╨╧рб▒с>■  ■   ■                                                                                                                                                                                                                                                                                                                                                                                                                                                   ¤   ■                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Root Entry            ■                                       Root Entry        ╨ь]а╛┬╥неа$╪БТ@xX╝Ыў└■   Contents            7                                ■   ¤    ■                                                                                                                                                                                                                                                                                                                                                                                                                                                                           {\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fnil\fprq1\fcharset0 Courier New;}{\f1\fmodern\fprq1\fcharset0 Courier New;}{\f2\fmodern\fprq1\fcharset0 Courier;}} {\colortbl ;\red0\green0\blue0;\red0\green128\blue0;} \viewkind4\uc1\pard\f0\fs18 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \tab\tab\tab \cf1\f1\fs48 \b STATISTICAL SIMULATION WITH S-PLUS \par \fs36 \par \par \par \par \par \par \par \par \par \tab BRANDT BALGOOYEN \par \tab JUNE 19, 2001 \par \par \par \tab\fs48 A Brief History of S-Plus\fs36 \par \par \tab\tab *In 1976, the S language is developed at the AT&T Bell Labs \par \par \par \tab\tab *Credit for the development of the S language \par \tab\tab\tab -Rick Becker \par \tab\tab\tab -John Chambers \par \tab\tab\tab -Later Allan Wilks joined the core team \par \par \par \tab\tab *Early 80's mark a strong interest in S amongst statisticians \par \par \par \tab\tab *In 1987, Statsci, Inc. is founded as a division of MathSoft \par \tab\tab\tab -Founded by Douglas Martin from the University of Washington \par \par \par \tab\tab *In 1993, S-Plus is released for Windows \par \par \par \tab\fs48 S-Plus Demonstrations \par \fs36 \par \par \par \tab\tab *t-test simulation using the case of 2 independent samples \par \par \tab\tab GOAL: \cf2 To explore power under varying conditions\cf1 \par \par \par \par \par \tab\tab *Randomization test using the case of 2 independent samples \par \par \tab\tab GOAL:\cf2 To see how a Randomization Test (this is a type \par \tab\tab\tab of Monte Carlo simulation) compares with the \par \tab\tab\tab\tab results from a t and Wilcoxon rank sum test \par \par \tab\tab \cf1 Note: \cf2 the Wilcoxon test is equivalent to \par \tab\tab\tab\tab the Mann-Whitney test \par \par \par \tab\cf1\fs44 t-Test for 2 Independent Samples \par \fs36 \par \par \tab\tab Assumptions for a 2 independent sample t-test \par \cf2\tab\tab\tab *Independent samples from normal distributions \par \tab\tab\tab with equal variances \par \tab \par \tab\tab\cf1 Note: \cf2 If X\fs28 1\fs36 and X\fs24 2\fs36 are normally distributed random variables, \par \cf1\tab\tab\tab\tab\cf2 the difference (X\fs24 1\fs36 -X\fs24 2\fs36 ) of two random variables \par \tab\tab\tab\tab will be normally distributed with Mean \par \tab\tab\tab\tab (Mu1-Mu2) and Variance (\fs28 (Sigma1^2)+(Sigma2^2)\fs36 ) \par \par \cf1\tab\tab Test Statistic: \par \par \tab\tab t =\ul (X\fs32 bar\fs36 )1-(X\fs32 bar\fs36 )2 - D\fs32 o\fs36 \ulnone \par \tab\tab\tab Sp*sqrt(1/n1 + 1/n2) \par \par \tab\tab Sp = sqrt(\ul (n1-1)*var1 + (n2-2)*var2\ulnone )\fs56 \par \fs36 \tab \tab sqrt( n1 + n2 - 2 ) \par \par \par \par \par \par \tab\fs44 GENERALIZED MONTE CARLO TEST \par \fs36\tab (Besag and Clifford 1989) \par \par \tab\tab i) an observed set of data is one of many sets that could \par \tab\tab\tab have occured \par \par \tab ii) all possible sets of data can be generated \par \par \tab iii) the null hyposthesis of interest states that all the \par \tab\tab\tab possible sets of data were equally likely to occur \par \par \tab iv) each possible set of data can be summarized by some \par \tab\tab\tab test statistic S. \par \par \par \par \par \par \par \par \tab\fs44 FURTHER APPLICATION\tab \par \tab\tab\fs36 *S-Plus simulations can be performed on virtually \par \tab\tab any classical statistical test: \par \tab\tab\tab -Hypothesis testing \par \tab\tab\tab -Confidence Intervals \par \tab\tab\tab -Linear Models \par \tab\tab\tab\tab regression \par \tab\tab\tab\tab anova \par \tab\tab\tab\tab etc. \par \tab\tab\tab -Statistical tests based on the multivariate normal dis'n \par \tab\tab\tab -time series \par \tab\tab\tab -etc. \par \par \tab\tab *S-Plus provides an ideal setting for Randomization, \par \tab\tab Bootstrapping, and other Monte Carlo methodologies \par \tab\tab \par \tab\tab *S-Plus can be used to study Bayesian statistics as well \par \tab\tab\tab -Strictly programming \par \fs44\tab\fs36 \par \par \par \par \par \par \par \par \par \par \tab\tab\tab\tab\fs56 QUESTION AND ANSWER SESSION \par \fs36 \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \par \f2\fs24 \par \f1\fs36 \par \par \par \tab \par \par \par }