Tag Archives: central limit theorem

My Second (R) Shiny App: Sampling Distributions & CLT

Image Credit: Doug Buckley of http://hyperactive.to

Image Credit: Doug Buckley of http://hyperactive.to

I was so excited about my initial foray into Shiny development using jennybc‘s amazing googlesheets package that I stayed up half the night last night (again) working on my second Shiny app: a Shiny-fied version of the function I shared in March to do simulations illustrating sampling distributions and the Central Limit Theorem using many different source distributions. (Note that Cauchy doesn’t play by the rules!) Hope this info is useful to all new Shiny developers.

If the app doesn’t work for you, it’s possible that I’ve exhausted my purchased hours at http://shinyapps.io — no idea how much traffic this post might generate. So if that happens to you, please try getting Shiny to work locally, cutting and pasting the code below into server.R and ui.R files, and then launching the simulation from your R console.

Here are some important lessons I learned on my 2nd attempt at Shiny development:

  • Creating a container (rv) for the server-side values that would change as a result of inputs from the UI was important. That container was then available to the portions of my Shiny code that prepared data for the UI, e.g. output$plotSample.
  • Because switch only takes arguments that are 1 character long, using radio buttons in the Shiny UI was really useful: I can map the label on each radio button to one character that will get passed into the data processing on the server side.
  • I was able to modify the CSS for the page by adding a couple lines to mainPanel() in my UI.
  • Although it was not mentally easy (for me) to convert from an R function to a Shiny app when initially presented with the problem, in retrospect, it was indeed straightforward. All I had to do was take the original function, split out the data processing from the presentation (par & hist commands), put the data processing code on the server side and the presentation code on the UI side, change the variable names on the server side so that they had the input$ prefix, and make sure the variable names were consistent between server and UI.
  • I originally tried writing one app.R file, but http://shinyapps.io did not seem to like that, so I put all the code that was not UI into the server side and tried deploying with server.R and ui.R, which worked. I don’t know what I did wrong.
  • If you want to publish to http://shinyapps.io, the directory name that hosts your files must be at least 4 characters long or you will get a “validation error” when you attempt to deployApp().
## Nicole's Second Shiny Demo App
## N. Radziwill, 12/6/2015, http://qualityandinnovation.com
## Used code from http://github.com/homerhanumat as a base
###########################################################
## ui
###########################################################

ui <- fluidPage(
titlePanel('Sampling Distributions and the Central Limit Theorem'),
sidebarPanel(
helpText('Choose your source distribution and number of items, n, in each
sample. 10000 replications will be run when you click "Sample Now".'),
h6(a("Read an article about this simulation at http://www.r-bloggers.com",
href="http://www.r-bloggers.com/sampling-distributions-and-central-limit-theorem-in-r/", target="_blank")),
sliderInput(inputId="n","Sample Size n",value=30,min=5,max=100,step=2),
radioButtons("src.dist", "Distribution type:",
c("Exponential: Param1 = mean, Param2 = not used" = "E",
"Normal: Param1 = mean, Param2 = sd" = "N",
"Uniform: Param1 = min, Param2 = max" = "U",
"Poisson: Param1 = lambda, Param2 = not used" = "P",
"Cauchy: Param1 = location, Param2 = scale" = "C",
"Binomial: Param1 = size, Param2 = success prob" = "B",
"Gamma: Param1 = shape, Param2 = scale" = "G",
"Chi Square: Param1 = df, Param2 = ncp" = "X",
"Student t: Param1 = df, Param2 = not used" = "T")),
numericInput("param1","Parameter 1:",10),
numericInput("param2","Parameter 2:",2),
actionButton("takeSample","Sample Now")
), # end sidebarPanel
mainPanel(
# Use CSS to control the background color of the entire page
tags$head(
tags$style("body {background-color: #9999aa; }")
),
plotOutput("plotSample")
) # end mainPanel
) # end UI

##############################################################
## server
##############################################################

library(shiny)
r <- 10000 # Number of replications... must be ->inf for sampling distribution!

palette(c("#E41A1C", "#377EB8", "#4DAF4A", "#984EA3",
"#FF7F00", "#FFFF33", "#A65628", "#F781BF", "#999999"))

server <- function(input, output) {
set.seed(as.numeric(Sys.time()))

# Create a reactive container for the data structures that the simulation
# will produce. The rv$variables will be available to the sections of your
# server code that prepare output for the UI, e.g. output$plotSample
rv <- reactiveValues(sample = NULL,
all.sums = NULL,
all.means = NULL,
all.vars = NULL)

# Note: We are giving observeEvent all the output connected to the UI actionButton.
# We can refer to input variables from our UI as input$variablename
observeEvent(input$takeSample,
{
my.samples <- switch(input$src.dist,
"E" = matrix(rexp(input$n*r,input$param1),r),
"N" = matrix(rnorm(input$n*r,input$param1,input$param2),r),
"U" = matrix(runif(input$n*r,input$param1,input$param2),r),
"P" = matrix(rpois(input$n*r,input$param1),r),
"C" = matrix(rcauchy(input$n*r,input$param1,input$param2),r),
"B" = matrix(rbinom(input$n*r,input$param1,input$param2),r),
"G" = matrix(rgamma(input$n*r,input$param1,input$param2),r),
"X" = matrix(rchisq(input$n*r,input$param1),r),
"T" = matrix(rt(input$n*r,input$param1),r))

# It was very important to make sure that rv contained numeric values for plotting:
rv$sample <- as.numeric(my.samples[1,])
rv$all.sums <- as.numeric(apply(my.samples,1,sum))
rv$all.means <- as.numeric(apply(my.samples,1,mean))
rv$all.vars <- as.numeric(apply(my.samples,1,var))
}
)

output$plotSample <- renderPlot({
# Plot only when user input is submitted by clicking "Sample Now"
if (input$takeSample) {
# Create a 2x2 plot area & leave a big space (5) at the top for title
par(mfrow=c(2,2), oma=c(0,0,5,0))
hist(rv$sample, main="Distribution of One Sample",
ylab="Frequency",col=1)
hist(rv$all.sums, main="Sampling Distribution of the Sum",
ylab="Frequency",col=2)
hist(rv$all.means, main="Sampling Distribution of the Mean",
ylab="Frequency",col=3)
hist(rv$all.vars, main="Sampling Distribution of the Variance",
ylab="Frequency",col=4)
mtext("Simulation Results", outer=TRUE, cex=3)
}
}, height=660, width=900) # end plotSample

} # end server

Sampling Distributions and Central Limit Theorem in R

The Central Limit Theorem (CLT), and the concept of the sampling distribution, are critical for understanding why statistical inference works. There are at least a handful of problems that require you to invoke the Central Limit Theorem on every ASQ Certified Six Sigma Black Belt (CSSBB) exam. The CLT says that if you take many repeated samples from a population, and calculate the averages or sum of each one, the collection of those averages will be normally distributed… and it doesn’t matter what the shape of the source distribution is! (Caveat: so long as the data comes from a distribution with finite variance… so that means the Cauchy distribution doesn’t count.)

I wrote some R code to help illustrate this principle for my students. This code allows you to choose a sample size (n), a source distribution, and parameters for that source distribution, and generate a plot of the sampling distributions of the mean, sum, and variance. (Note: the sampling distribution for the variance is a Chi-square distribution — if your source distribution is normal!)

sdm.sim <- function(n,src.dist=NULL,param1=NULL,param2=NULL) {
   r <- 10000  # Number of replications/samples - DO NOT ADJUST
   # This produces a matrix of observations with  
   # n columns and r rows. Each row is one sample:
   my.samples <- switch(src.dist,
	"E" = matrix(rexp(n*r,param1),r),
	"N" = matrix(rnorm(n*r,param1,param2),r),
	"U" = matrix(runif(n*r,param1,param2),r),
	"P" = matrix(rpois(n*r,param1),r),
        "B" = matrix(rbinom(n*r,param1,param2),r),
	"G" = matrix(rgamma(n*r,param1,param2),r),
	"X" = matrix(rchisq(n*r,param1),r),
	"T" = matrix(rt(n*r,param1),r))
   all.sample.sums <- apply(my.samples,1,sum)
   all.sample.means <- apply(my.samples,1,mean)   
   all.sample.vars <- apply(my.samples,1,var) 
   par(mfrow=c(2,2))
   hist(my.samples[1,],col="gray",main="Distribution of One Sample")
   hist(all.sample.sums,col="gray",main="Sampling Distribution\nof
	the Sum")
   hist(all.sample.means,col="gray",main="Sampling Distribution\nof the Mean")
   hist(all.sample.vars,col="gray",main="Sampling Distribution\nof
	the Variance")
}

There are 8 population distributions to choose from: exponential (E), normal (N), uniform (U), Poisson (P), binomial (B), gamma (G), Chi-Square (X), and the Student’s t distribution (T). Note also that you have to provide either one or two parameters, depending upon what distribution you are selecting. For example, a normal distribution requires that you specify the mean and standard deviation to describe where it’s centered, and how fat or thin it is (that’s two parameters). A Chi-square distribution requires that you specify the degrees of freedom (that’s only one parameter). You can find out exactly what distributions require what parameters by going here: http://en.wikibooks.org/wiki/R_Programming/Probability_Distributions.

Here is an example that draws from an exponential distribution with a mean of 1/1 (you specify the number you want in the denominator of the mean):

sdm.sim(50,src.dist="E",param1=1)

The code above produces this sequence of plots:

sd-blog-1

You aren’t allowed to change the number of replications in this simulation because of the nature of the sampling distribution: it’s a theoretical model that describes the distribution of statistics from an infinite number of samples. As a result, if you increase the number of replications, you’ll see the mean of the sampling distribution bounce around until it converges on the mean of the population. This is just an artifact of the simulation process: it’s not a characteristic of the sampling distribution, because to be a sampling distribution, you’ve got to have an infinite number of samples. Watkins et al. have a great description of this effect that all statistics instructors should be aware of. I chose 10,000 for the number of replications because 1) it’s close enough to infinity to ensure that the mean of the sampling distribution is the same as the mean of the population, but 2) it’s far enough away from infinity to not crash your computer, even if you only have 4GB or 8GB of memory.

Here are some more examples to try. You can see that as you increase your sample size (n), the shapes of the sampling distributions become more and more normal, and the variance decreases, constraining your estimates of the population parameters more and more.

sdm.sim(10,src.dist="E",1)
sdm.sim(50,src.dist="E",1)
sdm.sim(100,src.dist="E",1)
sdm.sim(10,src.dist="X",14)
sdm.sim(50,src.dist="X",14)
sdm.sim(100,src.dist="X",14)
sdm.sim(10,src.dist="N",param1=20,param2=3)
sdm.sim(50,src.dist="N",param1=20,param2=3)
sdm.sim(100,src.dist="N",param1=20,param2=3)
sdm.sim(10,src.dist="G",param1=5,param2=5)
sdm.sim(50,src.dist="G",param1=5,param2=5)
sdm.sim(100,src.dist="G",param1=5,param2=5)