Main analysis functions to estimate different types of models using MCMC sampling, while imputing missing values.

lm_imp(formula, data, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, models = NULL, no_model = NULL, shrinkage = FALSE,
ppc = TRUE, seed = NULL, inits = NULL, warn = TRUE, mess = TRUE,
...)

glm_imp(formula, family, data, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, models = NULL, no_model = NULL, shrinkage = FALSE,
ppc = TRUE, seed = NULL, inits = NULL, warn = TRUE, mess = TRUE,
...)

clm_imp(formula, data, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, nonprop = NULL, rev = NULL, models = NULL,
no_model = NULL, shrinkage = FALSE, ppc = TRUE, seed = NULL,
inits = NULL, warn = TRUE, mess = TRUE, ...)

lognorm_imp(formula, data, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, models = NULL, no_model = NULL, shrinkage = FALSE,
ppc = TRUE, seed = NULL, inits = NULL, warn = TRUE, mess = TRUE,
...)

betareg_imp(formula, data, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, models = NULL, no_model = NULL, shrinkage = FALSE,
ppc = TRUE, seed = NULL, inits = NULL, warn = TRUE, mess = TRUE,
...)

mlogit_imp(formula, data, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, models = NULL, no_model = NULL, shrinkage = FALSE,
ppc = TRUE, seed = NULL, inits = NULL, warn = TRUE, mess = TRUE,
...)

lme_imp(fixed, data, random, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, rd_vcov = "blockdiag", models = NULL,
no_model = NULL, shrinkage = FALSE, ppc = TRUE, seed = NULL,
inits = NULL, warn = TRUE, mess = TRUE, ...)

lmer_imp(fixed, data, random, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, rd_vcov = "blockdiag", models = NULL,
no_model = NULL, shrinkage = FALSE, ppc = TRUE, seed = NULL,
inits = NULL, warn = TRUE, mess = TRUE, ...)

glme_imp(fixed, data, random, family, n.chains = 3, n.adapt = 100,
n.iter = 0, thin = 1, monitor_params = c(analysis_main = TRUE),
auxvars = NULL, refcats = NULL, rd_vcov = "blockdiag", models = NULL,
no_model = NULL, shrinkage = FALSE, ppc = TRUE, seed = NULL,
inits = NULL, warn = TRUE, mess = TRUE, ...)

glmer_imp(fixed, data, random, family, n.chains = 3, n.adapt = 100,
n.iter = 0, thin = 1, monitor_params = c(analysis_main = TRUE),
auxvars = NULL, refcats = NULL, rd_vcov = "blockdiag", models = NULL,
no_model = NULL, shrinkage = FALSE, ppc = TRUE, seed = NULL,
inits = NULL, warn = TRUE, mess = TRUE, ...)

betamm_imp(fixed, random, data, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, rd_vcov = "blockdiag", models = NULL,
no_model = NULL, shrinkage = FALSE, ppc = TRUE, seed = NULL,
inits = NULL, warn = TRUE, mess = TRUE, ...)

lognormmm_imp(fixed, random, data, n.chains = 3, n.adapt = 100,
n.iter = 0, thin = 1, monitor_params = c(analysis_main = TRUE),
auxvars = NULL, refcats = NULL, rd_vcov = "blockdiag", models = NULL,
no_model = NULL, shrinkage = FALSE, ppc = TRUE, seed = NULL,
inits = NULL, warn = TRUE, mess = TRUE, ...)

clmm_imp(fixed, data, random, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, nonprop = NULL, rev = NULL, rd_vcov = "blockdiag",
models = NULL, no_model = NULL, shrinkage = FALSE, ppc = TRUE,
seed = NULL, inits = NULL, warn = TRUE, mess = TRUE, ...)

mlogitmm_imp(fixed, data, random, n.chains = 3, n.adapt = 100,
n.iter = 0, thin = 1, monitor_params = c(analysis_main = TRUE),
auxvars = NULL, refcats = NULL, rd_vcov = "blockdiag", models = NULL,
no_model = NULL, shrinkage = FALSE, ppc = TRUE, seed = NULL,
inits = NULL, warn = TRUE, mess = TRUE, ...)

survreg_imp(formula, data, n.chains = 3, n.adapt = 100, n.iter = 0,
thin = 1, monitor_params = c(analysis_main = TRUE), auxvars = NULL,
refcats = NULL, models = NULL, no_model = NULL, shrinkage = FALSE,
ppc = TRUE, seed = NULL, inits = NULL, warn = TRUE, mess = TRUE,
...)

coxph_imp(formula, data, df_basehaz = 6, n.chains = 3, n.adapt = 100,
n.iter = 0, thin = 1, monitor_params = c(analysis_main = TRUE),
auxvars = NULL, refcats = NULL, models = NULL, no_model = NULL,
shrinkage = FALSE, ppc = TRUE, seed = NULL, inits = NULL,
warn = TRUE, mess = TRUE, ...)

JM_imp(formula, data, df_basehaz = 6, n.chains = 3, n.adapt = 100,
n.iter = 0, thin = 1, monitor_params = c(analysis_main = TRUE),
auxvars = NULL, timevar = NULL, refcats = NULL,
rd_vcov = "blockdiag", models = NULL, no_model = NULL,
assoc_type = NULL, shrinkage = FALSE, ppc = TRUE, seed = NULL,
inits = NULL, warn = TRUE, mess = TRUE, ...)

## Arguments

formula

a two sided model formula (see formula) or a list of such formulas; (more details below).

data

a data.frame containing the original data (more details below)

n.chains

number of MCMC chains

number of iterations for adaptation of the MCMC samplers (see adapt)

n.iter

number of iterations of the MCMC chain (after adaptation; see coda.samples)

thin

thinning interval (integer; see window.mcmc). For example, thin = 1 (default) will keep the MCMC samples from all iterations; thin = 5 would only keep every 5th iteration.

monitor_params

named list or vector specifying which parameters should be monitored (more details below)

auxvars

optional; one-sided formula of variables that should be used as predictors in the imputation procedure (and will be imputed if necessary) but are not part of the analysis model(s). For more details with regards to the behaviour with non-linear effects see the vignette on Model Specification

refcats

optional; either one of "first", "last", "largest" (which sets the category for all categorical variables) or a named list specifying which category should be used as reference category per categorical variable. Options are the category label, the category number, or one of "first" (the first category), "last" (the last category) or "largest" (chooses the category with the most observations). Default is "first". If reference categories are specified for a subset of the categorical variables the default will be used for the remaining variables. (See also set_refcat)

models

optional; named vector specifying the types of models for (incomplete) covariates. This arguments replaces the argument meth used in earlier versions. If NULL (default) models will be determined automatically based on the class of the respective columns of data.

no_model

optional; vector of names of variables for which no model should be specified. Note that this is only possible for completely observed variables and implies the assumptions of independence between the excluded variable and the incomplete variables.

shrinkage

optional; either a character string naming the shrinkage method to be used for regression coefficients in all models or a named vector specifying the type of shrinkage to be used in the models given as names.

ppc

logical: should monitors for posterior predictive checks be set? (not yet used)

seed

optional; seed value (for reproducibility)

inits

optional; specification of initial values in the form of a list or a function (see jags.model). If omitted, starting values for the random number generator are created by JointAI, initial values are then generated by JAGS. It is an error to supply an initial value for an observed node.

warn

logical; should warnings be given? Default is TRUE.

mess

logical; should messages be given? Default is TRUE.

...

trunc

named list specifying limits of truncation for the distribution of the named incomplete variables (see the vignette ModelSpecification)

hyperpars

list of hyper-parameters, as obtained by default_hyperpars()

scale_vars

named vector of (continuous) variables that will be centred and scaled (such that mean = 0 and sd = 1) when they enter a linear predictor to improve convergence of the MCMC sampling. Default is that all numeric variables and integer variables with >20 different values will be scaled. If set to FALSE no scaling will be done.

custom

named list of JAGS model chunks (character strings) that replace the model for the given variable.

append_data_list

list that will be appended to the list containing the data that is passed to rjags (data_list). This may be necessary if additional data / variables are needed for custom (covariate) models.

progress.bar

character string specifying the type of progress bar. Possible values are "text" (default), "gui", and "none" (see update). Note: when sampling is performed in parallel it is not possible to display a progress bar.

quiet

logical; if TRUE then messages generated by rjags during compilation as well as the progress bar for the adaptive phase will be suppressed, (see jags.model)

keep_scaled_mcmc

should the "original" MCMC sample (i.e., the scaled version returned by coda.samples()) be kept? (The MCMC sample that is re-scaled to the scale of the data is always kept.)

modelname

character string specifying the name of the model file (including the ending, either .R or .txt). If unspecified a random name will be generated.

modeldir

directory containing the model file or directory in which the model file should be written. If unspecified a temporary directory will be created.

overwrite

logical; whether an existing model file with the specified <modeldir>/<modelname> should be overwritten. If set to FALSE and a model already exists, that model will be used. If unspecified (NULL) and a file exists, the user is asked for input on how to proceed.

keep_model

logical; whether the created JAGS model file should be saved or removed from (FALSE; default) when the sampling has finished.

family

only for glm_imp and glmm_imp/glmer_imp: a description of the distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. (For more details see below and family.)

nonprop

optional named list of one-sided formulas specifying covariates that have non-proportional effects in cumulative logit models. These covariates should also be part of the regular model formula, and the names of the list should be the names of the ordinal response variables.

rev

optional character vector; vector of ordinal outcome variable names for which the odds should be reversed, i.e., $$logit(y\le k)$$ instead of $$logit(y > k)$$.

fixed

a two sided formula describing the fixed-effects part of the model (see formula)

random

only for multi-level models: a one-sided formula of the form ~x1 + ... + xn | g, where x1 + ... + xn specifies the model for the random effects and g the grouping variable

rd_vcov

character string or list specifying the structure of the random effects variance covariance matrix, see details below.

df_basehaz

degrees of freedom for the B-spline used to model the baseline hazard in proportional hazards models (coxph_imp and JM_imp)

timevar

name of the variable indicating the time of the measurement of a time-varying covariate in a proportional hazards survival model (also in a joint model). The variable specified in "timevar" will automatically be added to "no_model".

assoc_type

named vector specifying the type of the association used for a time-varying covariate in the linear predictor of the survival model when using a "JM" model. Implemented options are "underl.value" (linear predictor; default for covariates modelled using a Gaussian, Gamma, beta or log-normal distribution) covariates) and "obs.value" (the observed/imputed value; default for covariates modelled using other distributions).

## Value

An object of class JointAI.

## Model formulas

### Random effects

It is possible to specify multi-level models as it is done in the package nlme, using fixed and random, or as it is done in the package lme4, using formula and specifying the random effects in brackets:

formula = y ~ x1 + x2 + x3 + (1 | id)

is equivalent to

fixed = y ~ x1 + x2 + x3, random = ~ 1|id


### Multiple levels of grouping

For multiple levels of grouping the specification using formula should be used. There is no distinction between nested and crossed random effects, i.e., ... + (1 | id) + (1 | center) is treated the same as ... + (1 | center/id).

### Nested vs crossed random effects

The distinction between nested and crossed random effects should come from the levels of the grouping variables, i.e., if id is nested in center, then there cannot be observations with the same id but different values for center.

### Modelling multiple models simultaneously & joint models

To fit multiple main models at the same time, a list of formula objects can be passed to the argument formula. Outcomes of one model may be contained as covariates in another model and it is possible to combine models for variables on different levels, for example:

formula = list(y ~ x1 + x2 + x3 + x4 + time + (time | id),
x2 ~ x3 + x4 + x5)

This principle is also used for the specification of a joint model for longitudinal and survival data.

Note that it is not possible to specify multiple models for the same outcome variable.

#### Random effects variance-covariance structure

(Note: This feature is new and has not been fully tested yet.)

By default, a block-diagonal structure is assumed for the variance-covariance matrices of the random effects in models with random effects. This means that per outcome and level random effects are assumed to be correlated, but random effects of different outcomes are modelled as independent. The argument rd_vcov allows the user specify different assumptions about these variance-covariance matrices. Implemented structures are full, blockdiag and indep (all off-diagonal elements are zero).

If rd_vcov is set to one of these options, the structure is assumed for all random effects variance-covariance matrices. Alternatively, it is possible to specify a named list of vectors, where the names are the structures and the vectors contain the names of the response variables which are included in this structure.

For example, for a multivariate mixed model with five outcomes y1, ..., y5, the specification could be:

rd_vcov = list(blockdiag = c("y1", "y2"),
full = c("y3", "y4"),
indep = "y5")

This would entail that the random effects for y3 and y4 are assumed to be correlated (within and across outcomes), random effects for y1 and y2 are assumed to be correlated within each outcome, and the random effects for y5 are assumed to be independent.

It is possible to have multiple sets of response variables for which separate full variance-covariance matrices are used, for example:

rd_vcov = list(full = c("y1", "y2", "y5"),
full = c("y3", "y4"))

In models with multiple levels of nesting, separate structures can be specified per level:

rd_vcov = list(id = list(blockdiag = c("y1", "y2"),
full = c("y3", "y4"),
indep = "y5"),
center = "indep")

### Survival models with frailties or time-varying covariates

Random effects specified in brackets can also be used to indicate a multi-level structure in survival models, as would, for instance be needed in a multi-centre setting, where patients are from multiple hospitals.

It also allows to model time-dependent covariates in a proportional hazards survival model (using coxph_imp), also in combination with additional grouping levels.

In time-dependent proportional hazards models, last-observation-carried-forward is used to fill in missing values in the time-varying covariates, and to determine the value of the covariate at the event time. Preferably, all time-varying covariates should be measured at baseline (timevar = 0). If a value for a time-varying covariate needs to be filled in and there is no previous observation, the next observation will be carried backward.

### Differences to basic regression models

It is not possible to specify transformations of outcome variables, i.e., it is not possible to use a model formula like

log(y) ~ x1 + x2 + ...

In the specific case of a transformation with the natural logarithm, a log-normal model can be used instead of a normal model.

Moreover, it is not possible to use . to indicate that all variables in a data.frame other than the outcome variable should be used as covariates. I.e., a formula y ~ . is not valid in JointAI.

## Data structure

For multi-level settings, the data must be in long format, so that repeated measurements are recorded in separate rows.

For survival data with time-varying covariates (coxph_imp and JM_imp) the data should also be in long format. The survival/censoring times and event indicator variables must be stored in separate variables in the same data and should be constant across all rows referring to the same subject.

During the pre-processing of the data the survival/censoring times will automatically be merged with the observation times of the time-varying covariates (which must be supplied via the argument timevar).

It is possible to have multiple time-varying covariates, which do not have to be measured at the same time points, but there can only be one timevar.

## Imputation methods / model types

Implemented model types that can be chosen in the argument models for baseline covariates (not repeatedly measured) are:

 lm linear (normal) model with identity link (alternatively: glm_gaussian_identity); default for continuous variables glm_gaussian_log linear (normal) model with log link glm_gaussian_inverse linear (normal) model with inverse link glm_logit logistic model for binary data (alternatively: glm_binomial_logit); default for binary variables glm_probit probit model for binary data (alternatively: glm_binomial_probit) glm_binomial_log binomial model with log link glm_binomial_cloglog binomial model with complementary log-log link glm_gamma_inverse gamma model with inverse link for skewed continuous data glm_gamma_identity gamma model with identity link for skewed continuous data glm_gamma_log gamma model with log link for skewed continuous data glm_poisson_log Poisson model with log link for count data glm_poisson_identity Poisson model with identity link for count data lognorm log-normal model for skewed continuous data beta beta model (with logit link) for skewed continuous data in (0, 1) mlogit multinomial logit model for unordered categorical variables; default for unordered factors with >2 levels clm cumulative logit model for ordered categorical variables; default for ordered factors

For repeatedly measured variables the following model types are available:

 lmm linear (normal) mixed model with identity link (alternatively: glmm_gaussian_identity); default for continuous variables glmm_gaussian_log linear (normal) mixed model with log link glmm_gaussian_inverse linear (normal) mixed model with inverse link glmm_logit logistic mixed model for binary data (alternatively: glmm_binomial_logit); default for binary variables glmm_probit probit model for binary data (alternatively: glmm_binomial_probit) glmm_binomial_log binomial mixed model with log link glmm_binomial_cloglog binomial mixed model with complementary log-log link glmm_gamma_inverse gamma mixed model with inverse link for skewed continuous data glmm_gamma_identity gamma mixed model with identity link for skewed continuous data glmm_gamma_log gamma mixed model with log link for skewed continuous data glmm_poisson_log Poisson mixed model with log link for count data glmm_poisson_identity Poisson mixed model with identity link for count data glmm_lognorm log-normal mixed model for skewed covariates glmm_beta beta mixed model for continuous data in (0, 1) mlogitmm multinomial logit mixed model for unordered categorical variables; default for unordered factors with >2 levels clmm cumulative logit mixed model for ordered factors; default for ordered factors

When models are specified for only a subset of the variables for which a model is needed, the default model choices (as indicated in the tables) are used for the unspecified variables.

## Parameters to follow (monitor_params)

Named vector specifying which parameters should be monitored. This can be done either directly by specifying the name of the parameter or indirectly by one of the key words selecting a set of parameters. Except for other, in which parameter names are specified directly, parameter (groups) are just set as TRUE or FALSE.

Models are divided into two groups, the main models, which are the models for which the user has explicitly specified a formula (via formula or fixed), and all other models, for which models were specified automatically.

If left unspecified, monitor_params = c("analysis_main" = TRUE) will be used.

 name/key word what is monitored analysis_main betas and sigma_main, tau_main (for beta regression) or shape_main (for parametric survival models), gamma_main (for cumulative logit models), codeD_main (for multi-level models) and basehaz in proportional hazards models) analysis_random ranef_main, D_main, invD_main, RinvD_main other_models alphas, tau_other, gamma_other, delta_other imps imputed values betas regression coefficients of the main analysis model tau_main precision of the residuals from the main analysis model(s) sigma_main standard deviation of the residuals from the main analysis model(s) gamma_main intercepts in ordinal main model(s) delta_main increments of ordinal main model(s) ranef_main random effects from the main analysis model(s) b D_main covariance matrix of the random effects from the main model(s) invD_main inverse(s) of D_main RinvD_main matrices in the priors for invD_main alphas regression coefficients in the covariate models tau_other precision parameters of the residuals from covariate models gamma_other intercepts in ordinal covariate models delta_other increments of ordinal intercepts ranef_other random effects from the other models b D_other covariance matrix of the random effects from the other models invD_other inverses of D_other RinvD_other matrices in the priors for invD_other other additional parameters

For example:
monitor_params = c(analysis_main = TRUE, tau_main = TRUE, sigma_main = FALSE) would monitor the regression parameters betas and the residual precision tau_main instead of the residual standard deviation sigma_main.

For a linear model, monitor_params = c(imps = TRUE) would monitor betas, and sigma_main (because analysis_main = TRUE by default) as well as the imputed values.

## Cumulative logit (mixed) models

In the default setting for cumulative logit models, i.e, rev = NULL, the odds for a variable $$y$$ with $$K$$ ordered categories are defined as $\log\left(\frac{P(y_i > k)}{P(y_i \leq k)}\right) = \gamma_k + \eta_i, \quad k = 1, \ldots, K-1,$ where $$\gamma_k$$ is a category specific intercept and $$\eta_i$$ the subject specific linear predictor.

To reverse the odds to $\log\left(\frac{P(y_i \leq k)}{P(y_i > k)}\right) = \gamma_k + \eta_i, \quad k = 1, \ldots, K-1,$ the name of the response variable has to be specified in the argument rev, e.g., rev = c("y").

By default, proportional odds are assumed and only the intercepts differ per category of the ordinal response. To allow for non-proportional odds, i.e., $\log\left(\frac{P(y_i > k)}{P(y_i \leq k)}\right) = \gamma_k + \eta_i + \eta_{ki}, \quad k = 1, \ldots, K-1,$ the argument nonprop can be specified. It takes a one-sided formula or a list of one-sided formulas. When a single formula is supplied, or a unnamed list with just one element, it is assumed that the formula corresponds to the main model. To specify non-proportional effects for linear predictors in models for ordinal covariates, the list has to be named with the names of the ordinal response variables.

For example, the following three specifications are equivalent and assume a non-proportional effect of C1 on O1, but C1 is assumed to have a proportional effect on the incomplete ordinal covariate O2:

clm_imp(O1 ~ C1 + C2 + B2 + O2, data = wideDF, nonprop = ~ C1)
clm_imp(O1 ~ C1 + C2 + B2 + O2, data = wideDF, nonprop = list(~ C1))
clm_imp(O1 ~ C1 + C2 + B2 + O2, data = wideDF, nonprop = list(O1 = ~ C1))

To specify non-proportional effects on O2, a named list has to be provided:

clm_imp(O1 ~ C1 + C2 + B2 + O2 + B1, data = wideDF,
nonprop = list(O1 = ~ C1,
O2 = ~ C1 + B1))

The variables for which a non-proportional effect is assumed also have to be part of the regular model formula.

## Custom model parts

(Note: This feature is experimental and has not been fully tested yet.)

Via the argument custom it is possible to provide custom sub-models that replace the sub-models that are automatically generated by JointAI.

Using this feature it is, for instance, possible to use the value of a repeatedly measured variable at a specific time point as covariate in another model. An example would be the use of "baseline" cholesterol (chol at day = 0) as covariate in a survival model.

First, the variable chol0 is added to the PBC data. For most patients the value of cholesterol at baseline is observed, but not for all. It is important that the data has a row with day = 0 for each patient.

PBC <- merge(PBC,
subset(PBC, day == 0, select = c("id", "chol")),
by = "id", suffixes = c("", "0"))

Next, the custom piece of JAGS model syntax needs to be specified. We loop here only over the patients for which the baseline cholesterol is missing.

calc_chol0 <- "
for (ii in 1:28) {
M_id[row_chol0_id[ii], 3] <- M_lvlone[row_chol0_lvlone[ii], 1]
}"

To be able to run the model with the custom imputation "model" for baseline cholesterol we need to provide the numbers of the rows in the data matrices that contain the missing values of baseline cholesterol and the rows that contain the imputed cholesterol at day = 0:

row_chol0_lvlone <- which(PBC$day == 0 & is.na(PBC$chol0))
row_chol0_id <- match(PBC$id, unique(PBC$id))[row_chol0_lvlone]

Then we pass both the custom sub-model and the additional data to the analysis function coxph_imp(). Note that we explicitly need to specify the model for chol.

coxph_imp(list(Surv(futime, status != "censored") ~ age + sex + chol0,
chol ~ age + sex + day + (day | id)),
no_model = "day", data = PBC,
append_data_list = list(row_chol0_lvlone = row_chol0_lvlone,
row_chol0_id = row_chol0_id),
custom = list(chol0 = calc_chol0))

## Note

### Coding of variables:

The default covariate (imputation) models are chosen based on the class of each of the variables, distinguishing between numeric, factor with two levels, unordered factor with >2 levels and ordered factor with >2 levels.

When a continuous variable has only two different values it is assumed to be binary and its coding and default (imputation) model will be changed accordingly. This behaviour can be overwritten specifying a model type via the argument models.

Variables of type logical are automatically converted to unordered factors.

#### Contrasts

JointAI version $$\geq$$ 1.0.0 uses the globally (via options("contrasts")) specified contrasts. However, for incomplete categorical variables, for which the contrasts need to be re-calculated within the JAGS model, currently only contr.treatment and contr.sum are possible. Therefore, when an in complete ordinal covariate is used and the default contrasts (contr.poly()) are set to be used for ordered factors, a warning message is printed and dummy coding (contr.treatment()) is used for that variable instead.

### Non-linear effects and transformation of variables:

JointAI handles non-linear effects, transformation of covariates and interactions the following way:
When, for instance, a model formula contains the function log(x) and x has missing values, x will be imputed and used in the linear predictor of models for which no formula was specified, i.e., it is assumed that the other variables have a linear association with x. The log() of the observed and imputed values of x is calculated and used in the linear predictor of the main analysis model.

If, instead of using log(x) in the model formula, a pre-calculated variable logx is used, this variable is imputed directly and used in the linear predictors of all models, implying that variables that have logx in their linear predictors have a linear association with logx but not with x.

When different transformations of the same incomplete variable are used in one model it is strongly discouraged to calculate these transformations beforehand and supply them as different variables. If, for example, a model formula contains both x and x2 (where x2 = x^2), they are treated as separate variables and imputed with separate models. Imputed values of x2 are thus not equal to the square of imputed values of x. Instead, x and I(x^2) should be used in the model formula. Then only x is imputed and x^2 is calculated from the imputed values of x internally.

The same applies to interactions involving incomplete variables.

### Sequence of models:

Models generated automatically (i.e., not mentioned in formula or fixed are specified in a sequence based on the level of the outcome of the respective model in the multi-level hierarchy and within each level according to the number of missing values. This means that level-1 variables have all level-2, level-3, ... variables in their linear predictor, and variables on the highest level only have variables from the same level in their linear predictor. Within each level, the variable with the most missing values has the most variables in its linear predictor.

### Not (yet) possible:

• prediction (using predict) conditional on random effects

• the use of splines for incomplete variables

• the use of (or equivalents for) pspline, or strata in survival models

• left censored or interval censored data

set_refcat, traceplot, densplot, summary.JointAI, MC_error, GR_crit, predict.JointAI, add_samples, JointAIObject, add_samples, parameters, list_models

Vignettes

## Examples

# Example 1: Linear regression with incomplete covariates
mod1 <- lm_imp(y ~ C1 + C2 + M1 + B1, data = wideDF, n.iter = 100)

# Example 2: Logistic regression with incomplete covariates
mod2 <- glm_imp(B1 ~ C1 + C2 + M1, data = wideDF,
family = binomial(link = "logit"), n.iter = 100)

if (FALSE) {

# Example 3: Linear mixed model with incomplete covariates
mod3 <- lme_imp(y ~ C1 + B2 + c1 + time, random = ~ time|id,
data = longDF, n.iter = 300)

# Example 4: Parametric Weibull survival model
mod4 <- survreg_imp(Surv(time, status) ~ age + sex + meal.cal + wt.loss,
data = survival::lung, n.iter = 100)

# Example 5: Proportional hazards survival model
mod5 <- coxph_imp(Surv(time, status) ~ age + sex + meal.cal + wt.loss,
data = survival::lung, n.iter = 200)

# Example 6: Joint model for longitudinal and survival data
mod6 <- JM_imp(list(Surv(futime, status != 'censored') ~ age + sex +
albumin + copper + trig + (1 | id),
albumin ~ day + age + sex + (day | id)),
timevar = 'day', data = PBC, n.iter = 100)

# Example 7: Proportional hazards  model with a time-dependent covariate
mod7 <- coxph_imp(Surv(futime, status != 'censored') ~ age + sex + copper +
trig + stage + (1 | id),
timevar = 'day', data = PBC, n.iter = 100)

# Example 8: Parallel computation
# If no strategy how the "future" should be handled is specified, the
# MCMC chains are run sequentially.
# To run MCMC chains in parallel, a strategy can be specified using the
# package \pkg{future} (see ?future::plan), for example:
future::plan(future::multisession, workers = 4)
mod8 <- lm_imp(y ~ C1 + C2 + B2, data = wideDF, n.iter = 500, n.chains = 8)
mod8$comp_info$future
# To re-set the strategy to sequential computation, the sequential strategy
# can be specified:
future::plan(future::sequential)

}