  Simulation studies are used to evaluate and compare the properties of
statistical methods in controlled experimental settings. In most cases,
performing a simulation study requires knowledge of the true value of the
parameter, or estimand, of interest. However, in many simulation designs, the
true value of the estimand is difficult to compute analytically. Here, we
illustrate the use of Monte Carlo integration to compute true estimand values
in simple and complex simulation designs. We provide general pseudocode that
can be replicated in any software program of choice to demonstrate key
principles in using Monte Carlo integration in two scenarios: a simple three
variable simulation where interest lies in the marginally adjusted odds ratio;
and a more complex causal mediation analysis where interest lies in the
controlled direct effect in the presence of mediator-outcome confounders
affected by the exposure. We discuss general strategies that can be used to
minimize Monte Carlo error, and to serve as checks on the simulation program to
avoid coding errors. R programming code is provided illustrating the
application of our pseudocode in these settings.
