Simulate a Node Using Linear Regression

Data from the parents is used to generate the node using linear regression by predicting the covariate specific mean and sampling from a normal distribution with that mean and a specified standard deviation.

Usage

node_gaussian(data, parents, formula=NULL, betas, intercept, error)

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.
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.
error: A single number specifying the sigma error that should be used when generating the node.

Details

Using the general linear regression equation, the observation-specific value that would be expected given the model is generated for every observation in the dataset generated thus far. We could stop here, but this would create a perfect fit for the node, which is unrealistic. Instead, we add an error term by taking one sample of a normal distribution for each observation with mean zero and standard deviation error. This error term is then added to the predicted mean.

Formal Description:

Formally, the data generation can be described as:

$$Y \sim \texttt{intercept} + \texttt{parents}_1 \cdot \texttt{betas}_1 + ... + \texttt{parents}_n \cdot \texttt{betas}_n+ N(0, \texttt{error}),$$

where $N(0, \texttt{error})$ denotes the normal distribution with mean 0 and a standard deviation of error and $n$ is the number of parents (length(parents)).

For example, given intercept=-15, parents=c("A", "B"), betas=c(0.2, 1.3) and error=2 the data generation process is defined as:

$$Y \sim -15 + A \cdot 0.2 + B \cdot 1.3 + N(0, 2).$$

Author

Robin Denz

Value

Returns a numeric vector of length nrow(data).

Examples

library(simDAG)

set.seed(12455432)

# define a DAG
dag <- empty_dag() +
  node("age", type="rnorm", mean=50, sd=4) +
  node("sex", type="rbernoulli", p=0.5) +
  node("bmi", type="gaussian", parents=c("sex", "age"),
       betas=c(1.1, 0.4), intercept=12, error=2)

# define the same DAG, but with a pretty formula for the child node
dag <- empty_dag() +
  node("age", type="rnorm", mean=50, sd=4) +
  node("sex", type="rbernoulli", p=0.5) +
  node("bmi", type="gaussian", error=2,
       formula= ~ 12 + sexTRUE*1.1 + age*0.4)

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