In this small vignette, we give more detailed examples on how best to
use the formula argument in the node() and
node_td() functions. This argument allows users to directly
specify the full structural equation that should be used to generate the
respective node in a clear and easy way, that does not directly rely on
the parents, betas and associated arguments.
Note that the formula argument may only be used with
certain node types, as mentioned in the documentation.
We will start with a very simple example. Suppose we want to generate some data from a simple DAG with no time-varying variables. Consider the following DAG:
library(simDAG)
dag <- empty_dag() +
node("A", type="rnorm", mean=0, sd=1) +
node("B", type="rbernoulli", p=0.5, output="numeric") +
node("C", type="rcategorical", probs=c(0.3, 0.2, 0.5),
output="factor", labels=c("low", "medium", "high"))This DAG contains only three root nodes of different types. \(A\) is normally distributed, \(B\) is Bernoulli distributed and \(C\) is a simple categorical variable with the levels “low”, “medium” and “high”. If we generate data from this DAG alone, it would look like this:
set.seed(23143)
dat <- sim_from_dag(dag, n_sim=10)
head(dat)
#> A B C
#> <num> <num> <fctr>
#> 1: -0.8041685 0 low
#> 2: 1.3390885 0 medium
#> 3: 0.9455804 0 high
#> 4: -2.3437852 1 low
#> 5: -0.9045554 1 medium
#> 6: 0.8532361 1 mediumSuppose we now want to generate an additional child node called \(D\) which should be based on a linear regression model of the form:
\[D \sim -8 + A \cdot 0.4 + B \cdot -2 + N(0, 1.5).\]
We could do this using the node() function, by supplying
appropriate values to the parents, betas,
intercept and error arguments. The following
code could be used:
dag_without_formula <- dag +
node("D", type="gaussian", parents=c("A", "B"), betas=c(0.4, -2),
intercept=-8, error=1.5)This does work just fine, but it may be a little cumbersome to
specify the DAG in this way. Since we want to use a linear regression
model, we could instead use the formula argument like
this:
Given the same random number generator seed, the same output will be produced from both DAGs, as shown below:
set.seed(34)
dat1 <- sim_from_dag(dag_without_formula, n_sim=100)
set.seed(34)
dat2 <- sim_from_dag(dag_with_formula, n_sim=100)
all.equal(dat1, dat2)
#> [1] TRUEFormulas should always start with a ~ sign and have
nothing else on the left hand side. All parts of the formula should be
connected by + signs, never - signs. The name
of the respective variable should always be connected to the associated
coefficient by a * sign. It does not matter whether the
name of the term or the coefficient go first, but it has to be
consistent in a formula. For example, ~ 1 + A*2 + B*3
works, and ~ 1 + 2*A + 3*B also works, but
~ 1 + 2*A + B*2 will produce an error. The formula may also
be supplied as a string and will produce the same output.
Apart from being easier to read, this also allows the user a lot more options. Through the use of formulas it is possible to specify nodes that have categorical parents. It is also possible to include any order of interaction effects and cubic terms using formulas, as shown below.
Suppose that \(D\) should additionally depend on \(C\), a categorical variable. For example, suppose this is the regression model we want to generate data from:
\[D \sim -8 + A \cdot 0.4 + B \cdot -2 + Cmedium \cdot -1 + Chigh \cdot -3 + N(0, 1.5).\]
In this model, the “low” category is used as a reference category. If
this is what we want to do, using the simple parents,
betas, intercept approach no longer works. We
have to use a formula. Fortunately, this is really simple to do using
the following code:
dag2 <- dag +
node("D", type="gaussian", error=1.5,
formula=~ -8 + A*0.4 + B*-2 + Cmedium*-1 + Chigh*-3,
parents=c("A", "B", "C"))Essentially, all we have to do is use the name of the categorical variable immediately followed by the category name. Note that if a different reference category should be used, the user needs to re-define the factor levels of the categorical variable accordingly first.
Note that we also defined the parents argument in this
case. This is not strictly necessary to generate the data in this case,
but it is recommended whenever categorical variables are used in a
formula for two reasons:
parents is not specified, the
sim_from_dag() function will not know that \(C\) is a parent of \(D\). If sort_dag=TRUE and/or
the nodes are not specified in a correctly topologically sorted order,
this may lead to errors when trying to generate the data.parents is not specified, other
functions that take DAG objects as input (such as the
plot.DAG() function) may produce incorrect output, because
they won’t know that \(C\) is a parent
of \(D\).Interactions of any sort may also be added to the DAG. Suppose we want to generate data from the following regression model:
\[D \sim -8 + A \cdot 0.4 + B \cdot -2 + A*B \cdot -5 + N(0, 1.5),\]
where \(A*B\) indicates the
interaction between \(A\) and \(B\). This can be specified in the
formula argument using the : sign:
Since both \(A\) and \(B\) are coded as numeric variables here, this works fine. If we instead want to include an interaction which includes a categorical variable, we again have to use the name with the respective category appended to it. For example, the following DAG includes an interaction between \(A\) and \(C\):
dag4 <- dag +
node("D", type="gaussian", error=1.5,
formula=~ -8 + A*0.4 + B*-2 + Cmedium*-1 + Chigh*-3 + A:Cmedium*0.3 +
A:Chigh*10,
parents=c("A", "B", "C"))Higher order interactions may be specified in exactly the same way,
just using more : symbols. It may not always be obvious in
which order the variables for the interaction need to be specified. If
the “wrong” order was used, the sim_from_dag() function
will return a helpful error message explaining which ones should be used
instead. For example, if we had used “Cmedium:A” instead of “A:Cmedium”,
this would not work because internally only the latter is recognized as
a valid column. Note that because \(C\)
is categorical, we also specified the parents argument here
just to be safe.
Sometimes we also want to include non-linear relationships between a continuous variable and the outcome in a data generation process. This can be done by including cubic terms of that variable in a formula. Suppose the regression model that we want to use has the following form:
\[D \sim -8 + A \cdot 0.4 + A^2 \cdot 0.02 + B \cdot -2 + N(0, 1.5).\]
The following code may be used to define such as node:
dag_with_formula <- dag +
node("D", type="gaussian", formula= ~ -8 + A*0.4 + I(A^2)*0.02 + B*-2,
error=1.5)Users may of course use as many cubic terms as they like.
There is also limited support for including functions in the formula as well. For example, it is allowed to call any function on the beta coefficients, which is useful to specify betas on a different scale (for example using Odds-Ratios instead of betas). For example:
is valid syntax. Any function can be used in the place of
log(), as long as it is a single function that is called on
a beta-coefficient.