An Introduction to R: Monte
Carlo Simulation
MWERA 2012
Emily A. Price, MS
Marsha Lewis, MPA
Dr. Gordon P. Brooks
Objectives and/or Goals
• Three main parts
– Data generation in R
– Basic Monte Carlo programming (e.g.
loops)
– Running simulations (e.g., investigating
Type I errors)
Why Use Monte Carlo Methods?
• According to Mooney (1997) Monte Carlo simulations
are useful to
– Make inferences when weak statistical theory exists for an
estimator
– Test null hypotheses under a variety of plausible conditions
– Assess the quality of an inference method
– Assess the robustness of parametric inference to
assumption violations
– Compare estimator’s properties
What are Monte Carlo Methods?
• Experiments composed of random numbers
to evaluate mathematical expressions
(Gentle, 2003)
• Empirically determine the sampling
distribution of a test statistic
• Computer-based methods for approximating
values and properties of random variables
(Braun & Murdoch, 2007)
Logic of Monte Carlo
• Mooney (1997) presents five steps
1.
2.
3.
4.
5.
Specify the pseudo-population in symbolic terms in such a way
that it can be used to generate samples. That is, writing code to
generate data in a specific manner.
Sample from the pseudo-population in ways that reflect the topic
of interest
Calculate θ in a pseudo-sample and store it in a vector
Repeat steps 2 and 3 t times where t is the number of trials
Construct a relative frequency distribution of resulting values
which is a Monte Carlo estimate of the sampling distribution of
under the conditions specified by the pseudo-population and the
sampling procedures
Practical Issues/ Considerations
•
•
•
•
What software to use?
How much time to run the simulation?
Reproducibility of results
Adequacy of random number generator
Why use R?
•
•
•
•
It’s FREE
It is a flexible language that can be controlled by the user
It uses a vector based approach
Depending on the package, there are built in commands which
the user can access and minimize the amount of programming
required for MC simulation
– Make sure to load the require packages at the beginning of
the session
• R community has a plethora of information: help websites,
listservs, textbooks, blogs
– Manuals for R available at http://cran.r-project.org/manuals.html
Part 1: Data Generation
• RNG and setting seed
– Purpose of the seed is to recovery results
•
•
•
•
Initialize all parameters of interest
Loops
Print results
Access output
Generating a Single Random Variable
• R has four parts: CDF, PDF, Quantile
function and simulation procedure
– dnorm, pnorm, qnorm, rnorm respectively
• rnorm(x,mean=0,sd=1)
• runif(20,min=2,max=5)
• Distributions: normal, uniform, poisson,
beta, gamma, chisquare, weibull,
exponential
Try it, you’ll like it!
• rnorm(x,mean=0,sd=1)
Generate a normal distribution of 50
values with a mean of 50 and sd of 10
• x <- sample(1:2,20,TRUE,prob=c(1/2,1/2))
Generate data that mimics rolling a die
Generating Correlated Data
• X~Normal (20, 5), Y~Normal (40, 10),
corr(X,Y) =0.6
– 4 inputs
• Sample size, mean, variance-covariance
matrix, and method
– 3 methods of data generation
• Eigenvalue (default), Singular Value, and
Cholesky
Try it, you’ll like it!
• rmvnorm(n, mean, sigma, method)
Generate data for 3 variables such that
X --Normal (20, 5), Y-- Normal (40, 10),
Z -- Normal (60,15) and Corr(X,Y) =0.6,
Corr(X,Z) = 0.7, Corr(Y,Z)=0.8
Part 2: Basic MC Programming
• Four steps (Braun & Murdoch, 2007)
1. Understand the problem
2. Work out a general idea how to solve it
•
Flow charts
3. Translate your general idea into a
detailed implementation
•
Turn the flowchart into code
4. Check: Does it work?
Programming Commands*
• Loops
– for, if, ifelse, while
• Statements
– repeat, break, next
* We can’t cover all programming aspects
but wanted to mention other commands
Functions
• They are “self-contained units with a well-defined
purpose” (Braun & Murdoch, 2007, p. 59)
– Take an input, do some calculations, and produce an output
• In R, functions are objects and can be manipulated
like other more common objects such as vectors,
matrices, and lists.
– R provides source code for its own functions
• R allows you to write your own functions
Part 3: Running Simulations
• Trimmed mean sampling distribution
• Replicating a published Monte Carlo
study in R.
– Zimmerman, D. W. (2004). A note on preliminary tests of
equality of variances. British Journal of Mathematical and
Statistical Psychology 57, 173–181.
Questions
• Thank you for your time
References
• Braun, W. J., & Murdoch, D. J. (2007). A first course in statistical
programming with R. New York: Cambridge University.
• Gentle, J. E. (2003). Random number generation and Monte
Carlo methods (2nd ed.). New York: Springer-Verlag.
• Mooney, C. Z. (1997). Monte Carlo simulation (Sage University
Paper series on Quantitative Applications in the Social
Sciences, series no. 07-116). Thousand Oaks, CA: Sage.
• Zimmerman, D. W. (2004). A note on preliminary tests of
equality of variances. British Journal of Mathematical and
Statistical Psychology 57, 173–181.
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