Author: Hugo Pedder
Date: 2025-02-06

# MBNMAdose: Calculating model predictions

#### Hugo Pedder

#### 2025-02-06

## Prediction

For calculating predictions from MBNMAdose 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`

After performing an MBNMA, responses can be predicted from the model parameter estimates using `predict()` on an `"mbnma"` object. A number of important arguments should be specified for prediction. See `?predict.mbnma` for detailed specification of these arguments.

`E0` This is the response at dose = 0 (equivalent to the placebo response). Since relative effects are the parameters estimated in MBNMA, the placebo response is not explicitly modelled and therefore must be provided by the user in some way. The simplest approach is to provide either a single numeric value for `E0` (deterministic approach), or a string representing a distribution for `E0` that can take any Random Number Generator (RNG) distribution for which a function exists in R (stochastic approach). Values should be given on the natural scale. For example for a binomial outcome:

- Deterministic: `E0 <- 0.2`
- Stochastic: `E0 <- "rbeta(n, shape1=2, shape2=10)"`

Another approach is to estimate `E0` from a set of studies. These would ideally be studies of untreated/placebo-treated patients that closely resemble the population for which predictions are desired, and the studies may be observational. However, synthesising results from the placebo arms of trials in the original network is also possible. For this, `E0` is assigned a data frame of studies in the long format (one row per study arm) with the variables `studyID`, and a selection of `y`, `se`, `r`, `n` and `E` (depending on the likelihood used in the MBNMA model). `synth` can be set to `"fixed"` or `"random"` to indicate whether this synthesis should be fixed or random effects.

    E0 <- triptans[triptans$dose == 0, ]

Additionally, it’s also necessary to specify the doses at which to predict responses. By default, `predict()` uses the maximum dose within the dataset for each agent, and predicts doses at a series of cut points. The number of cut points can be specified using `n.doses`, and the maximum dose to use for prediction for each agent can also be specified using `max.doses` (a named list of numeric values where element names correspond to agent names).

An alternative approach is to predict responses at specific doses for specific agents using the argument `exact.doses`. As with `max.doses`, this is a named list in which element names correspond to agent names, but each element is a numeric vector in which each value within the vector is a dose at which to predict a response for the given agent.

    # Predict 20 doses for each agent, with a stochastic distribution for E0
    doses <- list(Placebo = 0, eletriptan = 3, sumatriptan = 3, almotriptan = 3, zolmitriptan = 3,
        naratriptan = 3, rizatriptan = 3)

    pred <- predict(trip.emax, E0 = "rbeta(n, shape1=2, shape2=10)", max.doses = doses,
        n.dose = 20)

    # Predict exact doses for two agents, and estimate E0 from the data
    E0.data <- triptans[triptans$dose == 0, ]
    doses <- list(eletriptan = c(0, 1, 3), sumatriptan = c(0, 3))

    pred <- predict(trip.emax, E0 = E0.data, exact.doses = doses)
    #> Values for `agent` with dose = 0 have been recoded to `Placebo`
    #> agent is being recoded to enforce sequential numbering

An object of class `"mbnma.predict"` is returned, which is a list of summary tables and MCMC prediction matrices for each treatment (combination of dose and agent). The `summary()` method can be used to print mean posterior predictions at each time point for each treatment.

    summary(pred)
    #>         agent dose      mean          sd      2.5%       25%       50%
    #> 1     Placebo    0 0.1238319 0.003398588 0.1173367 0.1215236 0.1237340
    #> 2  eletriptan    0 0.1238319 0.003398588 0.1173367 0.1215236 0.1237340
    #> 3  eletriptan    1 0.4317285 0.018741568 0.3956369 0.4188110 0.4316759
    #> 4  eletriptan    3 0.5503016 0.027304234 0.4976447 0.5314575 0.5498709
    #> 5 sumatriptan    0 0.1238319 0.003398588 0.1173367 0.1215236 0.1237340
    #> 6 sumatriptan    3 0.3923071 0.018805738 0.3565498 0.3790666 0.3920169
    #>         75%     97.5%
    #> 1 0.1261012 0.1307333
    #> 2 0.1261012 0.1307333
    #> 3 0.4445404 0.4682948
    #> 4 0.5683603 0.6042371
    #> 5 0.1261012 0.1307333
    #> 6 0.4051519 0.4294165

### Plotting predictions

Predictions can also be plotted using the `plot()` method on an object of `class("mbnma.predict")`. The predicted responses are joined by a line to form the dose-response curve for each agent predicted, with 95% credible intervals (CrI). Therefore, when plotting the response it is important to predict a sufficient number of doses (using `n.doses`) to get a smooth curve.

    # Predict responses using default doses up to the maximum of each agent in the
    # dataset
    pred <- predict(trip.emax, E0 = 0.2, n.dose = 20)

    plot(pred)

Shaded counts of the number of studies in the original dataset that investigate each dose of an agent can be plotted over the 95% CrI for each treatment by setting `disp.obs = TRUE`, though this requires that the original `"mbnma.network"` object used to estimate the MBNMA be provided via `network`.

    plot(pred, disp.obs = TRUE)
    #> 66 placebo arms in the dataset are not shown within the plots

This can be used to identify any extrapolation/interpretation of the dose-response that might be occurring for a particular agent. As you can see, more observations typically leads to tighter 95% CrI for the predicted response at a particular point along the dose-response curve.

We can also plot the results of a “split” Network Meta-Analysis (NMA) in which all doses of an agent are assumed to be independent. As with `disp.obs` we also need to provide the original `mbnma.network` object to be able to estimate this, and we can also specify if we want to perform a common or random effects NMA using `method`. Treatments that are only connected to the network via the dose-response relationship (rather than by a direct head-to-head comparison) will not be included.

    alognet <- mbnma.network(alog_pcfb)
    alog.emax <- mbnma.run(alognet, fun = demax(), method = "random")
    pred <- predict(alog.emax, E0 = 0, n.dose = 20)
    plot(pred, overlay.split = TRUE, method = "random")

By plotting these, as well as observing how responses can be extrapolated/interpolated, we can also see which doses are likely to be providing most information to the dose-response relationship. The tighter 95% CrI on the predicted responses from the MBNMA also show that modelling the dose-response function also gives some additional precision even at doses for which there is information available.

More detailed documentation can be accessed using `?plot.mbnma.predict`.

### Ranking predicted responses

Predicted responses from an object of `class("mbnma.predict")` can also be ranked using the `rank()` method. As when applied to an object of `class("mbnma")`, this method will rank parameters (in this case predictions) in order from either highest to lowest (`direction=1`) or lowest to highest (`direction=-1`), and return an object of `class("mbnma.rank")`.

If there have been predictions at dose = 0 for several agents only one of these will be included in the rankings, in order to avoid duplication (since the predicted response at dose = 0 is the same for all agents).

    pred <- predict(trip.emax, E0 = 0.2, n.doses = 4, max.doses = list(eletriptan = 5,
        sumatriptan = 5, frovatriptan = 5, zolmitriptan = 5))

    ranks <- rank(pred)
    plot(ranks)

## References