MBNMAdose: Perform Network Meta-Regression

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  ≥ 2:

θi, k = μi + (f(x, βai, k) − f(x, βai, 1)) + (ψ1, ai, k − ψ1, ai, 1)

where θi, k is the linear predictor, μi is the baseline effect on arm 1, f(x, βai, k) is the dose-response function at dose x with dose-response parameters βai, k for agent a in arm k of study i. ψ1, ai, 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 ψ1, ai, 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] | -0.0924| -0.3600|  0.1629|
#> |escitalopram |beta.1[3] |  0.1228| -0.1396|  0.3956|
#> |fluoxetine   |beta.1[4] |  0.2685|  0.0361|  0.5053|
#> |paroxetine   |beta.1[5] | -0.1374| -0.4402|  0.1690|
#> |sertraline   |beta.1[6] |  1.4649| -7.7676| 10.5839|
#> 
#> 
#> beta.2 dose-response parameter results
#> 
#> Pooling: relative effects for each agent
#> 
#> |Agent        |Parameter |  Median|    2.5%|  97.5%|
#> |:------------|:---------|-------:|-------:|------:|
#> |citalopram   |beta.2[2] |  0.0678| -0.0037| 0.1401|
#> |escitalopram |beta.2[3] | -0.0057| -0.0930| 0.0784|
#> |fluoxetine   |beta.2[4] | -0.0654| -0.1334| 0.0018|
#> |paroxetine   |beta.2[5] |  0.0912| -0.0066| 0.1863|
#> |sertraline   |beta.2[6] | -0.1973| -1.1940| 0.7963|
#> 
#> 
#> 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] | -0.1602|  -0.3322|  0.0142|
#> |escitalopram |B.x.weeks[3] |  0.1632|  -0.0216|  0.3602|
#> |fluoxetine   |B.x.weeks[4] | -0.0287|  -0.1720|  0.1122|
#> |paroxetine   |B.x.weeks[5] |  0.0884|  -0.0242|  0.1956|
#> |sertraline   |B.x.weeks[6] |  1.8568| -17.6337| 21.0663|
#> 
#> 
#> Model Fit Statistics
#> Deviance = 890.2
#> Residual deviance = 190.5
#> Deviance Information Criterion (DIC) = 967.5

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, ψ̂, with a between-agent standard deviation of τψ:

ψ1, ai, k ∼ N(ψ̂, τψ2)

# 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] | -0.1103| -0.4700| 0.1732|
#> |escitalopram |beta.1[3] |  0.2579| -0.0151| 0.7551|
#> |fluoxetine   |beta.1[4] |  0.2391| -0.0372| 0.4998|
#> |paroxetine   |beta.1[5] | -0.0872| -0.4046| 0.2959|
#> |sertraline   |beta.1[6] |  0.6106|  0.0431| 1.1175|
#> 
#> 
#> beta.2 dose-response parameter results
#> 
#> Pooling: relative effects for each agent
#> 
#> |Agent        |Parameter |  Median|    2.5%|  97.5%|
#> |:------------|:---------|-------:|-------:|------:|
#> |citalopram   |beta.2[2] |  0.0736| -0.0027| 0.1730|
#> |escitalopram |beta.2[3] | -0.0408| -0.2114| 0.0504|
#> |fluoxetine   |beta.2[4] | -0.0560| -0.1340| 0.0250|
#> |paroxetine   |beta.2[5] |  0.0787| -0.0417| 0.1820|
#> |sertraline   |beta.2[6] | -0.1032| -0.2082| 0.0125|
#> 
#> 
#> 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 | 0.0267| -0.0676| 0.1219|
#> 
#> 
#> Standard deviation for random covariate-by-treatment effects
#> 
#> |Parameter    | Median|   2.5%|  97.5%|
#> |:------------|------:|------:|------:|
#> |sd.B.x.weeks | 0.0743| 0.0067| 0.2624|
#> 
#> 
#> Model Fit Statistics
#> Deviance = 890.2
#> Residual deviance = 190.5
#> Deviance Information Criterion (DIC) = 977.5

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:

ψ1, ai, k = ψ̂

# 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] | -0.0691| -0.3335| 0.2097|
#> |escitalopram |beta.1[3] |  0.2496|  0.0301| 0.4732|
#> |fluoxetine   |beta.1[4] |  0.2223| -0.0183| 0.4646|
#> |paroxetine   |beta.1[5] | -0.1144| -0.4138| 0.1934|
#> |sertraline   |beta.1[6] |  0.5965|  0.1323| 1.0675|
#> 
#> 
#> beta.2 dose-response parameter results
#> 
#> Pooling: relative effects for each agent
#> 
#> |Agent        |Parameter |  Median|    2.5%|   97.5%|
#> |:------------|:---------|-------:|-------:|-------:|
#> |citalopram   |beta.2[2] |  0.0594| -0.0155|  0.1295|
#> |escitalopram |beta.2[3] | -0.0297| -0.1072|  0.0476|
#> |fluoxetine   |beta.2[4] | -0.0569| -0.1250|  0.0119|
#> |paroxetine   |beta.2[5] |  0.0880| -0.0105|  0.1812|
#> |sertraline   |beta.2[6] | -0.1032| -0.2003| -0.0063|
#> 
#> 
#> 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 |  0.1480| -0.2519|  0.5494|
#> |Common effect     |B.r.weeks4  | -0.7300| -1.4624| -0.0377|
#> |Common effect     |B.r.weeks5  |  0.8164| -0.2470|  1.8572|
#> |Common effect     |B.r.weeks6  |  0.0533| -0.1328|  0.2522|
#> |Common effect     |B.r.weeks9  |  0.4120| -0.4235|  1.2815|
#> 
#> 
#> Model Fit Statistics
#> Deviance = 890.9
#> Residual deviance = 191.2
#> Deviance Information Criterion (DIC) = 967.2

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.