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.