Package 'missingHE' reference manual

Title:	Missing Outcome Data in Health Economic Evaluation
Description:	Contains a suite of functions for health economic evaluations with missing outcome data. The package can fit different types of statistical models under a fully Bayesian approach using the software 'JAGS' (which should be installed locally and which is loaded in 'missingHE' via the 'R' package 'R2jags'). Three classes of models can be fitted under a variety of missing data assumptions: selection models, pattern mixture models and hurdle models. In addition to model fitting, 'missingHE' provides a set of specialised functions to assess model convergence and fit, and to summarise the statistical and economic results using different types of measures and graphs. The methods implemented are described in Mason (2018) <doi:10.1002/hec.3793>, Molenberghs (2000) <doi:10.1007/978-1-4419-0300-6_18> and Gabrio (2019) <doi:10.1002/sim.8045>.
Authors:	Andrea Gabrio [aut, cre]
Maintainer:	Andrea Gabrio <[email protected]>
License:	GPL-2
Version:	1.5.0
Built:	2024-10-01 07:04:40 UTC
Source:	CRAN

An internal function to detect the random effects component from an object of class formula

Description

An internal function to detect the random effects component from an object of class formula

Usage

anyBars(term)
anyBars(term)

Arguments

term

formula to be processed

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

Extract regression coefficient estimates from objects in the class `missingHE`

Description

Produces a table printout with summary statistics for the regression coefficients of the health economic evaluation probabilistic model run using the function selection, selection_long, pattern or hurdle.

Usage

## S3 method for class 'missingHE'
coef(object, prob = c(0.025, 0.975), random = FALSE, time = 1, digits = 3, ...)
## S3 method for class 'missingHE'
coef(object, prob = c(0.025, 0.975), random = FALSE, time = 1, digits = 3, ...)

Arguments

`object`	A `missingHE` object containing the results of the Bayesian modelling and the economic evaluation
`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 estimates.
`random`	Logical. If `random` is `TRUE`, the estimates of the random effects parameters are printed, when available.
`time`	A number indicating the time point at which posterior results for the model coefficients should be reported (only for longitudinal models).
`digits`	Number of digits to be displayed for each estimate.
`...`	Additional arguments affecting the summary produced.

Value

Prints a table with some summary statistics, including posterior mean, standard deviation and lower and upper quantiles based on the values specified in prob, for the posterior distributions of the regression coefficients of the effects and costs models run using the function selection, selection_long, pattern or hurdle.

Author(s)

Andrea Gabrio

Examples

 
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
# 
# 
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
# 
#

A function to read and re-arrange the data in different ways for the hurdle model

Description

This internal function imports the data and outputs only those variables that are needed to run the hurdle model according to the information provided by the user.

Usage

data_read_hurdle(
  data,
  model.eff,
  model.cost,
  model.se,
  model.sc,
  se,
  sc,
  type,
  center
)
data_read_hurdle(
  data,
  model.eff,
  model.cost,
  model.se,
  model.sc,
  se,
  sc,
  type,
  center
)

Arguments

`data`	A data frame in which to find variables supplied in `model.eff`, `model.cost` (model formulas for effects and costs) and `model.se`, `model.sc` (model formulas for the structural effect and cost models) . Among these, effectiveness, cost and treatment indicator (only two arms) variables must always be provided and named 'e', 'c' and 't' respectively.
`model.eff`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics effectiveness outcome ('e') whose name must correspond to that used in `data`, and any covariates are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. Random effects can also be specified for each model parameter.
`model.cost`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics cost outcome ('c') whose name must correspond to that used in `data`, and any covariates are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`model.se`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics effectiveness outcome ('e') whose name must correspond to that used in `data`, and any covariates used to estimate the probability of structural effects are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. By default, covariates are placed on the "probability" parameter for the strcutural effects through a logistic-linear model. Random effects can also be specified for each model parameter.
`model.sc`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics cost outcome ('c') whose name must correspond to that used in `data`, and any covariates used to estimate the probability of structural costs are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. By default, covariates are placed on the "probability" parameter for the strcutural costs through a logistic-linear model. Random effects can also be specified for each model parameter.
`se`	Structural value to be found in the effect data defined in `data`. If set to `NULL`, no structural value is chosen and a standard model for the effects is run.
`sc`	Structural value to be found in the cost data defined in `data`. If set to `NULL`, no structural value is chosen and a standard model for the costs is run.
`type`	Type of structural value mechanism assumed, either 'SCAR' (Structural Completely At Random) or 'SAR' (Strcutural At Random).
`center`	Logical. If `center` is `TRUE` all the covariates in the model are centered.

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

A function to read and re-arrange the data in different ways

Description

This internal function imports the data and outputs only those variables that are needed to run the model according to the information provided by the user.

Usage

data_read_pattern(data, model.eff, model.cost, type, center)
data_read_pattern(data, model.eff, model.cost, type, center)

Arguments

`data`	A data frame in which to find variables supplied in `model.eff` and `model.cost`. Among these, effectiveness, cost and treatment indicator (only two arms) variables must always be provided and named 'e', 'c' and 't' respectively.
`model.eff`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics effectiveness outcome ('e') whose name must correspond to that used in `data`, and any covariates are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`model.cost`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics cost outcome ('c') whose name must correspond to that used in `data`, and any covariates are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR) and Missing Not At Random (MNAR).
`center`	Logical. If `center` is `TRUE` all the covariates in the model are centered.

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

A function to read and re-arrange the data in different ways

Description

This internal function imports the data and outputs only those variables that are needed to run the model according to the information provided by the user.

Usage

data_read_selection(
  data,
  model.eff,
  model.cost,
  model.me,
  model.mc,
  type,
  center
)
data_read_selection(
  data,
  model.eff,
  model.cost,
  model.me,
  model.mc,
  type,
  center
)

Arguments

`data`	A data frame in which to find variables supplied in `model.eff` and `model.cost`. Among these, effectiveness, cost and treatment indicator (only two arms) variables must always be provided and named 'e', 'c' and 't' respectively.
`model.eff`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics effectiveness outcome ('e') whose name must correspond to that used in `data`, and any covariates are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`model.cost`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics cost outcome ('c') whose name must correspond to that used in `data`, and any covariates are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`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 used to estimate the probability of missing effects are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`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 used to estimate the probability of missing costs should be given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR) and Missing Not At Random (MNAR).
`center`	Logical. If `center` is `TRUE` all the covariates in the model are centered.

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

A function to read and re-arrange the data in different ways

Description

This internal function imports the data and outputs only those variables that are needed to run the model according to the information provided by the user.

Usage

data_read_selection_long(
  data,
  model.eff,
  model.cost,
  model.mu,
  model.mc,
  type,
  center
)
data_read_selection_long(
  data,
  model.eff,
  model.cost,
  model.mu,
  model.mc,
  type,
  center
)

Arguments

`data`	A data frame in which to find variables supplied in `model.eff` and `model.cost`. Among these, effectiveness, cost and treatment indicator (only two arms) variables must always be provided and named 'u', 'c' and 't' respectively.
`model.eff`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics effectiveness outcome ('u') whose name must correspond to that used in `data`, and any covariates are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`model.cost`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economics cost outcome ('c') whose name must correspond to that used in `data`, and any covariates are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`model.mu`	A formula expression in conventional `R` linear modelling syntax. The response must be indicated with the term 'mu'(missing effects) and any covariates used to estimate the probability of missing effects are given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`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 used to estimate the probability of missing costs should be given on the right-hand side. If there are no covariates, specify `1` on the right hand side. 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.
`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR) and Missing Not At Random (MNAR).
`center`	Logical. If `center` is `TRUE` all the covariates in the model are centered.

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

Diagnostic checks for assessing MCMC convergence of Bayesian models fitted in `JAGS` using the function `selection`, `selection_long`, `pattern` or `hurdle`

Description

The focus is restricted to full Bayesian models in cost-effectiveness analyses based on the function selection, selection_long, pattern and hurdle, with convergence of the MCMC chains that is assessed through graphical checks of the posterior distribution of the parameters of interest, Examples are density plots, trace plots, autocorrelation plots, etc. Other types of posterior checks are related to some summary MCMC statistics that are able to detect possible issues in the convergence of the algorithm, such as the potential scale reduction factor or the effective sample size. Different types of diagnostic tools and statistics are used to assess model convergence using functions contained in the package ggmcmc and mcmcplots. Graphics and plots are managed using functions contained in the package ggplot2 and ggthemes.

Usage

diagnostic(x, type = "denplot", param = "all", theme = NULL, ...)
diagnostic(x, type = "denplot", param = "all", theme = NULL, ...)

Arguments

`x`	An object of class "missingHE" containing the posterior results of a full Bayesian model implemented using the function `selection`, `selection_long`, `pattern` or `hurdle`.
`type`	Type of diagnostic check to be plotted for the model parameter selected. Available choices include: 'histogram' for histogram plots, 'denplot' for density plots, 'traceplot' for trace plots, 'acf' for autocorrelation plots, 'running' for running mean plots, 'compare' for comparing the distribution of the whole chain with only its last part, 'cross' for crosscorrelation plots, 'Rhat' for the potential scale reduction factor, 'geweke' for the geweke diagnostic, 'pairs' for posterior correlation among the parameters,'caterpillar' for caterpillar plots. In addition the class 'summary' provides an overview of some of the most popular diagnostic checks for each parameter selected.
`param`	Name of the family of parameters to process, as given by a regular expression. For example the mean parameters for the effect and cost variables can be specified using 'mu.e' ('mu.u' for longitudinal models) and 'mu.c', respectively. Different types of models may have different parameters depending on the assumed distributions and missing data assumptions. To see a complete list of all possible parameters by types of models assumed see details.
`theme`	Type of ggplot theme among some pre-defined themes. For a full list of available themes see details.
`...`	Additional parameters that can be provided to manage the graphical output of `diagnostic`.

Details

Depending on the types of plots specified in the argument type, the output of diagnostic can produce different combinations of MCMC visual posterior checks for the family of parameters indicated in the argument param. For a full list of the available plots see the description of the argument type or see the corresponding plots in the package ggmcmc.

The parameters that can be assessed through diagnostic are only those included in the object x (see Arguments). Specific character names must be specified in the argument param according to the specific model implemented. If x contains the results from a longitudinal model, all parameter names indexed by "e" should be instead indexed by "u". The available names and the parameters associated with them are:

"mu.e" the mean parameters of the effect variables in the two treatment arms.
"mu.c" the mean parameters of the cost variables in the two treatment arms.
"mu.e.p" the pattern-specific mean parameters of the effect variables in the two treatment arms (only with the function pattern).
"mu.c.p" the pattern-specific mean parameters of the cost variables in the two treatment arms (only with the function pattern).
"sd.e" the standard deviation parameters of the effect variables in the two treatment arms.
"sd.c" the standard deviation parameters of the cost variables in the two treatment arms.
"alpha" the regression intercept and covariate coefficient parameters for the effect variables in the two treatment arms.
"beta" the regression intercept and covariate coefficient parameters for the cost variables in the two treatment arms.
"random.alpha" the regression random effects intercept and covariate coefficient parameters for the effect variables in the two treatment arms.
"random.beta" the regression random effects intercept and covariate coefficient parameters for the cost variables in the two treatment arms.
"p.e" the probability parameters of the missingness or structural values mechanism for the effect variables in the two treatment arms (only with the function selection, selection_long or hurdle).
"p.c" the probability parameters of the missingness or structural values mechanism for the cost variables in the two treatment arms (only with the function selection, selection_long or hurdle).
"gamma.e" the regression intercept and covariate coefficient parameters of the missingness or structural values mechanism for the effect variables in the two treatment arms (only with the function selection, selection_long or hurdle).
"gamma.c" the regression intercept and covariate coefficient parameters of the missingness or structural values mechanism for the cost variables in the two treatment arms (only with the function selection, selection_long or hurdle).
"random.gamma.e" the random effects regression intercept and covariate coefficient parameters of the missingness or structural values mechanism for the effect variables in the two treatment arms (only with the function selection, selection_long or hurdle).
"random.gamma.c" the random effects regression intercept and covariate coefficient parameters of the missingness or structural values mechanism for the cost variables in the two treatment arms (only with the function selection, selection_long or hurdle).
"pattern" the probabilities associated with the missingness patterns in the data (only with the function pattern).
"delta.e" the mnar parameters of the missingness mechanism for the effect variables in the two treatment arms (only with the function selection, selection_long, or pattern).
"delta.c" the mnar parameters of the missingness mechanism for the cost variables in the two treatment arms (only with the function selection, selection_long, or pattern).
"random.delta.e" the random effects mnar parameters of the missingness mechanism for the effect variables in the two treatment arms (only with the function selection or selection_long).
"random.delta.c" the random effects mnar parameters of the missingness mechanism for the cost variables in the two treatment arms (only with the function selection or selection_long).
"all" all available parameters stored in the object x.

When the object x is created using the function pattern, pattern-specific standard deviation ("sd.e", "sd.c") and regression coefficient parameters ("alpha", "beta") for both outcomes can be visualised. The parameters associated with a missingness mechanism can be accessed only when x is created using the function selection, selection_long, or pattern, while the parameters associated with the model for the structural values mechanism can be accessed only when x is created using the function hurdle.

The argument theme allows to customise the graphical output of the plots generated by diagnostic and allows to choose among a set of possible pre-defined themes taken form the package ggtheme. For a complete list of the available character names for each theme, see ggthemes.

Value

A ggplot object containing the plots specified in the argument type

Author(s)

Andrea Gabrio

References

Gelman, A. Carlin, JB., Stern, HS. Rubin, DB.(2003). Bayesian Data Analysis, 2nd edition, CRC Press.

Brooks, S. Gelman, A. Jones, JL. Meng, XL. (2011). Handbook of Markov Chain Monte Carlo, CRC/Chapman and Hall.

Examples

# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
#
#
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
#
#

An internal function to extract the random effects component from an object of class formula

Description

An internal function to extract the random effects component from an object of class formula

Usage

fb(term)
fb(term)

Arguments

term

formula to be processed

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

Full Bayesian Models to handle missingness in Economic Evaluations (Hurdle Models)

Description

Full Bayesian cost-effectiveness models to handle missing data in the outcomes using Hurdle models under a variatey of alternative parametric distributions for the effect and cost variables. Alternative assumptions about the mechanisms of the structural values are implemented using a hurdle approach. The analysis is performed using the BUGS language, which is implemented in the software JAGS using the function jags. The output is stored in an object of class 'missingHE'.

Usage

hurdle(
  data,
  model.eff,
  model.cost,
  model.se = se ~ 1,
  model.sc = sc ~ 1,
  se = 1,
  sc = 0,
  dist_e,
  dist_c,
  type,
  prob = c(0.025, 0.975),
  n.chains = 2,
  n.iter = 20000,
  n.burnin = floor(n.iter/2),
  inits = NULL,
  n.thin = 1,
  ppc = FALSE,
  save_model = FALSE,
  prior = "default",
  ...
)
hurdle(
  data,
  model.eff,
  model.cost,
  model.se = se ~ 1,
  model.sc = sc ~ 1,
  se = 1,
  sc = 0,
  dist_e,
  dist_c,
  type,
  prob = c(0.025, 0.975),
  n.chains = 2,
  n.iter = 20000,
  n.burnin = floor(n.iter/2),
  inits = NULL,
  n.thin = 1,
  ppc = FALSE,
  save_model = FALSE,
  prior = "default",
  ...
)

Arguments

`data`	A data frame in which to find the variables supplied in `model.eff`, `model.cost` (model formulas for effects and costs) and `model.se`, `model.sc` (model formulas for the structural effect and cost models). Among these, effectiveness, cost and treatment indicator (only two arms) variables must always be provided and named 'e', 'c' and 't', 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.se`	A formula expression in conventional `R` linear modelling syntax. The response must be indicated with the term 'se'(structural effects). 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 "probability" parameter for the structural 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.sc`	A formula expression in conventional `R` linear modelling syntax. The response must be indicated with the term 'sc'(structural costs). 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 "probability" parameter for the structural costs through a logistic-linear model. Random effects can also be specified for each model parameter. See details for how these can be specified.
`se`	Structural value to be found in the effect variables defined in `data`. If set to `NULL`, no structural value is chosen and a standard model for the effects is run.
`sc`	Structural value to be found in the cost variables defined in `data`. If set to `NULL`, no structural value is chosen and a standard model for the costs is run.
`dist_e`	Distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern').
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm').
`type`	Type of structural value mechanism assumed. Choices are Structural Completely At Random (SCAR), and Structural At Random (SAR).
`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.
`ppc`	Logical. If `ppc` is `TRUE`, the estimates of the parameters that can be used to generate replications from the model are saved.
`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 of length two containing the user-provided hyperprior 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(0, 100))`. 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.
`...`	Additional arguments that can be provided by the user. Examples are `d_e` and `d_c`, which should correspond to two binary indicator vectors with length equal to the number of rows of `data`. By default these variables are constructed within the function based on the observed data but it is possible for the user to directly provide them as a means to explore some Structural Not At Random (SNAR) mechanism assumptions about one or both outcomes. Individuals whose corresponding indicator value is set to `1` or `0` will be respectively associated with the structural or non-structural component in the model. Other optional arguments are `center = TRUE`, which centers all the covariates in the model or the additional arguments that can be provided to the function `bcea` to summarise the health economic evaluation results.

Details

Depending on the distributions specified for the outcome variables in the arguments dist_e and dist_c and the type of structural value mechanism specified in the argument type, different hurdle models are built and run in the background by the function hurdle. These are mixture models defined by two components: the first one is a mass distribution at the spike, while the second is a parametric model applied to the natural range of the relevant variable. Usually, a logistic regression is used to estimate the probability of incurring a "structural" value (e.g. 0 for the costs, or 1 for the effects); this is then used to weigh the mean of the "non-structural" values estimated in the second component. A simple example can be used to show how hurdle models are specified. Consider a data set comprising a response variable $y$ and a set of centered covariate $X_j$ .Specifically, for each subject in the trial $i = 1, ..., n$ we define an indicator variable $d_i$ taking value 1 if the $i$ -th individual is associated with a structural value and 0 otherwise. This is modelled as:

$d_i ~ Bernoulli(\pi_i)$

$logit(\pi_i) = \gamma_0 + \sum\gamma_j X_j$

where

$\pi_i$ is the individual probability of a structural value in $y$ .
$\gamma_0$ represents the marginal probability of a structural value in $y$ on the logit scale.
$\gamma_j$ represents the impact on the probability of a structural value in $y$ of the centered covariates $X_j$ .

When $\gamma_j = 0$ , the model assumes a 'SCAR' mechanism, while when $\gamma_j != 0$ the mechanism is 'SAR'. For the parameters indexing the structural value model, the default prior distributions assumed are the following:

$\gamma_0 ~ Logisitic(0, 1)$
$\gamma_j ~ Normal(0, 0.01)$

When user-defined hyperprior values are supplied via the argument prior in the function hurdle, the elements of this list (see Arguments) must be vectors of length 2 containing the user-provided hyperprior values and must take specific names according to the parameters they are associated with. Specifically, the names accepted by missingHE are the following:

location parameters $\alpha_0, \beta_0$ : "mean.prior.e"(effects) and/or "mean.prior.c"(costs)
auxiliary parameters $\sigma$ : "sigma.prior.e"(effects) and/or "sigma.prior.c"(costs)
covariate parameters $\alpha_j, \beta_j$ : "alpha.prior"(effects) and/or "beta.prior"(costs)
marginal probability of structural values $\gamma_0$ : "p.prior.e"(effects) and/or "p.prior.c"(costs)
covariate parameters in the model of the structural values $\gamma_j$ (if covariate data provided): "gamma.prior.e"(effects) and/or "gamma.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 structural 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 hurdle (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). Multiple random effects can be specified using the notation + (x1 + x2 | site) for each covariate that was included in the fixed effects formula. Random intercepts are included by default in the models if a random effects are specified but they can be removed by adding the term 0 within the random effects formula, e.g. + (0 + x | z).

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), the number of observed and missing individuals , the total number of individuals by treatment arm and the indicator vectors for the structural values
model_output: A list containing the output of a JAGS model generated from the functions jags, and the posterior samples for the main parameters of the model and the imputed values
cea: A list containing the output of the economic evaluation performed using the function bcea
type: A character variable that indicate which type of structural value mechanism has been used to run the model, either SCAR or SAR (see details)
data_format: A character variable that indicate which type of analysis was conducted, either using a wide or longitudinal dataset

Author(s)

Andrea Gabrio

References

Ntzoufras I. (2009). Bayesian Modelling Using WinBUGS, John Wiley and Sons.

Daniels, MJ. Hogan, JW. (2008). Missing Data in Longitudinal Studies: strategies for Bayesian modelling and sensitivity analysis, CRC/Chapman Hall.

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

Gelman, A. Carlin, JB., Stern, HS. Rubin, DB.(2003). Bayesian Data Analysis, 2nd edition, CRC Press.

Plummer, M. JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. (2003).

Examples

# Quick example to run using subset of MenSS dataset
MenSS.subset <- MenSS[50:100, ]

# Run the model using the hurdle function assuming a SCAR mechanism
# Use only 100 iterations to run a quick check
model.hurdle <- hurdle(data = MenSS.subset, model.eff = e ~ 1,model.cost = c ~ 1,
   model.se = se ~ 1, model.sc = sc ~ 1, se = 1, sc = 0, dist_e = "norm", dist_c = "norm",
   type = "SCAR", n.chains = 2, n.iter = 50,  ppc = FALSE)

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

# Use dic information criterion to assess model fit
pic.dic <- pic(model.hurdle, criterion = "dic", module = "total")
pic.dic
#

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



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

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

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


# Further examples which take longer to run
model.hurdle <- hurdle(data = MenSS, model.eff = e ~ u.0,model.cost = c ~ e,
   model.se = se ~ u.0, model.sc = sc ~ 1, se = 1, sc = 0, dist_e = "norm", dist_c = "norm",
   type = "SAR", n.chains = 2, n.iter = 500,  ppc = FALSE)
#
# Print results for all imputed values
print(model.hurdle, value.mis = TRUE)

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

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

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

# Summarise the CEA results
summary(model.hurdle)


#
#
# Quick example to run using subset of MenSS dataset
MenSS.subset <- MenSS[50:100, ]

# Run the model using the hurdle function assuming a SCAR mechanism
# Use only 100 iterations to run a quick check
model.hurdle <- hurdle(data = MenSS.subset, model.eff = e ~ 1,model.cost = c ~ 1,
   model.se = se ~ 1, model.sc = sc ~ 1, se = 1, sc = 0, dist_e = "norm", dist_c = "norm",
   type = "SCAR", n.chains = 2, n.iter = 50,  ppc = FALSE)

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

# Use dic information criterion to assess model fit
pic.dic <- pic(model.hurdle, criterion = "dic", module = "total")
pic.dic
#

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



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

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

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


# Further examples which take longer to run
model.hurdle <- hurdle(data = MenSS, model.eff = e ~ u.0,model.cost = c ~ e,
   model.se = se ~ u.0, model.sc = sc ~ 1, se = 1, sc = 0, dist_e = "norm", dist_c = "norm",
   type = "SAR", n.chains = 2, n.iter = 500,  ppc = FALSE)
#
# Print results for all imputed values
print(model.hurdle, value.mis = TRUE)

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

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

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

# Summarise the CEA results
summary(model.hurdle)


#
#

An internal function to detect the random effects component from an object of class formula

Description

An internal function to detect the random effects component from an object of class formula

Usage

isAnyArgBar(term)
isAnyArgBar(term)

Arguments

term

formula to be processed

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

An internal function to detect the random effects component from an object of class formula

Description

An internal function to detect the random effects component from an object of class formula

Usage

isBar(term)
isBar(term)

Arguments

term

formula to be processed

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

An internal function to summarise results from BUGS model

Description

This function hides missing data distribution from summary results of BUGS models

Usage

jagsresults(
  x,
  params,
  regex = FALSE,
  invert = FALSE,
  probs = c(0.025, 0.25, 0.5, 0.75, 0.975),
  signif,
  ...
)
jagsresults(
  x,
  params,
  regex = FALSE,
  invert = FALSE,
  probs = c(0.025, 0.25, 0.5, 0.75, 0.975),
  signif,
  ...
)

Arguments

`x`	The `rjags`, `rjags.parallel`, or `mcmc.list` object for which results will be printed.
`params`	Character vector or a regular expression pattern. The parameters for which results will be printed (unless `invert` is `FALSE`, in which case results for all parameters other than those given in `params` will be returned). If `regex` is `FALSE`, only those parameters that match `params` exactly will be returned. If `regex` is `TRUE`, `param` should be a character string giving the regular expression pattern to be matched.
`regex`	If `regex` is `TRUE`, then `param` is expected to be a single string giving a text pattern to be matched. Parameters with names matching the pattern will be returned (unless `invert` is `TRUE`, which results in all parameters that do not match the pattern being returned). Text pattern matching uses regular expressions (`regex`).
`invert`	Logical. If `invert` is `TRUE`, only those parameters that do not match elements of `params` will be returned.
`probs`	A numeric vector of probabilities within range [0, 1], representing the sample quantiles to be calculated and returned.
`signif`	If supplied, all columns other than `n.eff` will have their values rounded such that the most extreme value has the specified number of significant digits.
`...`	Additional arguments accepted by `grep`, e.g. `perl=TRUE`, to allow look-around pattern matching.

Examples

## Not run: 
## Data
N <- 100
temp <- runif(N)
rain <- runif(N)
wind <- runif(N)
a <- 0.13
beta.temp <- 1.3
beta.rain <- 0.86
beta.wind <- -0.44
sd <- 0.16
y <- rnorm(N, a + beta.temp*temp + beta.rain*rain + beta.wind*wind, sd)
dat <- list(N=N, temp=temp, rain=rain, wind=wind, y=y)

### bugs example
library(R2jags)

## Model
M <- function() {
  for (i in 1:N) {
    y[i] ~ dnorm(y.hat[i], sd^-2)
    y.hat[i] <- a + beta.temp*temp[i] + beta.rain*rain[i] + beta.wind*wind[i]
    resid[i] <- y[i] - y.hat[i]
  }
  sd ~ dunif(0, 100)
  a ~ dnorm(0, 0.0001)
  beta.temp ~ dnorm(0, 0.0001)
  beta.rain ~ dnorm(0, 0.0001)
  beta.wind ~ dnorm(0, 0.0001)
}

## Fit model
jagsfit <- jags(dat, inits=NULL, 
                parameters.to.save=c('a', 'beta.temp', 'beta.rain', 
                                     'beta.wind', 'sd', 'resid'), 
                model.file=M, n.iter=10000)

## Output
# model summary
jagsfit

# Results for beta.rain only
jagsresults(x=jagsfit, param='beta.rain')

# Results for 'a' and 'sd' only
jagsresults(x=jagsfit, param=c('a', 'sd'))
jagsresults(x=jagsfit, param=c('a', 'sd'), 
            probs=c(0.01, 0.025, 0.1, 0.25, 0.5, 0.75, 0.9, 0.975))

# Results for all parameters including the string 'beta'
jagsresults(x=jagsfit, param='beta', regex=TRUE)

# Results for all parameters not including the string 'beta'
jagsresults(x=jagsfit, param='beta', regex=TRUE, invert=TRUE)

# Note that the above is NOT equivalent to the following, which returns all
# parameters that are not EXACTLY equal to 'beta'.
jagsresults(x=jagsfit, param='beta', invert=TRUE)

# Results for all parameters beginning with 'b' or including 'sd'.
jagsresults(x=jagsfit, param='^b|sd', regex=TRUE)

# Results for all parameters not beginning with 'beta'.
# This is equivalent to using param='^beta' with invert=TRUE and regex=TRUE
jagsresults(x=jagsfit, param='^(?!beta)', regex=TRUE, perl=TRUE)

## End(Not run)
#
#

## Not run: 
## Data
N <- 100
temp <- runif(N)
rain <- runif(N)
wind <- runif(N)
a <- 0.13
beta.temp <- 1.3
beta.rain <- 0.86
beta.wind <- -0.44
sd <- 0.16
y <- rnorm(N, a + beta.temp*temp + beta.rain*rain + beta.wind*wind, sd)
dat <- list(N=N, temp=temp, rain=rain, wind=wind, y=y)

### bugs example
library(R2jags)

## Model
M <- function() {
  for (i in 1:N) {
    y[i] ~ dnorm(y.hat[i], sd^-2)
    y.hat[i] <- a + beta.temp*temp[i] + beta.rain*rain[i] + beta.wind*wind[i]
    resid[i] <- y[i] - y.hat[i]
  }
  sd ~ dunif(0, 100)
  a ~ dnorm(0, 0.0001)
  beta.temp ~ dnorm(0, 0.0001)
  beta.rain ~ dnorm(0, 0.0001)
  beta.wind ~ dnorm(0, 0.0001)
}

## Fit model
jagsfit <- jags(dat, inits=NULL, 
                parameters.to.save=c('a', 'beta.temp', 'beta.rain', 
                                     'beta.wind', 'sd', 'resid'), 
                model.file=M, n.iter=10000)

## Output
# model summary
jagsfit

# Results for beta.rain only
jagsresults(x=jagsfit, param='beta.rain')

# Results for 'a' and 'sd' only
jagsresults(x=jagsfit, param=c('a', 'sd'))
jagsresults(x=jagsfit, param=c('a', 'sd'), 
            probs=c(0.01, 0.025, 0.1, 0.25, 0.5, 0.75, 0.9, 0.975))

# Results for all parameters including the string 'beta'
jagsresults(x=jagsfit, param='beta', regex=TRUE)

# Results for all parameters not including the string 'beta'
jagsresults(x=jagsfit, param='beta', regex=TRUE, invert=TRUE)

# Note that the above is NOT equivalent to the following, which returns all
# parameters that are not EXACTLY equal to 'beta'.
jagsresults(x=jagsfit, param='beta', invert=TRUE)

# Results for all parameters beginning with 'b' or including 'sd'.
jagsresults(x=jagsfit, param='^b|sd', regex=TRUE)

# Results for all parameters not beginning with 'beta'.
# This is equivalent to using param='^beta' with invert=TRUE and regex=TRUE
jagsresults(x=jagsfit, param='^(?!beta)', regex=TRUE, perl=TRUE)

## End(Not run)
#
#

MenSS economic data on STIs

Description

Data from a pilot RCT trial (The MenSS trial) on youn men at risk of Sexually Trasmitted Infections (STIs). A total of 159 individuals were enrolled in trial: 75 in the control (t=1) and 84 in the active intervention (t=2). Clinical and health economic outcome data were collected via self-reported questionnaires at four time points throughout the study: baseline, 3 months, 6 months and 12 months follow-up. Health economic data include utility scores related to quality of life and costs, from which QALYs and total costs were then computed using the area under the curve method and by summing up the cost components at each time point. Clinical data include the total number of instances of unprotected sex and whether the individual was associated with an STI diagnosis or not. Baseline data are available for the utilities (no baseline costs collected), instances of unprotected sex, sti diagnosis, age, ethnicity and employment variables.

Usage

data(MenSS)
data(MenSS)

Format

A data frame with 159 rows and 12 variables

Details

id: id number
e: Quality Adjusted Life Years (QALYs)
c: Total costs in pounds
u.0: baseline utilities
age: Age in years
ethnicity: binary: white (1) and other (0)
employment: binary: working (1) and other (0)
t: Treatment arm indicator for the control (t=1) and the active intervention (t=2)
sex_inst.0: baseline number of instances of unprotected sex
sex_inst: number of instances of unprotected sex at 12 months follow-up
sti.0: binary : baseline sti diagnosis (1) and no baseline sti diagnosis (0)
sti: binary : sti diagnosis (1) and no sti diagnosis (0) at 12 months follow-up
site: site number

References

Bailey et al. (2016) Health Technology Assessment 20 (PubMed)

Examples

MenSS <- data(MenSS)
summary(MenSS)
str(MenSS)
MenSS <- data(MenSS)
summary(MenSS)
str(MenSS)

An internal function to separate the fixed and random effects components from an object of class formula

Description

An internal function to separate the fixed and random effects components from an object of class formula

Usage

nobars_(term)
nobars_(term)

Arguments

term

formula to be processed

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

Full Bayesian Models to handle missingness in Economic Evaluations (Pattern Mixture Models)

Description

Full Bayesian cost-effectiveness models to handle missing data in the outcomes under different missingness mechanism assumptions, using alternative parametric distributions for the effect and cost variables and a pattern mixture approach to identify the model. The analysis is performed using the BUGS language, which is implemented in the software JAGS using the function jags. The output is stored in an object of class 'missingHE'.

Usage

pattern(
  data,
  model.eff,
  model.cost,
  dist_e,
  dist_c,
  Delta_e,
  Delta_c,
  type,
  restriction = "CC",
  prob = c(0.025, 0.975),
  n.chains = 2,
  n.iter = 20000,
  n.burnin = floor(n.iter/2),
  inits = NULL,
  n.thin = 1,
  ppc = FALSE,
  save_model = FALSE,
  prior = "default",
  ...
)
pattern(
  data,
  model.eff,
  model.cost,
  dist_e,
  dist_c,
  Delta_e,
  Delta_c,
  type,
  restriction = "CC",
  prob = c(0.025, 0.975),
  n.chains = 2,
  n.iter = 20000,
  n.burnin = floor(n.iter/2),
  inits = NULL,
  n.thin = 1,
  ppc = FALSE,
  save_model = FALSE,
  prior = "default",
  ...
)

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 (only two arms) variables must always be provided and named 'e', 'c' and 't', 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 chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern').
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm').
`Delta_e`	Range of values for the prior on the sensitivity parameters used to identify the mean of the effects under MNAR. The value must be set to 0 under MAR.
`Delta_c`	Range of values for the prior on the sensitivity parameters used to identify the mean of the costs under MNAR. The value must be set to 0 under MAR.
`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 restrcition ('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.
`ppc`	Logical. If `ppc` is `TRUE`, the estimates of the parameters that can be used to generate replications from the model are saved.
`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 of length two containing the user-provided hyperprior values and must be named with the name of the corresponding parameter. For example, the hyperprior values for the standard deviation effect parameters can be provided using the list `prior = list('sigma.prior.e' = c(0, 100))`. 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.
`...`	Additional arguments that can be provided by the user. Examples are `center = TRUE` to center all the covariates in the model or the additional arguments that can be provided to the function `bcea` to summarise the health economic evaluation results.

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 covariate $X_j$ . 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. In general, a different number of patterns can be observed between the treatment groups and missingHE accounts for this possibility by modelling a different patterns' indicator variables for each arm. For simplicity, in this example, we assume that the same number of patterns is observed in both groups. $d_i$ is assigned a multinomial distribution, which probabilities are modelled using a Dirichlet prior (by default giving to each pattern the same weight). Next, the model specified in dist_e and dist_c is fitted in each pattern. The parameters that cannot be identified by the observed data in each pattern (d = 2, 3, 4), e.g. the means. $mu_e[d]$ and mu_c[d], can be identified using 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' departues for the effects and/or costs are explored assuming a Uniform prior distributions for the sensitivity parameters. The range of values for these priors is defined based on the boundaries specified in Delta_e and Delta_c (see Arguments), which must be provided by the user. 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 of length two containing the user-provided 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:

location parameters $\alpha_0$ and $\beta_0$ : "mean.prior.e"(effects) and/or "mean.prior.c"(costs)
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)

The only exception is the missingness patterns' probability $\pi$ , denoted with "patterns.prior", whose hyperprior values must be provided as a list formed by two elements. These must be vectors of the same length equal to the number of patterns in the control (first element) and intervention (second element) group.

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), 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 jags, and the posterior samples for the main parameters of the model and the imputed values
cea: A list containing the output of the economic evaluation performed using the function bcea
type: A character variable that indicate which type of missingness assumption has been used to run the model, either MAR or MNAR (see details)
data_format: A character variable that indicate which type of analysis was conducted, either using a wide or longitudinal dataset

Author(s)

Andrea Gabrio

References

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

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

Gelman, A. Carlin, JB., Stern, HS. Rubin, DB.(2003). Bayesian Data Analysis, 2nd edition, CRC Press.

Plummer, M. JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. (2003).

Examples

# Quck example to run using subset of MenSS dataset
MenSS.subset <- MenSS[50:100, ]

# Run the model using the pattern function assuming a SCAR mechanism
# Use only 100 iterations to run a quick check
model.pattern <- pattern(data = MenSS.subset,model.eff = e~1,model.cost = c~1,
   dist_e = "norm", dist_c = "norm",type = "MAR", Delta_e = 0, Delta_c = 0, 
   n.chains = 2, n.iter = 100, ppc = FALSE)

# 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", module = "total")
pic.dic
#

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



# 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)


# Further examples which take longer to run
model.pattern <- pattern(data = MenSS, model.eff = e ~ u.0,model.cost = c ~ e,
   Delta_e = 0, Delta_c = 0, dist_e = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 500, ppc = FALSE)
#
# Print results for all imputed values
print(model.pattern, value.mis = TRUE)

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

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

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

# Summarise the CEA results
summary(model.pattern)


#
#
# Quck example to run using subset of MenSS dataset
MenSS.subset <- MenSS[50:100, ]

# Run the model using the pattern function assuming a SCAR mechanism
# Use only 100 iterations to run a quick check
model.pattern <- pattern(data = MenSS.subset,model.eff = e~1,model.cost = c~1,
   dist_e = "norm", dist_c = "norm",type = "MAR", Delta_e = 0, Delta_c = 0, 
   n.chains = 2, n.iter = 100, ppc = FALSE)

# 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", module = "total")
pic.dic
#

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



# 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)


# Further examples which take longer to run
model.pattern <- pattern(data = MenSS, model.eff = e ~ u.0,model.cost = c ~ e,
   Delta_e = 0, Delta_c = 0, dist_e = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 500, ppc = FALSE)
#
# Print results for all imputed values
print(model.pattern, value.mis = TRUE)

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

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

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

# Summarise the CEA results
summary(model.pattern)


#
#

PBS economic data on intellectual disability and challenging behaviour

Description

Longitudinal data from a cluster RCT trial (The PBS trial) on people suffering from intellectual disability and challenging behaviour. A total of 244 individuals across 23 sites were enrolled in the trial: 136 in the control (t=1) and 108 in the active intervention (t=2). Health economic outcome data were collected via self-reported questionnaires at three time points throughout the study: baseline (time=1), 6 months (time=2) and 12 months (time=3) follow-up, and included utility scores related to quality of life and costs. Baseline data are available for age, gender, ethnicity, living status, type of carer, marital status, and disability level variables.

Usage

data(PBS)
data(PBS)

Format

A data frame with 732 rows and 16 variables

Details

id: id number
time: time indicator
u: utilities
c: costs (in pounds)
age: Age in years
gender: binary: male (1) and female (0)
ethnicity: binary: white (1) and other (0)
carer: binary: paid carer (1) and family carer (0)
marital: binary: single (1) and married (0)
living: categorical: alone (1), with partner (2) and with parents (3)
disability: categorical: mild (1), moderate (2) and severe (3)
site: site number

References

Hassiotis et al. (2014) BMC Psychiatry 14 (PubMed)

Examples

PBS <- data(PBS)
summary(PBS)
str(PBS)
PBS <- data(PBS)
summary(PBS)
str(PBS)

Predictive information criteria for Bayesian models fitted in `JAGS` using the funciton `selection`, `selection_long`, `pattern` or `hurdle`

Description

Efficient approximate leave-one-out cross validation (LOO), deviance information criterion (DIC) and widely applicable information criterion (WAIC) for Bayesian models, calculated on the observed data.

Usage

pic(x, criterion = "dic", module = "total")
pic(x, criterion = "dic", module = "total")

Arguments

`x`	A `missingHE` object containing the results of a Bayesian model fitted in cost-effectiveness analysis using the function `selection`, `selection_long`, `pattern` or `hurdle`.
`criterion`	type of information criteria to be produced. Available choices are `'dic'` for the Deviance Information Criterion, `'waic'` for the Widely Applicable Information Criterion, and `'looic'` for the Leave-One-Out Information Criterion.
`module`	The modules with respect to which the information criteria should be computed. Available choices are `'total'` for the whole model, `'e'` for the effectiveness variables only (`'u'` for longitudinal models), `'c'` for the cost variables only, and `'both'` for both outcome variables.

Details

The Deviance Information Criterion (DIC), Leave-One-Out Information Criterion (LOOIC) and the Widely Applicable Information Criterion (WAIC) are methods for estimating out-of-sample predictive accuracy from a Bayesian model using the log-likelihood evaluated at the posterior simulations of the parameters. If x contains the results from a longitudinal model, all parameter names indexed by "e" should be instead indexed by "u". In addition, for longitudinal models information criteria results are displayed by time and only a general approximation to the total value of the criteria and pD is given as the sum of the corresponding measures computed at each time point. DIC is computationally simple to calculate but it is known to have some problems, arising in part from it not being fully Bayesian in that it is based on a point estimate. LOOIC can be computationally expensive but can be easily approximated using importance weights that are smoothed by fitting a generalised Pareto distribution to the upper tail of the distribution of the importance weights. For more details about the methods used to compute LOOIC see the PSIS-LOO section in loo-package. WAIC is fully Bayesian and closely approximates Bayesian cross-validation. Unlike DIC, WAIC is invariant to parameterisation and also works for singular models. In finite cases, WAIC and LOO give similar estimates, but for influential observations WAIC underestimates the effect of leaving out one observation.

Value

A named list containing different predictive information criteria results and quantities according to the value of criterion. In all cases, the measures are computed on the observed data for the specific modules of the model selected in module.

d_bar: Posterior mean deviance (only if criterion is 'dic').
pD: Effective number of parameters calculated with the formula used by JAGS (only if criterion is 'dic')

dic: Deviance Information Criterion calculated with the formula used by JAGS (only if criterion is 'dic')

d_hat: Deviance evaluated at the posterior mean of the parameters and calculated with the formula used by JAGS (only if criterion is 'dic')
elpd, elpd_se: Expected log pointwise predictive density and standard error calculated on the observed data for the model nodes indicated in module (only if criterion is 'waic' or 'loo').
p, p_se: Effective number of parameters and standard error calculated on the observed data for the model nodes indicated in module (only if criterion is 'waic' or 'loo').
looic, looic_se: The leave-one-out information criterion and standard error calculated on the observed data for the model nodes indicated in module (only if criterion is 'loo').
waic, waic_se: The widely applicable information criterion and standard error calculated on the observed data for the model nodes indicated in module (only if criterion is 'waic').
pointwise: A matrix containing the pointwise contributions of each of the above measures calculated on the observed data for the model nodes indicated in module (only if criterion is 'waic' or 'loo').
pareto_k: A vector containing the estimates of the shape parameter $k$ for the generalised Pareto fit to the importance ratios for each leave-one-out distribution calculated on the observed data for the model nodes indicated in module (only if criterion is 'loo'). See loo for details about interpreting $k$ .
sum_dic: DIC value calculated by summing up all model dic evaluated at each time point (only for longitudinal models). Similar estimates can are obtained also for the other criteria, either sum_waic or sum_looic.
sum_pdic: DIC value calculated by summing up all model effective number of parameter estimates based on dic evaluated at each time point (only for longitudinal models). Similar estimates can are obtained also for the other criteria, either sum_pwaic or sum_plooic.

Author(s)

Andrea Gabrio

References

Plummer, M. JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. (2003).

Vehtari, A. Gelman, A. Gabry, J. (2016a) Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. Advance online publication.

Vehtari, A. Gelman, A. Gabry, J. (2016b) Pareto smoothed importance sampling. ArXiv preprint.

Gelman, A. Hwang, J. Vehtari, A. (2014) Understanding predictive information criteria for Bayesian models. Statistics and Computing 24, 997-1016.

Watanable, S. (2010). Asymptotic equivalence of Bayes cross validation and widely application information criterion in singular learning theory. Journal of Machine Learning Research 11, 3571-3594.

Examples

 
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
# 
# 
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
# 
#

Plot method for the imputed data contained in the objects of class `missingHE`

Description

Produces a plot of the observed and imputed values (with credible intervals) for the effect and cost outcomes from a Bayesian cost-effectiveness analysis model with two treatment arms, implemented using the function selection, selection_long, pattern or hurdle. The graphical layout is obtained from the functions contained in the package ggplot2 and ggthemes.

Usage

## S3 method for class 'missingHE'
plot(
  x,
  prob = c(0.025, 0.975),
  class = "scatter",
  outcome = "all",
  time_plot = NULL,
  theme = NULL,
  ...
)
## S3 method for class 'missingHE'
plot(
  x,
  prob = c(0.025, 0.975),
  class = "scatter",
  outcome = "all",
  time_plot = NULL,
  theme = NULL,
  ...
)

Arguments

`x`	A `missingHE` object containing the results of the Bayesian model for cost-effectiveness analysis.
`prob`	A numeric vector of probabilities representing the upper and lower CI sample quantiles to be calculated and returned for the imputed values.
`class`	Type of the plot comparing the observed and imputed outcome data. Available choices are 'histogram' and 'scatter' for a histogram or a scatter plot of the observed and imputed outcome data, respectively.
`outcome`	The outcome variables that should be displayed. Options are: 'all' (default) which shows the plots for both treatment arms, time points (only for longitudinal models) and types of outcome variables; 'effects' and 'costs' which show the plots for the corresponding outcome variables in both arms; 'arm1' and 'arm2' which show the plots by the selected treatment arm. To select the plots for a specific outcome in a specific treatment arm the options that can be used are 'effects_arm1', 'effects_arm2', 'costs_arm1' or 'costs_arm2'.
`time_plot`	Time point for which plots should be displayed (only for longitudinal models).
`theme`	Type of ggplot theme among some pre-defined themes, mostly taken from the package ggthemes. For a full list of available themes see details.
`...`	Additional parameters that can be provided to manage the output of `plot.missingHE`.

Details

The function produces a plot of the observed and imputed effect and cost data in a two-arm based cost-effectiveness model implemented using the function selection, selection_long, pattern or hurdle. The purpose of this graph is to visually compare the outcome values for the fully-observed individuals with those imputed by the model for the missing individuals. For the scatter plot, imputed values are also associated with the credible intervals specified in the argument prob. The argument theme allows to customise the graphical aspect of the plots generated by plot.missingHE and allows to choose among a set of possible pre-defined themes taken form the package ggtheme. For a complete list of the available character names for each theme and scheme set, see ggthemes and bayesplot.

Value

A ggplot object containing the plots specified in the argument class.

Author(s)

Andrea Gabrio

References

Daniels, MJ. Hogan, JW. (2008) Missing Data in Longitudinal Studies: strategies for Bayesian modelling and sensitivity analysis, CRC/Chapman Hall.

Molenberghs, G. Fitzmaurice, G. Kenward, MG. Tsiatis, A. Verbeke, G. (2015) Handbook of Missing Data Methodology, CRC/Chapman Hall.

Examples

# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
#
#
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
#
#

Posterior predictive checks for assessing the fit to the observed data of Bayesian models implemented in `JAGS` using the function `selection`, `selection_long`, `pattern` or `hurdle`

Description

The focus is restricted to full Bayesian models in cost-effectiveness analyses based on the function selection, selection_long, pattern and hurdle, with the fit to the observed data being assessed through graphical checks based on the posterior replications generated from the model. Examples include the comparison of histograms, density plots, intervals, test statistics, evaluated using both the observed and replicated data. Different types of posterior predictive checks are implemented to assess model fit using functions contained in the package bayesplot. Graphics and plots are managed using functions contained in the package ggplot2 and ggthemes.

Usage

ppc(
  x,
  type = "histogram",
  outcome = "all",
  ndisplay = 15,
  time_plot = NULL,
  theme = NULL,
  scheme_set = NULL,
  legend = "top",
  ...
)
ppc(
  x,
  type = "histogram",
  outcome = "all",
  ndisplay = 15,
  time_plot = NULL,
  theme = NULL,
  scheme_set = NULL,
  legend = "top",
  ...
)

Arguments

`x`	An object of class "missingHE" containing the posterior results of a full Bayesian model implemented using the function `selection`, `selection_long`, `pattern` or `hurdle`.
`type`	Type of posterior predictive check to be plotted for assessing model fit. Available choices include: 'histogram', 'boxplot', 'freqpoly', 'dens', 'dens_overlay' and ecdf_overlay', which compare the empirical and repicated distributions of the data; 'stat' and 'stat_2d', which compare the value of some statistics evaluated on the observed data with the replicated values for those statistics from the posterior predictions; 'error_hist', 'error_scatter', 'error_scatter_avg' and 'error_binned', which display the predictive errors of the model; 'intervals' and 'ribbon', which compare medians and central interval estimates of the replications with the observed data overlaid; 'scatter' and 'scatter_avg', which display scatterplots of the observed and replicated data.
`outcome`	The outcome variables that should be displayed. Use the names 'effects_arm1' and effects_arm2' for the effectiveness in the control and intervention arm; use costs_arm1' or 'costs_arm2' for the costs; use "effects" or "costs" for the respective outcome in both arms; use "all" for all outcomes.
`ndisplay`	Number of posterior replications to be displayed in the plots.
`time_plot`	Time point for which posterior predictive checks should be displayed (only for longitudinal models).
`theme`	Type of ggplot theme among some pre-defined themes, mostly taken from the package ggthemes. For a full list of available themes see details.
`scheme_set`	Type of scheme sets among some pre-defined schemes, mostly taken from the package bayesplot. For a full list of available themes see details.
`legend`	Position of the legend: available choices are: "top", "left", "right", "bottom" and "none".
`...`	Additional parameters that can be provided to manage the output of `ppc`. For more details see bayesplot.

Details

The funciton produces different types of graphical posterior predictive checks using the estimates from a Bayesian cost-effectiveness model implemented with the function selection, selection_long, pattern or hurdle. The purpose of these checks is to visually compare the distribution (or some relevant quantity) of the observed data with respect to that from the replicated data for both effectiveness and cost outcomes in each treatment arm. Since predictive checks are meaningful only with respect to the observed data, only the observed outcome values are used to assess the fit of the model. The arguments theme and scheme_set allow to customise the graphical aspect of the plots generated by ppc and allow to choose among a set of possible pre-defined themes and scheme sets taken form the package ggtheme and bayesplot. For a complete list of the available character names for each theme and scheme set, see ggthemes and bayesplot.

Value

A ggplot object containing the plots specified in the argument type.

Author(s)

Andrea Gabrio

References

Gelman, A. Carlin, JB., Stern, HS. Rubin, DB.(2003). Bayesian Data Analysis, 2nd edition, CRC Press.

Examples

# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
#
#
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
#
#

Print method for the posterior results contained in the objects of class `missingHE`

Description

Prints the summary table for the model fitted, with the estimate of the parameters and/or missing values.

Usage

## S3 method for class 'missingHE'
print(x, value.mis = FALSE, only.means = TRUE, ...)
## S3 method for class 'missingHE'
print(x, value.mis = FALSE, only.means = TRUE, ...)

Arguments

`x`	A `missingHE` object containing the results of the Bayesian model run using the function `selection`, `selection_long`, `pattern` or `hurdle`.
`value.mis`	Logical. If `value.mis` is `TRUE`, the model results displayed contain also the imputed values, else if `value.mis` is `FALSE` the missing values are hidden.
`only.means`	Logical. If `only.means` is `TRUE`, then the `print` function only shows the summary statistics for the mean effectiveness and costs. Defaults at `FALSE` (in which case, shows the summary statistics for all parameters in the model).
`...`	additional arguments affecting the printed output produced. For example: `digits=` number of significant digits to be shown in the printed table (default=3). Not available if `value.mis=`TRUE.

Author(s)

Andrea Gabrio

Examples

 
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}}, 
# \code{\link{pattern}} or \code{\link{hurdle}}
# 
# 
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}}, 
# \code{\link{pattern}} or \code{\link{hurdle}}
# 
#

An internal function to change the hyperprior parameters in the hurdle model provided by the user depending on the type of structural value mechanism and outcome distributions assumed

Description

This function modifies default hyper prior parameter values in the type of hurdle model selected according to the type of structural value mechanism and distributions for the outcomes assumed.

Usage

prior_hurdle(
  type,
  dist_e,
  dist_c,
  pe_fixed,
  pc_fixed,
  ze_fixed,
  zc_fixed,
  model_e_random,
  model_c_random,
  model_se_random,
  model_sc_random,
  pe_random,
  pc_random,
  ze_random,
  zc_random,
  se,
  sc
)
prior_hurdle(
  type,
  dist_e,
  dist_c,
  pe_fixed,
  pc_fixed,
  ze_fixed,
  zc_fixed,
  model_e_random,
  model_c_random,
  model_se_random,
  model_sc_random,
  pe_random,
  pc_random,
  ze_random,
  zc_random,
  se,
  sc
)

Arguments

`type`	Type of structural value mechanism assumed. Choices are Structural Completely At Random (SCAR), and Structural At Random (SAR). For a complete list of all available hyper parameters and types of models see the manual.
`dist_e`	distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern')
`dist_c`	distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm')
`pe_fixed`	Number of fixed effects for the effectiveness model
`pc_fixed`	Number of fixed effects for the cost model
`ze_fixed`	Number of fixed effects or the structural indicators model for the effectiveness
`zc_fixed`	Number of fixed effects or the structural indicators model for the costs
`model_e_random`	Random effects formula for the effectiveness model
`model_c_random`	Random effects formula for the costs model
`model_se_random`	Random effects formula for the structural indicators model for the effectiveness
`model_sc_random`	Random effects formula for the structural indicators model for the costs
`pe_random`	Number of random effects for the effectiveness model
`pc_random`	Number of random effects for the cost model
`ze_random`	Number of random effects or the structural indicators model for the effectiveness
`zc_random`	Number of random effects or the structural indicators model for the costs
`se`	Structural value for the effectiveness
`sc`	Structural value for the costs

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

An internal function to change the hyperprior parameters in the selection model provided by the user depending on the type of missingness mechanism and outcome distributions assumed

Description

This function modifies default hyper prior parameter values in the type of selection model selected according to the type of missingness mechanism and distributions for the outcomes assumed.

Usage

prior_pattern(
  type,
  dist_e,
  dist_c,
  pe_fixed,
  pc_fixed,
  model_e_random,
  model_c_random,
  pe_random,
  pc_random,
  d_list,
  restriction
)
prior_pattern(
  type,
  dist_e,
  dist_c,
  pe_fixed,
  pc_fixed,
  model_e_random,
  model_c_random,
  pe_random,
  pc_random,
  d_list,
  restriction
)

Arguments

`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR), Missing Not At Random for the effects (MNAR_eff), Missing Not At Random for the costs (MNAR_cost), and Missing Not At Random for both (MNAR). For a complete list of all available hyper parameters and types of models see the manual.
`dist_e`	distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern')
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm')
`pe_fixed`	Number of fixed effects for the effectiveness model
`pc_fixed`	Number of fixed effects for the cost model
`model_e_random`	Random effects formula for the effectiveness model
`model_c_random`	Random effects formula for the costs model
`pe_random`	Number of random effects for the effectiveness model
`pc_random`	Number of random effects for the cost model
`d_list`	a list of the number and types of patterns in the data
`restriction`	type of identifying restriction to be imposed

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

An internal function to change the hyperprior parameters in the selection model provided by the user depending on the type of missingness mechanism and outcome distributions assumed

Description

This function modifies default hyper prior parameter values in the type of selection model selected according to the type of missingness mechanism and distributions for the outcomes assumed.

Usage

prior_selection(
  type,
  dist_e,
  dist_c,
  pe_fixed,
  pc_fixed,
  ze_fixed,
  zc_fixed,
  model_e_random,
  model_c_random,
  model_me_random,
  model_mc_random,
  pe_random,
  pc_random,
  ze_random,
  zc_random
)
prior_selection(
  type,
  dist_e,
  dist_c,
  pe_fixed,
  pc_fixed,
  ze_fixed,
  zc_fixed,
  model_e_random,
  model_c_random,
  model_me_random,
  model_mc_random,
  pe_random,
  pc_random,
  ze_random,
  zc_random
)

Arguments

`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR), Missing Not At Random for the effects (MNAR_eff), Missing Not At Random for the costs (MNAR_cost), and Missing Not At Random for both (MNAR). For a complete list of all available hyper parameters and types of models see the manual.
`dist_e`	distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern')
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm')
`pe_fixed`	Number of fixed effects for the effectiveness model
`pc_fixed`	Number of fixed effects for the cost model
`ze_fixed`	Number of fixed effects or the missingness indicators model for the effectiveness
`zc_fixed`	Number of fixed effects or the missingness indicators model for the costs
`model_e_random`	Random effects formula for the effectiveness model
`model_c_random`	Random effects formula for the costs model
`model_me_random`	Random effects formula for the missingness indicators model for the effectiveness
`model_mc_random`	Random effects formula for the missingness indicators model for the costs
`pe_random`	Number of random effects for the effectiveness model
`pc_random`	Number of random effects for the cost model
`ze_random`	Number of random effects or the missingness indicators model for the effectiveness
`zc_random`	Number of random effects or the missingness indicators model for the costs

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

An internal function to change the hyperprior parameters in the selection model provided by the user depending on the type of missingness mechanism and outcome distributions assumed

Description

This function modifies default hyper prior parameter values in the type of selection model selected according to the type of missingness mechanism and distributions for the outcomes assumed.

Usage

prior_selection_long(
  type,
  dist_u,
  dist_c,
  pu_fixed,
  pc_fixed,
  zu_fixed,
  zc_fixed,
  model_u_random,
  model_c_random,
  model_mu_random,
  model_mc_random,
  pu_random,
  pc_random,
  zu_random,
  zc_random
)
prior_selection_long(
  type,
  dist_u,
  dist_c,
  pu_fixed,
  pc_fixed,
  zu_fixed,
  zc_fixed,
  model_u_random,
  model_c_random,
  model_mu_random,
  model_mc_random,
  pu_random,
  pc_random,
  zu_random,
  zc_random
)

Arguments

`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR), Missing Not At Random for the effects (MNAR_eff), Missing Not At Random for the costs (MNAR_cost), and Missing Not At Random for both (MNAR). For a complete list of all available hyper parameters and types of models see the manual.
`dist_u`	distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern')
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm')
`pu_fixed`	Number of fixed effects for the effectiveness model
`pc_fixed`	Number of fixed effects for the cost model
`zu_fixed`	Number of fixed effects or the missingness indicators model for the effectiveness
`zc_fixed`	Number of fixed effects or the missingness indicators model for the costs
`model_u_random`	Random effects formula for the effectiveness model
`model_c_random`	Random effects formula for the costs model
`model_mu_random`	Random effects formula for the missingness indicators model for the effectiveness
`model_mc_random`	Random effects formula for the missingness indicators model for the costs
`pu_random`	Number of random effects for the effectiveness model
`pc_random`	Number of random effects for the cost model
`zu_random`	Number of random effects or the missingness indicators model for the effectiveness
`zc_random`	Number of random effects or the missingness indicators model for the costs

Examples

#Internal function only
#no examples
#
#
#Internal function only
#no examples
#
#

An internal function to execute a JAGS hurdle model and get posterior results

Description

This function fits a JAGS using the jags funciton and obtain posterior inferences.

Usage

run_hurdle(type, dist_e, dist_c, inits, se, sc, sde, sdc, ppc)
run_hurdle(type, dist_e, dist_c, inits, se, sc, sde, sdc, ppc)

Arguments

`type`	Type of structural value mechanism assumed. Choices are Structural Completely At Random (SCAR), and Structural At Random (SAR).
`dist_e`	distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern')
`dist_c`	distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm').
`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 BUGS model, or a function creating (possibly random) initial values. If inits is NULL, JAGS will generate initial values for parameters
`se`	Structural value to be found in the effect data. If set to `NULL`, no structural value is chosen and a standard model for the effects is run.
`sc`	Structural value to be found in the cost data. If set to `NULL`, no structural value is chosen and a standard model for the costs is run.
`sde`	hyper-prior value for the standard deviation of the distribution of the structural effects. The default value is `1.0E-6` to approximate a point mass at the structural value provided by the user.
`sdc`	hyper-prior value for the standard deviation of the distribution of the structural costs. The default value is `1.0E-6` to approximate a point mass at the structural value provided by the user.
`ppc`	Logical. If `ppc` is `TRUE`, the estimates of the parameters that can be used to generate replications from the model are saved.

Examples

#Internal function only
#No examples
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#Internal function only
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An internal function to execute a JAGS pattern mixture model and get posterior results

Description

This function fits a JAGS using the jags funciton and obtain posterior inferences.

Usage

run_pattern(type, dist_e, dist_c, inits, d_list, d1, d2, restriction, ppc)
run_pattern(type, dist_e, dist_c, inits, d_list, d1, d2, restriction, ppc)

Arguments

`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR), Missing Not At Random for the effects (MNAR_eff), Missing Not At Random for the costs (MNAR_cost), and Missing Not At Random for both (MNAR).
`dist_e`	distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern').
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm').
`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 BUGS model, or a function creating (possibly random) initial values. If inits is NULL, JAGS will generate initial values for parameters.
`d_list`	a list of the number and types of patterns in the data.
`d1`	Patterns in the control.
`d2`	Patterns in the intervention.
`restriction`	type of identifying restriction to be imposed.
`ppc`	Logical. If `ppc` is `TRUE`, the estimates of the parameters that can be used to generate replications from the model are saved.

Examples

#Internal function only
#No examples
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#Internal function only
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An internal function to execute a JAGS selection model and get posterior results

Description

This function fits a JAGS using the jags funciton and obtain posterior inferences.

Usage

run_selection(type, dist_e, dist_c, inits, ppc)
run_selection(type, dist_e, dist_c, inits, ppc)

Arguments

`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR), Missing Not At Random for the effects (MNAR_eff), Missing Not At Random for the costs (MNAR_cost), and Missing Not At Random for both (MNAR).
`dist_e`	distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern').
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm').
`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 BUGS model, or a function creating (possibly random) initial values. If inits is NULL, JAGS will generate initial values for parameters.
`ppc`	Logical. If `ppc` is `TRUE`, the estimates of the parameters that can be used to generate replications from the model are saved.

Examples

#Internal function only
#No examples
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#Internal function only
#No examples
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An internal function to execute a JAGS selection model and get posterior results

Description

This function fits a JAGS using the jags funciton and obtain posterior inferences.

Usage

run_selection_long(type, dist_u, dist_c, inits, ppc)
run_selection_long(type, dist_u, dist_c, inits, ppc)

Arguments

`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR), Missing Not At Random for the effects (MNAR_eff), Missing Not At Random for the costs (MNAR_cost), and Missing Not At Random for both (MNAR).
`dist_u`	distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern').
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm').
`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 BUGS model, or a function creating (possibly random) initial values. If inits is NULL, JAGS will generate initial values for parameters.
`ppc`	Logical. If `ppc` is `TRUE`, the estimates of the parameters that can be used to generate replications from the model are saved.

Examples

#Internal function only
#No examples
#
#
#Internal function only
#No examples
#
#

Full Bayesian Models to handle missingness in Economic Evaluations (Selection Models)

Description

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 selection model approach to identify the model. The analysis is performed using the BUGS language, which is implemented in the software JAGS using the function jags The output is stored in an object of class 'missingHE'.

Usage

selection(
  data,
  model.eff,
  model.cost,
  model.me = me ~ 1,
  model.mc = mc ~ 1,
  dist_e,
  dist_c,
  type,
  prob = c(0.025, 0.975),
  n.chains = 2,
  n.iter = 20000,
  n.burnin = floor(n.iter/2),
  inits = NULL,
  n.thin = 1,
  ppc = FALSE,
  save_model = FALSE,
  prior = "default",
  ...
)
selection(
  data,
  model.eff,
  model.cost,
  model.me = me ~ 1,
  model.mc = mc ~ 1,
  dist_e,
  dist_c,
  type,
  prob = c(0.025, 0.975),
  n.chains = 2,
  n.iter = 20000,
  n.burnin = floor(n.iter/2),
  inits = NULL,
  n.thin = 1,
  ppc = FALSE,
  save_model = FALSE,
  prior = "default",
  ...
)

Arguments

`data`	A data frame in which to find the 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 and treatment indicator (only two arms) variables must always be provided and named 'e', 'c' and 't', 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 chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern').
`dist_c`	Distribution assumed for the costs. Current available chocies 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).
`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.
`ppc`	Logical. If `ppc` is `TRUE`, the estimates of the parameters that can be used to generate replications from the model are saved.
`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 of length two containing the user-provided hyperprior values and must be named with the name of the corresponding parameter. For example, the hyperprior values for the standard deviation effect parameters can be provided using the list `prior = list('sigma.prior.e' = c(0, 100))`. 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.
`...`	Additional arguments that can be provided by the user. Examples are `center = TRUE` to center all the covariates in the model or the additional arguments that can be provided to the function `bcea` to summarise the health economic evaluation results.

Details

$m_i ~ Bernoulli(\pi_i)$

$logit(\pi_i) = \gamma_0 + \sum\gamma_j X_j + \delta(y)$

where

$\pi_i$ is the individual probability of a missing value in $y$
$\gamma_0$ represents the marginal probability of a missing value in $y$ on the logit scale.
$\gamma_j$ represents the impact on the probability of a missing value in $y$ of the centered covariates $X_j$ .
$\delta$ represents the impact on the probability of a missing value in $y$ of the missing value itself.

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 the following:

$\gamma_0 ~ Logisitc(0, 1)$
$\gamma_j ~ Normal(0, 0.01)$
$\delta ~ Normal(0, 1)$

When user-defined hyperprior values are supplied via the argument prior in the function selection, the elements of this list (see Arguments) must be vectors of length two containing the user-provided 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:

location parameters $\alpha_0$ and $\beta_0$ : "mean.prior.e"(effects) and/or "mean.prior.c"(costs)
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)
marginal probability of missing values $\gamma_0$ : "p.prior.e"(effects) and/or "p.prior.c"(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 selection (see Arguments).

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), 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 jags, and the posterior samples for the main parameters of the model and the imputed values
cea: A list containing the output of the economic evaluation performed using the function bcea
type: A character variable that indicate which type of missingness mechanism has been used to run the model, either MAR or MNAR (see details)
data_format: A character variable that indicate which type of analysis was conducted, either using a wide or longitudinal dataset

Author(s)

Andrea Gabrio

References

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

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

Gelman, A. Carlin, JB., Stern, HS. Rubin, DB.(2003). Bayesian Data Analysis, 2nd edition, CRC Press.

Plummer, M. JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. (2003).

Examples

# Quck example to run using subset of MenSS dataset
MenSS.subset <- MenSS[50:100, ]

# Run the model using the selection function assuming a MAR mechanism
# Use only 100 iterations to run a quick check
model.selection <- selection(data = MenSS.subset, model.eff = e ~ 1,model.cost = c ~ 1,
   model.me = me ~ 1, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 100, ppc = TRUE)

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

# Use dic information criterion to assess model fit
pic.dic <- pic(model.selection, criterion = "dic", module = "total")
pic.dic
#

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



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

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

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


# Further examples which take longer to run
model.selection <- selection(data = MenSS, model.eff = e ~ u.0,model.cost = c ~ e,
   model.se = me ~ u.0, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 500, ppc = FALSE)
#
# Print results for all imputed values
print(model.selection, value.mis = TRUE)

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

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

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

# Summarise the CEA results
summary(model.selection)


#
#
# Quck example to run using subset of MenSS dataset
MenSS.subset <- MenSS[50:100, ]

# Run the model using the selection function assuming a MAR mechanism
# Use only 100 iterations to run a quick check
model.selection <- selection(data = MenSS.subset, model.eff = e ~ 1,model.cost = c ~ 1,
   model.me = me ~ 1, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 100, ppc = TRUE)

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

# Use dic information criterion to assess model fit
pic.dic <- pic(model.selection, criterion = "dic", module = "total")
pic.dic
#

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



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

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

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


# Further examples which take longer to run
model.selection <- selection(data = MenSS, model.eff = e ~ u.0,model.cost = c ~ e,
   model.se = me ~ u.0, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 500, ppc = FALSE)
#
# Print results for all imputed values
print(model.selection, value.mis = TRUE)

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

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

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

# Summarise the CEA results
summary(model.selection)


#
#

Full Bayesian Models to handle missingness in Economic Evaluations (Selection Models)

Description

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 and using a selection model approach to identify the model. The analysis is performed using the BUGS language, which is implemented in the software JAGS using the function jags The output is stored in an object of class 'missingHE'.

Usage

selection_long(
  data,
  model.eff,
  model.cost,
  model.mu = mu ~ 1,
  model.mc = mc ~ 1,
  dist_u,
  dist_c,
  type,
  prob = c(0.025, 0.975),
  time_dep = "AR1",
  n.chains = 2,
  n.iter = 20000,
  n.burnin = floor(n.iter/2),
  inits = NULL,
  n.thin = 1,
  ppc = FALSE,
  save_model = FALSE,
  prior = "default",
  ...
)
selection_long(
  data,
  model.eff,
  model.cost,
  model.mu = mu ~ 1,
  model.mc = mc ~ 1,
  dist_u,
  dist_c,
  type,
  prob = c(0.025, 0.975),
  time_dep = "AR1",
  n.chains = 2,
  n.iter = 20000,
  n.burnin = floor(n.iter/2),
  inits = NULL,
  n.thin = 1,
  ppc = FALSE,
  save_model = FALSE,
  prior = "default",
  ...
)

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.mu`, `model.mc` (model formulas for the missing effect and cost models). Among these, effectiveness, cost, time and treatment indicator (only two arms) variables must always be provided and named 'u', 'c' , 'time' and 't', respectively.
`model.eff`	A formula expression in conventional `R` linear modelling syntax. The response must be a health economic effectiveness outcome ('u') 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.mu`	A formula expression in conventional `R` linear modelling syntax. The response must be indicated with the term 'mu' (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_u`	Distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern').
`dist_c`	Distribution assumed for the costs. Current available chocies 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).
`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.
`time_dep`	Type of dependence structure assumed between effectiveness and cost outcomes. Current choices include: autoregressive structure of order one ('AR1') - default - and independence ('none').
`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.
`ppc`	Logical. If `ppc` is `TRUE`, the estimates of the parameters that can be used to generate replications from the model are saved.
`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 of length two containing the user-provided hyperprior values and must be named with the name of the corresponding parameter. For example, the hyperprior values for the standard deviation effect parameters can be provided using the list `prior = list('sigma.prior.u' = c(0, 100))`. 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.
`...`	Additional arguments that can be provided by the user. Examples are `center = TRUE` to center all the covariates in the model or the additional arguments that can be provided to the function `bcea` to summarise the health economic evaluation results. Users may also provide, using the argument `qaly_calc` and `tcost_calc`, the weights to be used for computing the posterior mean QALYs (Area Under the Curve approach) and Total cost (sum over follow-up costs) quantities which are then used to generate any cost-effectiveness decision output (e.g. Cost-Effectiveness Plane). If these are not provided, default values of '0.5' and '1' are used, respectively.

Details

Depending on the distributions specified for the outcome variables in the arguments dist_u and dist_c and the type of missingness mechanism specified in the argument type, different selection models are built and run in the background by the function selection. These models consist in multinomial logistic regressions that are used to estimate the probability of a missingness dropout pattern k (completers, intermittent, dropout) in one or both the longitudinal outcomes. A simple example can be used to show how these selection models are specified. Consider a longitudinal data set comprising a response variable $y$ measures at S occasions and a set of centered covariate $X_j$ . 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 missingness. This is modelled as:

$m_i ~ Multinomial(\pi^k_i)$

$\pi^k_i = \phi^k_i/\sum\phi_i$

$log(\phi^k_i) = \gamma^k_0 + \sum\gamma^k_j X_j + \delta^k (y)$

where

$\pi_i$ is the individual probability of a missing value in $y$ for pattern $k$ at a given time point.
$\gamma^k_0$ represents the marginal probability of a missingness dropout pattern in $y$ for pattern $k$ on the log scale at a given time point.
$\gamma^k_j$ represents the impact on the probability of a specific missingness dropout pattern in $y$ of the centered covariates $X_j$ for pattern $k$ at a given time point.
$\delta^k$ represents the impact on the probability of a specific missingness dropout pattern $k$ in $y$ of the missing pattern itself at a given time point.

$\gamma^k_0 ~ Logisitc(0, 1)$
$\gamma^k_j ~ Normal(0, 0.01)$
$\delta^k ~ Normal(0, 1)$

When user-defined hyperprior values are supplied via the argument prior in the function selection_long, the elements of this list (see Arguments) must be vectors of length two containing the user-provided 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:

location parameters $\alpha_0$ and $\beta_0$ : "mean.prior.u"(effects) and/or "mean.prior.c"(costs)
auxiliary parameters $\sigma$ : "sigma.prior.u"(effects) and/or "sigma.prior.c"(costs)
covariate parameters $\alpha_j$ and $\beta_j$ : "alpha.prior"(effects) and/or "beta.prior"(costs)
marginal probability of missing values for pattern $k$ $\gamma^k_0$ : "p.prior.u"(effects) and/or "p.prior.c"(costs)
covariate parameters in the missingness model for pattern $k$ $\gamma^k_j$ (if covariate data provided): "gamma.prior.u"(effects) and/or "gamma.prior.c"(costs)
mnar parameter for pattern $k$ $\delta^k$ : "delta.prior.u"(effects) and/or "delta.prior.c"(costs)

Value

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

data_set: A list containing the original longitudinal data set provided in data (see Arguments), the number of observed and missing individuals , the total number of individuals by treatment arm and the indicator vectors for the missing values for each time point
model_output: A list containing the output of a JAGS model generated from the functions jags, and the posterior samples for the main parameters of the model and the imputed values
cea: A list containing the output of the economic evaluation performed using the function bcea
type: A character variable that indicate which type of missingness mechanism has been used to run the model, either MAR or MNAR (see details)
data_format: A character variable that indicate which type of analysis was conducted, either using a wide or longitudinal dataset
time_dep: A character variable that indicate which type of time dependence assumption was made, either none or AR1

Author(s)

Andrea Gabrio

References

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.

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

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

Gelman, A. Carlin, JB., Stern, HS. Rubin, DB.(2003). Bayesian Data Analysis, 2nd edition, CRC Press.

Plummer, M. JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. (2003).

Examples

# Quck example to run using subset of PBS dataset

# Load longitudinal dataset

PBS.long <- PBS


# Run the model using the selection function assuming a MAR mechanism
# Use only 100 iterations to run a quick check
model.selection.long <- selection_long(data = PBS.long, model.eff = u ~ 1,model.cost = c ~ 1,
   model.mu = mu ~ 1, model.mc = mc ~ 1, dist_u = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 100, ppc = TRUE, time_dep = "none")

# Print the results of the JAGS model
print(model.selection.long)
#

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

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

# Further examples which take longer to run
model.selection.long <- selection_long(data = PBS.long, model.eff = u ~ 1,model.cost = c ~ u,
   model.se = mu ~ 1, model.mc = mc ~ 1, dist_u = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 500, ppc = FALSE, time_dep = "none")
#
# Print results for all imputed values
print(model.selection.long, value.mis = TRUE)

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

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

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

# Summarise the CEA results
summary(model.selection.long)


#
#
# Quck example to run using subset of PBS dataset

# Load longitudinal dataset

PBS.long <- PBS


# Run the model using the selection function assuming a MAR mechanism
# Use only 100 iterations to run a quick check
model.selection.long <- selection_long(data = PBS.long, model.eff = u ~ 1,model.cost = c ~ 1,
   model.mu = mu ~ 1, model.mc = mc ~ 1, dist_u = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 100, ppc = TRUE, time_dep = "none")

# Print the results of the JAGS model
print(model.selection.long)
#

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

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

# Further examples which take longer to run
model.selection.long <- selection_long(data = PBS.long, model.eff = u ~ 1,model.cost = c ~ u,
   model.se = mu ~ 1, model.mc = mc ~ 1, dist_u = "norm", dist_c = "norm",
   type = "MAR", n.chains = 2, n.iter = 500, ppc = FALSE, time_dep = "none")
#
# Print results for all imputed values
print(model.selection.long, value.mis = TRUE)

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

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

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

# Summarise the CEA results
summary(model.selection.long)


#
#

Summary method for objects in the class `missingHE`

Description

Produces a table printout with some summary results of the health economic evaluation probabilistic model run using the function selection, selection_long, pattern or hurdle.

Usage

## S3 method for class 'missingHE'
summary(object, ...)
## S3 method for class 'missingHE'
summary(object, ...)

Arguments

`object`	A `missingHE` object containing the results of the Bayesian modelling and the economic evaluation
`...`	Additional arguments affecting the summary produced.

Value

Prints a table with some information on the health economic model based on the assumption selected for the missingness using the function selection, selection_long, pattern or hurdle. Summary information on the main parameters of interests is provided.

Author(s)

Andrea Gabrio

References

Baio, G.(2012). Bayesian Methods in Health Economcis. CRC/Chapman Hall, London.

Examples

# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
#
#
# For examples see the function \code{\link{selection}}, \code{\link{selection_long}},
# \code{\link{pattern}} or \code{\link{hurdle}}
#
#

An internal function to select which type of hurdle model to execute for both effectiveness and costs. Alternatives vary depending on the type of distribution assumed for the effect and cost variables, type of structural value mechanism assumed and independence or joint modelling This function selects which type of model to execute.

Description

An internal function to select which type of hurdle model to execute for both effectiveness and costs. Alternatives vary depending on the type of distribution assumed for the effect and cost variables, type of structural value mechanism assumed and independence or joint modelling This function selects which type of model to execute.

Usage

write_hurdle(
  dist_e,
  dist_c,
  type,
  pe_fixed,
  pc_fixed,
  ze_fixed,
  zc_fixed,
  ind_fixed,
  pe_random,
  pc_random,
  ze_random,
  zc_random,
  ind_random,
  model_e_random,
  model_c_random,
  model_se_random,
  model_sc_random,
  se,
  sc
)
write_hurdle(
  dist_e,
  dist_c,
  type,
  pe_fixed,
  pc_fixed,
  ze_fixed,
  zc_fixed,
  ind_fixed,
  pe_random,
  pc_random,
  ze_random,
  zc_random,
  ind_random,
  model_e_random,
  model_c_random,
  model_se_random,
  model_sc_random,
  se,
  sc
)

Arguments

`dist_e`	Distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern')
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm')
`type`	Type of structural value mechanism assumed. Choices are Structural Completely At Random (SCAR) and Structural At Random (SAR)
`pe_fixed`	Number of fixed effects for the effectiveness model
`pc_fixed`	Number of fixed effects for the cost model
`ze_fixed`	Number of fixed effects or the structural indicators model for the effectiveness
`zc_fixed`	Number of fixed effects or the structural indicators model for the costs
`ind_fixed`	Logical; if TRUE independence at the level of the fixed effects between effectiveness and costs is assumed, else correlation is accounted for
`pe_random`	Number of random effects for the effectiveness model
`pc_random`	Number of random effects for the cost model
`ze_random`	Number of random effects or the structural indicators model for the effectiveness
`zc_random`	Number of random effects or the structural indicators model for the costs
`ind_random`	Logical; if TRUE independence at the level of the random effects between effectiveness and costs is assumed, else correlation is accounted for
`model_e_random`	Random effects formula for the effectiveness model
`model_c_random`	Random effects formula for the costs model
`model_se_random`	Random effects formula for the structural indicators model for the effectiveness
`model_sc_random`	Random effects formula for the structural indicators model for the costs
`se`	Structural value for the effectiveness
`sc`	Structural value for the costs

Examples

#Internal function only
#No examples
#
#
#Internal function only
#No examples
#
#

An internal function to select which type of pattern mixture model to execute. Alternatives vary depending on the type of distribution assumed for the effect and cost variables, type of missingness mechanism assumed and independence or joint modelling This function selects which type of model to execute.

Description

An internal function to select which type of pattern mixture model to execute. Alternatives vary depending on the type of distribution assumed for the effect and cost variables, type of missingness mechanism assumed and independence or joint modelling This function selects which type of model to execute.

Usage

write_pattern(
  type,
  dist_e,
  dist_c,
  pe_fixed,
  pc_fixed,
  ind_fixed,
  pe_random,
  pc_random,
  ind_random,
  model_e_random,
  model_c_random,
  d_list,
  d1,
  d2,
  restriction
)
write_pattern(
  type,
  dist_e,
  dist_c,
  pe_fixed,
  pc_fixed,
  ind_fixed,
  pe_random,
  pc_random,
  ind_random,
  model_e_random,
  model_c_random,
  d_list,
  d1,
  d2,
  restriction
)

Arguments

`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR), Missing Not At Random for the effects (MNAR_eff), Missing Not At Random for the costs (MNAR_cost), and Missing Not At Random for both (MNAR)
`dist_e`	Distribution assumed for the effects. Current available choices are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern')
`dist_c`	Distribution assumed for the costs. Current available choices are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm')
`pe_fixed`	Number of fixed effects for the effectiveness model
`pc_fixed`	Number of fixed effects for the cost model
`ind_fixed`	Logical; if TRUE independence between effectiveness and costs is assumed, else correlation is accounted for
`pe_random`	Number of random effects for the effectiveness model
`pc_random`	Number of random effects for the cost model
`ind_random`	Logical; if TRUE independence at the level of the random effects between effectiveness and costs is assumed, else correlation is accounted for
`model_e_random`	Random effects formula for the effectiveness model
`model_c_random`	Random effects formula for the costs model
`d_list`	Number and type of patterns
`d1`	Pattern indicator in the control
`d2`	Pattern indicator in the intervention
`restriction`	type of identifying restriction to be imposed

Examples

# Internal function only
# No examples
#
#
# Internal function only
# No examples
#
#

An internal function to select which type of selection model to execute. Alternatives vary depending on the type of distribution assumed for the effect and cost variables, type of missingness mechanism assumed and independence or joint modelling This function selects which type of model to execute.

Description

An internal function to select which type of selection model to execute. Alternatives vary depending on the type of distribution assumed for the effect and cost variables, type of missingness mechanism assumed and independence or joint modelling This function selects which type of model to execute.

Usage

write_selection(
  dist_e,
  dist_c,
  type,
  pe_fixed,
  pc_fixed,
  ze_fixed,
  zc_fixed,
  ind_fixed,
  pe_random,
  pc_random,
  ze_random,
  zc_random,
  ind_random,
  model_e_random,
  model_c_random,
  model_me_random,
  model_mc_random
)
write_selection(
  dist_e,
  dist_c,
  type,
  pe_fixed,
  pc_fixed,
  ze_fixed,
  zc_fixed,
  ind_fixed,
  pe_random,
  pc_random,
  ze_random,
  zc_random,
  ind_random,
  model_e_random,
  model_c_random,
  model_me_random,
  model_mc_random
)

Arguments

`dist_e`	Distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern')
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm')
`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR), Missing Not At Random for the effects (MNAR_eff), Missing Not At Random for the costs (MNAR_cost), and Missing Not At Random for both (MNAR)
`pe_fixed`	Number of fixed effects for the effectiveness model
`pc_fixed`	Number of fixed effects for the cost model
`ze_fixed`	Number of fixed effects or the missingness indicators model for the effectiveness
`zc_fixed`	Number of fixed effects or the missingness indicators model for the costs
`ind_fixed`	Logical; if TRUE independence between effectiveness and costs is assumed, else correlation is accounted for
`pe_random`	Number of random effects for the effectiveness model
`pc_random`	Number of random effects for the cost model
`ze_random`	Number of random effects or the missingness indicators model for the effectiveness
`zc_random`	Number of random effects or the missingness indicators model for the costs
`ind_random`	Logical; if TRUE independence at the level of the random effects between effectiveness and costs is assumed, else correlation is accounted for
`model_e_random`	Random effects formula for the effectiveness model
`model_c_random`	Random effects formula for the costs model
`model_me_random`	Random effects formula for the missingness indicators model for the effectiveness
`model_mc_random`	Random effects formula for the missingness indicators model for the costs

Examples

#Internal function only
#No examples
#
#
#Internal function only
#No examples
#
#

An internal function to select which type of selection model to execute. Alternatives vary depending on the type of distribution assumed for the effect and cost variables, type of missingness mechanism assumed and independence or joint modelling This function selects which type of model to execute.

Description

Usage

write_selection_long(
  dist_u,
  dist_c,
  type,
  pu_fixed,
  pc_fixed,
  zu_fixed,
  zc_fixed,
  ind_fixed,
  ind_time_fixed,
  pu_random,
  pc_random,
  zu_random,
  zc_random,
  ind_random,
  model_u_random,
  model_c_random,
  model_mu_random,
  model_mc_random
)
write_selection_long(
  dist_u,
  dist_c,
  type,
  pu_fixed,
  pc_fixed,
  zu_fixed,
  zc_fixed,
  ind_fixed,
  ind_time_fixed,
  pu_random,
  pc_random,
  zu_random,
  zc_random,
  ind_random,
  model_u_random,
  model_c_random,
  model_mu_random,
  model_mc_random
)

Arguments

`dist_u`	Distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weibull'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('nbinom') or Bernoulli ('bern')
`dist_c`	Distribution assumed for the costs. Current available chocies are: Normal ('norm'), Gamma ('gamma') or LogNormal ('lnorm')
`type`	Type of missingness mechanism assumed. Choices are Missing At Random (MAR), Missing Not At Random for the effects (MNAR_eff), Missing Not At Random for the costs (MNAR_cost), and Missing Not At Random for both (MNAR)
`pu_fixed`	Number of fixed effects for the effectiveness model
`pc_fixed`	Number of fixed effects for the cost model
`zu_fixed`	Number of fixed effects or the missingness indicators model for the effectiveness
`zc_fixed`	Number of fixed effects or the missingness indicators model for the costs
`ind_fixed`	Logical; if TRUE independence between effectiveness and costs at the same time is assumed, else correlation is accounted for
`ind_time_fixed`	Logical; if TRUE independence between effectiveness and costs over time is assumed, else an AR1 correlation structure is accounted for
`pu_random`	Number of random effects for the effectiveness model
`pc_random`	Number of random effects for the cost model
`zu_random`	Number of random effects or the missingness indicators model for the effectiveness
`zc_random`	Number of random effects or the missingness indicators model for the costs
`ind_random`	Logical; if TRUE independence at the level of the random effects between effectiveness and costs is assumed, else correlation is accounted for
`model_u_random`	Random effects formula for the effectiveness model
`model_c_random`	Random effects formula for the costs model
`model_mu_random`	Random effects formula for the missingness indicators model for the effectiveness
`model_mc_random`	Random effects formula for the missingness indicators model for the costs

Examples

#Internal function only
#No examples
#
#
#Internal function only
#No examples
#
#

Package 'missingHE'

Help Index

An internal function to detect the random effects component from an object of class formula

Description

Usage

Arguments

Examples

Extract regression coefficient estimates from objects in the class missingHE

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

A function to read and re-arrange the data in different ways for the hurdle model

Description

Usage

Arguments

Examples

A function to read and re-arrange the data in different ways

Description

Usage

Arguments

Examples

A function to read and re-arrange the data in different ways

Description

Usage

Arguments

Examples

A function to read and re-arrange the data in different ways

Description

Usage

Arguments

Examples

Diagnostic checks for assessing MCMC convergence of Bayesian models fitted in JAGS using the function selection, selection_long, pattern or hurdle

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

An internal function to extract the random effects component from an object of class formula

Description

Usage

Arguments

Examples

Full Bayesian Models to handle missingness in Economic Evaluations (Hurdle Models)

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

An internal function to detect the random effects component from an object of class formula

Description

Usage

Arguments

Examples

An internal function to detect the random effects component from an object of class formula

Description

Usage

Arguments

Examples

An internal function to summarise results from BUGS model

Description

Usage

Arguments

Examples

MenSS economic data on STIs

Description

Usage

Format

Details

Extract regression coefficient estimates from objects in the class `missingHE`

Diagnostic checks for assessing MCMC convergence of Bayesian models fitted in `JAGS` using the function `selection`, `selection_long`, `pattern` or `hurdle`

Predictive information criteria for Bayesian models fitted in `JAGS` using the funciton `selection`, `selection_long`, `pattern` or `hurdle`

Plot method for the imputed data contained in the objects of class `missingHE`

Posterior predictive checks for assessing the fit to the observed data of Bayesian models implemented in `JAGS` using the function `selection`, `selection_long`, `pattern` or `hurdle`

Print method for the posterior results contained in the objects of class `missingHE`