0.08     2.38     2.49     2.54     2.60
    #> d[SEC_150]                              2.14    0.00 0.11     1.92     2.07     2.14     2.22
    #> d[SEC_300]                              2.45    0.00 0.12     2.22     2.37     2.44     2.52
    #> lp__                                -1576.31    0.08 3.44 -1583.87 -1578.39 -1575.99 -1573.87
    #>                                        97.5% n_eff Rhat
    #> beta[durnpso]                           0.16  5380    1
    #> beta[prevsys]                           0.18  4640    1
    #> beta[bsa]                               0.77  5090    1
    #> beta[weight]                            0.09  5161    1
    #> beta[psa]                               0.24  5850    1
    #> beta[durnpso:.trtclassTNFa blocker]     0.11  5119    1
    #> beta[durnpso:.trtclassIL blocker]       0.12  6475    1
    #> beta[prevsys:.trtclassTNFa blocker]     0.54  4709    1
    #> beta[prevsys:.trtclassIL blocker]       0.40  5621    1
    #> beta[bsa:.trtclassTNFa blocker]         1.05  5151    1
    #> beta[bsa:.trtclassIL blocker]           1.24  6214    1
    #> beta[weight:.trtclassTNFa blocker]     -0.10  6012    1
    #> beta[weight:.trtclassIL blocker]       -0.04  6372    1
    #> beta[psa:.trtclassTNFa blocker]         0.35  5873    1
    #> beta[psa:.trtclassIL blocker]           0.38  6348    1
    #> d[ETN]                                  1.71  3885    1
    #> d[IXE_Q2W]                              3.13  4930    1
    #> d[IXE_Q4W]                              2.70  4535    1
    #> d[SEC_150]                              2.36  4780    1
    #> d[SEC_300]                              2.68  5108    1
    #> lp__                                -1570.61  1641    1
    #>
    #> Samples were drawn using NUTS(diag_e) at Fri Mar 14 16:01:22 2025.
    #> 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(pso_fit_FE, pars = c("d", "beta", "mu"))

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

    plot_prior_posterior(pso_fit_FE, prior = c("intercept", "trt", "reg"))

We now recommend assessing sufficient accuracy of the numerical integration by running half of the chains with `n_int / 2` integration points and half with the full `n_int`. Any `Rhat` or `n_eff` diagnostic warnings can then either be attributed to insufficient MCMC iterations (argument `iter` in `nma()`) or to insufficient integration points (`n_int` in `add_integration()`), depending on whether they occur within the two groups of chains or for all chains combined. This feature is enabled by default (`int_check = TRUE`). In this case, there are no warnings and so we are content with both the number of iterations and with the number of integration points.

(Phillippo et al. (2020) used an alternative approach based on saving cumulative integration points and plotting the empirical integration error, which can be achieved by setting `int_thin` in `nma()` and using the `plot_integration_error()` function.)

### Random effects ML-NMR

We now fit a RE model. Again, we specify weakly-informative \(N(0, 10^2)\) priors on each parameter, and now specify a \(\textrm{half-N}(0, 2.5^2)\) prior for the heterogeneity standard deviation \(\tau\). The range of parameter values implied by these prior distributions can be checked using 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(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 model uses the same call to `nma()` as before, except now with `trt_effects = "random"`.

    pso_fit_RE <- nma(pso_net,
                      trt_effects = "random",
                      link = "probit",
                      likelihood = "bernoulli2",
                      regression = ~(durnpso + prevsys + bsa + weight + psa)*.trt,
                      class_interactions = "common",
                      prior_intercept = normal(scale = 10),
                      prior_trt = normal(scale = 10),
                      prior_reg = normal(scale = 10),
                      prior_het = half_normal(scale = 2.5),
                      init_r = 0.1,
                      QR = TRUE)
    #> Note: Setting "PBO" as the network reference treatment.
    #> Warning: There were 11 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

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

    print(pso_fit_RE)
    #> A random effects ML-NMR with a bernoulli2 likelihood (probit link).
    #> Regression model: ~(durnpso + prevsys + bsa + weight + psa) * .trt.
    #> Centred covariates at the following overall mean values:
    #>   durnpso   prevsys       bsa    weight       psa
    #> 1.8134535 0.6450416 0.2909089 8.9369318 0.2147914
    #> Inference for Stan model: binomial_2par.
    #> 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%
    #> beta[durnpso]                           0.05    0.00 0.06    -0.07     0.01     0.05     0.09
    #> beta[prevsys]                          -0.13    0.00 0.16    -0.44    -0.24    -0.13    -0.02
    #> beta[bsa]                              -0.10    0.01 0.45    -1.02    -0.39    -0.09     0.20
    #> beta[weight]                            0.04    0.00 0.03    -0.01     0.03     0.04     0.06
    #> beta[psa]                              -0.06    0.00 0.17    -0.40    -0.17    -0.06     0.05
    #> beta[durnpso:.trtclassTNFa blocker]    -0.03    0.00 0.07    -0.17    -0.08    -0.03     0.02
    #> beta[durnpso:.trtclassIL blocker]      -0.02    0.00 0.07    -0.15    -0.06    -0.02     0.03
    #> beta[prevsys:.trtclassTNFa blocker]     0.18    0.00 0.19    -0.20     0.05     0.19     0.32
    #> beta[prevsys:.trtclassIL blocker]       0.05    0.00 0.18    -0.31    -0.07     0.05     0.17
    #> beta[bsa:.trtclassTNFa blocker]         0.08    0.01 0.52    -0.91    -0.27     0.07     0.43
    #> beta[bsa:.trtclassIL blocker]           0.33    0.01 0.49    -0.59    -0.01     0.33     0.65
    #> beta[weight:.trtclassTNFa blocker]     -0.17    0.00 0.03    -0.24    -0.20    -0.17    -0.15
    #> beta[weight:.trtclassIL blocker]       -0.10    0.00 0.03    -0.16    -0.13    -0.10    -0.08
    #> beta[psa:.trtclassTNFa blocker]        -0.07    0.00 0.20    -0.45    -0.21    -0.08     0.07
    #> beta[psa:.trtclassIL blocker]          -0.01    0.00 0.19    -0.37    -0.13    -0.01     0.11
    #> d[ETN]                                  1.56    0.00 0.15     1.27     1.48     1.55     1.64
    #> d[IXE_Q2W]                              2.98    0.00 0.16     2.68     2.89     2.97     3.07
    #> d[IXE_Q4W]                              2.57    0.00 0.15     2.29     2.48     2.56     2.65
    #> d[SEC_150]                              2.12    0.01 0.23     1.63     1.99     2.12     2.25
    #> d[SEC_300]                              2.42    0.01 0.23     1.92     2.29     2.42     2.55
    #> lp__                                -1580.26    0.16 4.77 -1590.29 -1583.34 -1580.04 -1576.92
    #> tau                                     0.19    0.01 0.12     0.02     0.10     0.17     0.25
    #>                                        97.5% n_eff Rhat
    #> beta[durnpso]                           0.17  4545 1.00
    #> beta[prevsys]                           0.19  4460 1.00
    #> beta[bsa]                               0.73  4314 1.00
    #> beta[weight]                            0.10  4233 1.00
    #> beta[psa]                               0.26  4463 1.00
    #> beta[durnpso:.trtclassTNFa blocker]     0.11  4360 1.00
    #> beta[durnpso:.trtclassIL blocker]       0.12  5138 1.00
    #> beta[prevsys:.trtclassTNFa blocker]     0.56  4587 1.00
    #> beta[prevsys:.trtclassIL blocker]       0.40  5275 1.00
    #> beta[bsa:.trtclassTNFa blocker]         1.13  4645 1.00
    #> beta[bsa:.trtclassIL blocker]           1.34  4972 1.00
    #> beta[weight:.trtclassTNFa blocker]     -0.10  4623 1.00
    #> beta[weight:.trtclassIL blocker]       -0.04  4810 1.00
    #> beta[psa:.trtclassTNFa blocker]         0.33  4706 1.00
    #> beta[psa:.trtclassIL blocker]           0.36  4769 1.00
    #> d[ETN]                                  1.89  1153 1.00
    #> d[IXE_Q2W]                              3.31  1156 1.00
    #> d[IXE_Q4W]                              2.88  1099 1.00
    #> d[SEC_150]                              2.58  1462 1.01
    #> d[SEC_300]                              2.87  1336 1.00
    #> lp__                                -1571.57   935 1.01
    #> tau                                     0.48   498 1.01
    #>
    #> Samples were drawn using NUTS(diag_e) at Fri Mar 14 16:06:37 2025.
    #> 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", "beta", "tau", "mu", "delta"))

There are a number of divergent transitions, which we can investigate using the `pairs()` method:

    pairs(pso_fit_RE, pars = c("delta[UNCOVER-2: ETN]", "d[ETN]", "tau", "lp__"))

The divergent transition errors (red crosses) seem to be concentrated in the upper tail of the heterogeneity standard deviation parameter. This suggests that the information to identify the heterogeneity parameter is weak - we have only four studies in the network - and that a more informative prior distribution might aid estimation.

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

    plot_prior_posterior(pso_fit_RE, prior = c("intercept", "trt", "reg", "het"))

## Model comparison

The model fit under the FE and RE models can be checked using the `dic()` function.

    (pso_dic_FE <- dic(pso_fit_FE))
    #> Residual deviance: 3129.4 (on 3858 data points)
    #>                pD: 24.1
    #>               DIC: 3153.5
    (pso_dic_RE <- dic(pso_fit_RE))
    #> Residual deviance: 3123.5 (on 3858 data points)
    #>                pD: 28.1
    #>               DIC: 3151.6

The DIC is similar between the FE and RE models, suggesting that there is little evidence for any residual heterogeneity.

## Producing relative effects and event probabilities

Parameter estimates can be plotted using the `plot()` method, for example to examine the estimated regression coefficients:

    plot(pso_fit_FE,
         pars = "beta",
         stat = "halfeye",
         ref_line = 0)

Plots of posterior summaries are based on the `ggdist` package, which allows a great degree of flexibility, and can be further customised using `ggplot2` commands. In the above command we specify a `"halfeye"` plot, which shows the posterior density along with posterior medians (points) and 95% Credible Intervals (thin line) with 66% inner bands (thicker line) by default. For more details on the plotting options see `?plot.nma_summary`.

We can produce population-adjusted relative effects for each study population in the network using the `relative_effects()` function.

    (pso_releff_FE <- relative_effects(pso_fit_FE))
    #> ---------------------------------------------------------------- Study: FIXTURE ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>      1.6    0.62 0.34   8.34 0.14
    #>
    #>                     mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[FIXTURE: ETN]     1.66 0.09 1.49 1.60 1.66 1.72  1.83     3979     3396    1
    #> d[FIXTURE: IXE_Q2W] 3.03 0.10 2.83 2.96 3.02 3.09  3.23     5133     3418    1
    #> d[FIXTURE: IXE_Q4W] 2.62 0.09 2.43 2.55 2.62 2.68  2.81     4659     3247    1
    #> d[FIXTURE: SEC_150] 2.22 0.12 1.99 2.14 2.21 2.29  2.44     4509     3388    1
    #> d[FIXTURE: SEC_300] 2.52 0.12 2.29 2.44 2.52 2.60  2.76     4968     3101    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-1 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>        2    0.73 0.28   9.24 0.28
    #>
    #>                       mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[UNCOVER-1: ETN]     1.51 0.08 1.34 1.45 1.50 1.56  1.67     3860     3076    1
    #> d[UNCOVER-1: IXE_Q2W] 2.92 0.09 2.76 2.86 2.92 2.98  3.09     4707     2767    1
    #> d[UNCOVER-1: IXE_Q4W] 2.51 0.08 2.35 2.45 2.51 2.57  2.67     4504     3272    1
    #> d[UNCOVER-1: SEC_150] 2.11 0.12 1.87 2.03 2.11 2.19  2.34     4788     3192    1
    #> d[UNCOVER-1: SEC_300] 2.42 0.12 2.17 2.33 2.41 2.50  2.66     5374     3159    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-2 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.87    0.64 0.27   9.17 0.24
    #>
    #>                       mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[UNCOVER-2: ETN]     1.51 0.08 1.35 1.45 1.50 1.56  1.66     3891     2912    1
    #> d[UNCOVER-2: IXE_Q2W] 2.92 0.09 2.76 2.86 2.92 2.98  3.09     4945     3229    1
    #> d[UNCOVER-2: IXE_Q4W] 2.51 0.08 2.35 2.46 2.51 2.56  2.67     4687     3240    1
    #> d[UNCOVER-2: SEC_150] 2.11 0.12 1.88 2.03 2.11 2.19  2.34     4888     3330    1
    #> d[UNCOVER-2: SEC_300] 2.41 0.12 2.18 2.33 2.41 2.50  2.65     5449     3150    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-3 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight psa
    #>     1.78    0.59 0.28   9.01 0.2
    #>
    #>                       mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[UNCOVER-3: ETN]     1.53 0.08 1.37 1.47 1.53 1.58  1.68     3898     3151    1
    #> d[UNCOVER-3: IXE_Q2W] 2.94 0.09 2.77 2.88 2.94 3.00  3.11     5123     3253    1
    #> d[UNCOVER-3: IXE_Q4W] 2.53 0.08 2.37 2.47 2.53 2.58  2.68     4798     3268    1
    #> d[UNCOVER-3: SEC_150] 2.13 0.11 1.90 2.05 2.13 2.21  2.35     4826     3391    1
    #> d[UNCOVER-3: SEC_300] 2.43 0.12 2.20 2.35 2.43 2.51  2.67     5386     3148    1
    plot(pso_releff_FE, ref_line = 0)

Predicted probabilities of achieving PASI 75 in each study population on each treatment are produced using the `predict()` method. The argument `type = "reponse"` specifies that we want predicted probabilities, rather than probit probabilities.

    (pso_pred_FE <- predict(pso_fit_FE, type = "response"))
    #> ---------------------------------------------------------------- Study: FIXTURE ----
    #>
    #>                        mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[FIXTURE: PBO]     0.04 0.01 0.03 0.04 0.04 0.05  0.06     4020     3300    1
    #> pred[FIXTURE: ETN]     0.46 0.02 0.41 0.44 0.46 0.47  0.51     7974     3670    1
    #> pred[FIXTURE: IXE_Q2W] 0.89 0.02 0.85 0.88 0.89 0.90  0.92     5353     3010    1
    #> pred[FIXTURE: IXE_Q4W] 0.80 0.03 0.74 0.78 0.80 0.82  0.84     7023     3519    1
    #> pred[FIXTURE: SEC_150] 0.67 0.03 0.62 0.65 0.67 0.69  0.72     8100     2812    1
    #> pred[FIXTURE: SEC_300] 0.77 0.02 0.72 0.75 0.77 0.79  0.81     8619     3155    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-1 ----
    #>
    #>                          mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[UNCOVER-1: PBO]     0.06 0.01 0.04 0.05 0.06 0.06  0.08     5240     2858    1
    #> pred[UNCOVER-1: ETN]     0.46 0.03 0.41 0.44 0.46 0.48  0.52     5856     3136    1
    #> pred[UNCOVER-1: IXE_Q2W] 0.90 0.01 0.88 0.89 0.90 0.91  0.92     7646     2574    1
    #> pred[UNCOVER-1: IXE_Q4W] 0.81 0.02 0.78 0.80 0.81 0.82  0.84     7338     2292    1
    #> pred[UNCOVER-1: SEC_150] 0.69 0.04 0.60 0.66 0.69 0.72  0.77     6927     2944    1
    #> pred[UNCOVER-1: SEC_300] 0.78 0.04 0.71 0.76 0.78 0.81  0.85     7257     3330    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-2 ----
    #>
    #>                          mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[UNCOVER-2: PBO]     0.05 0.01 0.03 0.04 0.05 0.05  0.06     4697     3057    1
    #> pred[UNCOVER-2: ETN]     0.42 0.02 0.38 0.41 0.42 0.43  0.46     7304     2814    1
    #> pred[UNCOVER-2: IXE_Q2W] 0.88 0.01 0.86 0.87 0.88 0.89  0.91     6045     3262    1
    #> pred[UNCOVER-2: IXE_Q4W] 0.78 0.02 0.75 0.77 0.78 0.79  0.81     7057     3038    1
    #> pred[UNCOVER-2: SEC_150] 0.65 0.04 0.57 0.62 0.65 0.68  0.73     7090     3258    1
    #> pred[UNCOVER-2: SEC_300] 0.75 0.04 0.68 0.73 0.75 0.78  0.82     7876     3319    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-3 ----
    #>
    #>                          mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[UNCOVER-3: PBO]     0.08 0.01 0.06 0.07 0.08 0.08  0.10     4903     3060    1
    #> pred[UNCOVER-3: ETN]     0.53 0.02 0.49 0.52 0.53 0.54  0.57     7369     3233    1
    #> pred[UNCOVER-3: IXE_Q2W] 0.93 0.01 0.91 0.92 0.93 0.93  0.94     6120     3195    1
    #> pred[UNCOVER-3: IXE_Q4W] 0.85 0.01 0.83 0.85 0.86 0.86  0.88     8259     3034    1
    #> pred[UNCOVER-3: SEC_150] 0.75 0.03 0.68 0.72 0.75 0.77  0.81     7444     3268    1
    #> pred[UNCOVER-3: SEC_300] 0.83 0.03 0.77 0.81 0.83 0.85  0.88     7894     3197    1
    plot(pso_pred_FE, ref_line = c(0, 1))

We can produce population-adjusted ranks, rank probabilities, and cumulative rank probabilities in each study population using the `posterior_ranks()` and `posterior_rank_probs()` functions (although here the ranks are unchanged between populations, as the distributions of effect modifiers are similar). We specify `lower_better = FALSE`, since a higher outcome is better (higher chance of achieving PASI 75).

    (pso_ranks_FE <- posterior_ranks(pso_fit_FE, lower_better = FALSE))
    #> ---------------------------------------------------------------- Study: FIXTURE ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>      1.6    0.62 0.34   8.34 0.14
    #>
    #>                        mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> rank[FIXTURE: PBO]     6.00 0.00    6   6   6   6     6       NA       NA   NA
    #> rank[FIXTURE: ETN]     5.00 0.00    5   5   5   5     5       NA       NA   NA
    #> rank[FIXTURE: IXE_Q2W] 1.00 0.00    1   1   1   1     1       NA       NA   NA
    #> rank[FIXTURE: IXE_Q4W] 2.22 0.41    2   2   2   2     3     4218     4016    1
    #> rank[FIXTURE: SEC_150] 4.00 0.03    4   4   4   4     4     4022       NA    1
    #> rank[FIXTURE: SEC_300] 2.79 0.41    2   3   3   3     3     4261     4021    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-1 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>        2    0.73 0.28   9.24 0.28
    #>
    #>                          mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> rank[UNCOVER-1: PBO]     6.00 0.00    6   6   6   6     6       NA       NA   NA
    #> rank[UNCOVER-1: ETN]     5.00 0.00    5   5   5   5     5       NA       NA   NA
    #> rank[UNCOVER-1: IXE_Q2W] 1.00 0.00    1   1   1   1     1       NA       NA   NA
    #> rank[UNCOVER-1: IXE_Q4W] 2.22 0.41    2   2   2   2     3     4218     4016    1
    #> rank[UNCOVER-1: SEC_150] 4.00 0.03    4   4   4   4     4     4022       NA    1
    #> rank[UNCOVER-1: SEC_300] 2.79 0.41    2   3   3   3     3     4261     4021    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-2 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.87    0.64 0.27   9.17 0.24
    #>
    #>                          mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> rank[UNCOVER-2: PBO]     6.00 0.00    6   6   6   6     6       NA       NA   NA
    #> rank[UNCOVER-2: ETN]     5.00 0.00    5   5   5   5     5       NA       NA   NA
    #> rank[UNCOVER-2: IXE_Q2W] 1.00 0.00    1   1   1   1     1       NA       NA   NA
    #> rank[UNCOVER-2: IXE_Q4W] 2.22 0.41    2   2   2   2     3     4218     4016    1
    #> rank[UNCOVER-2: SEC_150] 4.00 0.03    4   4   4   4     4     4022       NA    1
    #> rank[UNCOVER-2: SEC_300] 2.79 0.41    2   3   3   3     3     4261     4021    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-3 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight psa
    #>     1.78    0.59 0.28   9.01 0.2
    #>
    #>                          mean   sd 2.5% 25% 50% 75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> rank[UNCOVER-3: PBO]     6.00 0.00    6   6   6   6     6       NA       NA   NA
    #> rank[UNCOVER-3: ETN]     5.00 0.00    5   5   5   5     5       NA       NA   NA
    #> rank[UNCOVER-3: IXE_Q2W] 1.00 0.00    1   1   1   1     1       NA       NA   NA
    #> rank[UNCOVER-3: IXE_Q4W] 2.22 0.41    2   2   2   2     3     4218     4016    1
    #> rank[UNCOVER-3: SEC_150] 4.00 0.03    4   4   4   4     4     4022       NA    1
    #> rank[UNCOVER-3: SEC_300] 2.79 0.41    2   3   3   3     3     4261     4021    1
    plot(pso_ranks_FE)

    (pso_rankprobs_FE <- posterior_rank_probs(pso_fit_FE, lower_better = FALSE))
    #> ---------------------------------------------------------------- Study: FIXTURE ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>      1.6    0.62 0.34   8.34 0.14
    #>
    #>                     p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
    #> d[FIXTURE: PBO]             0      0.00      0.00         0         0         1
    #> d[FIXTURE: ETN]             0      0.00      0.00         0         1         0
    #> d[FIXTURE: IXE_Q2W]         1      0.00      0.00         0         0         0
    #> d[FIXTURE: IXE_Q4W]         0      0.78      0.21         0         0         0
    #> d[FIXTURE: SEC_150]         0      0.00      0.00         1         0         0
    #> d[FIXTURE: SEC_300]         0      0.22      0.78         0         0         0
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-1 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>        2    0.73 0.28   9.24 0.28
    #>
    #>                       p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
    #> d[UNCOVER-1: PBO]             0      0.00      0.00         0         0         1
    #> d[UNCOVER-1: ETN]             0      0.00      0.00         0         1         0
    #> d[UNCOVER-1: IXE_Q2W]         1      0.00      0.00         0         0         0
    #> d[UNCOVER-1: IXE_Q4W]         0      0.78      0.21         0         0         0
    #> d[UNCOVER-1: SEC_150]         0      0.00      0.00         1         0         0
    #> d[UNCOVER-1: SEC_300]         0      0.22      0.78         0         0         0
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-2 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.87    0.64 0.27   9.17 0.24
    #>
    #>                       p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
    #> d[UNCOVER-2: PBO]             0      0.00      0.00         0         0         1
    #> d[UNCOVER-2: ETN]             0      0.00      0.00         0         1         0
    #> d[UNCOVER-2: IXE_Q2W]         1      0.00      0.00         0         0         0
    #> d[UNCOVER-2: IXE_Q4W]         0      0.78      0.21         0         0         0
    #> d[UNCOVER-2: SEC_150]         0      0.00      0.00         1         0         0
    #> d[UNCOVER-2: SEC_300]         0      0.22      0.78         0         0         0
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-3 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight psa
    #>     1.78    0.59 0.28   9.01 0.2
    #>
    #>                       p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
    #> d[UNCOVER-3: PBO]             0      0.00      0.00         0         0         1
    #> d[UNCOVER-3: ETN]             0      0.00      0.00         0         1         0
    #> d[UNCOVER-3: IXE_Q2W]         1      0.00      0.00         0         0         0
    #> d[UNCOVER-3: IXE_Q4W]         0      0.78      0.21         0         0         0
    #> d[UNCOVER-3: SEC_150]         0      0.00      0.00         1         0         0
    #> d[UNCOVER-3: SEC_300]         0      0.22      0.78         0         0         0
    plot(pso_rankprobs_FE)

    (pso_cumrankprobs_FE <- posterior_rank_probs(pso_fit_FE, lower_better = FALSE, cumulative = TRUE))
    #> ---------------------------------------------------------------- Study: FIXTURE ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>      1.6    0.62 0.34   8.34 0.14
    #>
    #>                     p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
    #> d[FIXTURE: PBO]             0      0.00         0         0         0         1
    #> d[FIXTURE: ETN]             0      0.00         0         0         1         1
    #> d[FIXTURE: IXE_Q2W]         1      1.00         1         1         1         1
    #> d[FIXTURE: IXE_Q4W]         0      0.78         1         1         1         1
    #> d[FIXTURE: SEC_150]         0      0.00         0         1         1         1
    #> d[FIXTURE: SEC_300]         0      0.22         1         1         1         1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-1 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>        2    0.73 0.28   9.24 0.28
    #>
    #>                       p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
    #> d[UNCOVER-1: PBO]             0      0.00         0         0         0         1
    #> d[UNCOVER-1: ETN]             0      0.00         0         0         1         1
    #> d[UNCOVER-1: IXE_Q2W]         1      1.00         1         1         1         1
    #> d[UNCOVER-1: IXE_Q4W]         0      0.78         1         1         1         1
    #> d[UNCOVER-1: SEC_150]         0      0.00         0         1         1         1
    #> d[UNCOVER-1: SEC_300]         0      0.22         1         1         1         1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-2 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.87    0.64 0.27   9.17 0.24
    #>
    #>                       p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
    #> d[UNCOVER-2: PBO]             0      0.00         0         0         0         1
    #> d[UNCOVER-2: ETN]             0      0.00         0         0         1         1
    #> d[UNCOVER-2: IXE_Q2W]         1      1.00         1         1         1         1
    #> d[UNCOVER-2: IXE_Q4W]         0      0.78         1         1         1         1
    #> d[UNCOVER-2: SEC_150]         0      0.00         0         1         1         1
    #> d[UNCOVER-2: SEC_300]         0      0.22         1         1         1         1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-3 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight psa
    #>     1.78    0.59 0.28   9.01 0.2
    #>
    #>                       p_rank[1] p_rank[2] p_rank[3] p_rank[4] p_rank[5] p_rank[6]
    #> d[UNCOVER-3: PBO]             0      0.00         0         0         0         1
    #> d[UNCOVER-3: ETN]             0      0.00         0         0         1         1
    #> d[UNCOVER-3: IXE_Q2W]         1      1.00         1         1         1         1
    #> d[UNCOVER-3: IXE_Q4W]         0      0.78         1         1         1         1
    #> d[UNCOVER-3: SEC_150]         0      0.00         0         1         1         1
    #> d[UNCOVER-3: SEC_300]         0      0.22         1         1         1         1
    plot(pso_cumrankprobs_FE)

All of the above estimates (relative effects, predictions, rankings) can also be produced for a specific target population or populations by providing a suitable `newdata` argument to for function (and a `baseline` distribution for `predict()`).

To produce population-adjusted relative effects (and corresponding rankings) for a chosen target population, we require only the mean covariate values in that population. For example, `newdata` could provide the following mean covariate values:

    new_agd_means <- tibble(
      bsa = 0.6,
      prevsys = 0.1,
      psa = 0.2,
      weight = 10,
      durnpso = 3)

Population-adjusted relative effects in this target population are then calculated using the `relative_effects()` function, and can be plotted with the corresponding `plot()` method:

    (pso_releff_FE_new <- relative_effects(pso_fit_FE, newdata = new_agd_means))
    #> ------------------------------------------------------------------ Study: New 1 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys bsa weight psa
    #>        3     0.1 0.6     10 0.2
    #>
    #>                   mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[New 1: ETN]     1.25 0.23 0.82 1.10 1.25 1.40  1.72     5608     3332    1
    #> d[New 1: IXE_Q2W] 2.89 0.22 2.45 2.74 2.88 3.03  3.34     6226     3007    1
    #> d[New 1: IXE_Q4W] 2.47 0.22 2.04 2.33 2.46 2.61  2.92     6186     3053    1
    #> d[New 1: SEC_150] 2.07 0.22 1.64 1.92 2.07 2.22  2.53     5979     2865    1
    #> d[New 1: SEC_300] 2.38 0.23 1.96 2.22 2.38 2.53  2.83     5989     2950    1
    plot(pso_releff_FE_new, ref_line = 0)

For absolute predictions, we require information about the full covariate distribution in the target population, not just the mean values. If IPD are available for the target population, `newdata` is simply a data frame of the IPD. If AgD are available for the target population, `newdata` must be a data frame with added integration points created using the `add_integration()` function.

For example, suppose the aggregate target population introduced above had the following covariate means and standard deviations (for continuous covariates) or proportions (for discrete covariates):

    new_agd_int <- tibble(
      bsa_mean = 0.6,
      bsa_sd = 0.3,
      prevsys = 0.1,
      psa = 0.2,
      weight_mean = 10,
      weight_sd = 1,
      durnpso_mean = 3,
      durnpso_sd = 1
    )

We add integration points to this data frame in a similar manner to before. Again, we need to supply a correlation matrix for the joint covariate distribution; we use the same weighted mean correlation matrix computed earlier from the IPD in the network, which is stored in the network object as `int_cor`.

    new_agd_int <- add_integration(new_agd_int,
      durnpso = distr(qgamma, mean = durnpso_mean, sd = durnpso_sd),
      prevsys = distr(qbern, prob = prevsys),
      bsa = distr(qlogitnorm, mean = bsa_mean, sd = bsa_sd),
      weight = distr(qgamma, mean = weight_mean, sd = weight_sd),
      psa = distr(qbern, prob = psa),
      cor = pso_net$int_cor,
      n_int = 64)

Predicted probabilities of achieving PASI 75 in this target population, given a \(N(-1.75, 0.08^2)\) distribution on the baseline probit-probability of response on Placebo (at the reference levels of the covariates), are then produced using the `predict()` method:

    (pso_pred_FE_new <- predict(pso_fit_FE,
                                type = "response",
                                newdata = new_agd_int,
                                baseline = distr(qnorm, -1.75, 0.08)))
    #> ------------------------------------------------------------------ Study: New 1 ----
    #>
    #>                      mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[New 1: PBO]     0.06 0.03 0.03 0.04 0.06 0.08  0.12     4883     3389    1
    #> pred[New 1: ETN]     0.37 0.06 0.26 0.33 0.37 0.41  0.48     5997     3562    1
    #> pred[New 1: IXE_Q2W] 0.90 0.03 0.84 0.88 0.90 0.92  0.94     5212     3159    1
    #> pred[New 1: IXE_Q4W] 0.81 0.04 0.72 0.78 0.81 0.83  0.87     5165     3683    1
    #> pred[New 1: SEC_150] 0.68 0.06 0.56 0.64 0.68 0.72  0.79     5355     3860    1
    #> pred[New 1: SEC_300] 0.78 0.05 0.68 0.75 0.78 0.81  0.86     5608     3670    1
    plot(pso_pred_FE_new, ref_line = c(0, 1))

# Extended analysis

We now extend the network to include a further five studies (four AgD and one IPD), recreating the analysis of Phillippo et al. (2022). This larger network allows us to assess the key assumptions underlying population adjustment.

## Setup

### Preparing the data

We begin, as before, with some data transformations for each of the covariates and set up a treatment class variable `trtclass`.

    # IPD studies
    pso_ipd <- plaque_psoriasis_ipd %>%
      mutate(
        # Variable transformations
        bsa = bsa / 100,
        weight = weight / 10,
        durnpso = durnpso / 10,
        prevsys = as.numeric(prevsys),
        psa = as.numeric(psa),
        # Treatment classes
        trtclass = case_when(trtn == 1 ~ "Placebo",
                             trtn %in% c(2, 3, 5, 6) ~ "IL-17 blocker",
                             trtn == 4 ~ "TNFa blocker",
                             trtn == 7 ~ "IL-12/23 blocker"),
        # Check complete cases for covariates of interest
        is_complete = complete.cases(durnpso, prevsys, bsa, weight, psa)
      ) %>%
      arrange(studyc, trtn)

    # AgD studies
    pso_agd <- plaque_psoriasis_agd %>%
      mutate(
        # Variable transformations
        bsa_mean = bsa_mean / 100,
        bsa_sd = bsa_sd / 100,
        weight_mean = weight_mean / 10,
        weight_sd = weight_sd / 10,
        durnpso_mean = durnpso_mean / 10,
        durnpso_sd = durnpso_sd / 10,
        prevsys = prevsys / 100,
        psa = psa / 100,
        # Treatment classes
        trtclass = case_when(trtn == 1 ~ "Placebo",
                             trtn %in% c(2, 3, 5, 6) ~ "IL-17 blocker",
                             trtn == 4 ~ "TNFa blocker",
                             trtn == 7 ~ "IL-12/23 blocker")
        ) %>%
      arrange(studyc, trtn)

There are a very small number of individuals with missing values in the IPD, which we simply exclude from the analysis.

    pso_ipd %>%
      group_by(studyc) %>%
      summarise(n_total = n(),
                n_missing = sum(!is_complete),
                pct_missing = mean(!is_complete) * 100)
    #> # A tibble: 4 × 4
    #>   studyc    n_total n_missing pct_missing
    #>   <chr>       <int>     <int>       <dbl>
    #> 1 IXORA-S       260         0       0
    #> 2 UNCOVER-1    1296         0       0
    #> 3 UNCOVER-2    1221         2       0.164
    #> 4 UNCOVER-3    1341         2       0.149

    pso_ipd <- filter(pso_ipd, is_complete)

### Creating the network

Next we set up the network. We set the IPD with `set_ipd()` and AgD (arm-based) with `set_agd_arm()`, and combine these together using `combine_network()`. We specify an ordered categorical (multinomial) outcome using the `multi()` helper function. The outcome data are in “inclusive” format, i.e. the lowest category is the sample size (or 1 for IPD), the second category counts those achieving PASI 75 or greater (\(\ge 75\%\) reduction in symptoms), the third counts those achieving PASI 90 or greater (\(\ge 90\%\) reduction), and the final category counts those achieving PASI 100 (\(100\%\) reduction).2 We specify the treatment classes with `trt_class = trtclass`.

    pso_net <- combine_network(
      set_ipd(pso_ipd,
        study = studyc,
        trt = trtc,
        r = multi(r0 = 1,
                  PASI75 = pasi75,
                  PASI90 = pasi90,
                  PASI100 = pasi100,
                  type = "ordered", inclusive = TRUE),
        trt_class = trtclass),
      set_agd_arm(pso_agd,
        study = studyc,
        trt = trtc,
        r = multi(r0 = pasi75_n,
                  PASI75 = pasi75_r,
                  PASI90 = pasi90_r,
                  PASI100 = pasi100_r,
                  type = "ordered", inclusive = TRUE),
        trt_class = trtclass)
    )

    pso_net
    #> A network with 4 IPD studies, and 5 AgD studies (arm-based).
    #>
    #> ------------------------------------------------------------------- IPD studies ----
    #>  Study     Treatment arms
    #>  IXORA-S   2: IXE_Q2W | UST
    #>  UNCOVER-1 3: PBO | IXE_Q2W | IXE_Q4W
    #>  UNCOVER-2 4: PBO | IXE_Q2W | IXE_Q4W | ETN
    #>  UNCOVER-3 4: PBO | IXE_Q2W | IXE_Q4W | ETN
    #>
    #>  Outcome type: ordered (4 categories)
    #> ------------------------------------------------------- AgD studies (arm-based) ----
    #>  Study    Treatment arms
    #>  CLEAR    2: SEC_300 | UST
    #>  ERASURE  3: PBO | SEC_150 | SEC_300
    #>  FEATURE  3: PBO | SEC_150 | SEC_300
    #>  FIXTURE  4: PBO | ETN | SEC_150 | SEC_300
    #>  JUNCTURE 3: PBO | SEC_150 | SEC_300
    #>
    #>  Outcome type: ordered (4 categories)
    #> ------------------------------------------------------------------------------------
    #> Total number of treatments: 7, in 4 classes
    #> Total number of studies: 9
    #> Reference treatment is: PBO
    #> Network is connected

We create a network plot using the `plot()` function applied to the `pso_net` network object, choosing to scale the edges and nodes by the number of studies/sample size (`weight_edges` and `weight_nodes = TRUE`), colour the treatment nodes by class (`show_trt_class = TRUE`), and nudge the treatment names away from the nodes (`nudge = 0.1`). We further customise the plot using ggplot syntax to alter the colour scheme.

    class_pal <- c("#D95F02", "#7570B3", "#E7298A", "#E6AB02")

    plot(pso_net, weight_nodes = TRUE, weight_edges = TRUE, show_trt_class = TRUE, nudge = 0.1) +
      ggraph::scale_edge_colour_manual("Data",
                                       values = c(AgD = "#113259", IPD = "#55A480")) +
      scale_fill_manual("Treatment class",
                        values = class_pal,
                        aesthetics = c("fill", "colour")) +
      guides(edge_colour = guide_legend(override.aes = list(edge_width = 2)),
             fill = guide_legend(override.aes = list(size = 2)))
    #> Scale for edge_colour is already present.
    #> Adding another scale for edge_colour, which will replace the existing scale.
    #> Scale for fill is already present.
    #> Adding another scale for fill, which will replace the existing scale.

### Numerical integration for ML-NMR models

We add integration points to the AgD studies in the network using the `add_integration()` function, specifying the chosen marginal distribution for each covariate using the `distr()` function. As before, we specify Gamma distributions for weight and duration of psoriasis, a logit-Normal distribution for body surface area, and Bernoulli distributions for previous systemic treatment and psoriatic arthritis as binary covariates. Since we do not know the correlations between covariates in the AgD studies, we once again impute these with the weighted mean of the correlations in the IPD studies (the default option).

    pso_net <- add_integration(pso_net,
      durnpso = distr(qgamma, mean = durnpso_mean, sd = durnpso_sd),
      prevsys = distr(qbern, prob = prevsys),
      bsa = distr(qlogitnorm, mean = bsa_mean, sd = bsa_sd),
      weight = distr(qgamma, mean = weight_mean, sd = weight_sd),
      psa = distr(qbern, prob = psa),
      n_int = 64)
    #> Using weighted average correlation matrix computed from IPD studies.

## ML-NMR model

Using the `nma()` function, we fit a (fixed effect) ML-NMR model which includes main effects (prognostic terms) and covariate-treatment interactions (effect-modifying terms) for each of the five covariates. Ideally, we would fit independent interaction terms for each treatment; however, this requires either IPD or several AgD studies at a range of covariate values on each treatment. The data here are insufficient to fit independent interaction terms for each treatment, so we make the shared effect modifier assumption within each class of treatments (Phillippo et al. 2016) and specify common interaction terms within each treatment class (`class_interactions = "common"`). As before, we specify \(\mathrm{N}(0, 10^2)\) prior distributions on the study-specific intercepts, treatment effects, and regression parameters. However, since we now have an ordered multinomial likelihood we also need to specify priors for the differences between the latent cutoffs for each outcome category; we choose an improper flat prior \(\mathrm{U}(-\infty,\infty)\) which will automatically be truncated to meet the ordering constraints (`prior_aux = flat()`).

    pso_fit_FE <- nma(pso_net,
                      trt_effects = "fixed",
                      link = "probit",
                      regression = ~(durnpso + prevsys + bsa + weight + psa)*.trt,
                      class_interactions = "common",
                      prior_intercept = normal(scale = 10),
                      prior_trt = normal(scale = 10),
                      prior_reg = normal(scale = 10),
                      prior_aux = flat(),
                      QR = TRUE,
                      init_r = 0.5)

    #> Note: Setting "PBO" as the network reference treatment.

    pso_fit_FE

    #> A fixed effects ML-NMR with a ordered likelihood (probit link).
    #> Regression model: ~(durnpso + prevsys + bsa + weight + psa) * .trt.
    #> Centred covariates at the following overall mean values:
    #>   durnpso   prevsys       bsa    weight       psa
    #> 1.7947485 0.6504375 0.2973544 8.9165934 0.2074278
    #> 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%
    #> beta[durnpso]                               0.03    0.00 0.06    -0.09    -0.01     0.03     0.08
    #> beta[prevsys]                              -0.17    0.00 0.15    -0.46    -0.28    -0.18    -0.07
    #> beta[bsa]                                  -0.12    0.01 0.44    -1.02    -0.40    -0.11     0.18
    #> beta[weight]                                0.04    0.00 0.03    -0.01     0.03     0.04     0.06
    #> beta[psa]                                  -0.07    0.00 0.17    -0.43    -0.20    -0.07     0.04
    #> beta[durnpso:.trtclassTNFa blocker]        -0.02    0.00 0.07    -0.16    -0.07    -0.02     0.03
    #> beta[durnpso:.trtclassIL-12/23 blocker]    -0.06    0.00 0.10    -0.26    -0.13    -0.07     0.01
    #> beta[durnpso:.trtclassIL-17 blocker]       -0.02    0.00 0.06    -0.15    -0.07    -0.02     0.02
    #> beta[prevsys:.trtclassTNFa blocker]         0.20    0.00 0.18    -0.16     0.08     0.20     0.32
    #> beta[prevsys:.trtclassIL-12/23 blocker]     0.45    0.00 0.32    -0.20     0.24     0.46     0.68
    #> beta[prevsys:.trtclassIL-17 blocker]        0.17    0.00 0.16    -0.15     0.06     0.17     0.28
    #> beta[bsa:.trtclassTNFa blocker]             0.26    0.01 0.51    -0.71    -0.08     0.25     0.59
    #> beta[bsa:.trtclassIL-12/23 blocker]         0.61    0.01 0.66    -0.65     0.17     0.61     1.06
    #> beta[bsa:.trtclassIL-17 blocker]            0.29    0.01 0.46    -0.60    -0.02     0.28     0.58
    #> beta[weight:.trtclassTNFa blocker]         -0.16    0.00 0.03    -0.23    -0.18    -0.16    -0.14
    #> beta[weight:.trtclassIL-12/23 blocker]     -0.09    0.00 0.05    -0.18    -0.12    -0.09    -0.06
    #> beta[weight:.trtclassIL-17 blocker]        -0.13    0.00 0.03    -0.19    -0.15    -0.13    -0.11
    #> beta[psa:.trtclassTNFa blocker]            -0.06    0.00 0.20    -0.46    -0.19    -0.05     0.08
    #> beta[psa:.trtclassIL-12/23 blocker]         0.13    0.01 0.34    -0.53    -0.10     0.12     0.35
    #> beta[psa:.trtclassIL-17 blocker]            0.09    0.00 0.18    -0.26    -0.03     0.09     0.22
    #> d[ETN]                                      1.58    0.00 0.07     1.44     1.53     1.58     1.63
    #> d[IXE_Q2W]                                  2.91    0.00 0.07     2.77     2.86     2.91     2.96
    #> d[IXE_Q4W]                                  2.69    0.00 0.08     2.55     2.64     2.69     2.74
    #> d[SEC_150]                                  2.19    0.00 0.08     2.03     2.13     2.19     2.25
    #> d[SEC_300]                                  2.60    0.00 0.08     2.44     2.54     2.60     2.65
    #> d[UST]                                      2.13    0.00 0.11     1.91     2.06     2.13     2.21
    #> lp__                                    -7752.94    0.10 4.30 -7762.02 -7755.80 -7752.63 -7749.89
    #> cc[PASI75]                                  0.00     NaN 0.00     0.00     0.00     0.00     0.00
    #> cc[PASI90]                                  0.69    0.00 0.02     0.65     0.68     0.69     0.70
    #> cc[PASI100]                                 1.53    0.00 0.02     1.49     1.52     1.53     1.55
    #>                                            97.5% n_eff Rhat
    #> beta[durnpso]                               0.15  3278    1
    #> beta[prevsys]                               0.13  3239    1
    #> beta[bsa]                                   0.73  2993    1
    #> beta[weight]                                0.10  2717    1
    #> beta[psa]                                   0.25  3200    1
    #> beta[durnpso:.trtclassTNFa blocker]         0.12  3645    1
    #> beta[durnpso:.trtclassIL-12/23 blocker]     0.14  3762    1
    #> beta[durnpso:.trtclassIL-17 blocker]        0.11  3711    1
    #> beta[prevsys:.trtclassTNFa blocker]         0.55  3330    1
    #> beta[prevsys:.trtclassIL-12/23 blocker]     1.05  4366    1
    #> beta[prevsys:.trtclassIL-17 blocker]        0.47  3593    1
    #> beta[bsa:.trtclassTNFa blocker]             1.30  3147    1
    #> beta[bsa:.trtclassIL-12/23 blocker]         1.89  3386    1
    #> beta[bsa:.trtclassIL-17 blocker]            1.22  3490    1
    #> beta[weight:.trtclassTNFa blocker]         -0.10  3342    1
    #> beta[weight:.trtclassIL-12/23 blocker]      0.00  4100    1
    #> beta[weight:.trtclassIL-17 blocker]        -0.07  3080    1
    #> beta[psa:.trtclassTNFa blocker]             0.34  3511    1
    #> beta[psa:.trtclassIL-12/23 blocker]         0.80  3769    1
    #> beta[psa:.trtclassIL-17 blocker]            0.46  3732    1
    #> d[ETN]                                      1.72  2114    1
    #> d[IXE_Q2W]                                  3.06  2708    1
    #> d[IXE_Q4W]                                  2.84  2949    1
    #> d[SEC_150]                                  2.36  2357    1
    #> d[SEC_300]                                  2.76  2456    1
    #> d[UST]                                      2.36  3349    1
    #> lp__                                    -7745.56  1701    1
    #> cc[PASI75]                                  0.00   NaN  NaN
    #> cc[PASI90]                                  0.72  4016    1
    #> cc[PASI100]                                 1.58  3777    1
    #>
    #> Samples were drawn using NUTS(diag_e) at Tue Jan  9 11:53:40 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).

## Assessing assumptions

In the first analysis, the small network made assessing assumptions difficult. With this larger network (although still only nine studies) we have greater opportunity to assess the key assumptions.

The key assumption made by ML-NMR (and indeed all population adjustment methods in connected networks) is the conditional constancy of relative effects assumption (Phillippo et al. 2016). This means that there are no unobserved effect modifiers, so that the relative effects are constant given the included effect-modifying covariates. This assumption implies that there is no residual heterogeneity or inconsistency, which can be assessed using standard network meta-analysis techniques. We assess residual heterogeneity using a random effects model, and residual inconsistency using an unrelated mean effects (UME) model.

### Assessing residual heterogeneity with a random effects ML-NMR

First, we fit a random effects model to assess residual heterogeneity. The call to the `nma()` function is identical to the fixed effect model above, except that now we specify `trt_effects = "random"` and need to provide a prior for the between-study heterogeneity (we choose a \(\textrm{half-N}(0, 2.5^2)\) prior with `prior_het = half_normal(scale = 2.5)`.

    pso_fit_RE <- nma(pso_net,
                      trt_effects = "random",
                      link = "probit",
                      regression = ~(durnpso + prevsys + bsa + weight + psa)*.trt,
                      class_interactions = "common",
                      prior_intercept = normal(scale = 10),
                      prior_trt = normal(scale = 10),
                      prior_reg = normal(scale = 10),
                      prior_aux = flat(),
                      prior_het = half_normal(scale = 2.5),
                      QR = TRUE,
                      init_r = 0.5)

    #> Note: Setting "PBO" 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

    pso_fit_RE

    #> A random effects ML-NMR with a ordered likelihood (probit link).
    #> Regression model: ~(durnpso + prevsys + bsa + weight + psa) * .trt.
    #> Centred covariates at the following overall mean values:
    #>   durnpso   prevsys       bsa    weight       psa
    #> 1.7947485 0.6504375 0.2973544 8.9165934 0.2074278
    #> 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%
    #> beta[durnpso]                               0.04    0.00 0.06    -0.09     0.00     0.04     0.08
    #> beta[prevsys]                              -0.16    0.00 0.16    -0.47    -0.27    -0.16    -0.05
    #> beta[bsa]                                  -0.13    0.01 0.47    -1.09    -0.43    -0.11     0.20
    #> beta[weight]                                0.05    0.00 0.03    -0.01     0.03     0.05     0.07
    #> beta[psa]                                  -0.07    0.00 0.18    -0.41    -0.19    -0.07     0.05
    #> beta[durnpso:.trtclassTNFa blocker]        -0.02    0.00 0.07    -0.17    -0.07    -0.03     0.03
    #> beta[durnpso:.trtclassIL-12/23 blocker]    -0.07    0.00 0.10    -0.26    -0.14    -0.07     0.00
    #> beta[durnpso:.trtclassIL-17 blocker]       -0.03    0.00 0.07    -0.16    -0.07    -0.03     0.02
    #> beta[prevsys:.trtclassTNFa blocker]         0.19    0.00 0.18    -0.16     0.07     0.19     0.31
    #> beta[prevsys:.trtclassIL-12/23 blocker]     0.44    0.00 0.33    -0.22     0.22     0.44     0.67
    #> beta[prevsys:.trtclassIL-17 blocker]        0.16    0.00 0.17    -0.17     0.05     0.16     0.27
    #> beta[bsa:.trtclassTNFa blocker]             0.25    0.01 0.53    -0.75    -0.12     0.24     0.59
    #> beta[bsa:.trtclassIL-12/23 blocker]         0.63    0.01 0.66    -0.65     0.18     0.62     1.07
    #> beta[bsa:.trtclassIL-17 blocker]            0.30    0.01 0.48    -0.60    -0.04     0.30     0.62
    #> beta[weight:.trtclassTNFa blocker]         -0.16    0.00 0.03    -0.23    -0.19    -0.16    -0.14
    #> beta[weight:.trtclassIL-12/23 blocker]     -0.09    0.00 0.05    -0.19    -0.12    -0.09    -0.06
    #> beta[weight:.trtclassIL-17 blocker]        -0.13    0.00 0.03    -0.19    -0.15    -0.13    -0.11
    #> beta[psa:.trtclassTNFa blocker]            -0.06    0.00 0.21    -0.47    -0.20    -0.06     0.08
    #> beta[psa:.trtclassIL-12/23 blocker]         0.11    0.00 0.34    -0.55    -0.12     0.11     0.34
    #> beta[psa:.trtclassIL-17 blocker]            0.08    0.00 0.19    -0.29    -0.04     0.08     0.21
    #> d[ETN]                                      1.59    0.00 0.11     1.38     1.52     1.59     1.65
    #> d[IXE_Q2W]                                  2.93    0.00 0.11     2.72     2.86     2.93     3.00
    #> d[IXE_Q4W]                                  2.71    0.00 0.11     2.48     2.64     2.71     2.78
    #> d[SEC_150]                                  2.22    0.00 0.12     2.00     2.14     2.21     2.29
    #> d[SEC_300]                                  2.64    0.00 0.12     2.42     2.56     2.63     2.71
    #> d[UST]                                      2.17    0.00 0.17     1.85     2.06     2.16     2.27
    #> lp__                                    -7761.18    0.20 6.16 -7774.24 -7765.11 -7760.94 -7756.88
    #> tau                                         0.14    0.00 0.07     0.03     0.09     0.13     0.17
    #> cc[PASI75]                                  0.00     NaN 0.00     0.00     0.00     0.00     0.00
    #> cc[PASI90]                                  0.69    0.00 0.02     0.65     0.68     0.69     0.70
    #> cc[PASI100]                                 1.53    0.00 0.02     1.49     1.52     1.53     1.55
    #>                                            97.5% n_eff Rhat
    #> beta[durnpso]                               0.16  4028    1
    #> beta[prevsys]                               0.15  3931    1
    #> beta[bsa]                                   0.75  3963    1
    #> beta[weight]                                0.10  4005    1
    #> beta[psa]                                   0.28  3835    1
    #> beta[durnpso:.trtclassTNFa blocker]         0.12  4080    1
    #> beta[durnpso:.trtclassIL-12/23 blocker]     0.13  4244    1
    #> beta[durnpso:.trtclassIL-17 blocker]        0.10  4440    1
    #> beta[prevsys:.trtclassTNFa blocker]         0.54  4143    1
    #> beta[prevsys:.trtclassIL-12/23 blocker]     1.07  4555    1
    #> beta[prevsys:.trtclassIL-17 blocker]        0.48  4517    1
    #> beta[bsa:.trtclassTNFa blocker]             1.33  4333    1
    #> beta[bsa:.trtclassIL-12/23 blocker]         1.96  4933    1
    #> beta[bsa:.trtclassIL-17 blocker]            1.30  4329    1
    #> beta[weight:.trtclassTNFa blocker]         -0.10  4309    1
    #> beta[weight:.trtclassIL-12/23 blocker]      0.00  4940    1
    #> beta[weight:.trtclassIL-17 blocker]        -0.07  4681    1
    #> beta[psa:.trtclassTNFa blocker]             0.35  3883    1
    #> beta[psa:.trtclassIL-12/23 blocker]         0.78  4810    1
    #> beta[psa:.trtclassIL-17 blocker]            0.44  4230    1
    #> d[ETN]                                      1.81  2539    1
    #> d[IXE_Q2W]                                  3.16  2657    1
    #> d[IXE_Q4W]                                  2.93  2394    1
    #> d[SEC_150]                                  2.47  2433    1
    #> d[SEC_300]                                  2.90  2193    1
    #> d[UST]                                      2.50  3305    1
    #> lp__                                    -7750.02   985    1
    #> tau                                         0.30   622    1
    #> cc[PASI75]                                  0.00   NaN  NaN
    #> cc[PASI90]                                  0.72  5611    1
    #> cc[PASI100]                                 1.58  5856    1
    #>
    #> Samples were drawn using NUTS(diag_e) at Tue Jan  9 12:36:43 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).

The estimated between-study heterogeneity standard deviation `tau` is small compared to the relative treatment effects. We compare the model fit using the DIC:

    (pso_dic_FE <- dic(pso_fit_FE))

    #> Residual deviance: 8811.2 (on 12387 data points)
    #>                pD: 36
    #>               DIC: 8847.3

    (pso_dic_RE <- dic(pso_fit_RE))

    #> Residual deviance: 8799.8 (on 12387 data points)
    #>                pD: 42.3
    #>               DIC: 8842.1

The DIC is lower for the RE model, indicating that there may be residual heterogeneity in the network and that the conditional constancy of relative effects assumption may be invalid—there may be additional effect modifiers that we have not accounted for. This result is different to the actual analysis reported by Phillippo et al. (2022), since here we are using synthetic IPD that have been simulated to closely resemble the original IPD. In the actual analysis the DIC was similar between the FE and RE models, so we might choose the more parsimonious FE model based on DIC alone, and there was no evidence for residual heterogeneity in this network.

### Assessing residual inconsistency with an unrelated mean effects ML-NMR

We assess residual inconsistency using an unrelated mean effects model (Dias et al. 2011). Again, the call to the `nma()` function is identical, except this time we specify `consistency = "ume"`. Node-splitting is also a possibility (with `consistency = "nodesplit"`), but this takes substantially longer since the model is re-run for each node-split comparison. Here we will proceed as in the analysis of Phillippo et al. (2022) and fit a fixed effect UME model (since there was no evidence for heterogeneity in the actual analysis); however, in our recreated analysis using synthetic IPD there was evidence of heterogeneity and we should really fit a random effects UME model instead.

    pso_fit_UME <- nma(pso_net,
                       trt_effects = "fixed",
                       consistency = "ume",
                       link = "probit",
                       regression = ~(durnpso + prevsys + bsa + weight + psa)*.trt,
                       class_interactions = "common",
                       prior_intercept = normal(scale = 10),
                       prior_trt = normal(scale = 10),
                       prior_reg = normal(scale = 10),
                       prior_aux = flat(),
                       QR = TRUE,
                       init_r = 0.5)

    #> Note: Setting "PBO" as the network reference treatment.

    pso_fit_UME

    #> A fixed effects ML-NMR with a ordered likelihood (probit link).
    #> An inconsistency model ('ume') was fitted.
    #> Regression model: ~(durnpso + prevsys + bsa + weight + psa) * .trt.
    #> Centred covariates at the following overall mean values:
    #>   durnpso   prevsys       bsa    weight       psa
    #> 1.7947485 0.6504375 0.2973544 8.9165934 0.2074278
    #> 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%
    #> beta[durnpso]                               0.04    0.00 0.06    -0.08     0.00     0.04     0.08
    #> beta[prevsys]                              -0.18    0.00 0.16    -0.48    -0.28    -0.18    -0.07
    #> beta[bsa]                                  -0.10    0.01 0.44    -0.99    -0.38    -0.09     0.20
    #> beta[weight]                                0.04    0.00 0.03    -0.01     0.03     0.04     0.06
    #> beta[psa]                                  -0.08    0.00 0.16    -0.40    -0.19    -0.08     0.03
    #> beta[durnpso:.trtclassTNFa blocker]        -0.02    0.00 0.07    -0.16    -0.07    -0.02     0.02
    #> beta[durnpso:.trtclassIL-12/23 blocker]    -0.06    0.00 0.10    -0.26    -0.13    -0.07     0.00
    #> beta[durnpso:.trtclassIL-17 blocker]       -0.02    0.00 0.06    -0.14    -0.07    -0.02     0.02
    #> beta[prevsys:.trtclassTNFa blocker]         0.20    0.00 0.18    -0.15     0.09     0.20     0.33
    #> beta[prevsys:.trtclassIL-12/23 blocker]     0.45    0.01 0.35    -0.28     0.23     0.47     0.70
    #> beta[prevsys:.trtclassIL-17 blocker]        0.17    0.00 0.16    -0.15     0.06     0.18     0.29
    #> beta[bsa:.trtclassTNFa blocker]             0.24    0.01 0.50    -0.74    -0.10     0.23     0.58
    #> beta[bsa:.trtclassIL-12/23 blocker]         0.60    0.01 0.66    -0.71     0.17     0.60     1.04
    #> beta[bsa:.trtclassIL-17 blocker]            0.27    0.01 0.46    -0.61    -0.05     0.26     0.57
    #> beta[weight:.trtclassTNFa blocker]         -0.16    0.00 0.03    -0.23    -0.19    -0.16    -0.14
    #> beta[weight:.trtclassIL-12/23 blocker]     -0.09    0.00 0.05    -0.18    -0.12    -0.09    -0.06
    #> beta[weight:.trtclassIL-17 blocker]        -0.13    0.00 0.03    -0.19    -0.15    -0.13    -0.11
    #> beta[psa:.trtclassTNFa blocker]            -0.05    0.00 0.19    -0.44    -0.18    -0.05     0.08
    #> beta[psa:.trtclassIL-12/23 blocker]         0.12    0.00 0.34    -0.53    -0.11     0.13     0.36
    #> beta[psa:.trtclassIL-17 blocker]            0.10    0.00 0.17    -0.24    -0.02     0.10     0.21
    #> d[ETN vs. PBO]                              1.58    0.00 0.07     1.44     1.53     1.58     1.63
    #> d[IXE_Q2W vs. PBO]                          2.91    0.00 0.07     2.77     2.86     2.91     2.96
    #> d[IXE_Q4W vs. PBO]                          2.69    0.00 0.07     2.55     2.64     2.69     2.74
    #> d[SEC_150 vs. PBO]                          2.19    0.00 0.08     2.03     2.14     2.19     2.25
    #> d[SEC_300 vs. PBO]                          2.60    0.00 0.08     2.43     2.54     2.60     2.66
    #> d[UST vs. IXE_Q2W]                         -0.79    0.00 0.16    -1.10    -0.90    -0.79    -0.68
    #> d[UST vs. SEC_300]                         -0.47    0.00 0.09    -0.65    -0.53    -0.47    -0.40
    #> lp__                                    -7756.43    0.11 4.32 -7765.88 -7759.08 -7756.13 -7753.33
    #> cc[PASI75]                                  0.00     NaN 0.00     0.00     0.00     0.00     0.00
    #> cc[PASI90]                                  0.69    0.00 0.02     0.65     0.67     0.69     0.70
    #> cc[PASI100]                                 1.53    0.00 0.02     1.48     1.52     1.53     1.55
    #>                                            97.5% n_eff Rhat
    #> beta[durnpso]                               0.15  3284    1
    #> beta[prevsys]                               0.12  3119    1
    #> beta[bsa]                                   0.74  3357    1
    #> beta[weight]                                0.10  3195    1
    #> beta[psa]                                   0.25  3296    1
    #> beta[durnpso:.trtclassTNFa blocker]         0.11  3451    1
    #> beta[durnpso:.trtclassIL-12/23 blocker]     0.14  4341    1
    #> beta[durnpso:.trtclassIL-17 blocker]        0.10  3853    1
    #> beta[prevsys:.trtclassTNFa blocker]         0.55  3201    1
    #> beta[prevsys:.trtclassIL-12/23 blocker]     1.08  4743    1
    #> beta[prevsys:.trtclassIL-17 blocker]        0.48  3578    1
    #> beta[bsa:.trtclassTNFa blocker]             1.23  3593    1
    #> beta[bsa:.trtclassIL-12/23 blocker]         1.85  4364    1
    #> beta[bsa:.trtclassIL-17 blocker]            1.20  3941    1
    #> beta[weight:.trtclassTNFa blocker]         -0.09  3555    1
    #> beta[weight:.trtclassIL-12/23 blocker]      0.00  4746    1
    #> beta[weight:.trtclassIL-17 blocker]        -0.07  3758    1
    #> beta[psa:.trtclassTNFa blocker]             0.32  3727    1
    #> beta[psa:.trtclassIL-12/23 blocker]         0.77  4746    1
    #> beta[psa:.trtclassIL-17 blocker]            0.45  3793    1
    #> d[ETN vs. PBO]                              1.73  2605    1
    #> d[IXE_Q2W vs. PBO]                          3.05  2642    1
    #> d[IXE_Q4W vs. PBO]                          2.84  2998    1
    #> d[SEC_150 vs. PBO]                          2.36  2650    1
    #> d[SEC_300 vs. PBO]                          2.77  2861    1
    #> d[UST vs. IXE_Q2W]                         -0.47  5596    1
    #> d[UST vs. SEC_300]                         -0.29  6888    1
    #> lp__                                    -7749.04  1498    1
    #> cc[PASI75]                                  0.00   NaN  NaN
    #> cc[PASI90]                                  0.72  3023    1
    #> cc[PASI100]                                 1.58  3164    1
    #>
    #> Samples were drawn using NUTS(diag_e) at Tue Jan  9 12:47:42 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).

We compare model fit to the FE ML-NMR model using the DIC.

    pso_dic_FE

    #> Residual deviance: 8811.2 (on 12387 data points)
    #>                pD: 36
    #>               DIC: 8847.3

    (pso_dic_UME <- dic(pso_fit_UME))

    #> Residual deviance: 8811.8 (on 12387 data points)
    #>                pD: 36.5
    #>               DIC: 8848.4

The DIC values are similar between the FE model (assuming consistency) and the UME (inconsistency) model, which suggests no evidence for inconsistency overall. It is also important to compare the residual deviance contributions under each model to see whether there are any points that fit better under the UME model, as this can also indicate inconsistency. Using the `plot()` function produces a “dev-dev” plot of the residual deviance contributions under either model.

    plot(pso_dic_FE, pso_dic_UME, show_uncertainty = FALSE) +
      xlab("Residual deviance - consistency model") +
      ylab("Residual deviance - inconsistency (UME) model")

All points lie on the line of equality, so there is no evidence of inconsistency. If random effects models had been fitted then the heterogeneity estimates should also be compared as a drop in `tau` for the UME model can also indicate inconsistency.

### Relaxing the shared effect modifier assumption

The treatment classes in the network are as follows:

    data.frame(classes = pso_net$classes, treatments = pso_net$treatments)
    #>            classes treatments
    #> 1          Placebo        PBO
    #> 2     TNFa blocker        ETN
    #> 3    IL-17 blocker    IXE_Q2W
    #> 4    IL-17 blocker    IXE_Q4W
    #> 5    IL-17 blocker    SEC_150
    #> 6    IL-17 blocker    SEC_300
    #> 7 IL-12/23 blocker        UST

We fitted common interaction terms within each treatment class, under the shared effect modifier assumption, in order to make the model estimable with the available data. Note that only the interleukin-17 blocker class has more than one treatment; etanercept and ustekinumab are in classes of their own and so are unaffected by specifying `class_interactions = "common"`. To assess this assumption we cannot simply fit independent interaction terms for all treatments and all effect modifiers at once as we do not have sufficient data. Instead, we relax this assumption one covariate at a time, estimating independent interactions for one covariate whilst keeping the shared effect modifier assumption (common interactions within each treatment class) for the other covariates.

To specify these relaxed models, we need to somehow mix `class_interactions = "common"` and `class_interactions = "independent"` for different covariates. The way we do this is with the `.trt` and `.trtclass` specials when specifying the regression model. To see how this works, first note that the model making the shared effect modifiers assumption

    regression = ~(durnpso + prevsys + bsa + weight + psa)*.trt,
    class_interactions = "common"

can be written equivalently using the `.trtclass` special as

    regression = ~(durnpso + prevsys + bsa + weight + psa)*.trtclass

The `.trtclass` special is essentially a factor variable containing the treatment classes, and is available any time treatment classes have been specified in the network; this regression formula therefore has a single interaction term for each covariate within each treatment class (the same result as specifying `class_interactions = "common"` above). Finally, to fit independent interactions for a single covariate, say `durnpso`, we split these out using the `.trt` special with `class_interactions = "independent"` (i.e. telling the model not to combine interactions for `.trt` within classes):

    regression = ~(prevsys + bsa + weight + psa)*.trtclass + durnpso*.trt,
    class_interactions = "independent"

Since we are fitting several of these models, let us set up a list of model specifications and iterate over these.

    noSEM_mods <- list(
      durnpso = ~(prevsys + bsa + weight + psa)*.trtclass + durnpso*.trt,
      prevsys = ~(durnpso + bsa + weight + psa)*.trtclass + prevsys*.trt,
      bsa = ~(durnpso + prevsys + weight + psa)*.trtclass + bsa*.trt,
      weight = ~(durnpso + prevsys + bsa + psa)*.trtclass + weight*.trt,
      psa = ~(durnpso + prevsys + bsa + weight)*.trtclass + psa*.trt
      )

    noSEM_fits <- noSEM_mods

    for (m in 1:length(noSEM_mods)) {
      cat("Fitting model with independent interactions for", names(noSEM_mods)[m], "\n")

      noSEM_fits[[m]] <-
        nma(pso_net,
            trt_effects = "fixed",
            link = "probit",
            regression = noSEM_mods[[m]],
            class_interactions = "independent",
            prior_intercept = normal(scale = 10),
            prior_trt = normal(scale = 10),
            prior_reg = normal(scale = 10),
            prior_aux = flat(),
            QR = TRUE,
            init_r = 0.5,
            # Using save_warmup = FALSE reduces memory footprint when
            # fitting many models in one session
            save_warmup = FALSE)
    }

    #> Fitting model with independent interactions for durnpso
    #> Note: Setting "PBO" as the network reference treatment.
    #> Fitting model with independent interactions for prevsys
    #> Note: Setting "PBO" as the network reference treatment.
    #> Fitting model with independent interactions for bsa
    #> Note: Setting "PBO" as the network reference treatment.
    #> Fitting model with independent interactions for weight
    #> Note: Setting "PBO" as the network reference treatment.
    #> Fitting model with independent interactions for psa
    #> Note: Setting "PBO" as the network reference treatment.

Comparing model fit using the DIC

    pso_dic_FE

    #> Residual deviance: 8811.2 (on 12387 data points)
    #>                pD: 36
    #>               DIC: 8847.3

    lapply(noSEM_fits, dic)

    #> $durnpso
    #> Residual deviance: 8812.4 (on 12387 data points)
    #>                pD: 37.7
    #>               DIC: 8850.1
    #>
    #> $prevsys
    #> Residual deviance: 8813 (on 12387 data points)
    #>                pD: 37.6
    #>               DIC: 8850.6
    #>
    #> $bsa
    #> Residual deviance: 8812.8 (on 12387 data points)
    #>                pD: 37.7
    #>               DIC: 8850.6
    #>
    #> $weight
    #> Residual deviance: 8807.3 (on 12387 data points)
    #>                pD: 38
    #>               DIC: 8845.3
    #>
    #> $psa
    #> Residual deviance: 8812 (on 12387 data points)
    #>                pD: 38.5
    #>               DIC: 8850.5

All of the models have similar or higher DIC to the original model making the shared effect modifier assumption for all covariates, with the only exception being the model with independent interactions for `weight` which has slightly lower DIC.

We also visually examine the differences between the estimated interaction terms under the original model (shared effect modifier assumption for all covariates) and the relaxed models (independent interactions, one covariate at a time).

    library(purrr)
    library(stringr)
    library(forcats)

    # Extract draws from relaxed models
    imap_dfr(noSEM_fits,
            ~as_tibble(as.matrix(.x, pars = "beta")) %>%
               pivot_longer(cols = everything(), names_to = "parameter", values_to = "value") %>%
               filter(str_detect(parameter, paste0("(IXE|SEC).+:", .y))) %>%
               mutate(model = .y)) %>%

      # Add in draws from the original model
      bind_rows(
        as_tibble(as.matrix(pso_fit_FE, pars = "beta")) %>%
        pivot_longer(cols = everything(), names_to = "parameter", values_to = "value") %>%
        filter(str_detect(parameter, ":.+IL\\-17 blocker")) %>%
        mutate(model = "all")
      ) %>%

      mutate(
        # Rescale BSA to per 10%
        value = if_else(str_detect(parameter, "bsa"), value / 10, value),
        # Create labels
        covariate = str_extract(parameter, "durnpso|prevsys|bsa|weight|psa"),
        covariatef = recode_factor(covariate,
                                   durnpso = "Duration of psoriasis, per 10 years",
                                   prevsys = "Previous systemic use",
                                   bsa = "Body surface area, per 10%",
                                   weight = "Weight, per 10 kg",
                                   psa = "Psoriatic arthritis"),
        treatment = str_remove(str_extract(parameter, "\\.trt(class)?.+?(?=[\\]:])"),
                               "\\.trt(class)?"),
        Interactions = fct_collapse(factor(model),
                                    Common = "all",
                                    other_level = "Independent")) %>%

    # Plot
    ggplot(aes(x = value, y = fct_rev(treatment), colour = Interactions, fill = Interactions)) +
      geom_vline(xintercept = 0, colour = "grey70") +
      ggdist::stat_halfeye(normalize = "panels", slab_alpha = 0.3, .width = c(0, 0.95)) +
      facet_wrap("covariatef", scales = "free") +
      xlab("Interaction effect (SMD)") +
      ylab("Treatment / Class") +
      scale_colour_manual(values = c(Common = "#7B3294", Independent = "#91D388"),
                          aesthetics = c("colour", "fill")) +
      theme_multinma() +
      theme(legend.position = c(0.85, 0.2))

The independent interaction estimates are very similar to the common interaction estimates, but with much more uncertainty—particularly for the secukinumab regimens which are estimated only from aggregate data. The only exception is for weight, where there is some suggestion that this covariate may interact differently with the secukinumab treatment regimens to the ixekizumab regimens. However, the credible intervals for the secukinumab interactions are wide and overlap those for the ixekizumab regimens and the common interaction. Overall, there is some weak evidence that the shared effect modifier assumption (for the class of interleukin-17 blockers) may be invalid for weight. Since we are fitting multiple models here we should be mindful of multiple testing and the possibility that such differences have occurred by chance. On the other hand, this approach is likely to have low power to detect violations of the shared effect modifier assumption, particularly when the data are lacking. In this case, results from the model relaxing the shared effect modifier assumption for weight are very similar to the original model (see Phillippo et al. 2022).

## Producing relative effects and event probabilities

### Study populations included in the network

Population-average treatment effects can be produced for all the study populations represented in the network using the `relative_effects()` function.

    (pso_releff_FE <- relative_effects(pso_fit_FE))

    #> ------------------------------------------------------------------ Study: CLEAR ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.73    0.66 0.32   8.74 0.16
    #>
    #>                   mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[CLEAR: ETN]     1.62 0.08 1.47 1.57 1.62 1.67  1.77     2191     2895    1
    #> d[CLEAR: IXE_Q2W] 2.94 0.08 2.78 2.89 2.94 2.99  3.10     2745     2706    1
    #> d[CLEAR: IXE_Q4W] 2.72 0.08 2.57 2.67 2.72 2.78  2.88     2975     2682    1
    #> d[CLEAR: SEC_150] 2.22 0.09 2.05 2.16 2.22 2.28  2.39     2365     2528    1
    #> d[CLEAR: SEC_300] 2.63 0.08 2.46 2.57 2.63 2.68  2.80     2411     2959    1
    #> d[CLEAR: UST]     2.17 0.11 1.95 2.09 2.16 2.24  2.38     3207     2921    1
    #>
    #> ---------------------------------------------------------------- Study: ERASURE ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.69    0.61 0.32   8.86 0.23
    #>
    #>                     mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[ERASURE: ETN]     1.59 0.07 1.44 1.54 1.59 1.64  1.73     2244     2968    1
    #> d[ERASURE: IXE_Q2W] 2.92 0.08 2.77 2.87 2.92 2.97  3.08     2777     2821    1
    #> d[ERASURE: IXE_Q4W] 2.70 0.08 2.56 2.65 2.70 2.75  2.86     3019     2884    1
    #> d[ERASURE: SEC_150] 2.20 0.08 2.04 2.14 2.20 2.26  2.37     2411     2749    1
    #> d[ERASURE: SEC_300] 2.61 0.08 2.45 2.55 2.61 2.66  2.77     2508     2891    1
    #> d[ERASURE: UST]     2.14 0.12 1.91 2.06 2.14 2.22  2.37     3493     3081    1
    #>
    #> ---------------------------------------------------------------- Study: FEATURE ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>      1.9    0.67 0.32   9.17 0.15
    #>
    #>                     mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[FEATURE: ETN]     1.55 0.07 1.41 1.50 1.55 1.60  1.70     2253     3017    1
    #> d[FEATURE: IXE_Q2W] 2.88 0.08 2.73 2.83 2.88 2.93  3.04     2881     2641    1
    #> d[FEATURE: IXE_Q4W] 2.66 0.08 2.51 2.61 2.66 2.71  2.82     3121     3016    1
    #> d[FEATURE: SEC_150] 2.16 0.08 2.00 2.10 2.16 2.22  2.33     2421     2566    1
    #> d[FEATURE: SEC_300] 2.57 0.08 2.41 2.51 2.57 2.62  2.73     2530     2830    1
    #> d[FEATURE: UST]     2.12 0.11 1.90 2.05 2.12 2.19  2.34     3432     3215    1
    #>
    #> ---------------------------------------------------------------- Study: FIXTURE ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>      1.6    0.62 0.34   8.34 0.14
    #>
    #>                     mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[FIXTURE: ETN]     1.69 0.08 1.53 1.63 1.68 1.74  1.85     2382     2730    1
    #> d[FIXTURE: IXE_Q2W] 2.99 0.09 2.82 2.93 2.99 3.05  3.17     2834     2794    1
    #> d[FIXTURE: IXE_Q4W] 2.77 0.09 2.61 2.72 2.77 2.83  2.95     3045     2875    1
    #> d[FIXTURE: SEC_150] 2.27 0.09 2.10 2.21 2.27 2.33  2.45     2406     2615    1
    #> d[FIXTURE: SEC_300] 2.68 0.09 2.51 2.62 2.68 2.74  2.86     2408     2992    1
    #> d[FIXTURE: UST]     2.20 0.12 1.97 2.12 2.20 2.28  2.43     3246     3097    1
    #>
    #> ---------------------------------------------------------------- Study: IXORA-S ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.67    0.92 0.32   8.78 0.13
    #>
    #>                     mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[IXORA-S: ETN]     1.67 0.10 1.48 1.60 1.67 1.73  1.86     2140     2711    1
    #> d[IXORA-S: IXE_Q2W] 2.98 0.10 2.79 2.91 2.97 3.04  3.17     2546     2835    1
    #> d[IXORA-S: IXE_Q4W] 2.76 0.10 2.57 2.69 2.76 2.82  2.96     2763     2864    1
    #> d[IXORA-S: SEC_150] 2.25 0.10 2.05 2.18 2.25 2.33  2.46     2373     3009    1
    #> d[IXORA-S: SEC_300] 2.66 0.10 2.47 2.59 2.66 2.73  2.87     2418     3124    1
    #> d[IXORA-S: UST]     2.28 0.13 2.04 2.19 2.27 2.36  2.53     2946     2530    1
    #>
    #> --------------------------------------------------------------- Study: JUNCTURE ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.99    0.55 0.27   9.17 0.23
    #>
    #>                      mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[JUNCTURE: ETN]     1.51 0.07 1.37 1.46 1.51 1.56  1.65     2419     3021    1
    #> d[JUNCTURE: IXE_Q2W] 2.85 0.07 2.71 2.80 2.85 2.90  3.00     3004     2766    1
    #> d[JUNCTURE: IXE_Q4W] 2.63 0.07 2.49 2.59 2.63 2.68  2.78     3295     3225    1
    #> d[JUNCTURE: SEC_150] 2.13 0.08 1.97 2.07 2.13 2.18  2.30     2559     2584    1
    #> d[JUNCTURE: SEC_300] 2.54 0.08 2.38 2.48 2.54 2.59  2.70     2737     2431    1
    #> d[JUNCTURE: UST]     2.04 0.13 1.78 1.94 2.03 2.12  2.31     3874     3205    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-1 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>        2    0.73 0.28   9.24 0.28
    #>
    #>                       mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[UNCOVER-1: ETN]     1.53 0.08 1.38 1.48 1.53 1.58  1.68     2019     2694    1
    #> d[UNCOVER-1: IXE_Q2W] 2.88 0.08 2.74 2.83 2.88 2.93  3.03     2590     2963    1
    #> d[UNCOVER-1: IXE_Q4W] 2.66 0.08 2.51 2.61 2.66 2.71  2.81     2771     2664    1
    #> d[UNCOVER-1: SEC_150] 2.16 0.09 1.99 2.10 2.16 2.22  2.33     2325     2735    1
    #> d[UNCOVER-1: SEC_300] 2.57 0.09 2.40 2.51 2.57 2.63  2.74     2452     2753    1
    #> d[UNCOVER-1: UST]     2.12 0.12 1.88 2.04 2.12 2.20  2.36     3351     3318    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-2 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.87    0.64 0.27   9.17 0.24
    #>
    #>                       mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[UNCOVER-2: ETN]     1.53 0.07 1.39 1.48 1.53 1.58  1.67     2151     2749    1
    #> d[UNCOVER-2: IXE_Q2W] 2.87 0.07 2.73 2.82 2.87 2.92  3.01     2708     2688    1
    #> d[UNCOVER-2: IXE_Q4W] 2.65 0.07 2.51 2.60 2.65 2.70  2.79     2977     2739    1
    #> d[UNCOVER-2: SEC_150] 2.15 0.08 1.99 2.09 2.15 2.20  2.31     2451     2480    1
    #> d[UNCOVER-2: SEC_300] 2.56 0.08 2.40 2.50 2.56 2.61  2.72     2566     2563    1
    #> d[UNCOVER-2: UST]     2.08 0.12 1.85 2.00 2.08 2.17  2.33     3490     3316    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-3 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight psa
    #>     1.78    0.59 0.28   9.01 0.2
    #>
    #>                       mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[UNCOVER-3: ETN]     1.55 0.07 1.41 1.50 1.55 1.60  1.69     2315     2954    1
    #> d[UNCOVER-3: IXE_Q2W] 2.88 0.07 2.74 2.83 2.88 2.93  3.03     2846     2790    1
    #> d[UNCOVER-3: IXE_Q4W] 2.67 0.07 2.52 2.62 2.67 2.72  2.81     3132     2866    1
    #> d[UNCOVER-3: SEC_150] 2.16 0.08 2.01 2.11 2.16 2.22  2.32     2481     2445    1
    #> d[UNCOVER-3: SEC_300] 2.57 0.08 2.41 2.52 2.57 2.62  2.73     2603     2581    1
    #> d[UNCOVER-3: UST]     2.08 0.12 1.85 2.00 2.08 2.16  2.33     3588     3148    1

These relative effects can then be plotted using the `plot()` function.

    plot(pso_releff_FE, ref_line = 0)

Similarly, average response probabilities on each treatment, in each study population, at each PASI cutoff can be produced using the `predict()` function. We specify `type = "response"` to produce predicted probabilities (rather than probit-probabilities).

    (pso_pred_FE <- predict(pso_fit_FE, type = "response"))

    #> ------------------------------------------------------------------ Study: CLEAR ----
    #>
    #>                               mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[CLEAR: PBO, PASI75]      0.09 0.02 0.07 0.08 0.09 0.10  0.13     3226     2898    1
    #> pred[CLEAR: PBO, PASI90]      0.02 0.01 0.01 0.02 0.02 0.03  0.04     3516     3005    1
    #> pred[CLEAR: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     3935     2940    1
    #> pred[CLEAR: ETN, PASI75]      0.61 0.03 0.54 0.58 0.61 0.63  0.67     6487     2944    1
    #> pred[CLEAR: ETN, PASI90]      0.35 0.03 0.29 0.32 0.35 0.37  0.41     6826     2581    1
    #> pred[CLEAR: ETN, PASI100]     0.11 0.02 0.08 0.10 0.11 0.12  0.15     7281     2823    1
    #> pred[CLEAR: IXE_Q2W, PASI75]  0.94 0.01 0.92 0.93 0.94 0.95  0.96     5425     3102    1
    #> pred[CLEAR: IXE_Q2W, PASI90]  0.81 0.02 0.76 0.80 0.81 0.83  0.86     5686     3102    1
    #> pred[CLEAR: IXE_Q2W, PASI100] 0.52 0.04 0.45 0.50 0.52 0.55  0.60     6050     3334    1
    #> pred[CLEAR: IXE_Q4W, PASI75]  0.91 0.02 0.88 0.90 0.91 0.92  0.94     5650     3038    1
    #> pred[CLEAR: IXE_Q4W, PASI90]  0.75 0.03 0.69 0.73 0.75 0.77  0.81     5883     2991    1
    #> pred[CLEAR: IXE_Q4W, PASI100] 0.44 0.04 0.37 0.41 0.44 0.46  0.51     6236     2898    1
    #> pred[CLEAR: SEC_150, PASI75]  0.80 0.02 0.76 0.79 0.80 0.82  0.85     6974     3030    1
    #> pred[CLEAR: SEC_150, PASI90]  0.57 0.03 0.51 0.55 0.57 0.59  0.63     7522     2931    1
    #> pred[CLEAR: SEC_150, PASI100] 0.26 0.03 0.21 0.24 0.26 0.28  0.31     7795     3102    1
    #> pred[CLEAR: SEC_300, PASI75]  0.90 0.01 0.87 0.89 0.90 0.90  0.92     7070     3340    1
    #> pred[CLEAR: SEC_300, PASI90]  0.72 0.02 0.68 0.71 0.72 0.73  0.76     8084     3407    1
    #> pred[CLEAR: SEC_300, PASI100] 0.40 0.02 0.36 0.39 0.40 0.42  0.45     9002     3557    1
    #> pred[CLEAR: UST, PASI75]      0.78 0.02 0.75 0.77 0.78 0.80  0.82     6525     3460    1
    #> pred[CLEAR: UST, PASI90]      0.55 0.02 0.51 0.54 0.55 0.57  0.59     7217     2959    1
    #> pred[CLEAR: UST, PASI100]     0.25 0.02 0.21 0.23 0.25 0.26  0.28     6991     3323    1
    #>
    #> ---------------------------------------------------------------- Study: ERASURE ----
    #>
    #>                                 mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[ERASURE: PBO, PASI75]      0.05 0.01 0.03 0.04 0.05 0.05  0.06     2649     2638    1
    #> pred[ERASURE: PBO, PASI90]      0.01 0.00 0.01 0.01 0.01 0.01  0.01     2816     2731    1
    #> pred[ERASURE: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     3172     2552    1
    #> pred[ERASURE: ETN, PASI75]      0.45 0.03 0.40 0.43 0.45 0.47  0.51     5025     2755    1
    #> pred[ERASURE: ETN, PASI90]      0.22 0.02 0.18 0.20 0.22 0.23  0.27     4758     3003    1
    #> pred[ERASURE: ETN, PASI100]     0.06 0.01 0.04 0.05 0.06 0.06  0.08     5151     3063    1
    #> pred[ERASURE: IXE_Q2W, PASI75]  0.88 0.02 0.85 0.87 0.88 0.89  0.91     4393     3184    1
    #> pred[ERASURE: IXE_Q2W, PASI90]  0.70 0.03 0.64 0.68 0.70 0.72  0.75     4403     3212    1
    #> pred[ERASURE: IXE_Q2W, PASI100] 0.38 0.03 0.32 0.36 0.38 0.40  0.44     4520     3034    1
    #> pred[ERASURE: IXE_Q4W, PASI75]  0.83 0.02 0.79 0.82 0.84 0.85  0.87     4583     3113    1
    #> pred[ERASURE: IXE_Q4W, PASI90]  0.62 0.03 0.55 0.60 0.62 0.64  0.68     4622     3175    1
    #> pred[ERASURE: IXE_Q4W, PASI100] 0.30 0.03 0.24 0.28 0.30 0.32  0.36     4734     3180    1
    #> pred[ERASURE: SEC_150, PASI75]  0.68 0.02 0.65 0.67 0.69 0.70  0.72     6563     2962    1
    #> pred[ERASURE: SEC_150, PASI90]  0.42 0.02 0.38 0.41 0.42 0.44  0.47     7123     3203    1
    #> pred[ERASURE: SEC_150, PASI100] 0.15 0.01 0.13 0.14 0.15 0.16  0.18     6746     3046    1
    #> pred[ERASURE: SEC_300, PASI75]  0.81 0.02 0.78 0.80 0.81 0.82  0.84     5243     2986    1
    #> pred[ERASURE: SEC_300, PASI90]  0.58 0.02 0.54 0.57 0.58 0.60  0.62     5377     2999    1
    #> pred[ERASURE: SEC_300, PASI100] 0.27 0.02 0.23 0.25 0.27 0.28  0.31     5602     2968    1
    #> pred[ERASURE: UST, PASI75]      0.66 0.04 0.58 0.63 0.66 0.68  0.73     7153     2959    1
    #> pred[ERASURE: UST, PASI90]      0.40 0.04 0.33 0.38 0.40 0.43  0.48     6931     3035    1
    #> pred[ERASURE: UST, PASI100]     0.15 0.02 0.11 0.13 0.15 0.16  0.20     6275     3460    1
    #>
    #> ---------------------------------------------------------------- Study: FEATURE ----
    #>
    #>                                 mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[FEATURE: PBO, PASI75]      0.05 0.01 0.03 0.04 0.05 0.06  0.08     3528     3111    1
    #> pred[FEATURE: PBO, PASI90]      0.01 0.00 0.01 0.01 0.01 0.01  0.02     3573     2996    1
    #> pred[FEATURE: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     3744     3288    1
    #> pred[FEATURE: ETN, PASI75]      0.45 0.04 0.37 0.42 0.45 0.48  0.54     4758     2641    1
    #> pred[FEATURE: ETN, PASI90]      0.22 0.03 0.16 0.20 0.22 0.24  0.29     4716     2394    1
    #> pred[FEATURE: ETN, PASI100]     0.06 0.01 0.04 0.05 0.06 0.06  0.08     4908     2778    1
    #> pred[FEATURE: IXE_Q2W, PASI75]  0.88 0.02 0.83 0.86 0.88 0.90  0.92     4349     2947    1
    #> pred[FEATURE: IXE_Q2W, PASI90]  0.69 0.04 0.61 0.67 0.69 0.72  0.77     4378     2871    1
    #> pred[FEATURE: IXE_Q2W, PASI100] 0.38 0.04 0.29 0.34 0.37 0.41  0.47     4556     2931    1
    #> pred[FEATURE: IXE_Q4W, PASI75]  0.83 0.03 0.77 0.81 0.83 0.85  0.89     4688     3086    1
    #> pred[FEATURE: IXE_Q4W, PASI90]  0.62 0.05 0.53 0.59 0.62 0.65  0.70     4720     3044    1
    #> pred[FEATURE: IXE_Q4W, PASI100] 0.30 0.04 0.22 0.27 0.30 0.33  0.38     4944     3268    1
    #> pred[FEATURE: SEC_150, PASI75]  0.68 0.03 0.61 0.66 0.68 0.71  0.75     5407     3219    1
    #> pred[FEATURE: SEC_150, PASI90]  0.42 0.04 0.35 0.40 0.42 0.45  0.50     5530     3094    1
    #> pred[FEATURE: SEC_150, PASI100] 0.15 0.02 0.11 0.14 0.15 0.17  0.20     5714     3379    1
    #> pred[FEATURE: SEC_300, PASI75]  0.81 0.03 0.75 0.79 0.81 0.83  0.86     5144     2976    1
    #> pred[FEATURE: SEC_300, PASI90]  0.58 0.04 0.51 0.55 0.58 0.61  0.65     5230     2987    1
    #> pred[FEATURE: SEC_300, PASI100] 0.27 0.03 0.21 0.24 0.27 0.29  0.33     5420     3102    1
    #> pred[FEATURE: UST, PASI75]      0.66 0.04 0.58 0.63 0.66 0.70  0.75     6147     3316    1
    #> pred[FEATURE: UST, PASI90]      0.41 0.05 0.32 0.38 0.41 0.44  0.50     5953     3395    1
    #> pred[FEATURE: UST, PASI100]     0.15 0.03 0.10 0.13 0.15 0.17  0.21     5912     3341    1
    #>
    #> ---------------------------------------------------------------- Study: FIXTURE ----
    #>
    #>                                 mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[FIXTURE: PBO, PASI75]      0.04 0.01 0.03 0.03 0.04 0.04  0.05     2405     2526    1
    #> pred[FIXTURE: PBO, PASI90]      0.01 0.00 0.00 0.01 0.01 0.01  0.01     2573     2779    1
    #> pred[FIXTURE: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     2939     2784    1
    #> pred[FIXTURE: ETN, PASI75]      0.44 0.02 0.40 0.43 0.44 0.46  0.49     5779     2950    1
    #> pred[FIXTURE: ETN, PASI90]      0.21 0.02 0.18 0.20 0.21 0.22  0.24     5838     3063    1
    #> pred[FIXTURE: ETN, PASI100]     0.05 0.01 0.04 0.05 0.05 0.06  0.07     6526     3466    1
    #> pred[FIXTURE: IXE_Q2W, PASI75]  0.87 0.02 0.84 0.86 0.87 0.88  0.90     4706     3270    1
    #> pred[FIXTURE: IXE_Q2W, PASI90]  0.68 0.03 0.63 0.66 0.68 0.70  0.73     4756     3306    1
    #> pred[FIXTURE: IXE_Q2W, PASI100] 0.36 0.03 0.31 0.34 0.36 0.37  0.41     5019     3429    1
    #> pred[FIXTURE: IXE_Q4W, PASI75]  0.82 0.02 0.78 0.81 0.82 0.84  0.86     4960     3445    1
    #> pred[FIXTURE: IXE_Q4W, PASI90]  0.60 0.03 0.54 0.58 0.60 0.62  0.65     5125     3367    1
    #> pred[FIXTURE: IXE_Q4W, PASI100] 0.28 0.03 0.23 0.26 0.28 0.30  0.33     5495     3586    1
    #> pred[FIXTURE: SEC_150, PASI75]  0.67 0.02 0.63 0.65 0.67 0.68  0.70     6709     2896    1
    #> pred[FIXTURE: SEC_150, PASI90]  0.40 0.02 0.36 0.39 0.40 0.42  0.44     7792     2801    1
    #> pred[FIXTURE: SEC_150, PASI100] 0.14 0.01 0.12 0.13 0.14 0.15  0.16     7645     2984    1
    #> pred[FIXTURE: SEC_300, PASI75]  0.80 0.01 0.77 0.79 0.80 0.81  0.82     7048     3072    1
    #> pred[FIXTURE: SEC_300, PASI90]  0.56 0.02 0.52 0.55 0.56 0.58  0.60     8080     2971    1
    #> pred[FIXTURE: SEC_300, PASI100] 0.25 0.02 0.22 0.24 0.25 0.26  0.28     8438     2901    1
    #> pred[FIXTURE: UST, PASI75]      0.64 0.03 0.57 0.61 0.64 0.66  0.70     8702     3157    1
    #> pred[FIXTURE: UST, PASI90]      0.38 0.03 0.31 0.36 0.38 0.40  0.45     8574     3069    1
    #> pred[FIXTURE: UST, PASI100]     0.13 0.02 0.10 0.12 0.13 0.14  0.17     7713     3267    1
    #>
    #> ---------------------------------------------------------------- Study: IXORA-S ----
    #>
    #>                                 mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[IXORA-S: PBO, PASI75]      0.05 0.01 0.03 0.04 0.05 0.05  0.07     3061     3007    1
    #> pred[IXORA-S: PBO, PASI90]      0.01 0.00 0.00 0.01 0.01 0.01  0.02     3191     2842    1
    #> pred[IXORA-S: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     3332     3061    1
    #> pred[IXORA-S: ETN, PASI75]      0.48 0.04 0.41 0.46 0.48 0.51  0.56     5617     3325    1
    #> pred[IXORA-S: ETN, PASI90]      0.24 0.03 0.19 0.22 0.24 0.26  0.30     5603     3080    1
    #> pred[IXORA-S: ETN, PASI100]     0.06 0.01 0.04 0.06 0.06 0.07  0.09     5527     3041    1
    #> pred[IXORA-S: IXE_Q2W, PASI75]  0.89 0.02 0.86 0.88 0.89 0.90  0.92     5451     3254    1
    #> pred[IXORA-S: IXE_Q2W, PASI90]  0.71 0.03 0.65 0.69 0.71 0.73  0.77     5597     3197    1
    #> pred[IXORA-S: IXE_Q2W, PASI100] 0.40 0.03 0.33 0.37 0.40 0.42  0.46     5615     3013    1
    #> pred[IXORA-S: IXE_Q4W, PASI75]  0.85 0.02 0.80 0.83 0.85 0.86  0.89     5613     2964    1
    #> pred[IXORA-S: IXE_Q4W, PASI90]  0.64 0.04 0.57 0.61 0.64 0.66  0.70     5772     2967    1
    #> pred[IXORA-S: IXE_Q4W, PASI100] 0.32 0.03 0.25 0.29 0.32 0.34  0.39     5831     2973    1
    #> pred[IXORA-S: SEC_150, PASI75]  0.70 0.04 0.63 0.68 0.70 0.73  0.77     4994     3394    1
    #> pred[IXORA-S: SEC_150, PASI90]  0.44 0.04 0.36 0.42 0.44 0.47  0.53     5162     3501    1
    #> pred[IXORA-S: SEC_150, PASI100] 0.17 0.03 0.12 0.15 0.16 0.18  0.22     5044     3639    1
    #> pred[IXORA-S: SEC_300, PASI75]  0.82 0.03 0.77 0.81 0.82 0.84  0.87     5587     3690    1
    #> pred[IXORA-S: SEC_300, PASI90]  0.60 0.04 0.53 0.58 0.60 0.63  0.67     5773     3365    1
    #> pred[IXORA-S: SEC_300, PASI100] 0.28 0.03 0.22 0.26 0.28 0.31  0.35     5646     3690    1
    #> pred[IXORA-S: UST, PASI75]      0.71 0.03 0.65 0.69 0.71 0.73  0.76     5639     2992    1
    #> pred[IXORA-S: UST, PASI90]      0.45 0.03 0.39 0.43 0.45 0.47  0.51     5732     2939    1
    #> pred[IXORA-S: UST, PASI100]     0.17 0.02 0.13 0.16 0.17 0.19  0.22     5739     3114    1
    #>
    #> --------------------------------------------------------------- Study: JUNCTURE ----
    #>
    #>                                  mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[JUNCTURE: PBO, PASI75]      0.06 0.01 0.03 0.05 0.05 0.06  0.08     4273     3268    1
    #> pred[JUNCTURE: PBO, PASI90]      0.01 0.00 0.01 0.01 0.01 0.01  0.02     4356     3124    1
    #> pred[JUNCTURE: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     4530     3119    1
    #> pred[JUNCTURE: ETN, PASI75]      0.45 0.04 0.37 0.42 0.45 0.48  0.54     5721     3097    1
    #> pred[JUNCTURE: ETN, PASI90]      0.22 0.03 0.16 0.20 0.22 0.24  0.29     5719     3291    1
    #> pred[JUNCTURE: ETN, PASI100]     0.06 0.01 0.03 0.05 0.06 0.06  0.09     5795     3200    1
    #> pred[JUNCTURE: IXE_Q2W, PASI75]  0.88 0.02 0.83 0.87 0.88 0.90  0.92     5095     3287    1
    #> pred[JUNCTURE: IXE_Q2W, PASI90]  0.70 0.04 0.61 0.67 0.70 0.72  0.77     5150     3371    1
    #> pred[JUNCTURE: IXE_Q2W, PASI100] 0.38 0.04 0.29 0.35 0.38 0.41  0.47     5350     3448    1
    #> pred[JUNCTURE: IXE_Q4W, PASI75]  0.83 0.03 0.77 0.81 0.84 0.85  0.88     5341     3326    1
    #> pred[JUNCTURE: IXE_Q4W, PASI90]  0.62 0.04 0.53 0.59 0.62 0.65  0.70     5399     3365    1
    #> pred[JUNCTURE: IXE_Q4W, PASI100] 0.30 0.04 0.22 0.27 0.30 0.33  0.39     5618     3134    1
    #> pred[JUNCTURE: SEC_150, PASI75]  0.68 0.03 0.61 0.66 0.68 0.71  0.75     5890     3193    1
    #> pred[JUNCTURE: SEC_150, PASI90]  0.42 0.04 0.35 0.40 0.42 0.45  0.50     6104     3324    1
    #> pred[JUNCTURE: SEC_150, PASI100] 0.16 0.02 0.11 0.14 0.15 0.17  0.20     6257     3386    1
    #> pred[JUNCTURE: SEC_300, PASI75]  0.81 0.03 0.75 0.79 0.81 0.83  0.86     6201     3114    1
    #> pred[JUNCTURE: SEC_300, PASI90]  0.58 0.04 0.51 0.56 0.58 0.61  0.65     6395     3145    1
    #> pred[JUNCTURE: SEC_300, PASI100] 0.27 0.03 0.21 0.25 0.27 0.29  0.33     6578     2844    1
    #> pred[JUNCTURE: UST, PASI75]      0.65 0.05 0.54 0.61 0.65 0.68  0.75     6638     3078    1
    #> pred[JUNCTURE: UST, PASI90]      0.39 0.05 0.29 0.36 0.39 0.43  0.50     6572     3031    1
    #> pred[JUNCTURE: UST, PASI100]     0.14 0.03 0.09 0.12 0.14 0.16  0.21     6285     3384    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-1 ----
    #>
    #>                                   mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[UNCOVER-1: PBO, PASI75]      0.06 0.01 0.04 0.05 0.06 0.06  0.07     2932     2882    1
    #> pred[UNCOVER-1: PBO, PASI90]      0.01 0.00 0.01 0.01 0.01 0.01  0.02     3181     2654    1
    #> pred[UNCOVER-1: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     3470     2886    1
    #> pred[UNCOVER-1: ETN, PASI75]      0.47 0.02 0.42 0.45 0.47 0.48  0.51     4589     2900    1
    #> pred[UNCOVER-1: ETN, PASI90]      0.23 0.02 0.20 0.22 0.23 0.24  0.26     4645     3166    1
    #> pred[UNCOVER-1: ETN, PASI100]     0.06 0.01 0.05 0.06 0.06 0.06  0.08     4571     2859    1
    #> pred[UNCOVER-1: IXE_Q2W, PASI75]  0.89 0.01 0.87 0.88 0.89 0.90  0.91     6322     3068    1
    #> pred[UNCOVER-1: IXE_Q2W, PASI90]  0.71 0.01 0.68 0.70 0.71 0.72  0.74     7204     3180    1
    #> pred[UNCOVER-1: IXE_Q2W, PASI100] 0.39 0.02 0.36 0.38 0.39 0.40  0.43     7586     3358    1
    #> pred[UNCOVER-1: IXE_Q4W, PASI75]  0.84 0.01 0.82 0.84 0.84 0.85  0.86     6326     3169    1
    #> pred[UNCOVER-1: IXE_Q4W, PASI90]  0.63 0.02 0.60 0.62 0.63 0.65  0.66     7231     3067    1
    #> pred[UNCOVER-1: IXE_Q4W, PASI100] 0.31 0.02 0.28 0.30 0.31 0.33  0.35     7674     3322    1
    #> pred[UNCOVER-1: SEC_150, PASI75]  0.70 0.03 0.64 0.68 0.70 0.72  0.75     4573     2962    1
    #> pred[UNCOVER-1: SEC_150, PASI90]  0.44 0.03 0.37 0.42 0.44 0.46  0.51     4631     3137    1
    #> pred[UNCOVER-1: SEC_150, PASI100] 0.17 0.02 0.13 0.15 0.16 0.18  0.21     4574     2811    1
    #> pred[UNCOVER-1: SEC_300, PASI75]  0.82 0.02 0.78 0.81 0.82 0.84  0.86     4833     3200    1
    #> pred[UNCOVER-1: SEC_300, PASI90]  0.60 0.03 0.54 0.58 0.60 0.62  0.66     4991     3389    1
    #> pred[UNCOVER-1: SEC_300, PASI100] 0.28 0.03 0.23 0.26 0.28 0.30  0.34     4873     3424    1
    #> pred[UNCOVER-1: UST, PASI75]      0.68 0.04 0.60 0.66 0.68 0.71  0.76     5260     3088    1
    #> pred[UNCOVER-1: UST, PASI90]      0.43 0.04 0.35 0.40 0.43 0.46  0.51     5378     3504    1
    #> pred[UNCOVER-1: UST, PASI100]     0.16 0.03 0.12 0.14 0.16 0.18  0.22     5219     3385    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-2 ----
    #>
    #>                                   mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[UNCOVER-2: PBO, PASI75]      0.05 0.01 0.04 0.05 0.05 0.06  0.06     3122     2522    1
    #> pred[UNCOVER-2: PBO, PASI90]      0.01 0.00 0.01 0.01 0.01 0.01  0.01     3356     2912    1
    #> pred[UNCOVER-2: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     3626     3266    1
    #> pred[UNCOVER-2: ETN, PASI75]      0.44 0.02 0.40 0.43 0.44 0.45  0.48     4685     3191    1
    #> pred[UNCOVER-2: ETN, PASI90]      0.21 0.01 0.18 0.20 0.21 0.22  0.24     5049     3053    1
    #> pred[UNCOVER-2: ETN, PASI100]     0.05 0.01 0.04 0.05 0.05 0.06  0.06     5048     3316    1
    #> pred[UNCOVER-2: IXE_Q2W, PASI75]  0.88 0.01 0.86 0.87 0.88 0.88  0.90     5035     2716    1
    #> pred[UNCOVER-2: IXE_Q2W, PASI90]  0.69 0.02 0.66 0.68 0.69 0.70  0.72     5739     2934    1
    #> pred[UNCOVER-2: IXE_Q2W, PASI100] 0.37 0.02 0.34 0.36 0.37 0.38  0.40     6147     3060    1
    #> pred[UNCOVER-2: IXE_Q4W, PASI75]  0.83 0.01 0.81 0.82 0.83 0.84  0.85     5474     2962    1
    #> pred[UNCOVER-2: IXE_Q4W, PASI90]  0.61 0.02 0.58 0.60 0.61 0.62  0.64     6371     3437    1
    #> pred[UNCOVER-2: IXE_Q4W, PASI100] 0.29 0.02 0.26 0.28 0.29 0.30  0.32     6973     3554    1
    #> pred[UNCOVER-2: SEC_150, PASI75]  0.68 0.03 0.61 0.66 0.68 0.70  0.73     5016     3385    1
    #> pred[UNCOVER-2: SEC_150, PASI90]  0.41 0.03 0.35 0.39 0.41 0.44  0.48     5125     3421    1
    #> pred[UNCOVER-2: SEC_150, PASI100] 0.15 0.02 0.11 0.13 0.15 0.16  0.19     5019     3539    1
    #> pred[UNCOVER-2: SEC_300, PASI75]  0.80 0.02 0.76 0.79 0.81 0.82  0.84     5077     3048    1
    #> pred[UNCOVER-2: SEC_300, PASI90]  0.57 0.03 0.51 0.55 0.57 0.60  0.63     5322     2906    1
    #> pred[UNCOVER-2: SEC_300, PASI100] 0.26 0.03 0.21 0.24 0.26 0.28  0.31     5359     3285    1
    #> pred[UNCOVER-2: UST, PASI75]      0.65 0.04 0.56 0.62 0.65 0.68  0.73     5584     3209    1
    #> pred[UNCOVER-2: UST, PASI90]      0.39 0.04 0.31 0.36 0.39 0.42  0.48     5564     3136    1
    #> pred[UNCOVER-2: UST, PASI100]     0.14 0.02 0.10 0.12 0.14 0.16  0.19     5238     3227    1
    #>
    #> -------------------------------------------------------------- Study: UNCOVER-3 ----
    #>
    #>                                   mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[UNCOVER-3: PBO, PASI75]      0.07 0.01 0.05 0.06 0.07 0.07  0.09     3408     2896    1
    #> pred[UNCOVER-3: PBO, PASI90]      0.02 0.00 0.01 0.01 0.01 0.02  0.02     3699     2795    1
    #> pred[UNCOVER-3: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     4199     3282    1
    #> pred[UNCOVER-3: ETN, PASI75]      0.51 0.02 0.47 0.49 0.51 0.52  0.54     5141     3015    1
    #> pred[UNCOVER-3: ETN, PASI90]      0.26 0.01 0.23 0.25 0.26 0.27  0.29     5478     3035    1
    #> pred[UNCOVER-3: ETN, PASI100]     0.07 0.01 0.06 0.07 0.07 0.08  0.09     5733     3307    1
    #> pred[UNCOVER-3: IXE_Q2W, PASI75]  0.91 0.01 0.89 0.90 0.91 0.91  0.92     4495     3217    1
    #> pred[UNCOVER-3: IXE_Q2W, PASI90]  0.74 0.01 0.71 0.73 0.74 0.75  0.77     5272     2943    1
    #> pred[UNCOVER-3: IXE_Q2W, PASI100] 0.43 0.02 0.40 0.42 0.43 0.44  0.46     5853     2938    1
    #> pred[UNCOVER-3: IXE_Q4W, PASI75]  0.87 0.01 0.85 0.86 0.87 0.87  0.88     5666     3558    1
    #> pred[UNCOVER-3: IXE_Q4W, PASI90]  0.67 0.02 0.64 0.66 0.67 0.68  0.70     6671     3608    1
    #> pred[UNCOVER-3: IXE_Q4W, PASI100] 0.35 0.02 0.32 0.34 0.35 0.36  0.38     8120     3670    1
    #> pred[UNCOVER-3: SEC_150, PASI75]  0.73 0.03 0.67 0.71 0.73 0.75  0.78     5136     3029    1
    #> pred[UNCOVER-3: SEC_150, PASI90]  0.48 0.03 0.41 0.46 0.48 0.50  0.54     5367     2968    1
    #> pred[UNCOVER-3: SEC_150, PASI100] 0.19 0.02 0.15 0.17 0.19 0.20  0.23     5276     2947    1
    #> pred[UNCOVER-3: SEC_300, PASI75]  0.84 0.02 0.80 0.83 0.84 0.86  0.88     5481     3182    1
    #> pred[UNCOVER-3: SEC_300, PASI90]  0.63 0.03 0.57 0.61 0.63 0.65  0.69     5782     2938    1
    #> pred[UNCOVER-3: SEC_300, PASI100] 0.31 0.03 0.26 0.29 0.31 0.33  0.37     5694     3154    1
    #> pred[UNCOVER-3: UST, PASI75]      0.70 0.04 0.62 0.67 0.70 0.73  0.77     5694     3534    1
    #> pred[UNCOVER-3: UST, PASI90]      0.45 0.04 0.37 0.42 0.45 0.48  0.53     5692     3345    1
    #> pred[UNCOVER-3: UST, PASI100]     0.17 0.03 0.13 0.16 0.17 0.19  0.23     5399     3632    1

Again, these can be plotted using the `plot()` function.

    plot(pso_pred_FE, ref_line = c(0, 1))

### External target populations

For the purposes of decision-making it is crucial that population-average estimates are produced for the decision target population of interest. The decision target population may not be represented by any of the study populations in the network, indeed it is likely best represented by an external registry or cohort study, or perhaps expert knowledge (Phillippo et al. 2016).

As an example, Phillippo et al. (2022) produce estimates for three external target populations represented by the PsoBest registry (Reich et al. 2015; Augustin et al. 2014), and the PROSPECT (Thaçi et al. 2019) and Chiricozzi 2019 (Chiricozzi et al. 2019) cohort studies. First of all, we need the covariate means and standard deviations in each of these populations:

    new_agd_means <- tibble::tribble(
                 ~study, ~covariate,  ~mean,   ~sd,
              "PsoBest",      "bsa",     24,  20.5,
              "PsoBest",  "durnpso",   18.2,  14.1,
              "PsoBest",  "prevsys",   0.54,    NA,
              "PsoBest",      "psa",  0.207,    NA,
              "PsoBest",   "weight",     85,  19.1,
             "PROSPECT",      "bsa",   18.7,  18.4,
             "PROSPECT",  "durnpso",   19.6,  13.5,
             "PROSPECT",  "prevsys", 0.9095,    NA,
             "PROSPECT",      "psa",  0.202,    NA,
             "PROSPECT",   "weight",   87.5,  20.3,
      "Chiricozzi 2019",      "bsa",     23, 16.79,
      "Chiricozzi 2019",  "durnpso",  16.93, 10.82,
      "Chiricozzi 2019",  "prevsys", 0.9061,    NA,
      "Chiricozzi 2019",      "psa", 0.2152,    NA,
      "Chiricozzi 2019",   "weight",   78.3, 15.87
      ) %>%
      # Tidy up
      pivot_wider(id_cols = study,
                  names_from = covariate,
                  values_from = c(mean, sd),
                  names_glue = "{covariate}_{.value}") %>%
      # Rescale as per analysis
      transmute(study,
                bsa_mean = bsa_mean / 100,
                bsa_sd = bsa_sd / 100,
                weight_mean = weight_mean / 10,
                weight_sd = weight_sd / 10,
                durnpso_mean = durnpso_mean / 10,
                durnpso_sd = durnpso_sd / 10,
                prevsys = prevsys_mean,
                psa = psa_mean)

To produce estimates of population-average treatment effects, we use the `relative_effects()` function with the data frame of covariate means in the target populations as the `newdata` argument. We only need the covariate means, with variable names matching those in the regression.

    (pso_releff_FE_new <- relative_effects(pso_fit_FE,
                                           newdata = transmute(new_agd_means,
                                                               study,
                                                               bsa = bsa_mean,
                                                               weight = weight_mean,
                                                               durnpso = durnpso_mean,
                                                               prevsys,
                                                               psa),
                                           study = study))

    #> -------------------------------------------------------- Study: Chiricozzi 2019 ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.69    0.91 0.23   7.83 0.22
    #>
    #>                             mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[Chiricozzi 2019: ETN]     1.79 0.11 1.59 1.72 1.79 1.86  2.01     2269     2766    1
    #> d[Chiricozzi 2019: IXE_Q2W] 3.08 0.10 2.88 3.01 3.08 3.15  3.28     2520     2614    1
    #> d[Chiricozzi 2019: IXE_Q4W] 2.86 0.10 2.65 2.79 2.86 2.93  3.07     2693     2604    1
    #> d[Chiricozzi 2019: SEC_150] 2.36 0.11 2.14 2.28 2.36 2.43  2.58     2435     2679    1
    #> d[Chiricozzi 2019: SEC_300] 2.77 0.11 2.55 2.69 2.77 2.84  2.99     2479     2858    1
    #> d[Chiricozzi 2019: UST]     2.31 0.15 2.02 2.21 2.31 2.41  2.60     3003     2920    1
    #>
    #> --------------------------------------------------------------- Study: PROSPECT ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight psa
    #>     1.96    0.91 0.19   8.75 0.2
    #>
    #>                      mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[PROSPECT: ETN]     1.63 0.11 1.42 1.56 1.63 1.70  1.84     2256     2690    1
    #> d[PROSPECT: IXE_Q2W] 2.94 0.10 2.74 2.87 2.94 3.01  3.14     2650     2950    1
    #> d[PROSPECT: IXE_Q4W] 2.72 0.10 2.53 2.66 2.72 2.79  2.92     2829     2750    1
    #> d[PROSPECT: SEC_150] 2.22 0.11 2.00 2.14 2.22 2.29  2.44     2590     3014    1
    #> d[PROSPECT: SEC_300] 2.63 0.11 2.41 2.55 2.63 2.70  2.85     2679     2654    1
    #> d[PROSPECT: UST]     2.18 0.15 1.90 2.08 2.18 2.29  2.48     3181     3182    1
    #>
    #> ---------------------------------------------------------------- Study: PsoBest ----
    #>
    #> Covariate values:
    #>  durnpso prevsys  bsa weight  psa
    #>     1.82    0.54 0.24    8.5 0.21
    #>
    #>                     mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> d[PsoBest: ETN]     1.61 0.08 1.46 1.56 1.61 1.66  1.77     2455     2879    1
    #> d[PsoBest: IXE_Q2W] 2.93 0.08 2.78 2.88 2.93 2.98  3.09     2824     2943    1
    #> d[PsoBest: IXE_Q4W] 2.71 0.08 2.56 2.66 2.71 2.77  2.87     3100     2982    1
    #> d[PsoBest: SEC_150] 2.21 0.09 2.04 2.15 2.21 2.27  2.38     2533     2357    1
    #> d[PsoBest: SEC_300] 2.62 0.09 2.45 2.56 2.62 2.67  2.79     2647     2672    1
    #> d[PsoBest: UST]     2.08 0.14 1.82 1.99 2.08 2.17  2.35     3616     3150    1

These estimates are plotted using the `plot()` function.

    plot(pso_releff_FE_new, ref_line = 0) + facet_wrap("Study")

Estimates of average event probabilities are produced by integrating predictions over the joint covariate distribution in each population. Since we have marginal summary statistics available, rather than full IPD, we create integration points using the `add_integration()` function by specifying the forms of the marginal distributions and the correlation matrix. We choose to use the same forms of the marginal distributions that we used when specifying integration points for the AgD studies in the network, and the weighted correlation matrix from the IPD studies.

    new_agd_int <- add_integration(filter(new_agd_means, study != "PsoBest"),
                                   durnpso = distr(qgamma, mean = durnpso_mean, sd = durnpso_sd),
                                   prevsys = distr(qbern, prob = prevsys),
                                   bsa = distr(qlogitnorm, mean = bsa_mean, sd = bsa_sd),
                                   weight = distr(qgamma, mean = weight_mean, sd = weight_sd),
                                   psa = distr(qbern, prob = psa),
                                   n_int = 64,
                                   cor = pso_net$int_cor)

We then use the `predict()` function to produce the average event probabilities (`type = "response"`, and `level = "aggregate"` which is the default) in each of the target populations. To do so, we also need to specify a distribution for the baseline event probabilities (i.e. probability of achieving PASI 75 response) in each of the target populations. PASI 75 event counts for individuals receiving secukinumab 300 mg treatment were available from PROSPECT (1156 achieved PASI 75 out of 1509) and Chiricozzi 2019 (243 out of 330), which we use to construct beta distributions on the baseline average response probabilities (we specify `baseline_level = "aggregate"` as these are population averages, rather than specific to a reference individual, and `baseline_type = "response"` as these are probabilities rather than transformed probit probabilities). No information on baseline response was available from PsoBest, so no predictions of absolute response rates could be made.

    (pso_pred_FE_new <- predict(pso_fit_FE,
            type = "response",
            newdata = new_agd_int,
            study = study,
            baseline = list(PROSPECT = distr(qbeta, 1156, 1509-1156),
                            "Chiricozzi 2019" = distr(qbeta, 243, 330-243)),
            baseline_type = "response",
            baseline_level = "aggregate",
            baseline_trt = "SEC_300"))

    #> -------------------------------------------------------- Study: Chiricozzi 2019 ----
    #>
    #>                                         mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[Chiricozzi 2019: PBO, PASI75]      0.02 0.01 0.01 0.01 0.02 0.02  0.03     2774     2898    1
    #> pred[Chiricozzi 2019: PBO, PASI90]      0.00 0.00 0.00 0.00 0.00 0.00  0.01     2838     3324    1
    #> pred[Chiricozzi 2019: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     2893     3086    1
    #> pred[Chiricozzi 2019: ETN, PASI75]      0.37 0.04 0.29 0.35 0.37 0.40  0.46     4735     3515    1
    #> pred[Chiricozzi 2019: ETN, PASI90]      0.16 0.03 0.11 0.14 0.16 0.18  0.22     4671     3820    1
    #> pred[Chiricozzi 2019: ETN, PASI100]     0.03 0.01 0.02 0.03 0.03 0.04  0.05     4874     3910    1
    #> pred[Chiricozzi 2019: IXE_Q2W, PASI75]  0.83 0.03 0.77 0.81 0.83 0.85  0.88     3937     4088    1
    #> pred[Chiricozzi 2019: IXE_Q2W, PASI90]  0.60 0.04 0.52 0.57 0.60 0.63  0.69     3975     3974    1
    #> pred[Chiricozzi 2019: IXE_Q2W, PASI100] 0.28 0.04 0.21 0.26 0.28 0.31  0.36     4029     3977    1
    #> pred[Chiricozzi 2019: IXE_Q4W, PASI75]  0.77 0.03 0.69 0.74 0.77 0.79  0.83     4139     3729    1
    #> pred[Chiricozzi 2019: IXE_Q4W, PASI90]  0.52 0.05 0.43 0.49 0.52 0.55  0.61     4171     4113    1
    #> pred[Chiricozzi 2019: IXE_Q4W, PASI100] 0.22 0.03 0.16 0.19 0.22 0.24  0.29     4273     4046    1
    #> pred[Chiricozzi 2019: SEC_150, PASI75]  0.59 0.04 0.51 0.57 0.59 0.62  0.66     4475     4038    1
    #> pred[Chiricozzi 2019: SEC_150, PASI90]  0.33 0.04 0.26 0.30 0.33 0.35  0.40     4604     4025    1
    #> pred[Chiricozzi 2019: SEC_150, PASI100] 0.10 0.02 0.07 0.09 0.10 0.11  0.14     4509     3701    1
    #> pred[Chiricozzi 2019: SEC_300, PASI75]  0.74 0.02 0.68 0.72 0.74 0.75  0.78     3565     3798    1
    #> pred[Chiricozzi 2019: SEC_300, PASI90]  0.48 0.03 0.42 0.46 0.48 0.50  0.54     3838     3636    1
    #> pred[Chiricozzi 2019: SEC_300, PASI100] 0.19 0.02 0.15 0.17 0.19 0.20  0.23     3745     3395    1
    #> pred[Chiricozzi 2019: UST, PASI75]      0.57 0.05 0.47 0.54 0.57 0.61  0.68     4619     3313    1
    #> pred[Chiricozzi 2019: UST, PASI90]      0.32 0.05 0.23 0.28 0.32 0.35  0.42     4616     3415    1
    #> pred[Chiricozzi 2019: UST, PASI100]     0.10 0.02 0.06 0.08 0.10 0.11  0.15     4635     3362    1
    #>
    #> --------------------------------------------------------------- Study: PROSPECT ----
    #>
    #>                                  mean   sd 2.5%  25%  50%  75% 97.5% Bulk_ESS Tail_ESS Rhat
    #> pred[PROSPECT: PBO, PASI75]      0.03 0.01 0.02 0.03 0.03 0.04  0.05     2934     2914    1
    #> pred[PROSPECT: PBO, PASI90]      0.01 0.00 0.00 0.00 0.01 0.01  0.01     2952     2943    1
    #> pred[PROSPECT: PBO, PASI100]     0.00 0.00 0.00 0.00 0.00 0.00  0.00     3041     3053    1
    #> pred[PROSPECT: ETN, PASI75]      0.40 0.04 0.33 0.38 0.40 0.43  0.47     5801     3330    1
    #> pred[PROSPECT: ETN, PASI90]      0.18 0.02 0.14 0.16 0.18 0.20  0.23     5572     3383    1
    #> pred[PROSPECT: ETN, PASI100]     0.04 0.01 0.03 0.04 0.04 0.05  0.06     5504     3323    1
    #> pred[PROSPECT: IXE_Q2W, PASI75]  0.85 0.02 0.81 0.84 0.85 0.86  0.89     5040     3313    1
    #> pred[PROSPECT: IXE_Q2W, PASI90]  0.64 0.03 0.57 0.62 0.64 0.66  0.70     5008     3529    1
    #> pred[PROSPECT: IXE_Q2W, PASI100] 0.32 0.03 0.26 0.30 0.32 0.34  0.38     5095     3461    1
    #> pred[PROSPECT: IXE_Q4W, PASI75]  0.79 0.03 0.74 0.78 0.79 0.81  0.84     5273     3203    1
    #> pred[PROSPECT: IXE_Q4W, PASI90]  0.56 0.04 0.49 0.53 0.56 0.58  0.63     5255     3293    1
    #> pred[PROSPECT: IXE_Q4W, PASI100] 0.25 0.03 0.19 0.23 0.24 0.27  0.31     5121     3195    1
    #> pred[PROSPECT: SEC_150, PASI75]  0.63 0.03 0.57 0.61 0.63 0.64  0.68     6002     3692    1
    #> pred[PROSPECT: SEC_150, PASI90]  0.36 0.03 0.31 0.34 0.36 0.38  0.42     5902     3841    1
    #> pred[PROSPECT: SEC_150, PASI100] 0.12 0.01 0.09 0.11 0.12 0.13  0.15     5461     3642    1
    #> pred[PROSPECT: SEC_300, PASI75]  0.77 0.01 0.74 0.76 0.77 0.77  0.79     3985     3918    1
    #> pred[PROSPECT: SEC_300, PASI90]  0.52 0.02 0.49 0.51 0.52 0.53  0.55     4033     3889    1
    #> pred[PROSPECT: SEC_300, PASI100] 0.22 0.01 0.19 0.21 0.22 0.22  0.24     3920     3878    1
    #> pred[PROSPECT: UST, PASI75]      0.61 0.05 0.52 0.58 0.62 0.65  0.70     5757     3516    1
    #> pred[PROSPECT: UST, PASI90]      0.36 0.05 0.27 0.32 0.35 0.39  0.45     5644     3496    1
    #> pred[PROSPECT: UST, PASI100]     0.12 0.02 0.07 0.10 0.12 0.13  0.17     5586     3549    1

Again, we then plot these estimates using the `plot()` function, here with some customisation using ggplot syntax.

    plot(pso_pred_FE_new, ref_line = c(0, 1)) +
      facet_grid(rows = "Study") +
      aes(colour = Category) +
      scale_colour_brewer(palette = "Blues")

# References

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1. The convergence rate of QMC is typically \(\mathcal{O}(1/n)\), whereas the expected convergence rate of standard MC is \(\mathcal{O}(1/n^\frac{1}{2})\) (Caflisch 1998; Niederreiter 1978).↩︎

2. The alternative is “exclusive” format, where the lowest category counts those not achieving any higher outcomes (i.e. failure to achieve PASI 75, \(<75\%\) reduction in symptoms), the second counts those achieving PASI 75 but not PASI 90 or 100 (\(\ge 75\%\) and \(<90\%\) reduction), the third counts those achieving PASI 90 but not PASI 100 (\(\ge 90\%\) and \(<100\%\) reduction), and the final category counts those achieving PASI 100 (\(100\%\) reduction).↩︎