1. An Introduction to R: Monte Carlo Simulation MWERA 2012 Emily A. Price, MS Marsha Lewis, MPA Dr. Gordon P. Brooks

2. 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)

3. 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

4. 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)

5. Logic of Monte Carlo Mooney (1997) presents five steps 1.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. 2.Sample from the pseudo-population in ways that reflect the topic of interest 3.Calculate θ in a pseudo-sample and store it in a vector 4.Repeat steps 2 and 3 t times where t is the number of trials 5.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

6. Practical Issues/ Considerations What software to use? How much time to run the simulation? Reproducibility of results Adequacy of random number generator

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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?

14. Programming Commands* Loops –for, if, ifelse, while Statements –repeat, break, next * We can’t cover all programming aspects but wanted to mention other commands

15. 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

16. 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.

17. Questions Thank you for your time

18. 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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