Simulate a Node Using (Mixed) Poisson Regression

Data from the parents is used to generate the node using poisson regression by predicting the covariate specific lambda and sampling from a poisson distribution accordingly. Allows inclusion of arbitrary random effects and slopes.

Usage

node_poisson(data, parents, formula=NULL, betas, intercept,
             var_corr=NULL)

Arguments

data: A data.table (or something that can be coerced to a data.table) containing all columns specified by parents.
parents: A character vector specifying the names of the parents that this particular child node has. If non-linear combinations or interaction effects should be included, the user may specify the formula argument instead.
formula: An optional formula object to describe how the node should be generated or NULL (default). If supplied it should start with ~, having nothing else on the left hand side. The right hand side may contain any valid formula syntax, such as A + B or A + B + I(A^2), allowing non-linear effects. If this argument is defined, there is no need to define the parents argument. For example, using parents=c("A", "B") is equal to using formula= ~ A + B. May contain random effects and random slopes, in which case the simr package is used to generate the data. See details.
betas: A numeric vector with length equal to parents, specifying the causal beta coefficients used to generate the node.
intercept: A single number specifying the intercept that should be used when generating the node.
var_corr: Variances and covariances for random effects. Only used when formula contains mixed model syntax. If there are multiple random effects, their parameters should be supplied as a named list. More complex structures are also supported. This argument is directly passed to the makeLmer function of the simr package. Please consult the documentation of that package for more information on how mixed models should be specified. Some guidance can also be found in the "Issues" section of the official simr github page.

Details

Essentially, this function simply calculates the linear predictor defined by the betas-coefficients, the intercept and the values of the parents. The exponential function is then applied to this predictor and the result is passed to the rpois function. The result is a draw from a subject-specific poisson distribution, resembling the user-defined poisson regression model.

Formal Description:

Formally, the data generation can be described as:

$$Y \sim Poisson(\lambda),$$

where $Poisson()$ means that the variable is Poisson distributed with:

$$P_\lambda(k) = \frac{\lambda^k e^{-\lambda}}{k!}.$$

Here, $k$ is the count and $e$ is eulers number. The parameter $\lambda$ is determined as:

$$\lambda = \exp(\texttt{intercept} + \texttt{parents}_1 \cdot \texttt{betas}_1 + ... + \texttt{parents}_n \cdot \texttt{betas}_n),$$

where $n$ is the number of parents (length(parents)).

For example, given intercept=-15, parents=c("A", "B"), betas=c(0.2, 1.3) the data generation process is defined as:

$$Y \sim Poisson(\exp(-15 + A \cdot 0.2 + B \cdot 1.3)).$$

Random Effects and Random Slopes:

This function also allows users to include arbitrary amounts of random slopes and random effects using the formula argument. If this is done, the formula, and data arguments are passed to the variables of the same name in the makeGlmer function of the simr package. The fixef argument of that function will be passed the numeric vector c(intercept, betas) and the VarCorr argument receives the var_corr argument as input. If used as a node type in a DAG, all of this is taken care of behind the scenes. Users can simply use the regular enhanced formula interface of the node function to define these formula terms, as shown in detail in the formula vignette (vignette(topic="v_using_formulas", package="simDAG")). Please consult that vignette for examples. Also, please note that inclusion of random effects or random slopes usually results in significantly longer computation times.

Author

Robin Denz

Value

Returns a numeric vector of length nrow(data).

Examples

library(simDAG)

set.seed(345345)

dag <- empty_dag() +
  node("age", type="rnorm", mean=50, sd=4) +
  node("sex", type="rbernoulli", p=0.5) +
  node("smoking", type="poisson",
       formula= ~ -2 + sexTRUE*1.1 + age*0.4)

sim_dat <- sim_from_dag(dag=dag, n_sim=100)

## an example using a random effect
if (requireNamespace("simr")) {

library(simr)

dag_mixed <- empty_dag() +
  node("School", type="rcategorical", probs=rep(0.1, 10),
       labels=LETTERS[1:10]) +
  node("Age", type="rnorm", mean=12, sd=2) +
  node("Grade", type="poisson", formula= ~ -2 + Age*1.2 + (1|School),
       var_corr=0.3)

sim_dat <- sim_from_dag(dag=dag_mixed, n_sim=20)
}