# Model Specification

#### 2020-06-20

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.

## Analysis model type

JointAI has several main functions (which are abbreviated with `*_imp()`):

- `lm_imp()`: linear regression
- `glm_imp()`: generalized linear regression
- `lognorm_imp()`: log-normal regression
- `betareg_imp()`: beta regression
- `clm_imp()`: cumulative logit models for ordinal data
- `mlogit_imp()`: multinomial logit models for unordered factors
- `lme_imp()` / `lmer_imp()`: linear mixed effects regression
- `glme_imp()` / `glmer_imp()`: generalized linear mixed effects regression
- `lognormmm_imp()`: log-normal mixed effects regression
- `betamm_imp()`: beta mixed effects regression
- `clmm_imp()`: cumulative logit mixed models
- `mlogitmm_imp()`: multinomial logit mixed models
- `survreg_imp()`: parametric (Weibull) survival models
- `coxph_imp()`: Proportional hazards survival models
- `JM_imp()`: Joint model for longitudinal and survival data

Specification 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).

### Model family and link functions

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: logit

To use, for instance, a “probit” link instead, this needs to be specified explicitly:

    mod1d <- glm_imp(educ ~ age + gender + creat, data = NHANES,
                   family = binomial(link = 'probit'), n.adapt = 0)

    mod1d$analysis_type
    #> [1] "glm"
    #> attr(,"family")
    #>
    #> Family: binomial
    #> Link function: probit

## Model formula

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

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

#### Interaction with multiple variables

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]

#### Higher level interactions

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.

### Non-linear functional forms

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()`
- polynomials (using `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)

#### What happens inside JointAI?

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 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, 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, I(WC^2)
    #>
    #>
    #> 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, I(WC^2)
    #>
    #>
    #> Linear model for "WC"
    #>    family: gaussian
    #>    link: identity
    #> * Predictor variables:
    #>   (Intercept), age, genderfemale

Incomplete variables are always imputed on their original scale, i.e.,

- in `mod3b` the variable `bili` is imputed and the quadratic and cubic versions calculated from the imputed values.
- Likewise, `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`.

#### Functions with restricted support

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`.

#### Functions that are not available in R

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)

#### Nested functions

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:

    # define the complementary log log transformation
    cloglog <- function(x) log(-log(1 - x))

    # define the inverse logit (in case you have not done it in the previous example)
    ilogit <- plogis

    # nest ilogit inside cloglog
    mod6a <- lm_imp(SBP ~ age + gender + cloglog(ilogit(creat)), data = NHANES)

## Multi-level structure & longitudinal covariates

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 effects

`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 variable
- `random = ~ 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:

    <fixed effects> + (1 | id) + (1 | center)

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.

### Longitudinal covariates

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.

#### Why do we need models for completely observed covariates?

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.

## Covariate model types

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`).

### Specification of covariate model types

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.

### Order of the sequence of imputation models

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, genderfemale

## Auxiliary variables

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

- associated with an incomplete variable of interest, or are
- associated with the missingness of that variable, and
- do not have too many missing values themselves. Importantly, they should be observed for a large proportion of the cases that have a missing value in the variable to be imputed.

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, educhigh

### Functions of variables as auxiliary variables

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(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, 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 mis-specified which may lead to biased results (similar to leaving a confounding variable out of a model).

## Categorical covariates: coding and reference categories

### Coding

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.

### Reference categories

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`.

#### Setting reference categories for all variables

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')".

#### Setting reference categories for individual variables

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

- a specified model formula and/or
- a vector of auxiliary variables, or
- a vector of naming covariates

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: 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 = '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.

## Hyper-parameters

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        0.001

To 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`.

## Scaling

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.

## Shrinkage

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.

1. Note: these examples are chosen to demonstrate functionality and may not fit the data.↩︎

2. 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.↩︎

3. Van Buuren, S. (2012). Flexible imputation of missing data. Chapman and Hall/CRC. See also the second edition online.↩︎