Bayesian Longitudinal Missing Data Models

Full Bayesian cost-effectiveness models to handle missing data in longitudinal outcomes under different missing data mechanism assumptions, using alternative parametric distributions for the effect and cost variables. The analysis is performed using the BUGS language, which is implemented in the software JAGS using the function R2jags::jags() The output is stored in an object of class missingHE.

Usage

lmdm(
  data,
  model.eff,
  model.cost,
  model.me = me ~ 1,
  model.mc = mc ~ 1,
  dist_e,
  dist_c,
  type,
  time_dep = "AR1",
  prob = c(0.025, 0.975),
  n.chains = 2,
  n.iter = 10000,
  n.burnin = floor(n.iter/2),
  inits = NULL,
  n.thin = 1,
  save_model = FALSE,
  prior = "default",
  center = FALSE,
  ...
)

Arguments

data: A data frame in which to find the longitudinal variables supplied in model.eff, model.cost (model formulas for effects and costs) and model.me, model.mc (model formulas for the missing effect and cost models). Among these, effectiveness, cost, time and treatment indicator variables must always be provided and named e, c, time and trt, respectively.
model.eff: A formula expression in conventional R linear modelling syntax. The response must be a health economic effectiveness outcome (e) whose name must correspond to that used in data. Any covariates in the model must be provided on the right-hand side of the formula. If there are no covariates, 1 should be specified on the right hand side of the formula. By default, covariates are placed on the "location" parameter of the distribution through a linear model. Random effects can also be specified for each model parameter. See details for how these can be specified.
model.cost: A formula expression in conventional R linear modelling syntax. The response must be a health economic cost outcome (c) whose name must correspond to that used in data. Any covariates in the model must be provided on the right-hand side of the formula. If there are no covariates, 1 should be specified on the right hand side of the formula. By default, covariates are placed on the "location" parameter of the distribution through a linear model. A joint bivariate distribution for effects and costs can be specified by including e on the right-hand side of the formula for the costs model. Random effects can also be specified for each model parameter. See details for how these can be specified.
model.me: A formula expression in conventional R linear modelling syntax. The response must be indicated with the term me (missing effects) and any covariates must be provided on the right-hand side of the formula. If there are no covariates, 1 should be specified on the right hand side of the formula. By default, covariates are placed on the "probability" parameter for the missing effects through a logistic-linear model. Random effects can also be specified for each model parameter. See details for how these can be specified.
model.mc: A formula expression in conventional R linear modelling syntax. The response must be indicated with the term mc (missing costs) and any covariates must be provided on the right-hand side of the formula. If there are no covariates, 1 should be specified on the right hand side of the formula. By default, covariates are placed on the "probability" parameter for the missing costs through a logistic-linear model. Random effects can also be specified for each model parameter. See details for how these can be specified.
dist_e: Distribution assumed for the effects. Current available choices are: Normal (norm), Beta (beta), Gamma (gamma), Exponential (exp), Weibull (weib), Logistic (logis), Poisson (pois), Negative Binomial (negbin) or Bernoulli (bern).
dist_c: Distribution assumed for the costs. Current available choices are: Normal ('norm'), Gamma (gamma) or LogNormal (lnorm).
type: Type of missingness mechanism assumed. Choices are Missing At Random (MAR) and Missing Not At Random (MNAR).
time_dep: Type of dependence structure assumed between effectiveness and cost outcomes. Current choices include: autoregressive structure of order one (AR1) - default, bivariate at each time (biv) and independence (none).
prob: A numeric vector of probabilities within the range (0, 1), representing the upper and lower CI sample quantiles to be calculated and returned for the imputed values.
n.chains: Number of chains.
n.iter: Number of iterations.
n.burnin: Number of warmup iterations.
inits: A list with elements equal to the number of chains selected; each element of the list is itself a list of starting values for the JAGS model, or a function creating (possibly random) initial values. If inits is NULL, JAGS will generate initial values for all the model parameters.
n.thin: Thinning interval.
save_model: Logical. If save_model is TRUE, a txt file containing the model code is printed in the current working directory.
prior: A list containing the hyperprior values provided by the user. Each element of this list must be a vector containing the user-provided prior distribution and parameter values and must be named with the name of the corresponding parameter. For example, the hyperprior values for the standard deviation parameter for the effects can be provided using the list prior = list("sigma.prior.e" = c("unif", 0, 10)). For more information about how to provide prior hypervalues for different types of parameters and models see details. If prior is set to default, the default values will be used.
center: Logical. If center is TRUE, all covariates in both the effect and cost models will be centred.
...: Additional arguments that can be provided by the user. Examples are the additional arguments that can be provided to the function BCEA::bcea() to summarise the health economic evaluation results.

Value

An object of the class 'missingHE' containing the following elements:

data_set: A list containing the original data set provided in data (see Arguments). Additional information is also included about, among others, the number of observed and missing individuals, the total number of individuals by treatment arm and the indicator vectors for the missing values.
model_output: A list containing the output of a JAGS model generated from the functions R2jags::jags(), and the posterior samples for the main parameters of the model.
cea: A list containing the output of the economic evaluation performed using the function BCEA::bcea().
type: A character variable that indicate which type of missing value mechanism used to run the model, either MAR or MNAR (see details).
data_format: A character variable that indicates which type of analysis was conducted, either using a wide or long dataset.
time_dep: A character variable that indicate which type of time dependence assumption was made, either none or AR1.
w_e: A numeric vector containing the weight values used to calculate aggregated effectiveness mean parameters used in the cea output.
w_c: A numeric vector containing the weight values used to calculate aggregated cost mean parameters used in the cea output.
call: The matched call.
prior: The custom prior choices specified by the user. If no custom prior specified, default priors are used (see details).

Details

Depending on the distributions specified for the outcome variables in the arguments dist_e and dist_c and the type of missingness mechanism specified in the argument type, different models are built and run in the background by the function lmdm. These models fit multinomial logistic regressions to estimate the probability of a given missing data pattern k (1 = completers, 2 = intermittent, 3 = dropout) in one or both longitudinal outcomes. A simple example can be used to show how longitudinal missing data models are specified. Consider a longitudinal data set comprising a response variable $y$ measured at $S$ occasions and a set of centered covariate $X_j$, for $i = j, ..., J$. For each subject in the trial $i = 1, ..., n$ and time $s = 1, ..., S$ we define an indicator variable $m_i$ taking value k = 1 if the $i$-th individual is associated with no missing value (completer), a value k = 2 for intermittent missingness over the study period, and a value k = 3 for dropout. This is modelled as:

$$m_i ~ Multinomial(\pi^k_i)$$ $$\pi^k_i = \phi^k_i/\sum\phi_i$$ $$log(\phi^k_i) = \sum\gamma^k_j X_j + \delta^k y_i$$ where

$\pi^k_i$ is the individual probability of a missing value in $y$ for pattern $k$ at a given time.
$\gamma^k_j$ represents the impact on the missing data probability in $y$ of the covariate $X_j$ for pattern $k$ at a given time.
$\delta^k$ represents the impact on the missing data probability in $y$ for pattern $k$ of missingness itself at a given time.

When $\delta = 0$ the model assumes a MAR mechanism, while when $\delta != 0$ the mechanism is MNAR. For the parameters indexing the missingness model, the default prior distributions assumed are:

$\gamma^k_j ~ Normal(0, 0.01)$.
$\delta^k ~ Normal(0, 1)$.

When user-defined hyperprior values are supplied via the argument prior in the function lmdm, the elements of this list (see Arguments) must be vectors containing the user-provided distribution name and hyperprior values and must take specific names according to the parameters they are associated with. Specifically, the names for the parameters indexing the model which are accepted by missingHE are the following:

auxiliary parameters $\sigma$: sigma.prior.e (effects) and/or sigma.prior.c (costs).
covariate parameters $\alpha_j$ and $\beta_j$: alpha.prior (effects) and/or beta.prior (costs).
covariate parameters in the missingness model $\gamma_j$ (if covariate data provided): gamma.prior.e (effects) and/or gamma.prior.c (costs).
mnar parameter $\delta$: delta.prior.e (effects) and/or delta.prior.c (costs).

For simplicity, here we have assumed that the set of covariates $X_j$ used in the models for the effects/costs and in the model of the missing effect/cost values is the same. However, it is possible to specify different sets of covariates for each model using the arguments in the function lmdm (see Arguments).

For each model, random effects can also be specified for each parameter by adding the term + (x | z) to each model formula, where x is the fixed regression coefficient for which also the random effects are desired and z is the clustering variable across which the random effects are specified (must be the name of a factor variable in the dataset).

For an overview of the general modeling framework and additional package utilities, see missingHE-package.

References

Gabrio, A., Daniels, MJ. Baio, G. (2019). A Bayesian parametric approach to handle missing longitudinal outcome data in trial-based health economic evaluations. Journal of the Royal Statistical Society Series A: Statistics in Society, 183(2), 607-629 <10.1111/rssa.12522>

Mason, AJ. Gomes, M. Carpenter, J. Grieve, R. (2021). Flexible Bayesian longitudinal models for cost‐effectiveness analyses with informative missing data. Health economics, 30(12), 3138-3158. <10.1002/hec.4408>

Daniels, MJ. Hogan, JW. Missing Data in Longitudinal Studies: strategies for Bayesian modelling and sensitivity analysis, CRC/Chapman Hall. doi:10.1201/9781420011180

Baio, G.(2012). Bayesian Methods in Health Economics. CRC/Chapman Hall, London. doi:10.1201/b13099

Plummer, M. JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. (2003). https://www.r-project.org/conferences/DSC-2003/Proceedings/Plummer.pdf

Author

Andrea Gabrio

Examples

if (isTRUE(tryCatch(requireNamespace("rjags", quietly = TRUE), error = function(e) FALSE))) {
# Quick example to run using subset of PBS dataset

# Load longitudinal dataset

PBS.long <- PBS

# \donttest{
# Run the model using the long_miss function assuming a MAR mechanism
# Use only 100 iterations to run a quick check
model.long <- lmdm(data = PBS.long, model.eff = e ~ trt, model.cost = c ~ trt,
   model.me = me ~ 1, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 100, time_dep = "none")

# Extract regression coefficient estimates
coef(model.long)
#

# Summarise the CEA information from the model
summary(model.long)

# Further examples which take longer to run
model.long <- lmdm(data = PBS.long, model.eff = e ~ trt, model.cost = c ~ trt + age,
   model.me = me ~ 1, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 500, time_dep = "none")

# Use looic to assess model fit
pic.looic <- pic(model.long, criterion = "looic")
pic.looic

# Show density plots for all parameters
diag.den <- diagnostic(model.long, type = "denplot", param = "alpha")

# Plots of imputations for all effect data
p1 <- plot(model.long, class = "scatter", outcome = "effects", time.plot = "all")

# Summarise the CEA results
summary(model.long)

# }
}
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 2852
#>    Unobserved stochastic nodes: 112
#>    Total graph size: 16272
#> 
#> Initializing model
#> 
#> $Effects
#>                Mean    SD    QL    QU
#> (Intercept).1 0.475 0.035 0.408 0.533
#> trt2.1        0.090 0.046 0.012 0.190
#> (Intercept).2 0.501 0.031 0.443 0.556
#> trt2.2        0.138 0.047 0.039 0.230
#> (Intercept).3 0.478 0.029 0.419 0.526
#> trt2.3        0.138 0.040 0.064 0.207
#> 
#> $Costs
#>                  Mean     SD      QL      QU
#> (Intercept).1 147.799 48.826  62.582 248.354
#> trt2.1        100.570 39.491  34.235 167.534
#> (Intercept).2  73.607 33.988  17.269 139.259
#> trt2.2         45.972 28.579 -23.516  94.459
#> (Intercept).3  49.996 30.584 -21.641 104.961
#> trt2.3         30.893 31.002 -21.537  95.492
#> 
#> 
#>  Cost-effectiveness analysis summary 
#>  
#>  CE parameter estimates under MAR assumption
#>  
#>  Mean effects by intervention 
#>    Mean    SD    QL    QU
#> 1 0.491 0.032 0.439 0.554
#> 2 0.615 0.031 0.554 0.673
#> 
#>  
#>  Mean costs by intervention 
#>   Mean  SD   QL   QU
#> 1  244 435 -555 1108
#> 2  451 450 -467 1340
#> 
#>  
#>  Mean net monetary benefit by intervention and wtp = 50000 
#>    Mean   SD    QL    QU
#> 1 24292 1669 21540 27643
#> 2 30293 1551 27081 33234
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 2852
#>    Unobserved stochastic nodes: 115
#>    Total graph size: 16835
#> 
#> Initializing model
#> 
#> 
#>  Cost-effectiveness analysis summary 
#>  
#>  CE parameter estimates under MAR assumption
#>  
#>  Mean effects by intervention 
#>    Mean    SD    QL    QU
#> 1 0.489 0.022 0.449 0.531
#> 2 0.612 0.023 0.564 0.656
#> 
#>  
#>  Mean costs by intervention 
#>   Mean  SD   QL   QU
#> 1 5395 343 4746 6078
#> 2 5834 352 5173 6526
#> 
#>  
#>  Mean net monetary benefit by intervention and wtp = 50000 
#>    Mean   SD    QL    QU
#> 1 19042 1129 16809 21224
#> 2 24779 1207 22298 27224
#