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Sampling Distribution of Various Statistics: Shiny app at http://www.statistics.calpoly.edu/shiny
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
## 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)
Title: Sampling Distributions of Various Statistics
Author: Gail Potter
AuthorUrl: http://www.gailpotter.org
License: MIT
DisplayMode: Normal
Tags: Sampling Distributions
Type: Shiny
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.
# ------------------
# 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 = '<br/>'))
}
})
})
.shiny-progress {
top: 50% !important;
left: 50% !important;
margin-top: -220px !important;
margin-left: 50px !important;
}
# ------------------
# 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")
))
)
))
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