lmdm: Full Bayesian Models to handle missingness in Economic...
In missingHE: Missing Outcome Data in Health Economic Evaluation
lmdm R Documentation
Full Bayesian Models to handle missingness in Economic Evaluations (Longitudinal Missing Data 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. 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
lmdm(
data,
model.eff,
model.cost,
model.me = me ~ 1,
model.mc = mc ~ 1,
dist_e,
dist_c,
type,
time_dep = "AR1",
prob = c(0.025, 0.975),
n.chains = 2,
n.iter = 10000,
n.burnin = floor(n.iter/2),
inits = NULL,
n.thin = 1,
save_model = FALSE,
prior = "default",
center = FALSE,
...
)
Arguments
data
A data frame in which to find the longitudinal variables supplied in model.eff, model.cost (model formulas for effects and costs)
and model.me, model.mc (model formulas for the missing effect and cost models). Among these,
effectiveness, cost, time and treatment indicator (only two arms) variables must always be provided and named 'e', 'c', 'time' and 'trt', respectively.
model.eff
A formula expression in conventional R linear modelling syntax. The response must be a health economic
effectiveness outcome ('e') whose name must correspond to that used in data. Any covariates in the model must be provided on the right-hand side of the formula.
If there are no covariates, 1 should be specified on the right hand side of the formula. By default, covariates are placed on the "location" parameter of the distribution through a linear model.
Random effects can also be specified for each model parameter. See details for how these can be specified.
model.cost
A formula expression in conventional R linear modelling syntax. The response must be a health economic
cost outcome ('c') whose name must correspond to that used in data. Any covariates in the model must be provided on the right-hand side of the formula.
If there are no covariates, 1 should be specified on the right hand side of the formula. By default, covariates are placed on the "location" parameter of the distribution through a linear model.
A joint bivariate distribution for effects and costs can be specified by including 'e' on the right-hand side of the formula for the costs model.
Random effects can also be specified for each model parameter. See details for how these can be specified.
model.me
A formula expression in conventional R linear modelling syntax. The response must be indicated with the
term 'me' (missing effects) and any covariates must be provided on the right-hand side of the formula. If there are no covariates, 1 should be specified on the right hand side of the formula.
By default, covariates are placed on the "probability" parameter for the missing effects through a logistic-linear model.
Random effects can also be specified for each model parameter. See details for how these can be specified.
model.mc
A formula expression in conventional R linear modelling syntax. The response must be indicated with the term 'mc' (missing costs) and any covariates must be provided on the right-hand side of the formula.
If there are no covariates, 1 should be specified on the right hand side of the formula. By default, covariates are placed on the "probability" parameter for the missing costs through a logistic-linear model.
Random effects can also be specified for each model parameter. See details for how these can be specified.
dist_e
Distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'),
Weibull ('weib'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('negbin') or Bernoulli ('bern').
dist_c
Distribution assumed for the costs. Current available 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).
time_dep
Type of dependence structure assumed between effectiveness and cost outcomes.
Current choices include: autoregressive structure of order one ('AR1') - default, bivariate at each time ('biv') and independence ('none').
prob
A numeric vector of probabilities within the range (0, 1), representing the upper and lower
CI sample quantiles to be calculated and returned for the imputed values.
n.chains
Number of chains.
n.iter
Number of iterations.
n.burnin
Number of warmup iterations.
inits
A list with elements equal to the number of chains selected; each element of the list is itself a list of starting values for the
JAGS model, or a function creating (possibly random) initial values. If inits is NULL, JAGS
will generate initial values for all the model parameters.
n.thin
Thinning interval.
save_model
Logical. If save_model is TRUE, a txt file containing the model code is printed
in the current working directory.
prior
A list containing the hyperprior values provided by the user. Each element of this list must be a vector
containing the user-provided prior distribution and parameter values and must be named with the name of the corresponding parameter. For example, the hyperprior
values for the standard deviation parameter for the effects can be provided using the list prior = list('sigma.prior.e' = c("unif", 0, 10)). For more information about how
to provide prior hypervalues for different types of parameters and models see details. If prior is set to 'default', the default values will be used.
center
Logical. If center is TRUE, all covariates in both the effect and cost models will be centred.
...
Additional arguments that can be provided by the user. Examples are the additional arguments that can be provided to the function bcea 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 models
are built and run in the background by the function lmdm. These models fit multinomial logistic regressions to estimate
the probability of a given missing data pattern k (1 = completers, 2 = intermittent, 3 = dropout) in one or both longitudinal outcomes. A simple example can be used
to show how longitudinal missing data models are specified. Consider a longitudinal data set comprising a response variable y measured at S occasions
and a set of centered covariate X_j, for i = j, ..., J. For each subject in the trial i = 1, ..., n and time s = 1, ..., S we define
an indicator variable m_i taking value k = 1 if the i-th individual is associated with no missing value (completer), a value k = 2
for intermittent missingness over the study period, and a value k = 3 for dropout.
This is modelled as:
m_i ~ Multinomial(\pi^k_i)
\pi^k_i = \phi^k_i/\sum\phi_i
log(\phi^k_i) = \sum\gamma^k_j X_j + \delta^k y_i
where
-
\pi^k_i is the individual probability of a missing value in y for pattern k at a given time.
-
\gamma^k_j represents the impact on the missing data probability in y of the covariate X_j for pattern k at a given time.
-
\delta^k represents the impact on the missing data probability in y for pattern k of missingness itself at a given time.
When \delta = 0 the model assumes a 'MAR' mechanism, while when \delta != 0 the mechanism is 'MNAR'. For the parameters indexing the missingness model,
the default prior distributions assumed are:
-
\gamma^k_j ~ Normal(0, 0.01)
-
\delta^k ~ Normal(0, 1)
When user-defined hyperprior values are supplied via the argument prior in the function lmdm, the elements of this list (see Arguments)
must be vectors containing the user-provided distribution name and hyperprior values and must take specific names according to the parameters they are associated with.
Specifically, the names for the parameters indexing the model which are accepted by missingHE are the following:
auxiliary parameters \sigma: "sigma.prior.e"(effects) and/or "sigma.prior.c"(costs)
covariate parameters \alpha_j and \beta_j: "alpha.prior"(effects) and/or "beta.prior"(costs)
covariate parameters in the missingness model \gamma_j (if covariate data provided): "gamma.prior.e"(effects) and/or "gamma.prior.c"(costs)
mnar parameter \delta: "delta.prior.e"(effects) and/or "delta.prior.c"(costs)
For simplicity, here we have assumed that the set of covariates X_j used in the models for the effects/costs and in the
model of the missing effect/cost values is the same. However, it is possible to specify different sets of covariates for each model
using the arguments in the function lmdm (see Arguments).
For each model, random effects can also be specified for each parameter by adding the term + (x | z) to each model formula,
where x is the fixed regression coefficient for which also the random effects are desired and z is the clustering variable across which
the random effects are specified (must be the name of a factor variable in the dataset).
Value
An object of the class 'missingHE' containing the following elements
- data_set
A list containing the original data set provided in data (see Arguments). Additional information is also included about, among others,
the number of observed and missing individuals, the total number of individuals by treatment arm and the indicator vectors for the missing values
- model_output
A list containing the output of a JAGS model generated from the functions jags, and
the posterior samples for the main parameters of the model
- cea
A list containing the output of the economic evaluation performed using the function bcea
- type
A character variable that indicate which type of missing value mechanism used to run the model, either MAR or MNAR (see details)
- data_format
A character variable that indicates which type of analysis was conducted, either using a wide or long dataset
- time_dep
A character variable that indicate which type of time dependence assumption was made, either none or AR1
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).
See Also
jags, bcea
Examples
# Quick example to run using subset of PBS dataset
# Load longitudinal dataset
PBS.long <- PBS
# Run the model using the long_miss function assuming a MAR mechanism
# Use only 100 iterations to run a quick check
model.long <- lmdm(data = PBS.long, model.eff = e ~ trt, model.cost = c ~ trt,
model.me = me ~ 1, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
type = "MAR", n.chains = 2, n.iter = 100, time_dep = "none")
# Extract regression coefficient estimates
coef(model.long)
#
# Summarise the CEA information from the model
summary(model.long)
# Further examples which take longer to run
model.long <- lmdm(data = PBS.long, model.eff = e ~ trt, model.cost = c ~ trt + age,
model.me = me ~ 1, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
type = "MAR", n.chains = 2, n.iter = 500, time_dep = "none")
# Use looic to assess model fit
pic.looic <- pic(model.long, criterion = "looic")
pic.looic
# Show density plots for all parameters
diag.den <- diagnostic(model.long, type = "denplot", param = "alpha")
# Plots of imputations for all effect data
p1 <- plot(model.long, class = "scatter", outcome = "effects", time.plot = "all")
# Summarise the CEA results
summary(model.long)
#
#
missingHE documentation built on March 19, 2026, 5:06 p.m.
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| lmdm | R Documentation |
Full Bayesian Models to handle missingness in Economic Evaluations (Longitudinal Missing Data 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. 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
lmdm(
data,
model.eff,
model.cost,
model.me = me ~ 1,
model.mc = mc ~ 1,
dist_e,
dist_c,
type,
time_dep = "AR1",
prob = c(0.025, 0.975),
n.chains = 2,
n.iter = 10000,
n.burnin = floor(n.iter/2),
inits = NULL,
n.thin = 1,
save_model = FALSE,
prior = "default",
center = FALSE,
...
)
Arguments
data |
A data frame in which to find the longitudinal variables supplied in |
model.eff |
A formula expression in conventional |
model.cost |
A formula expression in conventional |
model.me |
A formula expression in conventional |
model.mc |
A formula expression in conventional |
dist_e |
Distribution assumed for the effects. Current available chocies are: Normal ('norm'), Beta ('beta'), Gamma ('gamma'), Exponential ('exp'), Weibull ('weib'), Logistic ('logis'), Poisson ('pois'), Negative Binomial ('negbin') or Bernoulli ('bern'). |
dist_c |
Distribution assumed for the costs. Current available 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). |
time_dep |
Type of dependence structure assumed between effectiveness and cost outcomes. Current choices include: autoregressive structure of order one ('AR1') - default, bivariate at each time ('biv') and independence ('none'). |
prob |
A numeric vector of probabilities within the range (0, 1), representing the upper and lower CI sample quantiles to be calculated and returned for the imputed values. |
n.chains |
Number of chains. |
n.iter |
Number of iterations. |
n.burnin |
Number of warmup iterations. |
inits |
A list with elements equal to the number of chains selected; each element of the list is itself a list of starting values for the
|
n.thin |
Thinning interval. |
save_model |
Logical. If |
prior |
A list containing the hyperprior values provided by the user. Each element of this list must be a vector
containing the user-provided prior distribution and parameter values and must be named with the name of the corresponding parameter. For example, the hyperprior
values for the standard deviation parameter for the effects can be provided using the list |
center |
Logical. If |
... |
Additional arguments that can be provided by the user. Examples are the additional arguments that can be provided to the function |
Details
Depending on the distributions specified for the outcome variables in the arguments dist_e and
dist_c and the type of missingness mechanism specified in the argument type, different models
are built and run in the background by the function lmdm. These models fit multinomial logistic regressions to estimate
the probability of a given missing data pattern k (1 = completers, 2 = intermittent, 3 = dropout) in one or both longitudinal outcomes. A simple example can be used
to show how longitudinal missing data models are specified. Consider a longitudinal data set comprising a response variable y measured at S occasions
and a set of centered covariate X_j, for i = j, ..., J. For each subject in the trial i = 1, ..., n and time s = 1, ..., S we define
an indicator variable m_i taking value k = 1 if the i-th individual is associated with no missing value (completer), a value k = 2
for intermittent missingness over the study period, and a value k = 3 for dropout.
This is modelled as:
m_i ~ Multinomial(\pi^k_i)
\pi^k_i = \phi^k_i/\sum\phi_i
log(\phi^k_i) = \sum\gamma^k_j X_j + \delta^k y_i
where
-
\pi^k_iis the individual probability of a missing value inyfor patternkat a given time. -
\gamma^k_jrepresents the impact on the missing data probability inyof the covariateX_jfor patternkat a given time. -
\delta^krepresents the impact on the missing data probability inyfor patternkof missingness itself at a given time.
When \delta = 0 the model assumes a 'MAR' mechanism, while when \delta != 0 the mechanism is 'MNAR'. For the parameters indexing the missingness model,
the default prior distributions assumed are:
-
\gamma^k_j ~ Normal(0, 0.01) -
\delta^k ~ Normal(0, 1)
When user-defined hyperprior values are supplied via the argument prior in the function lmdm, the elements of this list (see Arguments)
must be vectors containing the user-provided distribution name and hyperprior values and must take specific names according to the parameters they are associated with.
Specifically, the names for the parameters indexing the model which are accepted by missingHE are the following:
auxiliary parameters
\sigma: "sigma.prior.e"(effects) and/or "sigma.prior.c"(costs)covariate parameters
\alpha_jand\beta_j: "alpha.prior"(effects) and/or "beta.prior"(costs)covariate parameters in the missingness model
\gamma_j(if covariate data provided): "gamma.prior.e"(effects) and/or "gamma.prior.c"(costs)mnar parameter
\delta: "delta.prior.e"(effects) and/or "delta.prior.c"(costs)
For simplicity, here we have assumed that the set of covariates X_j used in the models for the effects/costs and in the
model of the missing effect/cost values is the same. However, it is possible to specify different sets of covariates for each model
using the arguments in the function lmdm (see Arguments).
For each model, random effects can also be specified for each parameter by adding the term + (x | z) to each model formula, where x is the fixed regression coefficient for which also the random effects are desired and z is the clustering variable across which the random effects are specified (must be the name of a factor variable in the dataset).
Value
An object of the class 'missingHE' containing the following elements
- data_set
A list containing the original data set provided in
data(see Arguments). Additional information is also included about, among others, the number of observed and missing individuals, the total number of individuals by treatment arm and the indicator vectors for the missing values- model_output
A list containing the output of a
JAGSmodel generated from the functionsjags, and the posterior samples for the main parameters of the model- cea
A list containing the output of the economic evaluation performed using the function
bcea- type
A character variable that indicate which type of missing value mechanism used to run the model, either
MARorMNAR(see details)- data_format
A character variable that indicates which type of analysis was conducted, either using a
wideorlongdataset- time_dep
A character variable that indicate which type of time dependence assumption was made, either
noneorAR1
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).
See Also
jags, bcea
Examples
# Quick example to run using subset of PBS dataset
# Load longitudinal dataset
PBS.long <- PBS
# Run the model using the long_miss function assuming a MAR mechanism
# Use only 100 iterations to run a quick check
model.long <- lmdm(data = PBS.long, model.eff = e ~ trt, model.cost = c ~ trt,
model.me = me ~ 1, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
type = "MAR", n.chains = 2, n.iter = 100, time_dep = "none")
# Extract regression coefficient estimates
coef(model.long)
#
# Summarise the CEA information from the model
summary(model.long)
# Further examples which take longer to run
model.long <- lmdm(data = PBS.long, model.eff = e ~ trt, model.cost = c ~ trt + age,
model.me = me ~ 1, model.mc = mc ~ 1, dist_e = "norm", dist_c = "norm",
type = "MAR", n.chains = 2, n.iter = 500, time_dep = "none")
# Use looic to assess model fit
pic.looic <- pic(model.long, criterion = "looic")
pic.looic
# Show density plots for all parameters
diag.den <- diagnostic(model.long, type = "denplot", param = "alpha")
# Plots of imputations for all effect data
p1 <- plot(model.long, class = "scatter", outcome = "effects", time.plot = "all")
# Summarise the CEA results
summary(model.long)
#
#
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