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)

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)

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)

Arguments

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, pattern or hurdle.

Usage

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

Arguments

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, pattern or hurdle.

Author(s)

Andrea Gabrio

See Also

selection pattern hurdle diagnostic plot.missingHE

Examples

 
# For examples see the function \code{\link{selection}}, 
# \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
)

Arguments

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)

Arguments

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
)

Arguments

Examples

#Internal function only
#no examples
#
#

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

Description

The focus is restricted to full Bayesian models in cost-effectiveness analyses based on the function selection, 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. Graphics and plots are managed using functions contained in the package ggplot2 and ggthemes.

Usage

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

Arguments

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. 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 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 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 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 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 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 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 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 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).

• "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).

• "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 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.

See Also

ggs selection selection hurdle.

Examples

#For examples see the function selection, pattern or 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)

Arguments

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",
  ...
)

Arguments

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)

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).

See Also

jags, bcea

Examples

# Quck 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 = 100,  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)

Arguments

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)

Arguments

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,
  ...
)

Arguments

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)
#
#

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)

Arguments

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",
  ...
)

Arguments

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.

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 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)

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).

See Also

jags, bcea

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)


#
#

Predictive information criteria for Bayesian models fitted in JAGS using the funciton selection, 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")

Arguments

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. 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. 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 esitmates, 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.

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.

See Also

jags, loo, waic

Examples

 
#For examples see the function selection, pattern or 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, 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",
  theme = NULL,
  ...
)

Arguments

Details

The funciton 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, 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.

See Also

selection pattern hurdle diagnostic

Examples

#For examples see the function selection, pattern or hurdle
#
#

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

Description

The focus is restricted to full Bayesian models in cost-effectiveness analyses based on the function selection, 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,
  theme = NULL,
  scheme_set = NULL,
  legend = "top",
  ...
)

Arguments

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, 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.

See Also

selection pattern hurdle diagnostic

Examples

# For examples see the function \code{\link{selection}}, 
# \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, ...)

Arguments

Author(s)

Andrea Gabrio

Examples

 
# For examples see the function \code{\link{selection}}, 
# \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
)

Arguments

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
)

Arguments

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
)

Arguments

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)

Arguments

Examples

#Internal function only
#No examples
#
#

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)

Arguments

Examples

#Internal function only
#No examples
#
#

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)

Arguments

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",
  ...
)

Arguments

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 selection models are built and run in the background by the function selection. These models consist in logistic regressions that are used to estimate the probability of missingness in one or both the outcomes. A simple example can be used to show how selection models are specified. Consider a data set comprising a response variable y and a set of centered covariate X_j. For each subject in the trial i = 1, ..., n we define an indicator variable m_i taking value 1 if the i-th individual is associated with a missing value and 0 otherwise. This is modelled as:

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).

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 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)

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).

See Also

jags, bcea

Examples

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

# Run the model using the selection function assuming a SCAR 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)


#
#

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, pattern or hurdle.

Usage

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

Arguments

Value

Prints a table with some information on the health economic model based on the assumption selected for the missingness using the function selection, 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.

See Also

selection pattern hurdle diagnostic plot.missingHE

Examples

#For examples see the function selection, pattern or 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
)

Arguments

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
)

Arguments

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
)

Arguments

Examples

#Internal function only
#No examples
#
#