Author: Hugo Pedder
Date: 2025-02-06

# MBNMAdose: Perform a Model-Based Network Meta-Analysis (MBNMA)

#### Hugo Pedder

#### 2025-02-06

# Analysis using `mbnma.run()`

MBNMA is performed in `MBNMAdose` by applying `mbnma.run()`. A `"mbnma.network"` object must be provided as the data for `mbnma.run()`. The key arguments within `mbnma.run()` involve specifying the functional form used to model the dose-response, and the dose-response parameters that comprise that functional form.

## Dose-response functions

Various functional forms are implemented within `MBNMAdose`, that allow a variety of parameterizations and dose-response shapes. These are provided as an object of class `"dosefun"` to the `fun` argument in `mbnma.run()`. The interpretation of the dose-response parameter estimates will therefore depend on the dose-response function used. In previous versions of `MBNMAdose` (prior to version 0.4.0), wrapper functions were provided for each of the commonly used dose-response functions in `mbnma.run()`. For example, `mbnma.emax()` is equivalent to `mbnma.run(fun=demax())`. This will be deprecated in future versions.

For the following functions \(x_{i,k}\) refers to the dose and \(t_{i,k}\) to the agent in arm \(k\) of study \(i\).

Log-linear (`dloglin()`) \[f(x_{i,k}, t_{i,k})=\lambda_{t_{i,k}} \times ln(x_{i,k} + 1)\] where \(lambda\) controls the gradient of the dose-response relationship.

Exponential (`dexp()`) \[f(x_{i,k}, t_{i,k})=Emax_{t_{i,k}} (1 - e^{-x_{i,k}})\] where \(Emax\) is the maximum efficacy of an agent.

Emax (`demax()`) \[f(x_{i,k}, t_{i,k})=\dfrac{Emax_{t_{i,k}} \times {x_{i,k}^{\gamma_{t_{i,k}}}}} {ED50_{t_{i,k}}^{\gamma_{t_{i,k}}} + x_{i,k}^{\gamma_{t_{i,k}}}}\] where \(Emax\) is the maximum efficacy that can be achieved, \(ED50\) is the dose at which 50% of the maximum efficacy is achieved, and \(\gamma\) is the Hill parameter that controls the sigmoidal shape of the function. By default, `demax()` fits a 2-parameter Emax function in which \(\gamma_{t_{i,k}}=1\) (`hill=NULL` in function argument).

Polynomial (e.g. linear) (`dpoly()`) \[f(x_{i,k}, t_{i,k})=\beta_{1_{t_{i,k}}}x_{i,k}+...+\beta_{p_{t_{i,k}}}x^p_{i,k}\] where \(p\) is the degree of the polynomial (e.g. 1 for linear) and \(\beta_p\) are the coefficients

Fractional polynomial (`dfpoly()`) \[f(x_{i,k}, t_{i,k})=\beta_{1_{t_{i,k}}}x_{i,k}^{\gamma_1}+...+\beta_{p_{t_{i,k}}}x^{\gamma_p}_{i,k}\] where \(p\) is the degree of the polynomial, \(\beta_p\) are the coefficients, and \(x_{i,k}^{\gamma_p}\) is a regular power except where \(\gamma_p=0\) where \({x_{i,k}^{(0)}=ln(x_{i,k})}\). If a fractional polynomial power \({\gamma_p}\) repeats within the function it is multiplied by another \({ln(x_{i,k})}\).

Spline functions (`dspline()`) B-splines (`type="bs"`), natural cubic splines (`type="ns"`) and piecewise linear splines (`type="ls"`) can be fitted. \[f(x_{i,k}, t_{i,k})=\sum_{p=1}^{P} \beta_{p,t_{i,k}} X_{p,i,k}\] where \(\beta_{p,t_{i,k}}\) is the regression coefficient for the \(p^{th}\) spline and \(X_{1:P,i,k}\) is the basis matrix for the spline, defined by the spline type.

Non-parametric monotonic function (`dnonparam()`) Follows the approach of Owen et al. (2015). The `direction` can bespecified as `"increasing"` or `"decreasing"`.

User-defined function (`duser()`) Any function that can be explicitly defined by the user within (see User-defined dose-response function)

Agent-specific functions (`dmulti()`) Allows for a separate dose-response function to be fitted to each agent in the network (see Agent-specific dose-response functions)

## Dose-response parameters

Dose-response parameters can be specified in different ways which affects the key parameters estimated by the model and implies different modelling assumptions. Three different specifications are available for each parameter:

- `"rel"` indicates that relative effects should be pooled for this dose-response parameter separately for each agent in the network. This preserves randomisation within included studies and is likely to vary less between studies (only due to effect modification).
- `"common"` indicates that a single absolute value for this dose-response parameter should be estimated across the whole network that does not vary by agent. This is particularly useful for parameters expected to be constant (e.g. Hill parameters in `demax()` or fractional polynomial power parameters in `fpoly()`).
- `"random"` indicates that a single absolute value should be estimated separately for each agent, but that all the agent values vary randomly around a single mean absolute network effect. It is similar to `"common"` but makes slightly less strong assumptions.
- `numeric()` Assigned a numeric value - this is similar to assigning `"common"`, but the single absolute value is assigned as a numeric value by the user, rather than estimated from the data.

In `mbnma.run()`, an additional argument, `method`, indicates what method to use for pooling relative effects and can take either the values `"common"`, implying that all studies estimate the same true effect (akin to a “fixed effect” meta-analysis), or `"random"`, implying that all studies estimate a separate true effect, but that each of these true effects vary randomly around a true mean effect. This approach allows for modelling of between-study heterogeneity.

## Output

`mbnma.run()` returns an object of class `c("rjags", "mbnma")`. `summary()` provides posterior medians and 95% credible intervals (95%CrI) for different parameters in the model, naming them by agent and giving some explanation of the way they have been specified in the model. `print()` can also be used to give full summary statistics of the posterior distributions for monitored nodes in the JAGS model. Estimates are automatically reported for parameters of interest depending on the model specification (unless otherwise specified in `parameters.to.save`).

Dose-response parameters will be named in the output, with a separate coefficient for each agent (if specified as `"rel"`). If class effects are modelled, parameters for classes are represented by the upper case name of the dose-response parameter they correspond to (e.g. `EMAX` will be the class effects on `emax`). The SD of the class effect (e.g. `sd.EMAX`, `sd.BETA.1`) is the SD of agents within a class for the dose-response parameter they correspond to.

`sd` corresponds to the between-study SD. However, `sd.` followed by a dose-response parameter name (e.g. `sd.emax`, `sd.beta.1`) is the between-agent SD for dose response parameters modeled using `"common"` or `"random"`.

`totresdev` is the residual deviance, and `deviance` the deviance of the model. Model fit statistics for `pD` (effective number of parameters) and `DIC` (Deviance Information Criterion) are also reported, with an explanation as to how they have been calculated.

## Examples

An example MBNMA of the triptans dataset using an Emax dose-response function and common treatment effects that pool relative effects for each agent separately on both Emax and ED50 parameters follows:

    # Prepare data using the triptans dataset
    tripnet <- mbnma.network(triptans)
    #> Values for `agent` with dose = 0 have been recoded to `Placebo`
    #> agent is being recoded to enforce sequential numbering

    # Run an Emax dose-response MBNMA
    mbnma <- mbnma.run(tripnet, fun = demax(emax = "rel", ed50 = "rel"), method = "random")
    #> `likelihood` not given by user - set to `binomial` based on data provided
    #> `link` not given by user - set to `logit` based on assigned value for `likelihood`

    # Print neat summary of output
    summary(mbnma)
    #> ========================================
    #> Dose-response MBNMA
    #> ========================================
    #>
    #> Likelihood: binomial
    #> Link function: logit
    #> Dose-response function: emax
    #>
    #> Pooling method
    #>
    #> Method: Random effects estimated for relative effects
    #>
    #> Parameter                    Median (95%CrI)
    #> -----------------------------------------------------------------------
    #> Between-study SD for relative effects        0.322 (0.232, 0.436)
    #>
    #> emax dose-response parameter results
    #>
    #> Pooling: relative effects for each agent
    #>
    #> |Agent        |Parameter |  Median|   2.5%|    97.5%|
    #> |:------------|:---------|-------:|------:|--------:|
    #> |eletriptan   |emax[2]   |  2.7748| 2.1261|   3.8787|
    #> |sumatriptan  |emax[3]   |  2.5217| 1.5432| 119.0245|
    #> |frovatriptan |emax[4]   | 74.2806| 5.5723| 185.1107|
    #> |almotriptan  |emax[5]   | 62.5136| 5.7424| 157.0455|
    #> |zolmitriptan |emax[6]   |  2.7553| 1.6047|   4.7243|
    #> |naratriptan  |emax[7]   | 46.6316| 7.4808| 124.1031|
    #> |rizatriptan  |emax[8]   | 90.7985| 4.6071| 221.3746|
    #>
    #>
    #> ed50 dose-response parameter results
    #>
    #> Pooling: relative effects for each agent
    #>
    #> |Agent        |Parameter |   Median|    2.5%|    97.5%|
    #> |:------------|:---------|--------:|-------:|--------:|
    #> |eletriptan   |ed50[2]   |   0.6507|  0.2638|   1.4121|
    #> |sumatriptan  |ed50[3]   |   1.5195|  0.3528| 160.6921|
    #> |frovatriptan |ed50[4]   |  78.0740|  3.7579| 190.3716|
    #> |almotriptan  |ed50[5]   |  76.2541|  5.0546| 189.6912|
    #> |zolmitriptan |ed50[6]   |   1.3661|  0.3292|   3.7776|
    #> |naratriptan  |ed50[7]   | 103.2924| 20.7800| 239.4765|
    #> |rizatriptan  |ed50[8]   |  56.0331|  1.8517| 142.3849|
    #>
    #>
    #> Model Fit Statistics
    #> Deviance = 1088.6
    #> Residual deviance = 185.5
    #> Deviance Information Criterion (DIC) = 1268.6

In this example the `emax` parameters are the maximum efficacy that can be achieved for each agent. The `ed50` parameters are the the dose at which 50% of the maximum efficacy is achieved for each agent. Results for ED50 are given on the log scale as it is constrained to be greater than zero. `sd` corresponds to the between-study SD (included because `method="random"`).

Instead of estimating a separate relative effect for each agent, a simpler dose-response model that makes stronger assumptions could estimate a single parameter across the whole network for ED50, but still estimate a separate effect for each agent for Emax:

    # Emax model with single parameter estimated for Emax
    emax <- mbnma.run(tripnet, fun = demax(emax = "rel", ed50 = "common"), method = "random")
    #> `likelihood` not given by user - set to `binomial` based on data provided
    #> `link` not given by user - set to `logit` based on assigned value for `likelihood`

    summary(emax)
    #> Warning in rhat.warning(object): The following parameters have Rhat values > 1.2
    #> which could be due to convergence issues:
    #> ed50
    #> emax[2]
    #> emax[3]
    #> emax[4]
    #> emax[5]
    #> emax[6]
    #> emax[7]
    #> emax[8]
    #> ========================================
    #> Dose-response MBNMA
    #> ========================================
    #>
    #> Likelihood: binomial
    #> Link function: logit
    #> Dose-response function: emax
    #>
    #> Pooling method
    #>
    #> Method: Random effects estimated for relative effects
    #>
    #> Parameter                    Median (95%CrI)
    #> -----------------------------------------------------------------------
    #> Between-study SD for relative effects        0.244 (0.159, 0.543)
    #>
    #> emax dose-response parameter results
    #>
    #> Pooling: relative effects for each agent
    #>
    #> |Agent        |Parameter | Median|   2.5%|    97.5%|
    #> |:------------|:---------|------:|------:|--------:|
    #> |eletriptan   |emax[2]   | 2.8924| 2.3821|  76.9274|
    #> |sumatriptan  |emax[3]   | 1.9316| 1.6332|  52.2024|
    #> |frovatriptan |emax[4]   | 2.1119| 1.4141|  73.9666|
    #> |almotriptan  |emax[5]   | 1.8224| 1.3995|  54.4346|
    #> |zolmitriptan |emax[6]   | 2.1589| 1.7359|  32.3098|
    #> |naratriptan  |emax[7]   | 1.0675| 0.4349|  28.4114|
    #> |rizatriptan  |emax[8]   | 2.7651| 2.2230| 112.4109|
    #>
    #>
    #> ed50 dose-response parameter results
    #>
    #> Pooling: single parameter across all agents in the network
    #>
    #> |Parameter | Median|   2.5%|   97.5%|
    #> |:---------|------:|------:|-------:|
    #> |ed50      |  0.727| 0.4474| 71.7133|
    #>
    #>
    #> Model Fit Statistics
    #> Deviance = 1093.3
    #> Residual deviance = 190.2
    #> Deviance Information Criterion (DIC) = 1277.2

In this example `ed50` only has a single parameter, which corresponds to the dose at which 50% of the maximum efficacy is achieved, assumed to be equal across all agents in the network.

## Parameter interpretation

Parameter interpretation depends both on the link scale on which the outcome is modeled, and on the dose-response function used in the model.

For example for a binomial outcome modeled using a logit link function and an Emax dose-response function, the `emax` parameter represents the maximum efficacy on the logit scale - it is modeled on the outcome scale and hence is dependent on the link function. As indicated in the help file (`?demax()`), the `ed50` parameter is modeled on the log-scale to ensure that it takes positive values on the natural scale, but is not modeled on the outcome scale so is not dependent on the link function. Therefore it can be interpreted as the log-dose at which 50% of the maximum efficacy is achieved. The `hill` parameter is also modeled on the log-scale, and it can be interpreted as a log-value that controls the sigmoidicity of the dose-response function for the outcome on the logit scale.

For a continuous outcome modeled using `link="smd"`, whilst not a true link function it implies modelling Standardised Mean Differences (SMD). For a linear dose-response function (`dpoly(degree=1)`), `beta.1` represents the change in SMD for each additional unit of dose. For a quadratic dose-response function (`dpoly(degree=2)`), `beta.2` represents the change in `beta.1` for each additional unit of dose.

With some dose-response functions (e.g. splines, fractional polynomials) parameter interpretation can be challenging. The `get.relative()` function can make this easier as this allows relative effects to be estimated between agents at specific doses, which is typically much more easily understandable and presentable (see [Estimating relative effects]).

## Additional arguments for `mbnma.run()`

Several additional arguments can be given to `mbnma.run()` that require further explanation.

### Link functions for `likelihood=="normal"`

For data with normal likelihood, users are likely to typically analyse data using an identity link function, the default given to the `link` argument in `mbnma.run()` when `likelihood="normal"`. This assumes an additive treatment effect (e.g. mean difference).

However, by specifying `link="log"` a user can model a log link and therefore assume a multiplicative treatment effect. For continuous data this models the treatment effect as a Ratio of Means (RoM) (Friedrich, Adhikari, and Beyene 2011). This also provides an advantage as the treatment effect is scale independent (i.e. studies measuring the same outcome using different measurement scales can be analysed simultaneously). However, within-study treatment effects must all be of the same direction (either positive or negative), and change from baseline measures must be adjusted so that they are also expressed as RoMs (log(follow-up) - log(baseline)) to avoid combining additive and multiplicative assumptions within the same analysis.

An alternative approach for modelling a measurement scale-independent treatment effect whilst still assuming additive treatment effects is to perform the analysis using Standardised Mean Differences (SMD). Whilst not strictly a different link function, this can be specified using `link="smd"`. The resulting treatment effects are reported in units of SD as SMDs.

By default, `MBNMAdose` standardises treatment effects using the pooled standard deviation (SD) in each study (“study-specific” SDs). However, a more robust approach to minimise bias from estimation of study-specific SD is to use a “reference” SD that is the same for each scale included in the dataset. Such a reference could be estimated from all the included studies within the dataset that report outcomes on the scale (an “internal reference SD”), but the best approach is to obtain a SD from an observational study that is specific to the target population of interest.

Once this has been obtained for each scale, it can be included in the original data frame as an additional variable, `standsd`. For the analysis to be meaningful, `stansd` should be the same for all arms within a study and for all studies that report the same outcome. If this has been included in the dataset used to create an `"mbnma.network"` object, then `sdscale=TRUE` argument can be specified in `mbnma.run()` to indicate that this variable should be used to standardise SMDs.

Note that for outputting results from a model using SMDs, results should be back-transformed to a clinically relevant scale to make robust interpretation. This can be done by multiplying the effect estimates or predictions by the standard deviation of the relevant scale in the population of interest. For further details of analysis of continuous data that include discussion of both RoM and SMD see (Daly et al. 2021).

### Class effects

Similar effects between agents within the network can be modelled using class effects. This requires assuming that different agents have some sort of common class effect, perhaps due to similar mechanisms of action. Advantages of this is that class effects can be used to connect agents that might otherwise be disconnected from the network, and they can also provide additional information on agents that might otherwise have insufficient data available to estimate a desired dose-response. The drawback is that this requires making additional assumptions regarding similarity of efficacy.

One difficult of this modelling aspect in particular is that the scales for dose-response parameters must be the same across different agents within a class for this assumption to be valid. For example, in an Emax model it may be reasonable to assume a class effect on the Emax parameter, as this is parameterised on the response scale and so could be similar across agents of the same class. However, the ED50 parameter is parameterised on the dose scale, which is likely to differ for each agent and so an assumption of similarity between agents for this parameter may be less valid. One way to try to account for this issue and make dose scales more consistent across agents is to standardise doses for each agent relative to its “common” dose (see Thorlund et al. (2014)), though we expect that this may lead to bias if the common dose is located at a different point along the dose-response curve.

Class effects can only be applied to dose-response parameters which vary by agent. In `mbnma.run()` they are supplied as a list, in which each element is named following the name of the corresponding dose-response parameter as defined in the dose-response function. The names will therefore differ when using wrapper functions for `mbnma.run()`. The class effect for each dose-response parameter can be either `"common"`, in which the effects for each agent within the same class are constrained to a common class effect, or `"random"`, in which the effects for each agent within the same class are assumed to be randomly distributed around a shared class mean.

When working with class effects in `MBNMAdose` a variable named `class` must be included in the original data frame provided to `mbnma.network()`. Below we assign a class for two similar agents in the dataset - Naproxcinod and Naproxen. We will estimate separate effects for all other agents, so we set their classes to be equal to their agents.

    # Using the osteoarthritis dataset
    pain.df <- osteopain

    # Set class equal to agent for all agents
    pain.df$class <- pain.df$class

    # Set a shared class (NSAID) only for Naproxcinod and Naproxen
    pain.df$class[pain.df$agent %in% c("Naproxcinod", "Naproxen")] <- "NSAID"

    # Run a restricted cubic spline MBNMA with a random class effect on beta.1
    painnet <- mbnma.network(pain.df)
    splines <- mbnma.run(painnet, fun = dspline(type = "bs", knots = 2), class.effect = list(beta.1 = "random"))

Mean class effects are given in the output as `D.ed50`/`D.1` parameters. These can be interpreted as the effect of each class for Emax parameters (`beta.1`). Note the number of `D.ed50` parameters is therefore equal to the number of classes defined in the dataset.

If we had specified that the class effects were `"random"`, each treatment effect for Emax (`beta.1`) would be assumed to be randomly distributed around its class mean with SD given in the output as `sd.D.ed50`/`sd.D.1`.

Mean class effects are represented in the output by the upper case name of the dose-response parameter they correspond to. In this case, `BETA.1` is the class effects on `beta.1`, the first spline coefficient. The SD of the class effect is the SD of agents within a class for the dose-response parameter they correspond to. In this case `sd.BETA.1` is the within-class SD for `beta.1`.

### User-defined dose-response function

Users can define their own dose-response function using `duser()` rather than using one of the functions provided in `MBNMAdose`. The dose-response is specified in terms of `beta` parameters and `dose`. This allows a huge degree of flexibility when defining the dose-response relationship.

The function assigned needs to fulfil a few criteria to be valid: * `dose` must always be included in the function * At least one `beta` dose-response parameter must be specified, up to a maximum of four. These must always be named `beta.1`, `beta.2`, `beta.3` and `beta.4`, and must be included sequentially (i.e. don’t include `beta.3` if `beta.2` is not included) * Indices used by JAGS should not be added (e.g. use `dose` rather than `dose[i,k]`) * Any mathematical/logical operators that can be implemented in JAGS can be added to the function (e.g. `exp()`, `ifelse()`). See the JAGS manual (2017) for further details.

    # Using the depression SSRI dataset
    depnet <- mbnma.network(ssri)

    # An example specifying a quadratic dose-response function
    quadfun <- ~(beta.1 * dose) + (beta.2 * (dose^2))

    quad <- mbnma.run(depnet, fun = duser(fun = quadfun, beta.1 = "rel", beta.2 = "rel"))

### Agent-specific dose-response functions

Different dose-response functions can be used for different agents within the network. This allows for the modelling of more complex dose-response functions in agents for which there are many doses available, and less complex functions in agents for which there are fewer doses available. Note that these models are typically less computationally stable than single dose-response function models, and they are likely to benefit less from modelling correlation between multiple dose-response parameters since there are fewer agents informing correlations between each dose-response parameter.

This can be modeled using the `dmulti()` dose-response function and assigning a list of objects of class `"dosefun"` to it. Each element in the list corresponds to an agent in the network (the order of which should be the same as the order of agents in the `"mbnma.network"` object). A dose-response function for Placebo should be included in the list, though which function is used is typically irrelevant since evaluating the function at dose=0 will typically equal 0.

    # Using the depression SSRI dataset
    depnet <- mbnma.network(ssri)

    dr.funs <- dmulti(list(Placebo = dfpoly(degree = 2), citalopram = dfpoly(degree = 2),
        escitalopram = dfpoly(degree = 2), fluoxetine = dspline(type = "ns", knots = 2),
        paroxetine = dfpoly(degree = 2), sertraline = dspline(type = "ns", knots = 2)))

    multifun <- mbnma.run(depnet, fun = dr.funs, method = "common", n.iter = 50000)
    summary(multifun)

### Incorporating agents/interventions without a dose-response

Because an MBNMA model with a linear dose-response function (`dpoly(degree=1)`) is mathematically equivalent to a standard NMA model, using agent-specific dose-response functions allows analysis of datasets that both include multiple doses of different drugs and interventions for which a dose-response relationship is not realistic (e.g. surgery) or difficult to assume (e.g. psychotherapy, exercise interventions). Interventions without a dose-response relationship can be coded in the dataset as different agents, each of which should be assigned a dose of 1, and these can then be modeled using a linear dose-response relationship, whilst agents with a plausible dose-response can be assigned a function that appropriately captures their dose-response relationship. Further details of this approach can be found in the vignette on NMA in MBNMAdose

### Splines and `knots`

For a more flexible dose-response shape, various different splines can be fitted to the data by using `dspline()`. This model is very flexible and can allow for a variety of non-monotonic dose-response relationships, though parameters can be difficult to interpret and the resulting dose-response shape is often best visualised by calculating and plotting predictions (Predictions.

To fit this model, the number/location of `knots` (the points at which the different spline pieces meet) should be specified. If a single number is given, it represents the the number of knots to be equally spaced across the dose range of each agent. Alternatively several probabilities can be given that represent the quantiles of the dose range for each agent at which knots should be located. Note that by default, a boundary knot will be placed at the maximum of the dose range to limit the function extrapolating to extreme values.

    dspline(type = "bs", knots = 3)
    # ...is equivalent to
    dspline(type = "bs", knots = c(0.25, 0.5, 0.75))

    # Using a natural cubic spline on the SSRI dataset
    depnet <- mbnma.network(ssri)
    ns <- mbnma.run(depnet, fun = dspline(type = "ns", knots = c(0.25, 0.5, 0.75)))

### Correlation between dose-response parameters

`mbnma.run()` can model correlation between relative effects dose-response parameters, allowing information on one parameter to inform the other(s). This can be implemented by specifying `cor=TRUE` in `mbnma.run()`. The correlation is modeled using a vague Wishart prior. However, note that even a vague Wishart prior may be somewhat influential, given that there is typically limited data to inform the dose-response parameter correlations. The prior can be made more informative by specifying a scale matrix for it. This corresponds to the expectation of the Wishart prior. A different scale matrix can be given to the model in `omega`. Each row of the scale matrix corresponds to the 1st, 2nd, 3rd, etc. dose-response parameter that has been modeled using relative effects (as specified in the dose-response function).

### Priors

Default vague priors for the model are as follows:

\[ \begin{aligned} &d_{p,a} \sim N(0,10000)\\ &beta_{p} \sim N(0,10000)\\ &\sigma \sim N(0,400) \text{ limited to } x \in [0,\infty]\\ &\sigma_{p} \sim N(0,400) \text{ limited to } x \in [0,\infty]\\ &D_{p,c} \sim N(0,1000)\\ &\sigma^D_{p} \sim N(0,400) \text{ limited to } x \in [0,\infty]\\ \end{aligned} \]

…where \(p\) is an identifier for the dose-response parameter (e.g. 1 for Emax and 2 for ED50), \(a\) is an agent identifier and \(c\) is a class identifier.

For dose-response parameters that are constrained to be non-negative (`ed50`, `hill`, `onset`), default prior distributions are truncated (normal). An alternative is to model these parameters using a log-normal distribution (`"dlnorm(mean, prec)"`).

Users may wish to change these, perhaps in order to use more/less informative priors. For default prior distributions on some parameters this may lead to errors when compiling/updating models due to implausibly high values. Or in some cases they can constrain the posterior distribution when the user may wish to allow for a wider range of values.

If the model fails during compilation/updating (i.e. due to a problem in JAGS), `mbnma.run()` will generate an error and return a list of arguments that `mbnma.run()` used to generate the model. Within this (as within a model that has run successfully), the priors used by the model (in JAGS syntax) are stored within `"model.arg"`:

    print(mbnma$model.arg$priors)
    #> $mu
    #> [1] "dnorm(0,0.0001)"
    #>
    #> $ed50
    #> [1] "dnorm(0,0.0001) T(0,)"
    #>
    #> $emax
    #> [1] "dnorm(0,0.0001)"
    #>
    #> $sd
    #> [1] "dunif(0, 6.021)"

In this way a model can first be run with vague priors and then rerun with different priors, perhaps to allow successful computation, perhaps to provide more informative priors, or perhaps to run a sensitivity analysis with different priors. Increasing the precision of prior distributions only a little can also often improve convergence considerably.

To change priors within a model, a list of replacements can be provided to `priors` in `mbnma.run()`. The name of each element is the name of the parameter to change (without indices) and the value of the element is the JAGS distribution to use for the prior. This can include censoring or truncation if desired. Only the priors to be changed need to be specified - priors for parameters that aren’t specified will take default values.

For example, if we want to use tighter priors for the half-normal SD parameters we could increase the precision:

    # Define replacement prior
    new.priors <- list(sd = "dnorm(0, 1) T(0,)")

    # Run an MBNMA model with new priors
    emax <- mbnma.run(alognet, fun = demax(), method = "random", priors = new.priors)

Different prior distributions can be assigned for different indices of a parameter by specifying the list element for a parameter as a character vector. This allows (for example) for the user to fit specific priors for specific agents. The length of this vector must be equal to the number of indices of the parameter. The ordering will also be important - for example for agent-specific priors the order of the elements within the vector must match the order of the agents in the network.

For example for an Emax function we may have different prior beliefs about the ED50 parameter, as the dose scale will be different for different agents:

    ed50.priors <- list(ed50 = c(Celebrex = "dnorm(100, 0.0025) T(0,)", Etoricoxib = "dnorm(20, 0.01) T(0,)",
        Lumiracoxib = "dnorm(50, 0.0025) T(0,)", Naproxcinod = "dnorm(500, 0.0004) T(0,)",
        Naproxen = "dnorm(500, 0.0004) T(0,)", Rofecoxib = "dnorm(35, 0.04) T(0,)", Tramadol = "dnorm(200, 0.0004) T(0,)",
        Valdecoxib = "dnorm(4, 0.04) T(0,)"))

    # Using the osteoarthritis dataset
    mbnma <- mbnma.run(painnet, fun = demax(emax = "rel", ed50 = "rel"), priors = ed50.priors)

### pD (effective number of parameters)

The default value in for `pD` in `mbnma.run()` uses the Kullback-Leibler divergence to estimate pD (Plummer 2008). However, since it requires running additional MCMC samples, it can be set to `FALSE` to force the model to estimate pV, where `pv = var(deviance)/2`. Whilst this is easy to calculate, it is numerically less stable than the Kullback-Leibler divergence estimation, and may perform more poorly in certain conditions (Gelman, Hwang, and Vehtari 2014).

### Arguments to be sent to JAGS

In addition to the arguments specific to `mbnma.run()` it is also possible to use any arguments to be sent to `R2jags::jags()`. Most of these are likely to relate to improving the performance of MCMC simulations in JAGS and may help with parameter convergence (see [Convergence]). Some of the key arguments that may be of interest are:

- `n.chains` The number of Markov chains to run (default is 3)
- `n.iter` The total number of iterations per MCMC chain
- `n.burnin` The number of iterations that are discarded to ensure iterations are only saved once chains have converged
- `n.thin` The thinning rate which ensures that results are only saved for 1 in every `n.thin` iterations per chain. This can be increased to reduce autocorrelation

## Connecting networks via the dose-response relationship

One of the strengths of dose-response MBNMA is that it allows treatments to be connected in a network that might otherwise be disconnected, by linking up different doses of the same agent via the dose-response relationship (Pedder, Dias, Bennetts, et al. 2021). To illustrate this we can generate a version of the gout dataset which excludes placebo (to artificially disconnect the network):

    # Generate dataset without placebo
    noplac.gout <- gout[!gout$studyID %in% c(2001, 3102), ]  # Drop two-arm placebo studies
    noplac.gout <- noplac.gout[noplac.gout$agent != "Plac", ]  # Drop placebo arm from multi-arm studies

    # Create mbnma.network object
    noplac.net <- mbnma.network(noplac.gout)

    # Plot network
    plot(noplac.net, label.distance = 5)
    #> Warning in check.network(g): The following treatments/agents are not connected
    #> to the network reference:
    #> Allo_300
    #> Allo_400
    #> Arha_400
    #> Arha_600
    #> Benz_50
    #> Benz_200
    #> Febu_40
    #> Febu_80
    #> Febu_120
    #> RDEA_100
    #> RDEA_200
    #> RDEA_400

This results in a very disconnected network, and if we were to conduct a conventional “split” NMA (whereby different doses of an agent are considered to be independent), we would only be able to estimate relative effects for a very small number of treatments. However, if we assume a dose-response relationship then these different doses can be connected via this relationship, and we can connect up more treatments and agents in the network.

    # Network plot at the agent level illustrates how doses can connect using MBNMA
    plot(noplac.net, level = "agent", remove.loops = TRUE, label.distance = 4)
    #> Warning in check.network(g): The following treatments/agents are not connected
    #> to the network reference:
    #> Arha
    #> RDEA

There are still two agents that do not connect to the network because they involve comparisons of different doses of the same agent. However, multiple doses of an agent within a study allow us to estimate the dose-response relationship and tell us something about the placebo (dose = 0) response - the number of different doses of an agent within a study will determine the degrees of freedom with which we are able to estimate a given dose-response function. Although the placebo response is not estimated directly in the MBNMA framework (it is modelled as a nuisance parameter), it allows us to connect the dose-response function estimated for an agent in one study, with that in another.

To visualise this, we can use the `doselink` argument in `plot(mbnma.network)`. The integer given to this argument indicates the minimum number of doses from which a dose-response function could be estimated, and is equivalent to the number of parameters in the desired dose-response function plus one. For example for an exponential function, we would require at least two doses on a dose-response curve (including placebo), since this would allow one degree of freedom with which to estimate the one-parameter dose-response function. By modifying the `doselink` argument we can determine the complexity of a dose-response function that we might expect to be able to estimate whilst still connecting all agents within the network.

If placebo is not included in the original dataset then this argument will also add a node for placebo to illustrate the connection.

    # Network plot assuming connectivity via two doses Allows estimation of a
    # single-parameter dose-response function
    plot(noplac.net, level = "agent", remove.loops = TRUE, label.distance = 4, doselink = 2)
    #> Dose-response connections to placebo plotted based on a dose-response
    #>                    function with 1 degrees of freedom

    # Network plot assuming connectivity via three doses Allows estimation of a
    # two-parameter dose-response function
    plot(noplac.net, level = "agent", remove.loops = TRUE, label.distance = 4, doselink = 3)
    #> Warning in check.network(g): The following treatments/agents are not connected
    #> to the network reference:
    #> Allo
    #> Arha
    #> Benz
    #> Febu
    #> Dose-response connections to placebo plotted based on a dose-response
    #>                    function with 2 degrees of freedom

In this way we can fully connect up treatments in an otherwise disconnected network, though unless informative prior information is used this will be limited by the number of doses of agents within included studies. See Pedder et al. (2021) for more details on this.

## Non-parametric dose-response functions

In addition to the parametric dose-response functions described above, a non-parametric monotonic dose-response relationship can also be specified in `mbnma.run()`. `fun=dnonparam()` can be used to specify a monotonically increasing (`direction="increasing"`) or decreasing (`direction="decreasing"`) dose-response respectively. This is achieved in the model by imposing restrictions on the prior distributions of treatment effects that ensure that each increasing dose of an agent has an effect that is either the same or greater than the previous dose. The approach results in a similar model to that developed by Owen et al. (2015).

By making this assumption, this model is slightly more informative, and can lead to some slight gains in precision if relative effects are otherwise imprecisely estimated. However, because a functional form for the dose-response is not modeled, it cannot be used to connect networks that are disconnected at the treatment-level, unlike a parametric MBNMA.

In the case of MBNMA, it may be useful to compare the fit of a non-parametric model to that of a parametric dose-response function, to ensure that fitting a parametric dose-response function does not lead to significantly poorer model fit.

When fitting a non-parametric dose-response model, it is important to correctly choose the expected `direction` of the monotonic response, otherwise it can lead to computation error.

    nonparam <- mbnma.run(tripnet, fun = dnonparam(direction = "increasing"), method = "random")
    #> `likelihood` not given by user - set to `binomial` based on data provided
    #> `link` not given by user - set to `logit` based on assigned value for `likelihood`

    print(nonparam)
    #> Inference for Bugs model at "C:\Users\hp17602\AppData\Local\Temp\Rtmpesrqof\file80a41178cf5", fit using jags,
    #>  3 chains, each with 20000 iterations (first 10000 discarded), n.thin = 10
    #>  n.sims = 3000 iterations saved. Running time = 46.15 secs
    #>            mu.vect sd.vect     2.5%      25%      50%      75%    97.5%  Rhat
    #> d.1[1,1]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
    #> d.1[1,2]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
    #> d.1[2,2]     1.188   0.176    0.832    1.070    1.189    1.309    1.534 1.002
    #> d.1[3,2]     1.741   0.116    1.505    1.667    1.740    1.818    1.974 1.003
    #> d.1[4,2]     2.049   0.141    1.778    1.954    2.046    2.141    2.331 1.003
    #> d.1[1,3]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
    #> d.1[2,3]     0.948   0.141    0.633    0.862    0.964    1.049    1.178 1.001
    #> d.1[3,3]     1.117   0.084    0.952    1.062    1.117    1.173    1.283 1.004
    #> d.1[4,3]     1.256   0.111    1.047    1.180    1.252    1.332    1.476 1.002
    #> d.1[5,3]     1.456   0.084    1.296    1.400    1.456    1.513    1.622 1.005
    #> d.1[1,4]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
    #> d.1[2,4]     1.249   0.209    0.855    1.105    1.245    1.388    1.673 1.007
    #> d.1[3,4]     1.633   0.321    1.078    1.400    1.607    1.840    2.301 1.002
    #> d.1[1,5]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
    #> d.1[2,5]     0.618   0.237    0.132    0.461    0.628    0.788    1.047 1.006
    #> d.1[3,5]     1.034   0.121    0.804    0.953    1.034    1.112    1.269 1.001
    #> d.1[4,5]     1.434   0.211    1.043    1.285    1.429    1.567    1.887 1.002
    #> d.1[1,6]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
    #> d.1[2,6]     0.785   0.304    0.136    0.584    0.815    1.014    1.290 1.001
    #> d.1[3,6]     1.247   0.113    1.025    1.172    1.246    1.320    1.468 1.001
    #> d.1[4,6]     1.552   0.191    1.205    1.415    1.536    1.680    1.951 1.002
    #> d.1[5,6]     1.885   0.266    1.424    1.695    1.866    2.051    2.445 1.001
    #> d.1[6,6]     2.917   0.602    1.887    2.465    2.879    3.313    4.181 1.003
    #> d.1[1,7]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
    #> d.1[2,7]     0.551   0.207    0.146    0.409    0.548    0.694    0.961 1.002
    #> d.1[3,7]     0.994   0.312    0.453    0.773    0.967    1.189    1.683 1.003
    #> d.1[1,8]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
    #> d.1[2,8]     0.476   0.312    0.026    0.224    0.436    0.686    1.139 1.001
    #> d.1[3,8]     1.264   0.159    0.964    1.151    1.264    1.373    1.582 1.001
    #> d.1[4,8]     1.611   0.103    1.416    1.541    1.609    1.679    1.817 1.003
    #> sd           0.256   0.048    0.165    0.224    0.255    0.287    0.350 1.014
    #> totresdev  190.468  19.167  155.620  177.307  189.736  203.040  230.032 1.004
    #> deviance  1093.574  19.167 1058.726 1080.413 1092.842 1106.146 1133.138 1.004
    #>           n.eff
    #> d.1[1,1]      1
    #> d.1[1,2]      1
    #> d.1[2,2]   1800
    #> d.1[3,2]    950
    #> d.1[4,2]    830
    #> d.1[1,3]      1
    #> d.1[2,3]   3000
    #> d.1[3,3]    620
    #> d.1[4,3]   1400
    #> d.1[5,3]    770
    #> d.1[1,4]      1
    #> d.1[2,4]    330
    #> d.1[3,4]   1300
    #> d.1[1,5]      1
    #> d.1[2,5]   3000
    #> d.1[3,5]   3000
    #> d.1[4,5]   1000
    #> d.1[1,6]      1
    #> d.1[2,6]   3000
    #> d.1[3,6]   3000
    #> d.1[4,6]   1300
    #> d.1[5,6]   2500
    #> d.1[6,6]    800
    #> d.1[1,7]      1
    #> d.1[2,7]   1800
    #> d.1[3,7]    950
    #> d.1[1,8]      1
    #> d.1[2,8]   2300
    #> d.1[3,8]   3000
    #> d.1[4,8]    680
    #> sd          180
    #> totresdev   610
    #> deviance    640
    #>
    #> For each parameter, n.eff is a crude measure of effective sample size,
    #> and Rhat is the potential scale reduction factor (at convergence, Rhat=1).
    #>
    #> DIC info (using the rule: pV = var(deviance)/2)
    #> pV = 183.2 and DIC = 1276.8
    #> DIC info (using the rule: pD = Dbar-Dhat, computed via 'rjags::dic.samples')
    #> pD = 125.7 and DIC = 1220.3
    #> DIC is an estimate of expected predictive error (lower deviance is better).

In the output from non-parametric models, `d.1` parameters represent the relative effect for each treatment (specific dose of a specific agent) versus the reference treatment, similar to in a standard Network Meta-Analysis. The first index of `d` represents the dose identifier, and the second index represents the agent identifier. Information on the specific values of the doses is not included in the model, as only the ordering of them (lowest to highest) is important.

Note that some post-estimation functions (e.g. ranking, prediction) cannot be performed on non-parametric models within the package.

## Assessment of model fit

For looking at model fit we will demonstrate using results from an Emax MBNMA on the triptans dataset:

    tripnet <- mbnma.network(triptans)
    #> Values for `agent` with dose = 0 have been recoded to `Placebo`
    #> agent is being recoded to enforce sequential numbering
    trip.emax <- mbnma.run(tripnet, fun = demax(emax = "rel", ed50 = "rel"))
    #> `likelihood` not given by user - set to `binomial` based on data provided
    #> `link` not given by user - set to `logit` based on assigned value for `likelihood`

### Deviance plots

To assess how well a model fits the data, it can be useful to look at a plot of the contributions of each data point to the residual deviance. This can be done using `devplot()`. As individual deviance contributions are not automatically monitored in `parameters.to.save`, this might require the model to be automatically run for additional iterations.

Results can be plotted either as a scatter plot (`plot.type="scatter"`) or a series of boxplots (`plot.type="box"`).

    # Plot boxplots of residual deviance contributions (scatterplot is the default)
    devplot(trip.emax, plot.type = "box")
    #> `resdev` not monitored in mbnma$parameters.to.save.
    #> additional iterations will be run in order to obtain results for `resdev`

From these plots we can see that whilst the model fit does not seem to be systematically non-linear (which would suggest an alternative dose-response function may be a better fit), residual deviance is high at a dose of 1 for eletriptan, and at 2 for sumatriptan. This may indicate that fitting random effects may allow for additional variability in response which may improve the model fit.

If saved to an object, the output of `devplot()` contains the results for individual deviance contributions, and this can be used to identify any extreme outliers.

### Fitted values

Another approach for assessing model fit can be to plot the fitted values, using `fitplot()`. As with `devplot()`, this may require running additional model iterations to monitor `theta`.

    # Plot fitted and observed values with treatment labels
    fitplot(trip.emax)
    #> `theta` not monitored in mbnma$parameters.to.save.
    #> additional iterations will be run in order to obtain results

Fitted values are plotted as connecting lines and observed values in the original dataset are plotted as points. These plots can be used to identify if the model fits the data well for different agents and at different doses along the dose-response function.

## Model Selection

Detailed description of model selection based on statistical measures such as Deviance Information Criterion (DIC) and residual deviance is outside the scope of this vignette. However, the following approach for model identification and selection is recommended, which also gives details on model fit statistics used for comparison:

1. Identify candidate dose-response functions based on plotted results of split NMA (`plot.nma()`) and clinical/biological reasoning
2. Compare candidate dose-response functions fitted with common relative treatment effects (`mbnma.run(..., method="common")`)
3. If no candidate dose-response functions converge successfully, additional modelling assumptions can be made to strengthen inference, provided they are clinically justifiable (e.g. class effects, agent-specific dose-response functions, correlation between dose-response parameters)
4. Compare selected common effects model to the same model fitted with random effects (`mbnma.run(..., method="random")`)

Finally the validity of the consistency assumption should be explored in the selected final model (see [Consistency Testing] and Pedder et al. (2021)).

## MCMC Convergence

`MBNMAdose` runs Bayesian models in JAGS, which uses Markov Chain Monte Carlo (MCMC) simulation. However, the validity of results is reliant on the MCMC chains having converged successfully on the posterior densities of all parameters in the model. For highly parameterised models run on relatively limited data, as is often the case with MBNMA models, convergence can often be an challenge. Note that convergence is necessary to be able to compare models and evaluate model fit. However, successful convergence does not imply good model fit.

A full explanation of how to facilitate and to check for convergence is outside the scope of this vignette, but below are some simple steps for checking convergence. None of these steps alone ensures convergence has been successful, but interpretation of them jointly can provide strong evidence of it.

- Rhat values close to 1 (<1.1 is considered acceptable by some). These are shown for monitored parameters in the summary statistics table when an `"mbnma"` or `"nma"` object is printed.
- Trace plots that have a “fuzzy caterpillar” look that shows good mixing of MCMC chains
- Density plots that show a smooth posterior distribution that is similar across all chains
- Autocorrelation plots that show low autocorrelation between neighbouring MCMC iterations (i.e. low correlation at higher lags)

An HTML document with all the above convergence plots can easily be generated for all parameters in the model simultaneously using `mcmcplots::mcmcplot()`.

Two steps may improve convergence when using `MBNMAdose`. One step is to run models for more iterations (this can be done using the `n.iter` argument in `mbnma.run()`). It can take time for MCMC chains to converge, particularly for non-linear models with limited data. It is important to note that chains should only be monitored after they have converged - increasing the number of burn-in iterations ensures that this is the case (using the `n.burnin` argument in `mbnma.run()`). Another method to improve convergence is by providing more information to the model via informative priors.

For a detailed review of MCMC convergence assessment see Sinharay (2003).

## References

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Pedder, H., S. Dias, M. Bennetts, M. Boucher, and N. J. Welton. 2021. “Joining the Dots: Linking Disconnected Networks of Evidence Using Dose-Response Model-Based Network Meta-Analysis.” Journal Article. Medical Decision Making 41 (2): 194–208.
Pedder, H., S. Dias, M. Boucher, M. Bennetts, D. Mawdsley, and N. J. Welton. 2021. “Methods to Assess Evidence Consistency in Dose-Response Model Based Network Meta-Analysis.” Journal Article. Statistics in Medicine 41 (4): 625–44.
Plummer, M. 2008. “Penalized Loss Functions for Bayesian Model Comparison.” Journal Article. Biostatistics 9 (3): 523–39. https://pubmed.ncbi.nlm.nih.gov/18209015/.
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