Automatic Parallelized Monte Carlo Simulations.

The MonteCarlo Package

The MonteCarlo package allows to create simulation studies and to summarize their results in LaTeX tables quickly and easily. In the following, an example for the use of the MonteCarlo package is presented. In addition to that, there is a brief discussion of the more advanced features of the package. Further information is included in the package vignette.

There are only two main functions in the package:

1. MonteCarlo() runs a simulation study for a user defined parameter grid. It handles the generation of loops over these parameter grid and parallelizes the computation on a user specified number of CPU units.

2. MakeTable() creates LaTeX tables from the output of MonteCarlo(). It stacks high dimensional output arrays into tables with a user specified ordering of rows and columns.

To run a simulation study, the user has to nest both - the generation of a sample and the calculation of the desired statistics from this sample - in a single function. This function is passed to MonteCarlo(). No additional programming is required.

The idea behind this approach is to allow the user full control and flexibility with regard to the design of the Monte Carlo experiment. It also makes MonteCarlo() very versatile. Finally, it is very intuitive. The user formulates his experiment as if he/she was only interested in a single draw.

A First Example

The best way to get working with the MonteCarlo package is to look at an example. Suppose we want to evaluate the performance of a standard t-test for the hypothesis that the mean is equal to zero. We are interested to see how the size and power of the test change with the sample size (n), the distance from the null hypothesis (loc for location) and the standard deviation of the distribution (scale). The sample is generated from a normal distribution.

To conduct this analysis, we proceed as follows. First, we load the MonteCarlo package.

library(MonteCarlo)

Then define the following function.

#########################################
##      Example: t-test

# Define function that generates data and applies the method of interest

ttest<-function(n,loc,scale){
  
  # generate sample:
    sample<-rnorm(n, loc, scale)
  
  # calculate test statistic:
    stat<-sqrt(n)*mean(sample)/sd(sample)
  
  # get test decision:
    decision<-abs(stat)>1.96
  
  # return result:
    return(list("decision"=decision))
}

As discussed above, ttest() is formulated in a way as if we only want to generate a single test decision. The arguments of the function are the parameters we are interested in. ttest() carries out 4 steps:

1. Draw a sample of n observations from a normal distribution with mean loc and standard deviation scale.
2. Calculate the t-statistic.
3. Determine the test decision.
4. Return the desired result in form of a list.

We then define the combinations of parameters that we are interested in and collect them in a list. The elements of the lists must have the same names as the parameters for which we want to supply grids.

# define parameter grid:

  n_grid<-c(50,100,250,500)
  loc_grid<-seq(0,1,0.2)
  scale_grid<-c(1,2)

# collect parameter grids in list:
  param_list=list("n"=n_grid, "loc"=loc_grid, "scale"=scale_grid)

To run the simulation, the function ttest() and the parameter grid (param_list) are passed to MonteCarlo(), together with the desired number of Monte Carlo repetitions (nrep=1000).

# run simulation:

  MC_result<-MonteCarlo(func=ttest, nrep=1000, param_list=param_list)

There is no further coding required. All the mechanics of the Monte Carlo experiment are handled by the MonteCarlo() function.

Calling summary produces a short information on the simulation.

  summary(MC_result)

## Simulation of function: 
## 
## function(n,loc,scale){
##   
##   # generate sample:
##     sample<-rnorm(n, loc, scale)
##   
##   # calculate test statistic:
##     stat<-sqrt(n)*mean(sample)/sd(sample)
##   
##   # get test decision:
##     decision<-abs(stat)>1.96
##   
##   # return result:
##     return(list("decision"=decision))
## }
## 
## Required time: 12.33 secs for nrep = 1000  repetitions on 1 CPUs 
## 
## Parameter grid: 
## 
##      n : 50 100 250 500 
##    loc : 0 0.2 0.4 0.6 0.8 1 
##  scale : 1 2 
## 
##  
## 1 output arrays of dimensions: 4 6 2 1000

As one can see from the summary, the simulation results are stored in an array of dimension c(4,6,2,1000), where the Monte Carlo repetitions are collected in the last dimension of the array.

To summarize the results in a reasonable way and to include them as a table in a paper or report, we have to represent them in a matrix. This is handled by the MakeTable() function that stacks the submatrices collected in the array in the rows and columns of a matrix and prints the result in the form of code to generate a LaTeX table.

To determine in which order the results are stacked in rows and columns, we supply the function arguments rows and cols. These are vectors of the names of the parameters in the order in which we want them to appear in the table (sorted from the inside to the outside).

# generate table:

  MakeTable(output=MC_result, rows="n", cols=c("loc","scale"), digits=2, include_meta=FALSE)

## \begin{table}[h]
## \centering
## \resizebox{ 1 \textwidth}{!}{%
## \begin{tabular}{ rrrrrrrrrrrrrrr }
## \hline\hline\\\\
##  scale && \multicolumn{ 6 }{c}{ 1 } &  & \multicolumn{ 6 }{c}{ 2 } \\ 
## n/loc &  & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 &  & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\ 
##  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\ 
## 50 &  & 0.06 & 0.30 & 0.83 & 0.98 & 1.00 & 1.00 &  & 0.05 & 0.11 & 0.28 & 0.55 & 0.79 & 0.95 \\ 
## 100 &  & 0.05 & 0.51 & 0.98 & 1.00 & 1.00 & 1.00 &  & 0.05 & 0.16 & 0.54 & 0.84 & 0.97 & 1.00 \\ 
## 250 &  & 0.06 & 0.89 & 1.00 & 1.00 & 1.00 & 1.00 &  & 0.04 & 0.34 & 0.91 & 1.00 & 1.00 & 1.00 \\ 
## 500 &  & 0.06 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 &  & 0.04 & 0.58 & 0.99 & 1.00 & 1.00 & 1.00 \\ 
## \\
## \\
## \hline\hline
## \end{tabular}%
## }
## \caption{ decision  }
## \end{table}

To change the ordering, just change the vectors rows and cols.

# generate table:

  MakeTable(output=MC_result, rows=c("n","scale"), cols="loc", digits=2, include_meta=FALSE)

## \begin{table}[h]
## \centering
## \resizebox{ 1 \textwidth}{!}{%
## \begin{tabular}{ rrrrrrrrr }
## \hline\hline\\\\
## scale & n/loc &  & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\ 
##  &  &  &  &  &  &  &  &  \\ 
## \multirow{ 4 }{*}{ 1 } & 50 &  & 0.06 & 0.30 & 0.83 & 0.98 & 1.00 & 1.00 \\ 
##  & 100 &  & 0.05 & 0.51 & 0.98 & 1.00 & 1.00 & 1.00 \\ 
##  & 250 &  & 0.06 & 0.89 & 1.00 & 1.00 & 1.00 & 1.00 \\ 
##  & 500 &  & 0.06 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\ 
##  &  &  &  &  &  &  &  &  \\ 
## \multirow{ 4 }{*}{ 2 } & 50 &  & 0.05 & 0.11 & 0.28 & 0.55 & 0.79 & 0.95 \\ 
##  & 100 &  & 0.05 & 0.16 & 0.54 & 0.84 & 0.97 & 1.00 \\ 
##  & 250 &  & 0.04 & 0.34 & 0.91 & 1.00 & 1.00 & 1.00 \\ 
##  & 500 &  & 0.04 & 0.58 & 0.99 & 1.00 & 1.00 & 1.00 \\ 
## \\
## \\
## \hline\hline
## \end{tabular}%
## }
## \caption{ decision  }
## \end{table}

Now we can simply copy the code and add it to our paper, report or presentation. That is all. The user can focus on the design of the experiment he is interested in and the interpretation of the results.

Further Functionality of the Package

The MonteCarlo package provides several options to modify the behavior of MonteCarlo() and the appearance of tables generated by MakeTable(). To give a short overview, the tables below contain information from the help files about the function parameters.

The MonteCarlo() Function

As can be seen from the table below, the MonteCarlo() function has a number of optional arguments. The most important ones among them are export_also that is needed to export datasets if these are required in a parallezied simulation and time_n_test that produces an estimate of the time required for the desired simulation.

raw | Boolean that specifies whether the output should be averaged over the nrep repetitions. Default is raw=TRUE. | export_also | A list that specifies additional objects that are supposed to be exported to the cluster. This allows to export data or to bypass the automatic export of functions. Default is export_also=NULL.|

The MakeTable() Function

The MakeTable() function also comes with some optional arguments. Most importantly collapse and transform allow to influence how the results of a simulation are aggregated and presented. This makes the package much more versatile. Additional examples that demonstrate the use of these features can be found in the examples of the help file and in the vignette.