Description Usage Arguments Details Value Author(s) References See Also Examples
Full Bayesian costeffectiveness 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'.
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data 
A data frame in which to find the variables supplied in 
model.eff 
A formula expression in conventional 
model.cost 
A formula expression in conventional 
model.se 
A formula expression in conventional 
model.sc 
A formula expression in conventional 
se 
Structural value to be found in the effect variables defined in 
sc 
Structural value to be found in the cost variables defined in 
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

n.thin 
Thinning interval. 
ppc 
Logical. If 
save_model 
Logical. If 
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 userprovided 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 
... 
Additional arguments that can be provided by the user. Examples are 
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 "nonstructural" 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 ith individual is associated with a structural value and 0
otherwise.
This is modelled as:
d_i ~ Bernoulli(π_i)
logit(π_i) = γ_0 + ∑γ_j X_j
where
π_i is the individual probability of a structural value in y.
γ_0 represents the marginal probability of a structural value in y on the logit scale.
γ_j represents the impact on the probability of a structural value in y of the centered covariates X_j.
When γ_j = 0, the model assumes a 'SCAR' mechanism, while when γ_j != 0 the mechanism is 'SAR'. For the parameters indexing the structural value model, the default prior distributions assumed are the following:
γ_0 ~ Logisitic(0, 1)
γ_j ~ Normal(0, 0.01)
When userdefined 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 userprovided 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 α_0, β_0: "mean.prior.e"(effects) and/or "mean.prior.c"(costs)
auxiliary parameters σ: "sigma.prior.e"(effects) and/or "sigma.prior.c"(costs)
covariate parameters α_j, β_j: "alpha.prior"(effects) and/or "beta.prior"(costs)
marginal probability of structural values γ_0: "p.prior.e"(effects) and/or "p.prior.c"(costs)
covariate parameters in the model of the structural values γ_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).
An object of the class 'missingHE' containing the following elements
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
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
A list containing the output of the economic evaluation performed using the function bcea
A character variable that indicate which type of structural value mechanism has been used to run the model,
either SCAR
or SAR
(see details)
Andrea Gabrio
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).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62  # 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)
#
#

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