In this vignette, we use the NHANES data for examples in cross-sectional 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(): log-normal 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(): log-normal 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 Multi-level structure & longitudinal covariates.
Additionally, glm_imp(), glme_imp() and
glmer_imp() require the specification of the model family
(and link function).
Implemented families and links for 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: logitTo use, for instance, a “probit” link instead, this needs to be specified explicitly:
The arguments formula and fixed take a
two-sided 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
multi-state 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 * creatis equivalent to
SBP ~ age + gender + smoke + creat + smoke:creatInteractions 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 two-way 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 three-way 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 re-calculated within each iteration of the MCMC sampling, using the imputed values from the current iteration.
It is important not to pre-calculate 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 non-linear effect is more appropriate.
Non-linear 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 re-calculated 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, genderfemaleThe function list_models()
prints information on all sub-models 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, I(WC^2)
#>
#>
#> Linear model for "WC"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemaleWhen 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, I(WC^2)
#>
#>
#> Linear model for "WC"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemaleWhen 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, I(WC^2)
#>
#>
#> Linear model for "WC"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemaleIncomplete 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 log-normal 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(1e-5, NA)), data = NHANES)
# log-normal model for bili
mod4b <- lm_imp(SBP ~ age + gender + log(bili) + exp(creat),
models = c(bili = 'lognorm', creat = 'lm'), data = NHANES)
# gamma model with log-link for bili
mod4c <- lm_imp(SBP ~ age + gender + log(bili) + exp(creat),
models = c(bili = 'glm_gamma_log', creat = 'lm'), data = NHANES)If only one-sided 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 log-log transformation is implemented as
cloglog, but since this function does not exist in (base)
R, we first need to define it:
In multi-level 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 one-sided 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
multi-level 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 multi-level 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 2-level setting, if there are no missing values in level-2 variables, it is not necessary to specify models for completely observed level-1 variables. But if there are missing values in level-2 variables, models need to be specified for all level-1 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 (level-1) variables may have baseline (level-2) 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.
The automatically selected types for baseline (highest level) covariates are:
| 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 |
The default methods for lower level covariates are:
| 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 log-transformed variable | right-skewed variables >0 |
beta |
beta regression (with logit-link) | continuous variables with values in [0, 1] |
glm_<family>_<link> |
e.g. glm_gamma_inverse for Gamma
regression with an inverse-link |
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.
For longitudinal (lower-level) covariates corresponding model types are . available:
| name | model | variable type |
|---|---|---|
glmm_lognorm |
normal mixed model for the log-transformed variable | longitudinal right-skewed variables >0 |
glmm_beta |
beta regression (with logit-link) | continuous variables with values in [0, 1] |
glmm_<family>_<link> |
e.g. glmm_poisson_log for a poisson mixed
model with log-link |
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$models
#> SBP alc occup bili
#> "glm_gaussian_identity" "glm_binomial_logit" "mlogit" "glm_gamma_log"
#> smoke WC
#> "clm" "lognorm"When there is a “time” variable in the model, such as
age (age of the child at the time of the measurement) in
the simLong it may not be meaningful to specify a model for
that variable. Especially when the “time” variable is pre-specified by
the design of the study it can usually be assumed to be independent of
the covariates and a model for it has no useful interpretation.
The argument no_model allows us to exclude models for
such variables (as long as they are completely observed):
mod8b <- lme_imp(bmi ~ GESTBIR + ETHN + HEIGHT_M + SMOKE + hc + MARITAL +
ns(age, df = 2),
random = ~ns(age, df = 2) | ID, data = simLong,
no_model = "age", n.adapt = 0)
mod8b$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
#>
#>
#> Log-normal model for "WC"
#> family: lognorm
#> link: identity
#> * Predictor variables:
#> (Intercept), age, genderfemaleAuxiliary 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
one-sided 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, educhighAs 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(1e-10, 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, log(WC)
#>
#>
#> Cumulative logit model for "smoke"
#> * Predictor variables:
#> genderfemale, age, educhigh, log(WC)
#>
#>
#> Linear model for "WC"
#> family: gaussian
#> link: identity
#> * Predictor variables:
#> (Intercept), genderfemale, age, educhighNote:
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 mis-specified 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 re-coding 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: Non-Hispanic White
#> 4: Non-Hispanic Black
#> 5: otherAfter 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 = 'Non-Hispanic 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 hyper-parameters.
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 1e-04 1e-02 1e-02
#>
#> $gamma
#> mu_reg_gamma tau_reg_gamma shape_tau_gamma rate_tau_gamma
#> 0e+00 1e-04 1e-02 1e-02
#>
#> $beta
#> mu_reg_beta tau_reg_beta shape_tau_beta rate_tau_beta
#> 0e+00 1e-04 1e-02 1e-02
#>
#> $binom
#> mu_reg_binom tau_reg_binom
#> 0e+00 1e-04
#>
#> $poisson
#> mu_reg_poisson tau_reg_poisson
#> 0e+00 1e-04
#>
#> $multinomial
#> mu_reg_multinomial tau_reg_multinomial
#> 0e+00 1e-04
#>
#> $ordinal
#> mu_reg_ordinal tau_reg_ordinal mu_delta_ordinal tau_delta_ordinal
#> 0e+00 1e-04 0e+00 1e-04
#>
#> $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 0.001To change the hyper-parameters 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 de-bugging 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
sub-models (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.↩︎