# MBNMAdose: Perform Network Meta-Regression

#### Hugo Pedder

#### 2025-02-06

# Network meta-regression

Study-level covariates can be included in the model to adjust treatment effects following an approach for meta-regression outlined in NICE Technical Support Document 3 (Dias et al. 2011). This can be used to explore and account for potential effect modification.

Following the definition in NICE Technical Support Document 3, network meta-regression can be expressed as an interaction on the treatment effect in arms \(\geq2\):

\[\theta_{i,k}=\mu_i+(f(x,\beta_{a_{i,k}})-f(x,\beta_{a_{i,1}})) + (\psi_{1,a_{i,k}}-\psi_{1,a_{i,1}})\]

where \(\theta_{i,k}\) is the linear predictor, \(\mu_{i}\) is the baseline effect on arm 1, \(f(x,\beta_{a_{i,k}})\) is the dose-response function at dose \(x\) with dose-response parameters \(\beta_{a_{i,k}}\) for agent \(a\) in arm \(k\) of study \(i\). \(\psi_{1,a_{i,k}}\) is then the effect modifying interaction between the agent in arm \(k\) and the network reference agent (typically Placebo in a dose-response analysis).

## Data preparation

To improve estimation:

- continuous covariates should be centred around their mean
- binary/categorical variables should be recoded with the most commonly reported value as the reference category

    # Using the SSRI dataset
    ssri.reg <- ssri

    # For a continuous covariate
    ssri.reg <- ssri.reg %>%
        dplyr::mutate(x.weeks = weeks - mean(weeks, na.rm = TRUE))

    # For a categorical covariate
    table(ssri$weeks)  # Using 8 weeks as the reference
    ssri.reg <- ssri.reg %>%
        dplyr::mutate(r.weeks = factor(weeks, levels = c(8, 4, 5, 6, 9, 10)))

    # Create network object
    ssrinet <- mbnma.network(ssri.reg)
    #> Values for `agent` with dose = 0 have been recoded to `Placebo`
    #> agent is being recoded to enforce sequential numbering

## Modelling

For performing network meta-regression, different assumptions can be made regarding how the effect modification may be shared across agents:

### Independent, agent-specific interactions

The least constraining assumption available in `MBNMAdose` is to assume that the effect modifier acts on each agent independently, and separate \(\psi_{1,a_{i,k}}\) are therefore estimated for each agent in the network.

A slightly stronger assumption is to assume that agents within the same class share the same interaction effect, though classes must be specified within the dataset for this.

    # Regress for continuous weeks Separate effect modification for each agent vs
    # Placebo
    ssrimod.a <- mbnma.run(ssrinet, fun = dfpoly(degree = 2), regress = ~x.weeks, regress.effect = "agent")

    summary(ssrimod.a)
    #> ========================================
    #> Dose-response MBNMA
    #> ========================================
    #>
    #> Likelihood: binomial
    #> Link function: logit
    #> Dose-response function: fpoly
    #>
    #> Pooling method
    #>
    #> Method: Common (fixed) effects estimated for relative effects
    #>
    #>
    #> beta.1 dose-response parameter results
    #>
    #> Pooling: relative effects for each agent
    #>
    #> |Agent        |Parameter |  Median|    2.5%|   97.5%|
    #> |:------------|:---------|-------:|-------:|-------:|
    #> |citalopram   |beta.1[2] | -0.0912| -0.3705|  0.1815|
    #> |escitalopram |beta.1[3] |  0.1312| -0.1353|  0.3985|
    #> |fluoxetine   |beta.1[4] |  0.2665|  0.0457|  0.5083|
    #> |paroxetine   |beta.1[5] | -0.1387| -0.4479|  0.1517|
    #> |sertraline   |beta.1[6] |  1.2823| -8.3849| 10.5021|
    #>
    #>
    #> beta.2 dose-response parameter results
    #>
    #> Pooling: relative effects for each agent
    #>
    #> |Agent        |Parameter |  Median|    2.5%|  97.5%|
    #> |:------------|:---------|-------:|-------:|------:|
    #> |citalopram   |beta.2[2] |  0.0664| -0.0063| 0.1416|
    #> |escitalopram |beta.2[3] | -0.0071| -0.0936| 0.0770|
    #> |fluoxetine   |beta.2[4] | -0.0657| -0.1336| 0.0005|
    #> |paroxetine   |beta.2[5] |  0.0914|  0.0006| 0.1893|
    #> |sertraline   |beta.2[6] | -0.1832| -1.1860| 0.8738|
    #>
    #>
    #> power.1 dose-response parameter results
    #>
    #> Assigned a numeric value: 0
    #>
    #> power.2 dose-response parameter results
    #>
    #> Assigned a numeric value: 0
    #>
    #> Meta-regression
    #>
    #> Covariates interacting with study-level relative effects: x.weeks
    #> Common (identical) covariate-by-agent effects
    #>
    #>
    #> |Agent        |Parameter    |  Median|     2.5%|   97.5%|
    #> |:------------|:------------|-------:|--------:|-------:|
    #> |citalopram   |B.x.weeks[2] | -0.1617|  -0.3316|  0.0109|
    #> |escitalopram |B.x.weeks[3] |  0.1642|  -0.0311|  0.3644|
    #> |fluoxetine   |B.x.weeks[4] | -0.0253|  -0.1731|  0.1127|
    #> |paroxetine   |B.x.weeks[5] |  0.0817|  -0.0241|  0.1972|
    #> |sertraline   |B.x.weeks[6] |  1.3705| -19.1535| 20.8753|
    #>
    #>
    #> Model Fit Statistics
    #> Deviance = 890.8
    #> Residual deviance = 191.1
    #> Deviance Information Criterion (DIC) = 972.7

Within the output, a separate parameter (named `B.x.weeks[]`) has been estimated for each agent that corresponds to the effect of an additional week of study follow-up on the relative effect of the agent versus Placebo. Note that due to the inclusion of weeks as a continuous covariate, we are assuming a linear effect modification due to study follow-up.

### Random effect interaction

Alternatively, the effect modification for different agents versus the network reference agent can be assumed to be exchangeable/shared across the network about a common mean, \(\hat{\psi}\), with a between-agent standard deviation of \(\tau_\psi\):

\[\psi_{1,a_{i,k}} \sim N(\hat{\psi}, \tau^2_\psi)\]

    # Regress for continuous weeks Random effect modification across all agents vs
    # Placebo
    ssrimod.r <- mbnma.run(ssrinet, fun = dfpoly(degree = 2), regress = ~x.weeks, regress.effect = "random")

    summary(ssrimod.r)
    #> ========================================
    #> Dose-response MBNMA
    #> ========================================
    #>
    #> Likelihood: binomial
    #> Link function: logit
    #> Dose-response function: fpoly
    #>
    #> Pooling method
    #>
    #> Method: Common (fixed) effects estimated for relative effects
    #>
    #>
    #> beta.1 dose-response parameter results
    #>
    #> Pooling: relative effects for each agent
    #>
    #> |Agent        |Parameter |  Median|    2.5%|  97.5%|
    #> |:------------|:---------|-------:|-------:|------:|
    #> |citalopram   |beta.1[2] | -0.0845| -0.5706| 0.2247|
    #> |escitalopram |beta.1[3] |  0.2362| -0.2284| 0.5291|
    #> |fluoxetine   |beta.1[4] |  0.2578| -0.0170| 0.5143|
    #> |paroxetine   |beta.1[5] | -0.1442| -0.4958| 0.1754|
    #> |sertraline   |beta.1[6] |  0.6064| -0.0260| 1.2126|
    #>
    #>
    #> beta.2 dose-response parameter results
    #>
    #> Pooling: relative effects for each agent
    #>
    #> |Agent        |Parameter |  Median|    2.5%|  97.5%|
    #> |:------------|:---------|-------:|-------:|------:|
    #> |citalopram   |beta.2[2] |  0.0668| -0.0205| 0.1998|
    #> |escitalopram |beta.2[3] | -0.0329| -0.1329| 0.1188|
    #> |fluoxetine   |beta.2[4] | -0.0610| -0.1358| 0.0213|
    #> |paroxetine   |beta.2[5] |  0.0976| -0.0045| 0.2073|
    #> |sertraline   |beta.2[6] | -0.1011| -0.2208| 0.0333|
    #>
    #>
    #> power.1 dose-response parameter results
    #>
    #> Assigned a numeric value: 0
    #>
    #> power.2 dose-response parameter results
    #>
    #> Assigned a numeric value: 0
    #>
    #> Meta-regression
    #>
    #> Covariates interacting with study-level relative effects: x.weeks
    #> Random (exchangeable) covariate-by-treatment effects
    #>
    #>
    #> |Regression effect |Parameter | Median|    2.5%|  97.5%|
    #> |:-----------------|:---------|------:|-------:|------:|
    #> |Random effect     |B.x.weeks | 0.0279| -0.0658| 0.1335|
    #>
    #>
    #> Standard deviation for random covariate-by-treatment effects
    #>
    #> |Parameter    | Median|   2.5%|  97.5%|
    #> |:------------|------:|------:|------:|
    #> |sd.B.x.weeks |  0.061| 0.0035| 0.3098|
    #>
    #>
    #> Model Fit Statistics
    #> Deviance = 891.1
    #> Residual deviance = 191.4
    #> Deviance Information Criterion (DIC) = 982.2

In this case only a single regression paramter is estimated (`B.x.weeks`), which corresponds to the mean effect of an additional week of study follow-up on the relative effect of an active agent versus Placebo. A parameter is also estimated for the between-agent standard deviation, `sd.B.x.weeks`.

### Common effect interaction

This is the strongest assumption for network meta-regression, and it implies that effect modification is common (equal) for all agents versus the network reference agent:

\[\psi_{1,a_{i,k}} =\hat{\psi}\]

    # Regress for categorical weeks Common effect modification across all agents vs
    # Placebo
    ssrimod.c <- mbnma.run(ssrinet, fun = dfpoly(degree = 2), regress = ~r.weeks, regress.effect = "common")

    summary(ssrimod.c)
    #> ========================================
    #> Dose-response MBNMA
    #> ========================================
    #>
    #> Likelihood: binomial
    #> Link function: logit
    #> Dose-response function: fpoly
    #>
    #> Pooling method
    #>
    #> Method: Common (fixed) effects estimated for relative effects
    #>
    #>
    #> beta.1 dose-response parameter results
    #>
    #> Pooling: relative effects for each agent
    #>
    #> |Agent        |Parameter |  Median|    2.5%|  97.5%|
    #> |:------------|:---------|-------:|-------:|------:|
    #> |citalopram   |beta.1[2] | -0.0678| -0.3319| 0.1941|
    #> |escitalopram |beta.1[3] |  0.2473|  0.0147| 0.4744|
    #> |fluoxetine   |beta.1[4] |  0.2177| -0.0275| 0.4663|
    #> |paroxetine   |beta.1[5] | -0.1076| -0.4041| 0.1742|
    #> |sertraline   |beta.1[6] |  0.5788|  0.1109| 1.0601|
    #>
    #>
    #> beta.2 dose-response parameter results
    #>
    #> Pooling: relative effects for each agent
    #>
    #> |Agent        |Parameter |  Median|    2.5%|   97.5%|
    #> |:------------|:---------|-------:|-------:|-------:|
    #> |citalopram   |beta.2[2] |  0.0591| -0.0098|  0.1305|
    #> |escitalopram |beta.2[3] | -0.0288| -0.1099|  0.0510|
    #> |fluoxetine   |beta.2[4] | -0.0552| -0.1264|  0.0131|
    #> |paroxetine   |beta.2[5] |  0.0861| -0.0048|  0.1791|
    #> |sertraline   |beta.2[6] | -0.1004| -0.1969| -0.0049|
    #>
    #>
    #> power.1 dose-response parameter results
    #>
    #> Assigned a numeric value: 0
    #>
    #> power.2 dose-response parameter results
    #>
    #> Assigned a numeric value: 0
    #>
    #> Meta-regression
    #>
    #> Covariates interacting with study-level relative effects: r.weeks4, r.weeks5, r.weeks6, r.weeks9, r.weeks10
    #> Common (identical) covariate-by-treatment effects
    #>
    #>
    #> |Regression effect |Parameter   |  Median|    2.5%|   97.5%|
    #> |:-----------------|:-----------|-------:|-------:|-------:|
    #> |Common effect     |B.r.weeks10 |  0.1525| -0.2554|  0.5637|
    #> |Common effect     |B.r.weeks4  | -0.7260| -1.4185| -0.0548|
    #> |Common effect     |B.r.weeks5  |  0.8118| -0.1705|  1.8553|
    #> |Common effect     |B.r.weeks6  |  0.0587| -0.1343|  0.2453|
    #> |Common effect     |B.r.weeks9  |  0.4223| -0.4313|  1.2922|
    #>
    #>
    #> Model Fit Statistics
    #> Deviance = 891.4
    #> Residual deviance = 191.7
    #> Deviance Information Criterion (DIC) = 970.2

In this case we have performed the network meta-regression on study follow-up (weeks) as a categorical covariate. Therefore, although only a single parameter is estimated for each effect modifying term, there is a separate term for each category of week and a linear relationship for effect modification is no longer assumed.

### Alternative assumptions

Although this is beyond the capability of `MBNMAdose`, one could envision a more complex model in which the interaction effect also varied by a dose-response relationship, rather than assuming an effect by agent/class or across the whole network. This would in principle contain fewer parameters than a fully independent interaction model (in which a separate regression covariate is estimated for each treatment in the dataset).

### Aggregation bias

Note that adjusting for aggregated patient-level covariates (e.g. mean age, % males, etc.) whilst using a non-identity link function can introduce aggregation bias. This is a form of ecological bias that biases treatment effects towards the null and is typically more severe where treatment effects are strong and where the link function is highly non-linear (Dias et al. 2011). This can be resolved by performing a patient-level regression, but Individual Participant Data are required for this and such an analysis is outside the scope of `MBNMAdose`.

## Prediction using effect modifying covariates

Models fitted with meta-regression can also be used to make predictions for a specified set of covariate values. This includes when estimating relative effects using `get.relative()`. An additional argument `regress.vals` can be used to provide a named vector of covariate values at which to make predictions.

    # For a continuous covariate, make predictions at 5 weeks follow-up
    pred <- predict(ssrimod.a, regress.vals = c(x.weeks = 5))
    plot(pred)

Predictions are very uncertain for Sertraline, as studies only investigated this agent at 6 weeks follow-up and therefore the agent-specific effect modification is very poorly estimated.

    # For a categorical covariate, make predictions at 10 weeks follow-up
    regress.p <- c(r.weeks10 = 1, r.weeks4 = 0, r.weeks5 = 0, r.weeks6 = 0, r.weeks9 = 0)

    pred <- predict(ssrimod.c, regress.vals = regress.p)
    plot(pred)

Note that categorical covariates are modelled as multiple binary dummy covariates, and so a value for each of these must be included.

## References

Dias, S., A. J. Sutton, N. J. Welton, and A. E. Ades. 2011. “NICE DSU Technical Support Document 3: Heterogeneity: Subgroups, Meta-Regression, Bias and Bias, Adjustment.” Decision Support Unit, ScHARR, University of Sheffield. https://www.ncbi.nlm.nih.gov/books/NBK395886/pdf/Bookshelf_NBK395886.pdf.