In this vignette, we use the NHANES data for examples in crosssectional data and the dataset simLong for examples in longitudinal data. For more info on these datasets, check out the vignette Visualizing Incomplete Data, in which the distributions of variables and missing values in both sets is explored.
To learn more about the theoretical background of the statistical approach implemented in JointAI, check out the vignette Theoretical Background.
Note:
In some of the examples we use n.adapt = 0
(and n.iter = 0
, which is the default). This is to prevent the MCMC sampling and thereby reduce computational time when compiling this vignette.
JointAI has several main functions (which are abbreviated with *_imp()
):
lm_imp()
: linear regressionglm_imp()
: generalized linear regressionlognorm_imp()
: lognormal regressionbetareg_imp()
: beta regressionclm_imp()
: cumulative logit models for ordinal datamlogit_imp()
: multinomial logit models for unordered factorslme_imp()
/ lmer_imp()
: linear mixed effects regressionglme_imp()
/ glmer_imp()
: generalized linear mixed effects regressionlognormmm_imp()
: lognormal mixed effects regressionbetamm_imp()
: beta mixed effects regressionclmm_imp()
: cumulative logit mixed modelsmlogitmm_imp()
: multinomial logit mixed modelssurvreg_imp()
: parametric (Weibull) survival modelscoxph_imp()
: Proportional hazards survival modelsJM_imp()
: Joint model for longitudinal and survival dataSpecification of these functions is similar to the specification of the complete data versions lm()
, glm()
, lme()
(from package nlme) or lmer()
(from package lme4) and survreg()
and coxph()
(from package survival).
All functions require the arguments formula
(or fixed
and random
in for mixed models) and data
.
Specification of the (fixed effects) model formula is demonstrated in section Model formula, specification of the random random effects in section Multilevel structure & longitudinal covariates.
Additionally, glm_imp()
, glme_imp()
and glmer_imp()
require the specification of the model family
(and link
function).
glm_imp()
, glme_imp()
and glmer_imp()
are
family  

gaussian

with links: identity , log

binomial

with links: logit , probit , log , cloglog

Gamma

with links: inverse , identity , log

poisson

with links: log , identity

The argument family
can be provided as character string or as a function. If the link function is omitted, the default link is used.
Example:
The following three specifications are equal:
mod1a < glm_imp(educ ~ age + gender + creat, data = NHANES,
family = "binomial", n.adapt = 0)
mod1b < glm_imp(educ ~ age + gender + creat, data = NHANES,
family = binomial(), n.adapt = 0)
mod1c < glm_imp(educ ~ age + gender + creat, data = NHANES,
family = binomial(link = 'logit'), n.adapt = 0)
mod1a$analysis_type
#> [1] "glm"
#> attr(,"family")
#>
#> Family: binomial
#> Link function: logit
To use, for instance, a “probit” link instead, this needs to be specified explicitly:
The arguments formula
and fixed
take a twosided formula
object, where ~
separates the response (outcome / dependent variable) from the linear predictor, in which covariates (independent variables) are separated by +
. An intercept is added automatically (except in proportional hazard models or models for ordinal outcomes).
survreg_imp()
and coxph_imp()
expect a survival object (created with Surv()
) on the left hand side of the model formula. Currently, only right censored data can be handled and there can only be one type of event (i.e., no competing risks or multistate models).
Note: formula
and fixed
can not be specified together. You either need to provide the argument formula
or the arguments fixed
and random
.
Interactions between variables can be introduced using :
or *
, which adds the interaction term AND the main effects, i.e.,
SBP ~ age + gender + smoke * creat
is equivalent to
SBP ~ age + gender + smoke + creat + smoke:creat
Interactions between multiple variables can be specified using parentheses:
mod2a < glm_imp(educ ~ gender * (age + smoke + creat),
data = NHANES, family = binomial(), n.adapt = 0)
The function parameters()
returns a matrix off all parameters that are specified to be followed (column coef
) and, for regression coefficients, the name of the variable the coefficient relates to (varname
), the outcome variable of the respective model outcome
. For multinomial models, which have multiple linear predictors, the column outcat
identifies the category of the outcome the parameters refer to.
We use the function parameters()
here and in other vignettes to demonstrate the effect that different model specifications have.
parameters(mod2a)
#>
#> Note: "mod2a" does not contain MCMC samples.
#> outcome outcat varname coef
#> 1 educ <NA> (Intercept) beta[1]
#> 2 educ <NA> genderfemale beta[2]
#> 3 educ <NA> age beta[3]
#> 4 educ <NA> smokeformer beta[4]
#> 5 educ <NA> smokecurrent beta[5]
#> 6 educ <NA> creat beta[6]
#> 7 educ <NA> genderfemale:age beta[7]
#> 8 educ <NA> genderfemale:smokeformer beta[8]
#> 9 educ <NA> genderfemale:smokecurrent beta[9]
#> 10 educ <NA> genderfemale:creat beta[10]
#> 11 creat <NA> (Intercept) alpha[1]
#> 12 creat <NA> genderfemale alpha[2]
#> 13 creat <NA> age alpha[3]
#> 14 creat <NA> smokeformer alpha[4]
#> 15 creat <NA> smokecurrent alpha[5]
#> 16 smoke <NA> genderfemale alpha[6]
#> 17 smoke <NA> age alpha[7]
To specify interactions of a given level, ^
can be used:
# all twoway interactions:
mod2b < glm_imp(educ ~ gender + (age + smoke + creat)^2,
data = NHANES, family = binomial(), n.adapt = 0)
parameters(mod2b)
#> outcome outcat varname coef
#> 1 educ <NA> (Intercept) beta[1]
#> 2 educ <NA> genderfemale beta[2]
#> 3 educ <NA> age beta[3]
#> 4 educ <NA> smokeformer beta[4]
#> 5 educ <NA> smokecurrent beta[5]
#> 6 educ <NA> creat beta[6]
#> 7 educ <NA> age:smokeformer beta[7]
#> 8 educ <NA> age:smokecurrent beta[8]
#> 9 educ <NA> age:creat beta[9]
#> 10 educ <NA> smokeformer:creat beta[10]
#> 11 educ <NA> smokecurrent:creat beta[11]
#> 12 creat <NA> (Intercept) alpha[1]
#> 13 creat <NA> genderfemale alpha[2]
#> 14 creat <NA> age alpha[3]
#> 15 creat <NA> smokeformer alpha[4]
#> 16 creat <NA> smokecurrent alpha[5]
#> 17 smoke <NA> genderfemale alpha[6]
#> 18 smoke <NA> age alpha[7]
# all two and threeway interactions:
mod2c < glm_imp(educ ~ gender + (age + smoke + creat)^3,
data = NHANES, family = binomial(), n.adapt = 0)
parameters(mod2c)
#> outcome outcat varname coef
#> 1 educ <NA> (Intercept) beta[1]
#> 2 educ <NA> genderfemale beta[2]
#> 3 educ <NA> age beta[3]
#> 4 educ <NA> smokeformer beta[4]
#> 5 educ <NA> smokecurrent beta[5]
#> 6 educ <NA> creat beta[6]
#> 7 educ <NA> age:smokeformer beta[7]
#> 8 educ <NA> age:smokecurrent beta[8]
#> 9 educ <NA> age:creat beta[9]
#> 10 educ <NA> smokeformer:creat beta[10]
#> 11 educ <NA> smokecurrent:creat beta[11]
#> 12 educ <NA> age:smokeformer:creat beta[12]
#> 13 educ <NA> age:smokecurrent:creat beta[13]
#> 14 creat <NA> (Intercept) alpha[1]
#> 15 creat <NA> genderfemale alpha[2]
#> 16 creat <NA> age alpha[3]
#> 17 creat <NA> smokeformer alpha[4]
#> 18 creat <NA> smokecurrent alpha[5]
#> 19 smoke <NA> genderfemale alpha[6]
#> 20 smoke <NA> age alpha[7]
In JointAI, interactions between any variables, observed or incomplete, variables on different levels of a hierarchical structure, can be handled. When an incomplete variable is involved, the interaction term is recalculated within each iteration of the MCMC sampling, using the imputed values from the current iteration.
It is important not to precalculate interactions with incomplete variables as extra variables in the dataset, but to specify them in the model formula. Otherwise, imputed values of the main effect and interaction term will not match, and results may be incorrect.
In practice, associations between outcome and covariates do not always meet the standard assumption that all covariate effects are linear. Often, assuming a logarithmic, quadratic, or other nonlinear effect is more appropriate.
Nonlinear associations can be specified in the model formula using functions such as log()
(the natural logarithm), sqrt()
(the square root) or exp()
(the exponential function). It is also possible to use algebraic operations to calculate a new variable from one or more covariates. To indicate to R that the operators used in the formula should be interpreted as algebraic operators and not as formula operators, such calculations need to be wrapped in the function I()
.
For example, to include a quadratic effect of the variable x
we would have to use I(x^2)
. Just writing x^2
would be interpreted as the interaction of x
with itself, which simplifies to just x
.
For completely observed covariates, JointAI can handle any standard type of function implemented in R. This also includes splines, e.g., using ns()
or bs()
from the package splines (which is automatically installed with R).
Functions involving variables that have missing values need to be recalculated in each iteration of the MCMC sampling. Therefore, currently, only functions that can be interpreted by JAGS can be used for incomplete variables. Those functions include:
log()
, exp()
sqrt()
,abs()
sin()
, cos()
I()
) and other algebraic operations involving one or multiple (in)complete variables, as long as the formula can be interpreted by JAGS.The list of functions implemented in JAGS can be found in the JAGS user manual.
Some examples:^{1}
# Absolute difference between bili and creat
mod3a < lm_imp(SBP ~ age + gender + abs(bili  creat), data = NHANES)
# Using a natural cubic spline for age (completely observed) and a quadratic
# and a cubic effect for bili
library(splines)
mod3b < lm_imp(SBP ~ ns(age, df = 2) + gender + I(bili^2) + I(bili^3), data = NHANES)
# A function of creat and albu
mod3c < lm_imp(SBP ~ age + gender + I(creat/albu^2), data = NHANES,
models = c(creat = 'lognorm', albu = 'lognorm'))
# This function may make more sense to calculate BMI as weight/height^2, but
# we currently do not have those variables in the NHANES dataset.
# Using the sine and cosine
mod3d < lm_imp(SBP ~ bili + sin(creat) + cos(albu), data = NHANES)
When a model formula includes a function of a complete or incomplete variable, the main effect of that variable is automatically added as an auxiliary variable. (For more info on auxiliary variables, see the section “Auxiliary variables”.) In the linear predictors of models for covariates, usually, only the main effects are used.
In mod3b
from above, for example, the spline of age is used as predictor for SBP
, but in the imputation model for bili
, age
enters with a linear effect.
list_models(mod3b, priors = FALSE, regcoef = FALSE, otherpars = FALSE)
#> Linear model for "SBP"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), ns(age, df = 2)1, ns(age, df = 2)2, genderfemale, I(bili^2), I(bili^3)
#>
#>
#> Linear model for "bili"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale
The function list_models()
prints information on all submodels specified in a JointAI
object. This includes the model(s) specified by the user via the model formula and all models for covariates that JointAI has specified automatically. Since here we are only interested in the predictor variables, we suppress printing of information on prior distributions, regression coefficients and other parameters by setting priors
, regcoef
and otherpars
to FALSE
.
When a function of a variable is specified as an auxiliary variable, this function is used (as well) in the models for covariates. For example, in mod3e
, waist circumference (WC
) is not part of the model for SBP
, and the auxiliary variable I(WC^2)
is used in the linear predictor of the imputation model for bili
:
mod3e < lm_imp(SBP ~ age + gender + bili, auxvars = ~ I(WC^2), data = NHANES)
list_models(mod3e, priors = FALSE, regcoef = FALSE, otherpars = FALSE)
#> Linear model for "SBP"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale, bili
#>
#>
#> Linear model for "bili"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale
#>
#>
#> Linear model for "WC"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale
When WC
is used as predictor variable in the main model and a function of WC
is specified as auxiliary variable, both variables are used as predictors in the imputation models:
mod3f < lm_imp(SBP ~ age + gender + bili + WC, auxvars = ~ I(WC^2), data = NHANES)
list_models(mod3f, priors = FALSE, regcoef = FALSE, otherpars = FALSE)
#> Linear model for "SBP"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale, bili, WC
#>
#>
#> Linear model for "bili"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale, WC
#>
#>
#> Linear model for "WC"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale
When a function of a covariate is used in the linear predictor of the analysis model, and that function should also be used in the linear predictor of imputation models, it is necessary to also include that function in the argument auxvars
:
mod3g < lm_imp(SBP ~ age + gender + bili + I(WC^2), auxvars = ~ I(WC^2), data = NHANES)
list_models(mod3g, priors = FALSE, regcoef = FALSE, otherpars = FALSE)
#> Linear model for "SBP"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale, bili, I(WC^2)
#>
#>
#> Linear model for "bili"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale, WC
#>
#>
#> Linear model for "WC"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale
Incomplete variables are always imputed on their original scale, i.e.,
mod3b
the variable bili
is imputed and the quadratic and cubic versions calculated from the imputed values.creat
and albu
in mod3c
are imputed separately, and I(creat/albu^2)
calculated from the imputed (and observed) values.Important:
When different transformations of the same incomplete variable are used in one model, it is strongly discouraged to calculate these transformations beforehand and to supply them as separate variables. The same is the case for interactions.
If, for example, a model formula contains both x
and x2
(where x2
= x^2
), they are treated as separate variables and imputed with different models. Imputed values of x2
are thus not equal to the square of imputed values of x
. Instead, x + I(x^2)
should be used in the model formula. Then, only x
is imputed and used in the linear predictor of models for other incomplete variables, and x^2
is calculated from the imputed values of x
.
When a function has restricted support, e.g., log(x)
is only defined for x > 0
, the model used to impute x
needs to comply with these restrictions. This can either be achieved by truncating the distribution assumed for x
, using the argument trunc
, or by specifying a model for x
that meets the restrictions. For more information on imputation methods (models for covariates), see the section Imputation model types.
Example:
When using a log()
transformation for the covariate bili
, we can either use the default model for continuous variables, "lm"
, a linear model, i.e., assuming a normal distribution and truncate this distribution by specifying trunc = list(bili = c(<lower>, <upper>))
(where the lower and upper limits are the smallest and largest allowed values) or choose a model (using the argument models
; more details see the section on covariate model types) that only imputes positive values such as a lognormal distribution ("glm_lognorm"
) or a Gamma distribution (e.g., "glm_gamma_log"
):
# truncation of the distribution of bili
mod4a < lm_imp(SBP ~ age + gender + log(bili) + exp(creat),
trunc = list(bili = c(1e5, NA)), data = NHANES)
# lognormal model for bili
mod4b < lm_imp(SBP ~ age + gender + log(bili) + exp(creat),
models = c(bili = 'lognorm', creat = 'lm'), data = NHANES)
# gamma model with loglink for bili
mod4c < lm_imp(SBP ~ age + gender + log(bili) + exp(creat),
models = c(bili = 'glm_gamma_log', creat = 'lm'), data = NHANES)
If only onesided truncation is needed, the other limit can be provided as NA
.
It is possible to use functions that have different names in R and JAGS, or that do exist in JAGS, but not in R, by defining a new function in R that has the name of the function in JAGS.
Example:
In JAGS the inverse logit transformation is defined in the function ilogit
. In R, there is no function ilogit
, but the inverse logit is available as the distribution function of the logistic distribution plogis()
.
# Define the function ilogit
ilogit < plogis
# Use ilogit in the model formula
mod5a < lm_imp(SBP ~ age + gender + ilogit(creat), data = NHANES)
It is also possible to nest a function in another function.
Example:^{2}
The complementary log log transformation is restricted to values larger than 0 and smaller than 1. In order to use this function on a variable that exceeds this range, as is the case for creat
, a second transformation might be used, for instance the inverse logit from the previous example.
In JAGS, the complementary loglog transformation is implemented as cloglog
, but since this function does not exist in (base) R, we first need to define it:
In multilevel models, additional to the fixed effects structure specified by the argument fixed
a random effects structure needs to be provided via the argument random
. Alternatively, it is possible to provide a formula
that contains both the fixed and random effects structure (corresponding to the specification used in lme4).
random
takes a onesided formula starting with a ~
. Variables for which a random effect should be included are usually separated by a +
, and the grouping variable is separated by 
. A random intercept is added automatically and only needs to be specified in a random intercept only model.
A few examples:
random = ~ 1  id
: random intercept only, with id
as grouping variablerandom = ~ time  id
: random intercept and slope for variable time
random = ~ time + I(time^2)  id
: random intercept, slope and quadratic random effect for time
random = ~ time + x  id
random intercept, random slope for time
and random effect for variable x
The corresponding specifications using the argument formula
would be
<fixed effects> + (1  id)
<fixed effects> + (time  id)
<fixed effects> + (time + I(time^2)  id)
<fixed effects> + (time + x  id)
It is possible to use splines in the random effects structure (but only for completely observed variables), e.g.:
mod7a < lme_imp(bmi ~ GESTBIR + ETHN + HEIGHT_M + ns(age, df = 2),
random = ~ns(age, df = 2)  ID, data = simLong)
Since JointAI version 1.0.0 it is possible to model multilevel data with multiple levels of grouping In that setting, the formula
specification needs to be used:
It is possible to model both crossed and nested random effects, however the distinction between crossed and nested random effects must come from the coding of the id variables. For example, if patients are nested in hospitals, all observations that have the same patient id also need to have the same hospital id.
When this is not the case, i.e., some patients were measured at multiple hospitals, the random effects are crossed.
There is (theoretically) no restriction as to how many grouping levels are possible.
From JointAI version 0.5.0 onward imputation of longitudinal covariates is possible. For details the types of models that are available for covariates in a multilevel setting, see the section covariate model types below.
Note:
When incomplete baseline covariates (level > 1) are involved in the model it is usually necessary to specify models for all variables on lower levels, even if they are completely observed. This is done automatically by JointAI, but it may be necessary to change the default model types to models that better fit the distributions of the respective variables.
It is typically not necessary to specify models for variables on higher levels if there are no incomplete covariates on lower levels. For example, in a 2level setting, if there are no missing values in level2 variables, it is not necessary to specify models for completely observed level1 variables. But if there are missing values in level2 variables, models need to be specified for all level1 variables.
The joint distribution of an outcome \(y\), covariates \(x\), random effects \(b\) and parameters \(\theta\), \(p(y, x, b, \theta)\), is modelled as the product of univariate conditional distributions. To facilitate the specification of these distributions they are ordered so that longitudinal (level1) variables may have baseline (level2) variables in their linear predictors but not vice versa.
For example: \[\begin{align} p(y, x, b, \theta) = & p(y \mid x_1, ..., x_4, b_y, \theta_y) && \text{analysis model}\\ & p(x_1\mid \theta_{x1}) && \text{model for a complete baseline covariate}\\ & p(x_2\mid x_1, \theta_{x2}) && \text{model for an incomplete baseline covariate}\\ & p(x_3\mid x_1, x_2, b_{x3}, \theta_{x3}) && \text{model for a complete longitudinal covariate}\\ & p(x_4\mid x_1, x_2, x_3, b_{x4}, \theta_{x4}) && \text{model for an incomplete longitudinal covariate}\\ & p(b_y\theta_b) p(b_{x3}\theta_b) p(b_{x4}\theta_b) && \text{models for the random effects}\\ & p(\theta_y) p(\theta_{x1}) \ldots p(\theta_{x4}) p(\theta_b) && \text{prior distributions}\end{align}\]
Since the parameter vectors \(\theta_{x1}\), \(\theta_{x2}\), … are assumed to be a priori independent, and furthermore \(x_1\) is completely observed and modelled independently of incomplete variables, estimation of the other model parts is not affected by \(p(x_1\mid \theta_{x1})\) and, hence, this model can be omitted.
\(p(x_3 \mid x_1, x_2, b_{x3}, \theta_{x3})\), on the other hand is modelled conditional on the incomplete covariate \(x_2\) and can therefore not be omitted.
If there were no incomplete baseline covariates, i.e., if \(x_2\) was completely observed, \(p(x_3 \mid x_1, x_2, b_{x3}, \theta_{x3})\) would also not affect the estimation of parameters in the other parts of the model and could be omitted.
JointAI automatically selects models for all incomplete covariates (and if necessary also for some complete covariates). The type of model is selected automatically based on the class
of the variable and the number of levels.
name  model  variable type 

lm

linear regression  continuous variables 
logit

logistic regression  factors with two levels 
mlogit

multinomial logit model  unordered factors with >2 levels 
clm

cumulative logit model  ordered factors with >2 levels 
name  model  variable type 

lmm

linear mixed model  continuous longitudinal variables 
glmm_logit

logistic mixed model  longitudinal factors with two levels 
mlogitmm

multinomial logit mixed model  longitudinal unordered factors with >2 levels 
clmm

cumulative logit mixed model  longitudinal ordered factors with >2 levels 
The imputation models that are chosen by default may not necessarily be appropriate for the data at hand, especially for continuous variables, which often do not comply with the assumptions of (conditional) normality.
Therefore, alternative imputation methods are available for baseline covariates:name  model  variable type 

lognorm

normal regression of the logtransformed variable  rightskewed variables >0 
beta

beta regression (with logitlink)  continuous variables with values in [0, 1] 
glm_<family>_<link>

e.g. glm_gamma_inverse for Gamma regression with an inverselink

lognorm
assumes a normal distribution for the natural logarithm of the variable, but the variable enters the linear predictor of the analysis model (and possibly other imputation models) on its original scale.
name  model  variable type 

glmm_lognorm

normal mixed model for the logtransformed variable  longitudinal rightskewed variables >0 
glmm_beta

beta regression (with logitlink)  continuous variables with values in [0, 1] 
glmm_<family>_<link>

e.g. glmm_poisson_log for a poisson mixed model with loglink

longitudinal count variables 
Logistic (mixed) models can be abbreviated glm_logit
(glmm_logit
).
In models mod4b
and mod4c
we have already seen two examples in which the imputation model type was changed using the argument models
.
models
takes a named vector of (imputation/covariate) model types, where the names are the names of the covariates. When the vector supplied to models
only contains specifications for a subset of the covariates, default models are used for the remaining variables.
mod8a < lm_imp(SBP ~ age + gender + WC + alc + bili + occup + smoke,
models = c(bili = 'glm_gamma_log', WC = 'lognorm'),
data = NHANES, n.adapt = 0, progress.bar = 'none')
mod8a$models
#> SBP alc occup bili
#> "glm_gaussian_identity" "glm_binomial_logit" "mlogit" "glm_gamma_log"
#> smoke WC
#> "clm" "lognorm"
When there is a “time” variable in the model, such as age
(age of the child at the time of the measurement) in the simLong
it may not be meaningful to specify a model for that variable. Especially when the “time” variable is prespecified by the design of the study it can usually be assumed to be independent of the covariates and a model for it has no useful interpretation.
The argument no_model
allows us to exclude models for such variables (as long as they are completely observed):
mod8b < lme_imp(bmi ~ GESTBIR + ETHN + HEIGHT_M + SMOKE + hc + MARITAL +
ns(age, df = 2),
random = ~ns(age, df = 2)  ID, data = simLong,
no_model = "age", n.adapt = 0)
mod8b$models
#> bmi hc SMOKE MARITAL
#> "glmm_gaussian_identity" "lmm" "clm" "mlogit"
#> ETHN HEIGHT_M
#> "glm_binomial_logit" "lm"
By excluding the model for age
we implicitly assume that incomplete baseline variables are independent of age
.
Note:
When a continuous incomplete 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 when the imputation method for that variable is specified directly by the user.
Variables of type logical
are automatically converted to binary factors.
In JointAI, the models automatically specified for covariates are ordered by the hierarchical level of the respective response variable (descending). The linear predictor of each model contains the incomplete variables that are specified later in the sequence and all complete variables of the same or lower level.
Within each level, models are ordered by the proportion of missing values in the respective response variables, so that the variable with the most missing values has the most covariates in its linear predictor.
get_missinfo(mod8a)
#> $complete_cases
#> # %
#> lvlone 116 62.36559
#>
#> $miss_list
#> $miss_list$lvlone
#> # NA % NA
#> SBP 0 0.000000
#> age 0 0.000000
#> gender 0 0.000000
#> WC 2 1.075269
#> smoke 7 3.763441
#> bili 8 4.301075
#> occup 28 15.053763
#> alc 34 18.279570
# print information on the imputation models (and omit everything but the predictor variables)
list_models(mod8a, priors = FALSE, regcoef = FALSE, otherpars = FALSE, refcat = FALSE)
#> Linear model for "SBP"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale, WC, alc>=1, bili, occuplooking for work, occupnot
#> working, smokeformer, smokecurrent
#>
#>
#> Binomial model for "alc"
#> family: binomial
#> link: logit
#> * Predictor variables:
#> (Intercept), age, genderfemale, WC, bili, occuplooking for work, occupnot working,
#> smokeformer, smokecurrent
#>
#>
#> Multinomial logit model for "occup"
#> * Predictor variables:
#> (Intercept), age, genderfemale, WC, bili, smokeformer, smokecurrent
#>
#>
#> Gamma model for "bili"
#> family: Gamma
#> link: log
#> * Parametrization:
#>  shape: shape_bili = mu_bili^2 * tau_bili
#>  rate: rate_bili = mu_bili * tau_bili
#> * Predictor variables:
#> (Intercept), age, genderfemale, WC, smokeformer, smokecurrent
#>
#>
#> Cumulative logit model for "smoke"
#> * Predictor variables:
#> age, genderfemale, WC
#>
#>
#> Lognormal model for "WC"
#> family: lognorm
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemale
Auxiliary variables are variables that are not part of the analysis model, but should be considered as predictor variables in the imputation models because they can inform the imputation of unobserved values.
Good auxiliary variables are ^{3}
In the main functions, *_imp()
, auxiliary variables can be specified with the argument auxvars
, which is a onesided formula.
Example:
We might consider the variables educ
and smoke
as predictors for the imputation of occup
:
mod9a < lm_imp(SBP ~ gender + age + occup, auxvars = ~ educ + smoke,
data = NHANES, n.adapt = 0)
The variables educ
and smoke
are not used as predictors in the analysis model. They are, however, used as predictors in the imputation for occup
and imputed themselves (if they have missing values):
list_models(mod9a, priors = FALSE, regcoef = FALSE, otherpars = FALSE, refcat = FALSE)
#> Linear model for "SBP"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), genderfemale, age, occuplooking for work, occupnot working
#>
#>
#> Multinomial logit model for "occup"
#> * Predictor variables:
#> (Intercept), genderfemale, age, educhigh, smokeformer, smokecurrent
#>
#>
#> Cumulative logit model for "smoke"
#> * Predictor variables:
#> genderfemale, age, educhigh
As shown above in mod3e
and mod3f
, it is possible to specify functions of auxiliary variables. In that case, the auxiliary variable is not considered as linear effect but as specified by the function:
mod9b < lm_imp(SBP ~ gender + age + occup, data = NHANES,
auxvars = ~ educ + smoke + log(WC),
trunc = list(WC = c(1e10, 1e10)), n.adapt = 0)
list_models(mod9b, priors = FALSE, regcoef = FALSE, otherpars = FALSE, refcat = FALSE)
#> Linear model for "SBP"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), genderfemale, age, occuplooking for work, occupnot working
#>
#>
#> Multinomial logit model for "occup"
#> * Predictor variables:
#> (Intercept), genderfemale, age, educhigh, smokeformer, smokecurrent
#>
#>
#> Cumulative logit model for "smoke"
#> * Predictor variables:
#> genderfemale, age, educhigh
#>
#>
#> Linear model for "WC"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), genderfemale, age, educhigh
Note:
Omitting auxiliary variables from the analysis model implies that the outcome is independent of these variables, conditional on the other variables in the model. If this is not true, the model is misspecified which may lead to biased results (similar to leaving a confounding variable out of a model).
In practice, most often, dummy coding is used for categorical predictor variables, i.e., a binary variables is created for each category, except the reference category. These binary variables have value one when that category was observed and zero otherwise.
This is the default behaviour for unordered factors in R (contr.treatment()
). For ordered factors orthogonal polynomials (contr.poly()
) are used. The type of contrasts (i.e. “coding”) to be used for unordered and ordered factors can be checked and set in the global options:
options('contrasts')
#> $contrasts
#> unordered ordered
#> "contr.treatment" "contr.poly"
Since the imputation of incomplete factors is done in the original variable, the recoding from the original categorical variable into dummy variables or other contrasts needs to be done within the JAGS model. Currently, only dummy coding and reference coding (contr.sum()
) are possible for factors with missing values. This means that, if the default contr.poly
is set for ordinal factors, a warning message is printed and dummy coding is used for these variables instead.
In JointAI, the first category of a categorical variable is set to be the reference category when using for dummy or reference coding. However, this may not always allow the desired interpretation of the regression coefficients. Moreover, when categories are unbalanced, setting the largest group as reference may result in better mixing of the MCMC chains.
For unordered factors, it is possible to change the reference category directly in the data, for example using the base R function relevel()
. For ordinal variables, however, this is not possible, and especially when the ordinal variable needs imputation it is desirable to maintain the ordering in the categories.
Therefore, JointAI allows specification of the reference category separately for each variable, via the argument refcats
.
To specify the choice of reference category globally for all variables in the model, refcats
can be set as
refcats = "first"
refcats = "last"
refcats = "largest"
For example:
mod10a < lm_imp(SBP ~ gender + age + race + educ + occup + smoke,
refcats = "largest", data = NHANES, n.adapt = 0)
#> Warning:
#> It is currently not possible to use "contr.poly" for incomplete categorical covariates.
#> I will use "contr.treatment" instead. You can specify (globally) which types of
#> contrasts are used by changing "options('contrasts')".
Alternatively, refcats
takes a named vector, in which the reference category for each variable can be specified either by its number or its name, or one of the three global types: “first”, “last” or “largest”. For variables for which no reference category is specified in the list the default is used.
mod10b < lm_imp(SBP ~ gender + age + race + educ + occup + smoke,
refcats = list(occup = "not working", race = 3, educ = 'largest'),
data = NHANES, n.adapt = 0)
#> Warning:
#> It is currently not possible to use "contr.poly" for incomplete categorical covariates.
#> I will use "contr.treatment" instead. You can specify (globally) which types of
#> contrasts are used by changing "options('contrasts')".
To help to specify the reference category, the function set_refcat()
can be used. It prints the names of the categorical variables that are selected by
or all categorical variables in the data if only data
is provided, and asks a number of questions which the user needs to reply to by input of a number.
refs_mod10 < set_refcat(NHANES, formula = formula(mod10b))
#> The categorical variables are:
#>  "gender"
#>  "race"
#>  "educ"
#>  "occup"
#>  "smoke"
#>
#> How do you want to specify the reference categories?
#>
#> 1: Use the first category for each variable.
#> 2: Use the last category for each variabe.
#> 3: Use the largest category for each variable.
#> 4: Specify the reference categories individually.
When option 4 is chosen, a question for each categorical variable is asked, for example:
#> The reference category for “race” should be
#>
#> 1: Mexican American
#> 2: Other Hispanic
#> 3: NonHispanic White
#> 4: NonHispanic Black
#> 5: other
After specification of the reference categories for all categorical variables, the determined specification for the argument refcats
is printed:
#> In the JointAI model specify:
#> refcats = c(gender = 'female', race = 'NonHispanic White', educ = 'low',
#> occup = 'not working', smoke = 'never')
#>
#> or use the output of this function.
set_refcat()
also returns a named vector that can be passed to the argument refcats
:
mod10c < lm_imp(SBP ~ gender + age + race + educ + occup + smoke,
refcats = refs_mod10, data = NHANES, n.adapt = 0)
#> Warning:
#> It is currently not possible to use "contr.poly" for incomplete categorical covariates. I will use
#> "contr.treatment" instead. You can specify (globally) which types of contrasts are used by changing
#> "options('contrasts')".
Note:
Changing a reference category via the argument refcats
does not change the order of levels in the dataset or any of the data matrices inside JointAI. Only when, in the JAGS model, the categorical variables is converted into dummy variables, the reference category is used to determine for which levels the dummies are created.
In the Bayesian framework, parameters are random variables for which a distribution needs to be specified. These distributions depend on parameters themselves, i.e., on hyperparameters.
The function default_hyperpars()
returns a list containing the default hyper parameters used in a JointAI
model.
default_hyperpars()
#> $norm
#> mu_reg_norm tau_reg_norm shape_tau_norm rate_tau_norm
#> 0e+00 1e04 1e02 1e02
#>
#> $gamma
#> mu_reg_gamma tau_reg_gamma shape_tau_gamma rate_tau_gamma
#> 0e+00 1e04 1e02 1e02
#>
#> $beta
#> mu_reg_beta tau_reg_beta shape_tau_beta rate_tau_beta
#> 0e+00 1e04 1e02 1e02
#>
#> $binom
#> mu_reg_binom tau_reg_binom
#> 0e+00 1e04
#>
#> $poisson
#> mu_reg_poisson tau_reg_poisson
#> 0e+00 1e04
#>
#> $multinomial
#> mu_reg_multinomial tau_reg_multinomial
#> 0e+00 1e04
#>
#> $ordinal
#> mu_reg_ordinal tau_reg_ordinal mu_delta_ordinal tau_delta_ordinal
#> 0e+00 1e04 0e+00 1e04
#>
#> $ranef
#> shape_diag_RinvD rate_diag_RinvD KinvD_expr
#> "0.01" "0.001" "nranef + 1.0"
#>
#> $surv
#> mu_reg_surv tau_reg_surv
#> 0.000 0.001
To change the hyperparameters in a JointAI model, the list obtained from default_hyperpars()
can be edited and passed to the argument hyperpars
in the main functions *_imp()
.
mu_reg_*
and tau_reg_*
refer to the mean and precision in the distribution for regression coefficients.shape_tau_*
and rate_tau_*
are the shape and rate parameters of a Gamma distribution that is used has prior for precision parameters.KinvD
refers to the degrees of freedom in the Wishart prior used for the inverse of the random effects design matrix D
. KinvD_exp
should be a string that can be evaluated to calculate the number of degrees of freedom depending on the number of random effects nranef
(dimension of D
). By default, KinvD
will be set to the number of random effects plus one.shape_diag_RinvD
and rate_diag_RinvD
are the scale and rate parameters of the Gamma prior of the diagonal elements of RinvD
.In random effects models with only one random effect, instead of the Wishart distribution a Gamma prior is used for the inverse of D
.
When variables are measured on very different scales this can result in slow convergence and bad mixing. Therefore, JointAI includes scaling of continuous covariates in the JAGS model (i.e., instead of writing ... + covar + ...
in the linear predictor, ... + (covar  mean)/sd) + ...
is written). The scaling parameters (mean and standard deviation) are calculated based on the design matrices containing the original data. Results are transformed back to the original scale.
By setting the argument scale_vars = FALSE
the scaling can be prevented. If scale_vars
is a vector of variable names, scaling will only be done for those variables.
By default, only the MCMC samples that is scaled back to the scale of the data is stored in a JointAI
object. When the argument keep_scaled_mcmc = TRUE
also the scaled sample is kept. This is mainly for debugging purposes.
It is possible to use shrinkage priors to penalize large regression coefficients. This can be specified via the argument shrinkage
. At the moment, only ridge regression is implemented.
Setting shrinkage = 'ridge'
will impose ridge priors on all regression coefficients. To only use shrinkage for some of the submodels (main analysis model and covariate models), a vector can be provided that contains the names of the response variables of the models in which shrinkage should be applied, and the type of shrinkage for each of them.
For example, in mod11a
ridge regression is used for all models, and in modd11b
only in the models for SBP
and educ
:
mod11a < lm_imp(SBP ~ gender + age + race + educ + occup + smoke,
data = NHANES, shrinkage = 'ridge',
n.adapt = 0)
#>
#> Note: No MCMC sample will be created when n.iter is set to 0.
mod11b < lm_imp(SBP ~ gender + age + race + educ + occup + smoke,
data = NHANES, shrinkage = c(SBP = 'ridge', educ = 'ridge'),
n.adapt = 0)
#>
#> Note: No MCMC sample will be created when n.iter is set to 0.
Ridge regression is implemented as a \(\text{Ga}(0.01, 0.01)\) prior for the precision of the regression coefficients \(\beta\) instead of setting this precision to a fixed (small) value.
Note: these examples are chosen to demonstrate functionality and may not fit the data.↩︎
Again, this is just a demonstration of the possibilities in JointAI, but nesting transformations will most often result in coefficients that that do not have meaningful interpretation in practice.↩︎
Van Buuren, S. (2012). Flexible imputation of missing data. Chapman and Hall/CRC. See also the second edition online.↩︎