Network meta-analysis models

The nma function fits network meta-analysis and (multilevel) network meta-regression models in Stan.

Usage

nma(
  network,
  consistency = c("consistency", "ume", "nodesplit"),
  trt_effects = c("fixed", "random"),
  regression = NULL,
  class_interactions = c("common", "exchangeable", "independent"),
  likelihood = NULL,
  link = NULL,
  ...,
  nodesplit = get_nodesplits(network, include_consistency = TRUE),
  prior_intercept = .default(normal(scale = 100)),
  prior_trt = .default(normal(scale = 10)),
  prior_het = .default(half_normal(scale = 5)),
  prior_het_type = c("sd", "var", "prec"),
  prior_reg = .default(normal(scale = 10)),
  prior_aux = .default(),
  prior_aux_reg = .default(),
  aux_by = NULL,
  aux_regression = NULL,
  QR = FALSE,
  center = TRUE,
  adapt_delta = NULL,
  int_thin = 0,
  int_check = TRUE,
  mspline_degree = 3,
  n_knots = 7,
  knots = NULL,
  mspline_basis = NULL
)

Arguments

network: An nma_data object, as created by the functions set_*(), combine_network(), or add_integration()
consistency: Character string specifying the type of (in)consistency model to fit, either "consistency", "ume", or "nodesplit"
trt_effects: Character string specifying either "fixed" or "random" effects
regression: A one-sided model formula, specifying the prognostic and effect-modifying terms for a regression model. Any references to treatment should use the .trt special variable, for example specifying effect modifier interactions as variable:.trt (see details).
class_interactions: Character string specifying whether effect modifier interactions are specified as "common", "exchangeable", or "independent".
likelihood: Character string specifying a likelihood, if unspecified will be inferred from the data (see details)
link: Character string specifying a link function, if unspecified will default to the canonical link (see details)
...: Further arguments passed to sampling(), such as iter, chains, cores, etc.
nodesplit: For consistency = "nodesplit", the comparison(s) to split in the node-splitting model(s). Either a length 2 vector giving the treatments in a single comparison, or a 2 column data frame listing multiple treatment comparisons to split in turn. By default, all possible comparisons will be chosen (see get_nodesplits()).
prior_intercept: Specification of prior distribution for the intercept
prior_trt: Specification of prior distribution for the treatment effects
prior_het: Specification of prior distribution for the heterogeneity (if trt_effects = "random")
prior_het_type: Character string specifying whether the prior distribution prior_het is placed on the heterogeneity standard deviation \(\tau\) ("sd", the default), variance \(\tau^2\) ("var"), or precision \(1/\tau^2\) ("prec").
prior_reg: Specification of prior distribution for the regression coefficients (if regression formula specified)
prior_aux: Specification of prior distribution for the auxiliary parameter, if applicable (see details). For likelihood = "gengamma" this should be a list of prior distributions with elements sigma and k.
prior_aux_reg: Specification of prior distribution for the auxiliary regression parameters, if aux_regression is specified (see details).
aux_by: Vector of variable names listing the variables to stratify the auxiliary parameters by. Currently only used for survival models, see details. Cannot be used with aux_regression.
aux_regression: A one-sided model formula giving a regression model for the auxiliary parameters. Currently only used for survival models, see details. Cannot be used with aux_by.
QR: Logical scalar (default FALSE), whether to apply a QR decomposition to the model design matrix
center: Logical scalar (default TRUE), whether to center the (numeric) regression terms about the overall means
adapt_delta: See adapt_delta for details
int_thin: A single integer value, the thinning factor for returning cumulative estimates of integration error. Saving cumulative estimates is disabled by int_thin = 0, which is the default.
int_check: Logical, check sufficient accuracy of numerical integration by fitting half of the chains with n_int/2? When TRUE, Rhat and n_eff diagnostic warnings will be given if numerical integration has not sufficiently converged (suggesting increasing n_int in add_integration()). Default TRUE, but disabled (FALSE) when int_thin > 0.
mspline_degree: Non-negative integer giving the degree of the M-spline polynomial for likelihood = "mspline". Piecewise exponential hazards (likelihood = "pexp") are a special case with mspline_degree = 0.
n_knots: For mspline and pexp likelihoods, a non-negative integer giving the number of internal knots for partitioning the baseline hazard into intervals. The knot locations within each study will be determined by the corresponding quantiles of the observed event times, plus boundary knots at the earliest entry time (0 with no delayed entry) and the maximum event/censoring time. For example, with n_knots = 3, the internal knot locations will be at the 25%, 50%, and 75% quantiles of the observed event times. The default is n_knots = 7; overfitting is avoided by shrinking towards a constant hazard with a random walk prior (see details). If aux_regression is specified then a single set of knot locations will be calculated across all studies in the network. See make_knots() for more details on the knot positioning algorithms. Ignored when knots is specified.
knots: For mspline and pexp likelihoods, a named list of numeric vectors of knot locations (including boundary knots) for each of the studies in the network. Currently, each vector must have the same length (i.e. each study must use the same number of knots). Alternatively, a single numeric vector of knot locations can be provided which will be shared across all studies in the network. If unspecified (the default), the knots will be chosen based on n_knots as described above. If aux_regression is specified then knots should be a single numeric vector of knot locations which will be shared across all studies in the network. make_knots() can be used to help specify knots directly, or to investigate knot placement prior to model fitting.
mspline_basis: Instead of specifying mspline_degree and n_knots or knots, a named list of M-spline bases (one for each study) can be provided with mspline_basis which will be used directly. In this case, all other M-spline options will be ignored.

Value

nma() returns a stan_nma object, except when consistency = "nodesplit" when a nma_nodesplit or nma_nodesplit_df object is returned. nma.fit() returns a stanfit object.

Details

When specifying a model formula in the regression argument, the usual formula syntax is available (as interpreted by model.matrix()). The only additional requirement here is that the special variable .trt should be used to refer to treatment. For example, effect modifier interactions should be specified as variable:.trt. Prognostic (main) effects and interactions can be included together compactly as variable*.trt, which expands to variable + variable:.trt (plus .trt, which is already in the NMA model).

For the advanced user, the additional specials .study and .trtclass are also available, and refer to studies and (if specified) treatment classes respectively. When node-splitting models are fitted (consistency = "nodesplit") the special .omega is available, indicating the arms to which the node-splitting inconsistency factor is added.

See ?priors for details on prior specification. Default prior distributions are available, but may not be appropriate for the particular setting and will raise a warning if used. No attempt is made to tailor these defaults to the data provided. Please consider appropriate prior distributions for the particular setting, accounting for the scales of outcomes and covariates, etc. The function plot_prior_posterior() may be useful in examining the influence of the chosen prior distributions on the posterior distributions, and the summary() method for nma_prior objects prints prior intervals.

Likelihoods and link functions

Currently, the following likelihoods and link functions are supported for each data type:

Data type	Likelihood	Link function
Binary	`bernoulli`, `bernoulli2`	`logit`, `probit`, `cloglog`
Count	`binomial`, `binomial2`	`logit`, `probit`, `cloglog`
Rate	`poisson`	`log`
Continuous	`normal`	`identity`, `log`
Ordered	`ordered`	`logit`, `probit`, `cloglog`
Survival	`exponential`, `weibull`, `gompertz`, `exponential-aft`, `weibull-aft`, `lognormal`, `loglogistic`, `gamma`, `gengamma`, `mspline`, `pexp`	`log`

The bernoulli2 and binomial2 likelihoods correspond to a two-parameter Binomial likelihood for arm-based AgD, which more closely matches the underlying Poisson Binomial distribution for the summarised aggregate outcomes in a ML-NMR model than the typical (one parameter) Binomial distribution (see Phillippo et al. 2020) .

When a cloglog link is used, including an offset for log follow-up time (i.e. regression = ~offset(log(time))) results in a model on the log hazard (see Dias et al. 2011) .

For survival data, all accelerated failure time models (exponential-aft, weibull-aft, lognormal, loglogistic, gamma, gengamma) are parameterised so that the treatment effects and any regression parameters are log Survival Time Ratios (i.e. a coefficient of \(\log(2)\) means that the treatment or covariate is associated with a doubling of expected survival time). These can be converted to log Acceleration Factors using the relation \(\log(\mathrm{AF}) = -\log(\mathrm{STR})\) (or equivalently \(\mathrm{AF} = 1/\mathrm{STR}\)).

Further details on each likelihood and link function are given by Dias et al. (2011) .

Auxiliary parameters

Auxiliary parameters are only present in the following models.

Normal likelihood with IPD

When a Normal likelihood is fitted to IPD, the auxiliary parameters are the arm-level standard deviations \(\sigma_{jk}\) on treatment \(k\) in study \(j\).

Ordered multinomial likelihood

When fitting a model to \(M\) ordered outcomes, the auxiliary parameters are the latent cutoffs between each category, \(c_0 < c_1 < \dots < c_M\). Only \(c_2\) to \(c_{M-1}\) are estimated; we fix \(c_0 = -\infty\), \(c_1 = 0\), and \(c_M = \infty\). When specifying priors for these latent cutoffs, we choose to specify priors on the differences \(c_{m+1} - c_m\). Stan automatically truncates any priors so that the ordering constraints are satisfied.

Survival (time-to-event) likelihoods

All survival likelihoods except the exponential and exponential-aft likelihoods have auxiliary parameters. These are typically study-specific shape parameters \(\gamma_j>0\), except for the lognormal likelihood where the auxiliary parameters are study-specific standard deviations on the log scale \(\sigma_j>0\).

The gengamma likelihood has two sets of auxiliary parameters, study-specific scale parameters \(\sigma_j>0\) and shape parameters \(k_j\), following the parameterisation of Lawless (1980) , which permits a range of behaviours for the baseline hazard including increasing, decreasing, bathtub and arc-shaped hazards. This parameterisation is related to that discussed by Cox et al. (2007) and implemented in the flexsurv package with \(Q = k^{-0.5}\). The parameterisation used here effectively bounds the shape parameter \(k\) away from numerical instabilities as \(k \rightarrow \infty\) (i.e. away from \(Q \rightarrow 0\), the log-Normal distribution) via the prior distribution. Implicitly, this parameterisation is restricted to \(Q > 0\) and so certain survival distributions like the inverse-Gamma and inverse-Weibull are not part of the parameter space; however, \(Q > 0\) still encompasses the other survival distributions implemented in this package.

For the mspline and pexp likelihoods, the auxiliary parameters are the spline coefficients for each study. These form a unit simplex (i.e. lie between 0 and 1, and sum to 1), and are given a random walk prior distribution. prior_aux specifies the hyperprior on the random walk standard deviation \(\sigma\) which controls the level of smoothing of the baseline hazard, with \(\sigma = 0\) corresponding to a constant baseline hazard.

The auxiliary parameters can be stratified by additional factors through the aux_by argument. For example, to allow the shape of the baseline hazard to vary between treatment arms as well as studies, use aux_by = c(".study", ".trt"). (Technically, .study is always included in the stratification even if omitted from aux_by, but we choose here to make the stratification explicit.) This is a common way of relaxing the proportional hazards assumption. The default is equivalent to aux_by = ".study" which stratifies the auxiliary parameters by study, as described above.

A regression model may be specified on the auxiliary parameters using aux_regression. This is useful if we wish to model departures from non-proportionality, rather than allowing the baseline hazards to be completely independent using aux_by. This is necessary if absolute predictions (e.g. survival curves) are required in a population for unobserved combinations of covariates; for example, if aux_by = .trt then absolute predictions may only be produced for the observed treatment arms in each study population, whereas if aux_regression = ~.trt then absolute predictions can be produced for all treatments in any population. For mspline and pexp likelihoods, the regression coefficients are smoothed over time using a random walk prior to avoid overfitting: prior_aux_reg specifies the hyperprior for the random walk standard deviation. For other parametric likelihoods, prior_aux_reg specifies the prior for the auxiliary regression coefficients.

References

Cox C, Chu H, Schneider MF, Muñoz A (2007). “Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution.” Statistics in Medicine, 26(23), 4352–4374. doi:10.1002/sim.2836 .

Dias S, Welton NJ, Sutton AJ, Ades AE (2011). “NICE DSU Technical Support Document 2: A generalised linear modelling framework for pair-wise and network meta-analysis of randomised controlled trials.” National Institute for Health and Care Excellence. https://www.sheffield.ac.uk/nice-dsu.

Lawless JF (1980). “Inference in the Generalized Gamma and Log Gamma Distributions.” Technometrics, 22(3), 409–419. doi:10.1080/00401706.1980.10486173 .

Phillippo DM, Dias S, Ades AE, Belger M, Brnabic A, Schacht A, Saure D, Kadziola Z, Welton NJ (2020). “Multilevel Network Meta-Regression for population-adjusted treatment comparisons.” Journal of the Royal Statistical Society: Series A (Statistics in Society), 183(3), 1189–1210. doi:10.1111/rssa.12579 .