Bayesian Pattern Mixture Models

Full Bayesian cost-effectiveness models to handle missing data in the outcomes under different missing data mechanism assumptions, using alternative parametric distributions for the effect and cost variables and using a pattern mixture model approach to identify the model. 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

pattern(
  data,
  model.eff,
  model.cost,
  dist_e,
  dist_c,
  type,
  restriction = "CC",
  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 variables supplied in model.eff, model.cost (model formulas for effects and costs). Among these, effectiveness, cost and treatment indicator variables must always be provided and named e, c 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.
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).
restriction: type of identifying restriction to be imposed to identify the distributions of the missing data in each pattern. Available choices are: complete case restriction (CC) - default - or available case restriction (AC).
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.
restriction: A character name that indicates under which type of restrictions the model was identified (see details).
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 pattern mixture models are built and run in the background by the function pattern. The model for the outcomes is fitted in each missingness pattern and the parameters indexing the missing data distributions are identified using: the corresponding parameters identified from the observed data in other patterns (under MAR); or a combination of the parameters identified by the observed data and some sensitivity parameters (under MNAR). A simple example can be used to show how pattern mixture models are specified. Consider a data set comprising a response variable $y$ and a set of centered covariates $X_j$. We denote with $d_i$ the patterns' indicator variable for each subject in the trial $i = 1, ..., n$ such that: $d_i = 1$ indicates the completers (both e and c observed), $d_i = 2$ and $d_i = 3$ indicate that only the costs or effects are observed, respectively, while $d_i = 4$ indicates that neither of the two outcomes is observed. $d_i$ is assigned a multinomial distribution, which probabilities are modelled using a Dirichlet prior. Next, the outcomes model is fitted in each pattern and parameters that cannot be identified in each pattern (d = 2, 3, 4), e.g. $mu_e[d]$ and $mu_c[d]$, are identified using some restrictions based on the parameters estimated from other patterns. Two choices are currently available: the complete cases (CC) or available cases (AC). For example, using the CC restriction, the parameters indexing the distributions of the missing data are identified as:

$$mu_e[2] = \mu_e[4] = \mu_e[1] + \delta_e$$ $$mu_c[3] = \mu_c[4] = \mu_c[1] + \delta_c$$ where

$\mu_e[1]$ is the effects mean for the completers.
$\mu_c[1]$ is the costs mean for the completers.
$\delta_e$ is the sensitivity parameters associated with the marginal effects mean.
$\delta_c$ is the sensitivity parameters associated with the marginal costs mean.

If the AC restriction is chosen, only the parameters estimated from the observed data in pattern 2 (costs) and pattern 3 (effects) are used to identify those in the other patterns. When $\delta_e = 0$ and $\delta_c = 0$ the model assumes a MAR mechanism. When $\delta_e != 0$ and/or $\delta_c != 0$ MNAR departures for the effects and/or costs are explored with priors on sensitivity parameters that must be provided by the user (see Arguments).

When user-defined hyperprior values are supplied via the argument prior in the function pattern, 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).
sensitivity parameters $\delta$: delta.prior.e (effects) and/or delta.prior.c (costs).
missingness patterns' probabilities $\pi$: patterns.prior.

For each outcome 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

Molenberghs, G. Verbeke G. Pattern-Mixture Models, Linear Mixed Models for Longitudinal Data, 275-293 <10.1007/978-1-4419-0300-6_18>

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 MenSS dataset
MenSS.subset <- MenSS[50:100, ]

# Run the model using the pattern function assuming a MAR mechanism
# Use only 100 iterations to run a quick check
model.pattern <- pattern(data = MenSS.subset,model.eff = e ~ trt, model.cost = c ~ trt,
   dist_e = "norm", dist_c = "norm", type = "MAR", n.chains = 2, n.iter = 100)

# Print the results of the JAGS model
print(model.pattern)
#

# Use dic information criterion to assess model fit
pic.dic <- pic(model.pattern, criterion = "dic", cases = "cc")
pic.dic
#

# Extract regression coefficient estimates
coef(model.pattern)
#
DONTSHOW({
# Use waic information criterion to assess model fit
pic.waic <- pic(model.pattern, criterion = "waic", cases = "cc")
pic.waic
})

# Assess model convergence using graphical tools
# Produce histograms of the posterior samples for the mean effects
diag.hist <- diagnostic(model.pattern, type = "histogram", param = "mu.e")
#

# Compare observed effect data with imputations from the model
# using plots (posteiror means and credible intervals)
p1 <- plot(model.pattern, class = "scatter", outcome = "effects")
#

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

# \donttest{
# Further examples which take longer to run
model.pattern <- pattern(data = MenSS, model.eff = e ~ trt, model.cost = c ~ trt + e,
   dist_e = "norm", dist_c = "norm", type = "MAR", n.chains = 2, n.iter = 500)
#
# Print results for all imputed values
print(model.pattern)

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

# Show density plots for mean costs parameters
diag.den <- diagnostic(model.pattern, type = "denplot", param = "mu.c")

# Plots of imputations for all data
p1 <- plot(model.pattern, class = "scatter", outcome = "both")

# Summarise the CEA results
summary(model.pattern)

# }
}
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 81
#>    Unobserved stochastic nodes: 79
#>    Total graph size: 610
#> 
#> Initializing model
#> 
#> Call:
#> pattern(model.eff = e ~ trt, model.cost = c ~ trt, data = MenSS.subset, 
#>     dist_e = "norm", dist_c = "norm", type = "MAR", n.chains = 2, 
#>     n.iter = 100)
#> 
#> Coefficients:
#> $model.eff
#> (Intercept).1        trt2.1 (Intercept).2        trt2.2 
#>         0.907         0.065         0.907         0.065 
#> 
#> $model.cost
#> (Intercept).1        trt2.1 (Intercept).2        trt2.2 
#>        51.971        21.337        51.971        21.337 
#> 
#> $model.pattern
#>    p1    p2 
#> 0.296 0.704 
#> 
#> $Effects
#>                Mean    SD     QL    QU
#> (Intercept).1 0.907 0.036  0.844 0.983
#> trt2.1        0.907 0.036  0.844 0.983
#> (Intercept).2 0.065 0.072 -0.071 0.195
#> trt2.2        0.065 0.072 -0.071 0.195
#> 
#> $Costs
#>                 Mean     SD      QL      QU
#> (Intercept).1 51.971 26.820   3.901 104.555
#> trt2.1        51.971 26.820   3.901 104.555
#> (Intercept).2 21.337 31.658 -33.686  84.128
#> trt2.2        21.337 31.658 -33.686  84.128
#> 
#> 
#>  Cost-effectiveness analysis summary 
#>  
#>  CE parameter estimates under MAR assumption
#>  
#>  Mean effects by intervention 
#>    Mean    SD    QL    QU
#> 1 0.907 0.038 0.847 0.991
#> 2 0.971 0.063 0.849 1.103
#> 
#>  
#>  Mean costs by intervention 
#>   Mean   SD     QL  QU
#> 1 51.9 31.3 -0.692 107
#> 2 74.9 43.2  7.060 156
#> 
#>  
#>  Mean net monetary benefit by intervention and wtp = 50000 
#>    Mean   SD    QL    QU
#> 1 45306 1898 42299 49474
#> 2 48469 3164 42374 55110
#> Compiling model graph
#>    Resolving undeclared variables
#>    Allocating nodes
#> Graph information:
#>    Observed stochastic nodes: 251
#>    Unobserved stochastic nodes: 234
#>    Total graph size: 2197
#> 
#> Initializing model
#> 
#> Call:
#> pattern(model.eff = e ~ trt, model.cost = c ~ trt + e, data = MenSS, 
#>     dist_e = "norm", dist_c = "norm", type = "MAR", n.chains = 2, 
#>     n.iter = 500)
#> 
#> Coefficients:
#> $model.eff
#> (Intercept).1        trt2.1 (Intercept).2        trt2.2 
#>         0.904        -0.004         0.904        -0.004 
#> 
#> $model.cost
#> (Intercept).1        trt2.1 (Intercept).2        trt2.2           e.1 
#>        78.678        25.821        78.678        25.821        -7.551 
#>           e.2 
#>        -7.551 
#> 
#> $model.pattern
#>   p1   p2 
#> 0.29 0.71 
#> 
#> 
#>  Cost-effectiveness analysis summary 
#>  
#>  CE parameter estimates under MAR assumption
#>  
#>  Mean effects by intervention 
#>    Mean    SD    QL    QU
#> 1 0.905 0.023 0.859 0.949
#> 2 0.900 0.028 0.847 0.957
#> 
#>  
#>  Mean costs by intervention 
#>    Mean   SD   QL  QU
#> 1  78.5 25.8 25.0 124
#> 2 105.6 34.6 36.5 167
#> 
#>  
#>  Mean net monetary benefit by intervention and wtp = 50000 
#>    Mean   SD    QL    QU
#> 1 45154 1173 42889 47345
#> 2 44917 1418 42233 47780
#