Instantly share code, notes, and snippets.

calpolystat/#Sampling_Distribution.txt

Last active January 6, 2023 14:38
Show Gist options
• Save calpolystat/d7ed9873137267ee557b to your computer and use it in GitHub Desktop.
Sampling Distribution of Various Statistics: Shiny app at http://www.statistics.calpoly.edu/shiny
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 Sampling Distributions of Various Statistics Shiny App Base R code created by Gail Potter Shiny app files created by Gail Potter Cal Poly Statistics Dept Shiny Series http://statistics.calpoly.edu/shiny
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 ## WANT: FIND mu and sigma such that when ## X is defined by P(X<=x) = .5 Phi ((x+mu)/sigma)) + .5 Phi ((x-mu)/sigma)) ## we have Var[X] = 1 ## NOTE THAT WE will satisfy E[X] 0 since the means of the 2 normals are -mu and mu. ## We just need to find sigma so that Var[X]=1. ## Find the PDF: ## f(x) = d/dx F(x) = .5 psi ((x+mu)/sigma)) (1/sigma) + ## .5 psi ((x-mu)/sigma))(1/sigma), ## where psi is the PDF of the standard normal. pdf = function(x, mu, sigma) { .5* dnorm ((x+mu)/sigma) *(1/sigma) + .5 *dnorm ((x-mu)/sigma)*(1/sigma) } E[X] = \int_-infty ^ infty xf(x) dx E[X^2] = \int_-infty ^ infty x^2f(x) dx = \int x^2 .5 dnorm((x+mu)/sigma)(1/sigma) + \int x^2 .5 dnorm((x-mu)/sigma)(1/sigma) = .5*E[Y^2] + .5*E[Z^2] , where Y~normal ( -mu,sigma) and Z~normal(mu, sigma) = .5(2)(sigma^2 - mu^2) = sigma^2 - mu^2 Var[X] = E[X^2]-(E[X])^2 Var[Y] = sigma^2 = E[Y^2] - mu^2 E[Y]^2 = sigma^2 - mu^2 numsim = isolate(input$n)*isolate(input$nsim) numsim = 100000 mu = .92 sigma = sqrt(1-mu^2) "bimodal" = rnorm(numsim, mu*2*(rbinom(n=numsim, size=1, prob=.5)-.5), sd=sigma) ##, ncol=isolate(input$n))) hist(bimodal) sd(bimodal) mean(bimodal) ## Compute Q1, Q3: YES THEY ARE -mu and mu!!! x = -mu .5*pnorm(x, -mu, sigma) + .5*pnorm(x, mu, sigma) This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters  Title: Sampling Distributions of Various Statistics Author: Gail Potter AuthorUrl: http://www.gailpotter.org License: MIT DisplayMode: Normal Tags: Sampling Distributions Type: Shiny This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters  The MIT License (MIT) Copyright (c) 2015 Gail Potter Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters  # ------------------ # App Title: Sampling distribution demonstration # Author: Gail Potter # ------------------ Q1=function(x) quantile(x, .25) Q3=function(x) quantile(x, .75) CV=function(x) sd(x)/mean(x) ## Compute population parameters. Populations are standardized so that they all have mean =0, ## standard deviation = 1 parameters = data.frame( row.names= c("mean", "standard deviation", "Q1", "median", "Q3", "minimum", "maximum"), bimodal=rep(NA,7), normal=rep(NA,7), left.skewed=rep(NA,7), right.skewed=rep(NA,7),uniform=rep(NA,7)) parameters[1,]=0 parameters[2,]=1 ## normal quantiles parameters$normal[3:5] = qnorm(c(.25, .5, .75)) ## left-skewed quantiles parameters$left.skewed[3:5] = c( (10-qgamma(1-.25, shape=2, scale=5)) / (5*sqrt(2)), ## Q1 (10-qgamma(1-.5, shape=2, scale=5)) / (5*sqrt(2)) , ## Q2 (10-qgamma(1-.75, shape=2, scale=5)) / (5*sqrt(2))) ## Q3 ## right-skewed quantiles: parameters$right.skewed[3:5] = c( (qgamma(.25, shape=2, scale=5)-10 ) / (5*sqrt(2)), (qgamma(.5, shape=2, scale=5)-10 ) / (5*sqrt(2)), (qgamma(.75, shape=2, scale=5)-10 ) / (5*sqrt(2))) ## uniform quantiles parameters$uniform[3:5]=c(qunif(.25, -sqrt(3), sqrt(3)), 0, qunif(.75, -sqrt(3), sqrt(3))) parameters$bimodal[3:5]= c(-.92, 0, .92) parameters[6,] = -Inf parameters[7,] = Inf parameters[7, "left.skewed"] = 10/(5*sqrt(2)) parameters[6, "right.skewed"] = -10/(5*sqrt(2)) parameters[6:7, "uniform"] = c(-sqrt(3), sqrt(3)) shinyServer(function(input, output, session) { draw.sample <- reactiveValues() observe({ if (input$n > 0 & input$n <= 1000 & is.numeric(input$n) & (input$n %% 1==0) & !is.na(input$n)) return() showshinyalert(session, "shinyalert1", paste("Please enter an integer between 1 and 1000:")) }) observe({ if (input$nsim > 0 & input$nsim <= 100000 & is.numeric(input$nsim) & (input$nsim %% 1==0) & !is.na(input$nsim)) return() showshinyalert(session, "shinyalert2", paste("Please enter an integer between 1 and 100,000:")) }) observe({ if (is.numeric(input$popmean) & !is.na(input$popmean)) return() showshinyalert(session, "shinyalert3", paste("Please enter a number for the population mean:")) }) observe({ if (is.numeric(input$popsd) & !is.na(input$popsd)) return() showshinyalert(session, "shinyalert4", paste("Please enter a number for the population standard deviation:")) }) observe({ input$go x = switch(isolate(input$popdist), "normal"= matrix(rnorm(isolate(input$n)*isolate(input$nsim), 0,1), ncol=isolate(input$n)), "right.skewed" = matrix(rgamma(isolate(input$n)*isolate(input$nsim), shape=2, scale=5)/(5*sqrt(2))-10/(5*sqrt(2)), ncol=isolate(input$n)), "left.skewed" = matrix(10/(5*sqrt(2))-rgamma(isolate(input$n)*isolate(input$nsim), shape=2, scale=5)/(5*sqrt(2)), ncol=isolate(input$n)), "uniform" = matrix(runif(isolate(input$n)*isolate(input$nsim), -sqrt(3),sqrt(3)), ncol=isolate(input$n)), "bimodal" = matrix(rnorm(isolate(input$n)*isolate(input$nsim), 2*.92*(rbinom(n=isolate(input$n)*isolate(input$nsim), size=1, prob=.5)-.5), sd=sqrt(1-.92^2)), ncol=isolate(input$n))) x = isolate(input$popsd)*x + isolate(input$popmean) f=switch(isolate(input$statistic), mean=mean, median=median, Q1=Q1, Q3=Q3, "standard deviation"=sd, maximum=max, minimum=min, CV=CV) withProgress(session, { if(isolate(input$nsim)>1000) setProgress(message = "Calculating, please wait.", detail = " ", value=.5) sample.statistics = isolate(apply(x, 1, f)) draw.sample$sample.statistics <- c(sample.statistics, isolate(draw.sample$sample.statistics)) draw.sample$x = x[1,] }) }) observe({ input$n input$clear input$popdist input$statistic input$popmean input$popsd draw.sample$x<-NULL draw.sample$sample.statistics=NULL }) output$popdistn <- renderPlot({ popname = switch(input$popdist, "normal" = "Normal" , "left.skewed"= "Left-skewed", "uniform" = "Uniform", "right.skewed" = "Right-skewed" , "bimodal" = "Bimodal") pdf = switch(input$popdist, "normal"= dnorm, "right.skewed" = function(x) 5*sqrt(2)*dgamma(5*sqrt(2)*x+10, shape=2, scale=5), "left.skewed" = function(x) 5*sqrt(2)*dgamma(10-5*sqrt(2)*x, shape=2, scale=5), "uniform" = function(x) dunif(x, -sqrt(3), sqrt(3)), "bimodal" = function(x) (dnorm(x, mean=-.92, sd=sqrt(1-.92^2))+ dnorm(x, mean=.92, sd=sqrt(1-.92^2)))/2 ) xlim = switch(input$popdist, "normal"=c(-3,3), "right.skewed" = c(-3,3), "left.skewed" = c(-3,3), "uniform" = c(-2,2), "bimodal" = c(-2,2)) par(mfrow=c(1,2), mar=rep(2,4)) xlim = input$popsd*xlim + input$popmean parameters = input$popsd*parameters + input$popmean parameters[2,]=input$popsd title = paste(popname, "population,", input$statistic, "=", round(parameters[input$statistic, input$popdist], 2)) if (input$statistic=="standard deviation") title = paste(popname,", ", input$statistic, " = ", round(parameters[input$statistic, input$popdist], 2), sep="") curve(pdf((x-input$popmean)/input$popsd), xlim=xlim, xlab="", ylab="", main=title, cex=.75) pop.parameter = parameters[input$statistic, input$popdist] if (input$statistic=="standard deviation"){ height=.2 if (input$popdist=="uniform") height=.1 abline(v=input$popmean, lty=2, col="red") segments(input$popmean, height, (input$popmean+input$popsd), height, col="red") s=input$popsd text(input$popmean + .5*input$popsd, height+.05, expression(sigma==s), cex=1.25) } else abline(v=pop.parameter, col="red") }) output$dotplot <- renderPlot({ input$n x = draw.sample$x stats=draw.sample$sample.statistics this.statistic = stats[1] par(mfrow=c(1,2)) if (!is.null(x)){ ## Compute lower and upper limits for the histogram default.lower = -4*(input$popdist=="normal")+ (-1.5)*(input$popdist=="right.skewed")+ (-2)*(input$popdist=="uniform") + (-2.5)*(input$popdist=="bimodal")+ (-1.5)*(input$popdist=="left.skewed") default.lower = input$popsd*default.lower + input$popmean default.upper = 4*(input$popdist!="uniform" & input$popdist!= "bimodal" ) + 2*(input$popdist == "bimodal" | input$popdist=="uniform") default.upper = input$popsd*default.upper + input$popmean xmin = min(default.lower, floor(min(x)-.5)) xmax = max(default.upper, ceiling(max(x)+.5)) hist1.details = hist(x, col="slategray1", border="darkgray", main=paste("Histogram of sample",input$statistic,"=", round(this.statistic,2)), xlab="Data from a single sample",breaks=seq(xmin,xmax,length.out=20)) abline(h=0) height = max(hist1.details$counts)/2 if (input$statistic=="standard deviation") { abline(v=mean(x), lty=2, col="red") segments(mean(x), height, mean(x)+sd(x), height, lwd=2, col="red") text(mean(x)+.5*sd(x), height+.2, paste("s=", round(sd(x),2)), cex=1.25) } else if (input$statistic!="CV") abline(v=this.statistic, col="red", lwd=2) parameters = input$popsd*parameters + input$popmean parameters[2,]=input$popsd pop.parameter = parameters[input$statistic, input$popdist] sample.size = input$n xmin=min(pop.parameter - input$popsd, floor(min(stats)-.5)) xmax=max(pop.parameter + input$popsd, ceiling(max(stats)+.5)) hist.details = hist(draw.sample$sample.statistics, breaks=seq(xmin, xmax, length.out = 20), plot=FALSE) ylim = c(0, max(6, max(hist.details$counts)+2)) title.end = switch(isolate(input$statistic), mean="of the sample mean", median = "of the sample median", minimum = "of the sample minimum", maximum = "of the sample maximum", Q1 = "of the first quartile (Q1)", Q3 = "of the third quartile (Q3)", "standard deviation"= "of the standard deviation", CV = "of the coefficient of variation (CV)") hist2.details = hist(draw.sample$sample.statistics, col="tomato",#572, xlab=paste("Sample ", input$statistic, "s", sep=""), ylim=ylim, main=paste("Sampling distribution \n",title.end), breaks=seq(xmin,xmax,length.out=20) , border="darkgray") abline(h=0) if(input$display ){ n.stats = length(draw.sample$sample.statistics) height2 = (max(hist2.details$counts)/2) textheight = (max(hist2.details$counts)/2)*(n.stats>10)*1.1 + ((max(hist2.details$counts)/2)+1)*(n.stats<=10) abline(v=mean(stats), lty=2, lwd=1.25) segments(mean(stats), lwd=1.25, height2, mean(stats)+ sd(stats), height2) text(mean(stats)+.5*sd(stats),textheight, round(sd(stats),2), cex=1.2) text(mean(stats)+.5*sd(stats), max(max(hist2.details$counts)*.9, ylim[2]*.9), round(mean(stats),2), cex=1.25) } } }) output$numsims = renderText({ paste("Total samples drawn =", as.character(length(draw.sample$sample.statistics)), " ") }) output$display = renderText({ f=switch(isolate(input$statistic), mean="mean", median="median", Q1="Q1", Q3="Q3", "standard deviation"="sd", maximum="max", minimum="min", CV="CV") if (input$display) { str1 = paste("Mean of ", input$statistic, "s = ", round(mean(draw.sample$sample.statistics),2), sep="") str2 = paste("Standard deviation of ",input$statistic, "s = ", round(sd(draw.sample$sample.statistics),2), sep="") HTML(paste(str1, str2, sep = '
')) } }) })
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 .shiny-progress { top: 50% !important; left: 50% !important; margin-top: -220px !important; margin-left: 50px !important; }
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters. Learn more about bidirectional Unicode characters
 # ------------------ # App Title: Sampling distribution demonstration # Author: Gail Potter # ------------------ if (!require("devtools")) install.packages("devtools") if (!require("shinyBS")) install.packages("shinyBS") library(shinyBS) if (!require(shinyIncubator)) devtools::install_github("rstudio/shiny-incubator") library(shinyIncubator) if (!require("shinysky")) devtools::install_github("ShinySky","AnalytixWare") library(shinysky) shinyUI(fluidPage( includeCSS('styles.css'), progressInit(), tags$head(tags$link(rel = "icon", type = "image/x-icon", href = "https://webresource.its.calpoly.edu/cpwebtemplate/5.0.1/common/images_html/favicon.ico")), h3("Sampling distribution demonstration"), fluidRow( column(3, wellPanel( selectInput("popdist", label = h5("Population distribution"), choices = list("Normal" = "normal", "Left-skewed" = "left.skewed", "Uniform" = "uniform", "Right-skewed" = "right.skewed", "Bimodal"="bimodal"), selected = "normal"), br(), shinyalert("shinyalert3", TRUE, auto.close.after=5), numericInput("popmean", label = h5("Population mean"), value=0), br(), shinyalert("shinyalert4", TRUE, auto.close.after=5), numericInput("popsd", label = h5("Population standard deviation"), value=1), br(), shinyalert("shinyalert1", TRUE, auto.close.after=5), numericInput("n", label=h5("Sample size"), value=10, min=1, max=1000), selectInput("statistic", label = h5("Statistic"), choices = list("Mean" = "mean", "Median" = "median", "1st quartile (Q1)" = "Q1", "3rd quartile (Q3)" = "Q3", "Standard deviation" = "standard deviation", "Maximum"="maximum", "Minimum"="minimum"), selected = "mean"), div("Shiny app by", a(href="http://www.gailpotter.org",target="_blank", "Gail Potter"),align="right", style = "font-size: 8pt"), div("Base R code by", a(href="http://www.gailpotter.org",target="_blank", "Gail Potter"),align="right", style = "font-size: 8pt"), div("Shiny source files:", a(href="https://gist.github.com/calpolystat/d7ed9873137267ee557b", target="_blank","GitHub Gist"),align="right", style = "font-size: 8pt"), div(a(href="http://www.statistics.calpoly.edu/shiny",target="_blank", "Cal Poly Statistics Dept Shiny Series"),align="right", style = "font-size: 8pt")) ), tags\$style(type="text/css", ".shiny-output-error { visibility: hidden; }", ".shiny-output-error:before { visibility: hidden; }" ), column(9, wellPanel( p("In the left panel, specify a population shape, sample size, and statistic of interest. When you press the 'Draw samples' button, a sample from that population will be generated and plotted below left. The statistic will be calculated and added to the histogram at right. By generating many different samples, you can see how the statistic tends to vary from one sample to the next. That distribution is called the 'sampling distribution'. You can change the population distribution to see how that impacts your sample histogram as well as the sampling distribution."), shinyalert("shinyalert2", TRUE, auto.close.after=5), numericInput("nsim", label=h5("Number of samples"), value=1, min=1, max=1000000), actionButton("go", label = "Draw samples"), actionButton("clear",label="Clear"), bsCollapse(multiple = FALSE, open = NULL, id = "collapse1", bsCollapsePanel("Click here to display population characteristics. (Click again to hide.)", plotOutput("popdistn", height="200px"), id="popcurve", value="test3") ) , plotOutput("dotplot", height="290px"), textOutput("numsims"), checkboxInput("display", label="Display summaries of sampling distribution"), htmlOutput("display") )) ) ))