# 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.

If relative effects ("rel") are modelled on more than one dose-response parameter then by default, a correlation will be assumed between the dose-response parameters, which will typically improve estimation (provided the parameters are correlated…they usually are). This can be prevented by setting cor=FALSE.

## 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        0.339 (0.237, 0.458)
#>
#> emax dose-response parameter results
#>
#> Pooling: relative effects for each agent
#>
#> |Agent        |Parameter |   Median|    2.5%|    97.5%|
#> |:------------|:---------|--------:|-------:|--------:|
#> |eletriptan   |emax[2]   |   2.7755|  2.1027|   4.2419|
#> |sumatriptan  |emax[3]   |   5.3450|  1.5763| 135.2515|
#> |frovatriptan |emax[4]   |  77.3231| 16.6444| 197.2774|
#> |almotriptan  |emax[5]   |  76.0468| 13.5016| 173.5456|
#> |zolmitriptan |emax[6]   |   2.7990|  1.6852|   5.2800|
#> |naratriptan  |emax[7]   |  44.5333|  6.5365| 131.4252|
#> |rizatriptan  |emax[8]   | 105.8855| 26.8501| 230.4171|
#>
#>
#> ed50 dose-response parameter results
#>
#> Pooling: relative effects for each agent
#>
#> |Agent        |Parameter |   Median|    2.5%|    97.5%|
#> |:------------|:---------|--------:|-------:|--------:|
#> |eletriptan   |ed50[2]   |   0.6630|  0.2675|   1.8138|
#> |sumatriptan  |ed50[3]   |   5.3787|  0.3981| 182.1643|
#> |frovatriptan |ed50[4]   |  79.7400| 15.6928| 198.3528|
#> |almotriptan  |ed50[5]   |  91.5533| 14.8690| 209.8353|
#> |zolmitriptan |ed50[6]   |   1.4253|  0.3702|   4.4059|
#> |naratriptan  |ed50[7]   | 102.1475| 18.3188| 240.2148|
#> |rizatriptan  |ed50[8]   |  65.7051| 16.1266| 147.3789|
#>
#>
#> Model Fit Statistics
#> Effective number of parameters:
#> pD calculated using the Kullback-Leibler divergence = 138.2
#>
#> Deviance = 1089.7
#> Residual deviance = 186.6
#> Deviance Information Criterion (DIC) = 1227.9

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]
#> sd
#> ========================================
#> 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        0.505 (0.18, 0.637)
#>
#> emax dose-response parameter results
#>
#> Pooling: relative effects for each agent
#>
#> |Agent        |Parameter |   Median|   2.5%|    97.5%|
#> |:------------|:---------|--------:|------:|--------:|
#> |eletriptan   |emax[2]   |  69.9699| 2.5534| 151.2653|
#> |sumatriptan  |emax[3]   |  47.9562| 1.7285| 102.7305|
#> |frovatriptan |emax[4]   |  62.3579| 1.6174| 153.3086|
#> |almotriptan  |emax[5]   |  48.0610| 1.5019| 116.4098|
#> |zolmitriptan |emax[6]   |  30.4218| 1.8647|  65.3112|
#> |naratriptan  |emax[7]   |  19.3939| 0.3580|  79.5771|
#> |rizatriptan  |emax[8]   | 101.9199| 2.3889| 218.0980|
#>
#>
#> ed50 dose-response parameter results
#>
#> Pooling: single parameter across all agents in the network
#>
#> |Parameter |  Median|   2.5%|    97.5%|
#> |:---------|-------:|------:|--------:|
#> |ed50      | 66.3131| 0.5307| 144.0589|
#>
#>
#> Model Fit Statistics
#> Effective number of parameters:
#> pD calculated using the Kullback-Leibler divergence = 145.7
#>
#> Deviance = 1089.1
#> Residual deviance = 186
#> Deviance Information Criterion (DIC) = 1234.8

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.

### 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, but this can be made more informative by specifying a scale matrix for the prior. 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() is "pv", which uses the value automatically calculated in the R2jags package as pv = var(deviance)/2. Whilst this is easy to calculate, it is numerically less stable than pD and may perform more poorly in certain conditions (Gelman, Hwang, and Vehtari 2014).

A commonly-used approach for calculating pD is the plug-in method (pd="plugin") (Spiegelhalter et al. 2002). However, this can sometimes result in negative non-sensical values due to skewed posterior distributions for deviance contributions that can arise when fitting non-linear models.

Another approach that is more reliable than the plug-in method when modelling non-linear effects is using the Kullback-Leibler divergence (pd="pd.kl") (Plummer 2008). The disadvantage of this approach is that it requires running additional MCMC iterations, so can be slightly slower to calculate.

Finally, pD can also be calculated using an optimism adjustment (pd="popt") which allows for calculation of the penalized expected deviance (Plummer 2008). This adjustment allows for the fact that data used to estimate the model is the same as that used to assess its parsimony. It also requires running additional MCMC iterations.

### 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\RtmpwhT2NF\file311c76cb13b8", fit using jags,
#>  3 chains, each with 20000 iterations (first 10000 discarded), n.thin = 10
#>  n.sims = 3000 iterations saved
#>            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]     1.180   0.178    0.834    1.065    1.179    1.295    1.535 1.002
#> d.1[3,2]     1.739   0.119    1.510    1.657    1.739    1.820    1.970 1.003
#> d.1[4,2]     2.044   0.148    1.757    1.945    2.038    2.140    2.348 1.004
#> d.1[1,3]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
#> d.1[2,3]     0.945   0.144    0.629    0.859    0.958    1.047    1.191 1.001
#> d.1[3,3]     1.115   0.087    0.938    1.058    1.118    1.175    1.283 1.001
#> d.1[4,3]     1.257   0.115    1.039    1.174    1.254    1.335    1.486 1.002
#> d.1[5,3]     1.458   0.087    1.288    1.399    1.456    1.517    1.625 1.001
#> d.1[1,4]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
#> d.1[2,4]     1.243   0.213    0.844    1.093    1.242    1.380    1.677 1.003
#> d.1[3,4]     1.616   0.311    1.062    1.393    1.592    1.818    2.292 1.006
#> d.1[1,5]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
#> d.1[2,5]     0.617   0.245    0.127    0.447    0.620    0.797    1.064 1.001
#> d.1[3,5]     1.037   0.125    0.794    0.954    1.037    1.123    1.282 1.002
#> d.1[4,5]     1.443   0.210    1.046    1.294    1.437    1.585    1.858 1.004
#> d.1[1,6]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
#> d.1[2,6]     0.787   0.306    0.148    0.573    0.818    1.026    1.287 1.014
#> d.1[3,6]     1.242   0.117    1.004    1.165    1.244    1.318    1.471 1.006
#> d.1[4,6]     1.538   0.192    1.206    1.402    1.519    1.667    1.954 1.009
#> d.1[5,6]     1.862   0.273    1.402    1.664    1.848    2.033    2.438 1.008
#> d.1[6,6]     2.904   0.618    1.844    2.452    2.879    3.323    4.193 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]     0.553   0.207    0.154    0.410    0.552    0.695    0.960 1.006
#> d.1[3,7]     1.008   0.312    0.470    0.788    0.982    1.201    1.668 1.004
#> d.1[1,8]     0.000   0.000    0.000    0.000    0.000    0.000    0.000 1.000
#> d.1[2,8]     0.481   0.303    0.026    0.238    0.454    0.687    1.123 1.001
#> d.1[3,8]     1.261   0.164    0.940    1.152    1.263    1.375    1.573 1.001
#> d.1[4,8]     1.612   0.103    1.407    1.544    1.610    1.679    1.822 1.006
#> sd           0.261   0.045    0.176    0.230    0.261    0.291    0.355 1.007
#> totresdev  188.592  18.779  153.518  175.513  188.190  201.290  226.516 1.001
#> deviance  1091.698  18.779 1056.624 1078.619 1091.296 1104.396 1129.622 1.001
#>           n.eff
#> d.1[1,1]      1
#> d.1[1,2]      1
#> d.1[2,2]   3000
#> d.1[3,2]    670
#> d.1[4,2]    630
#> d.1[1,3]      1
#> d.1[2,3]   2100
#> d.1[3,3]   3000
#> d.1[4,3]   2000
#> d.1[5,3]   3000
#> d.1[1,4]      1
#> d.1[2,4]    710
#> d.1[3,4]    390
#> d.1[1,5]      1
#> d.1[2,5]   3000
#> d.1[3,5]   1000
#> d.1[4,5]    580
#> d.1[1,6]      1
#> d.1[2,6]    270
#> d.1[3,6]    370
#> d.1[4,6]    240
#> d.1[5,6]    270
#> d.1[6,6]    790
#> d.1[1,7]      1
#> d.1[2,7]   2200
#> d.1[3,7]    650
#> d.1[1,8]      1
#> d.1[2,8]   3000
#> d.1[3,8]   2300
#> d.1[4,8]    360
#> sd          300
#> totresdev  3000
#> deviance   3000
#>
#> 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, pD = var(deviance)/2)
#> pD = 128.3 and DIC = 1219.6
#> 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).

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