Hot Hand Phenomenon: Shiny app at http://www.statistics.calpoly.edu/shiny

#Hothand.txt

Hot Hand Phenomenon Shiny App
 
Base R code created by Kevin Ross
Shiny app files created by Kevin Ross

Cal Poly Statistics Dept Shiny Series 
http://statistics.calpoly.edu/shiny

app.R

# Shiny app to accompany
# Ross, K.J. (2017) "Classroom Investigations of Recent 
# Research Concerning the Hot Hand Phenomenon," 
# Journal of Statistics Education, 25(3)

if (!require("shiny")) install.packages("shiny")
if (!require("shinyBS")) install.packages("shinyBS")
if (!require("plyr")) install.packages("plyr")
if (!require("ggplot2")) install.packages("ggplot2")

library(shiny)
library(shinyBS)
library(plyr)
library(ggplot2)

# function that returns basic streak statistics given a sequence
source("streak_stats.r")

# List of descriptions of streak statistics
streak_stat_options = as.list(c("Proportion of S after streaks of S" = "phat_after_Sstreak",
                                "Difference in proportion of S (after streaks of S - other trials)" = "phat_Sstreak_vs_others",
                                "Difference in proportion of S (after streaks of S - after streaks of F)" = "phat_Sstreak_vs_Fstreak",
                                "Frequency of S streaks" = "Sstreak_frequency",
                                "Longest run of S" = "longest_run_success",
                                "Total number of runs" = "n_runs"
                                # "Number of runs of S" = "n_runs_success",
                                # "Proportion of S after streaks of F" = "phat_after_Fstreak",
                                # "Number of runs of F" = "n_runs_failure",
                                # "Longest run of F" = "longest_run_failure"
))

# global constants
max_n_repetitions = 20000  # maximum number of repetitions
max_n_dots = 113  # maximum number of dots to show before histogram
max_streak_length = 7  # maximum alloable streak length
observed_color = "#ffa500"  # orange
pvalue_color = "#00bfff"   # skyblue
`%then%` <- shiny:::`%OR%` # for Shiny validate (see validate help)


ui <- fluidPage(
      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")),
  theme = "bootswatch-cerulean.css",
  titlePanel("Randomization-based Analysis of the Hot Hand Phenomenon"),
  sidebarLayout(
    sidebarPanel(
      selectInput(inputId = "statistic",
                  label = "Choose a streak statistic:",
                  choices = streak_stat_options,
                  selected = "phat_after_Sstreak"
      ),
      numericInput(inputId = "streak_length",
                   label = "Define streak length:",
                   value = 1,
                   min = 1, max = max_streak_length, step = 1),
      radioButtons(inputId = "input_type",
                   label = "Select method to input observed data:",
                   choices = c("Input results sequence"= "input_data",
                               "Input summary statistics"="input_stat"),
                   selected = "input_data"),
      conditionalPanel(
        condition = "input.input_type == 'input_stat'",
        textInput(inputId = "n_trials",
                  label = "Number of trials:",
                  value = ""),
        textInput(inputId = "n_success",
                  label = "Total number of successes:", value = ""),
        textInput(inputId = "observed_stat",
                  label = "Observed value of streak statistic:",
                  value = "")
      ),
      conditionalPanel(
        condition = "input.input_type == 'input_data'",
        textInput(inputId = "observed_data",
                  label = "Enter observed results sequence",
                  value = "")
      ),
      actionButton(inputId = "run",
                   label="Accept inputs and initialize simulation"),
      tags$hr(style = "border-color: #000000;"),
      numericInput(inputId = "n_repetitions",
                   label = "Number of shuffles to simulate:",
                   value = 1,
                   min = 1, max = max_n_repetitions),
      actionButton(inputId = "more_repetitions",
                   label="Shuffle"),
      checkboxInput(inputId = "compute_pvalue",
                    label = "Compute p-value",
                    value = FALSE),
      checkboxInput(inputId = "recent_shuffle",
                    label = "Show most recent shuffle",
                    value = FALSE),
      actionButton(inputId = "reset",
                   label = "Clear plot"),
      div("Shiny app by",
          a(href="mailto:kjross@calpoly.edu",target="_blank",
            "Kevin Ross"),align="right", style = "font-size: 8pt"),
      div("Base R code by",
          a(href="mailto:kjross@calpoly.edu",target="_blank",
            "Kevin Ross"),align="right", style = "font-size: 8pt"),
      div("Shiny source files:",
          a(href="https://gist.github.com/calpolystat/d40a02fa87508ac5ac4b",
            target="_blank","GitHub Gist"), 
          align="right", style = "font-size: 8pt"),
      div(a(href="http://www.statistics.calpoly.edu/shiny",target="_blank",
            "Cal Poly Statistics Dept Shiny Series"),
          align="right", style = "font-size: 8pt")
    ),
    mainPanel(
      tabsetPanel(tabPanel(title = "Simulation app",
                           tags$b(tags$h4("Observed data",
                                          style = paste("color: ",
                                                        observed_color, ";"))),
                           tableOutput(outputId = "summary_stats"),
                           # bsAlert(anchorId = "alert_input_type"),
                           # bsAlert(anchorId = "alert_input_data"),
                           # bsAlert(anchorId = "alert_input_streak"),
                           bsAlert(anchorId = "alert_update_stat"),
                           tags$hr(style = "border-color: #000000;"),
                           bsAlert(anchorId = "alert_nrep"),
                           # bsAlert(anchorId = "alert_click_button"),
                           tags$b(tags$h4(textOutput(outputId = "plot_title"),
                                          style = "text-align: center;")),
                           textOutput(outputId = "recent_shuffle"),
                           textOutput(outputId = "recent_stat"),
                           plotOutput(outputId = "null_plot"),
                           tags$b(tags$h4(textOutput(outputId = "observed_stat"),
                                          style = paste("color: ",
                                                        observed_color, ";"))),
                           tags$b(tags$h4(textOutput(outputId = "pvalue"),
                                          style = paste("color: ",
                                                        pvalue_color, ";")))
      ),
      tabPanel(title = "Instructions and notes",
               includeHTML("instructions.html")),
      tabPanel(title = "References",
               includeHTML("references.html")),
      id = "current_tab",
      selected = "Simulation app"
      )
    )
  )
)

server <- function(input, output, session){
  
  ##### Validate inputs
  
  # choose statistic - just read input
  streak_stat_name = reactive({
    return(names(streak_stat_options)[which(streak_stat_options == input$statistic)])
  })
  
  observeEvent(c(input$statistic, input$streak_length), {
    if (input$input_type == 'input_stat' && input$observed_stat != ""){
      createAlert(session, anchorId = "alert_update_stat",
                  content = "Be sure to update the value of the observed statistic.")
    }
  })
  
  # length of streak - check integer between 1 and 7
  streak_length = reactive({
    validate(
      need(try(input$streak_length%%1 == 0),
           "Streak length must be a whole number")%then%
        need(try(input$streak_length >= 1),
             "Streak length must be at least 1")%then%
        need(try(input$streak_length <= max_streak_length),
             paste("Streak length cannot be more than ",max_streak_length))
    )
    return(input$streak_length)
  })
  
  # input data method - check that sequence is valid (just 1s and 0s)
  observed_results = reactive({
    if (input$input_type == 'input_data'){
      x = unlist(strsplit(input$observed_data, split=","))
      validate(
        need(setequal(x, c("0","1")),
             "Please enter observed results (1 for success, 0 for failure)
                  in sequence separated by commas
             (e.g. 0,1,1,1,0,0,1,0,1,1)")
      )
      return(as.numeric(x))
    }
  })
  
  # number of trials - check positive integer greater than streak length
  n_trials = reactive({
    if (input$input_type == 'input_stat'){
      validate(
        need(input$n_trials != "",
             "Please input number of trials")%then%
          need(try(as.numeric(input$n_trials) > streak_length()),
               "Number of trials must be greater than streak length")%then%
          need(try(as.numeric(input$n_trials)%%1 == 0),
               "Number of trials must be a whole number")
      )
      return(as.numeric(input$n_trials))
    }else{
      return(length(observed_results()))
    }
  })
  
  # number of successes - check positive integer less than number of trials
  n_success = reactive({
    if (input$input_type == 'input_stat'){
      validate(
        need(input$n_success != "",
             "Please input observed number of successes")%then%
          need(try(as.numeric(input$n_success) < n_trials()),
               "Number of successes must be less than number of trials")%then%
          need(try(as.numeric(input$n_success) >= streak_length()),
               "Number of successes must be greater than streak length")%then%
          need(try(as.numeric(input$n_success)%%1 == 0),
               "Number of successes must be a whole number")
      )
      return(as.numeric(input$n_success))
    }else{
      return(sum(observed_results()))
    }
  })
  
  
  #### Calculations
  
  # input data method - compute observed stats
  observed_streak_stats <- reactive({
    if (input$input_type == 'input_data'){
      return(streak_stats(observed_results(), streak_length()))
    }
  })
  
  # compute observed value of selected streak stat
  observed_stat <- reactive({
    if (input$input_type == 'input_data'){
      x = observed_streak_stats()[[input$statistic]]
    }else{
      x = as.numeric(input$observed_stat)
    }
    return(x)
  })
  
  # invalidate the simulation when any of the inputs change
  is_sim_valid <- reactiveValues(yes = TRUE)
  
  observeEvent(input$input_type, {
    is_sim_valid$yes = FALSE
    createAlert(session, anchorId = "alert_input_type",
                content = "Enter data and click Accept Inputs.")
  })
  
  observeEvent(c(input$n_trials, input$n_success, input$observed_data), {
    is_sim_valid$yes = FALSE
    createAlert(session, anchorId = "alert_input_data",
                content = "Observed data has changed.  Click Accept Inputs.")
  })
  
  observeEvent(input$streak_length, {
    is_sim_valid$yes = FALSE
    createAlert(session, anchorId = "alert_input_streak",
                content = "Streak length has changed.  Click Accept Inputs.")
  })
  
  # Simulate random permutations
  # Given # of trials and # of successes
  random_permutations = eventReactive(input$run, {
    is_sim_valid$yes = TRUE
    x = c(rep(1, n_success()), rep(0, n_trials() - n_success()))
    y = matrix(
      replicate(max_n_repetitions,
                sample(x, size = n_trials(), replace = F)),
      byrow=TRUE, nrow=max_n_repetitions)
    return(y)
  })
  
  # compute streak statistics for the random permutations
  null_dist = eventReactive(input$run, {
    withProgress(message = "Initializing simulation, please wait.",
                 detail = "When this message disappears you can use the simulate shuffles button.",
                 {
                   x = as.data.frame(
                     adply(random_permutations(), 
                           1, streak_stats, streak_length = streak_length()))
                 })
    return(x)
  })
  
  # update counter for producing the plot
  total_nrep <- reactiveValues(current = 0)
  observeEvent(input$more_repetitions, {
    if (is_sim_valid$yes == FALSE){
      createAlert(session, anchorId = "alert_click_button",
                  content = "You must click the Accept Inputs
                  button before simulating values.")
      total_nrep$current = 0
    }else{
      total_nrep$current = isolate(input$n_repetitions) + 
        isolate(total_nrep$current)
      if (total_nrep$current > max_n_repetitions){
        total_nrep$current = max_n_repetitions
        createAlert(session, anchorId = "alert_nrep",
                    content = paste("Note: ", max_n_repetitions,
                                    "is the maximum number of repetitions
                                  the app will run"))
      }
    }
  })
  
  # Reset the simulation when inputs change
  observe({
    input$run
    input$reset
    input$streak_length
    input$n_trials
    input$n_success
    input$observed_data
    total_nrep$current = 0
  })
  
  # all simulated values of the current statistic
  all_values <- reactive({
    y = null_dist()[[input$statistic]]
    return(y)
  })
  
  # values to plot; plot is updated incrementally
  plot_values <- reactive({
    if (total_nrep$current > 0){
      y = all_values()[1:total_nrep$current]
      return(y[!is.na(y)])
    }
  })
  
  # compute p-value (and numerator and denominator)
  pvalue <- reactive({
    if (input$statistic == "n_runs"){
      x = sum(plot_values() <= observed_stat())
    }else{
      x = sum(plot_values() >= observed_stat())
    }
    return(c("nrep" = length(plot_values()),
             "count" = x,
             "approx" = round(x / length(plot_values()), 4)))
  })
  
  #### Outputs
  
  # Table of summary stats
  output$summary_stats <- renderTable({
    x = data.frame(
      c("Number of trials", "Number of successes", streak_stat_name()),
      c(n_trials(), n_success(), observed_stat())
    )
    colnames(x) = c("Statistic", "Observed value")
    return(x)
  },
  include.rownames = FALSE, display = c("d","s","f") #, digits = 2
  )
  
  # Title of the plot of the null distribution (and pvalue calc)
  output$plot_title <- renderText({
    if (total_nrep$current>0){
      paste("Null distribution of streak statistic: ",
            streak_stat_name())}
  })
  
  # Most recent shuffle
  output$recent_shuffle <- renderText({
    if (input$recent_shuffle && total_nrep$current>0){
      paste("Most recent shuffle: ",
            toString(random_permutations()[total_nrep$current, ]))
    }
  })
  
  # Most recent value of streak statistic
  output$recent_stat <- renderText({
    if (input$recent_shuffle && total_nrep$current>0){
      if (!is.na(all_values()[total_nrep$current])){
        paste("Value of streak statistic for this shuffle: ",
              round(all_values()[total_nrep$current],4))
      }else{
        paste("Value of streak statistic cannot be computed for this shuffle\n
              since there are no streaks of length", streak_length())
      }
    }
  })
  
  # The plot of the null distribution and p-value
  output$null_plot <- renderPlot({
    x_axis_limits = range(c(all_values(),observed_stat()), na.rm=TRUE)*c(0.95,1.05)  ### Need c(.95, 1.05)???
    if (total_nrep$current>0){
      max_y_dots = all_values()[!is.na(all_values())]
      max_y_dots = max_y_dots[1:max_n_dots]
      if (input$statistic == "n_runs"){
        color_value = plot_values() <= observed_stat()
      }else{
        color_value = plot_values() >= observed_stat()
      }
      plot_data = data.frame(plot_value = plot_values(), color_value)
      null_plot = ggplot(data=plot_data, aes(x=plot_value, fill=color_value))
      if (pvalue()[["nrep"]] < max_n_dots){
        null_plot = null_plot +
          geom_dotplot(dotsize = .4,
                       na.rm = TRUE,
                       binwidth = (x_axis_limits[2]-x_axis_limits[1])/30) +
          scale_y_continuous(expand = c(0.01, 0.01),limits=c(0,max_n_dots/2), name="", breaks = NULL)
      }else{
        null_plot = null_plot +
          geom_histogram(colour="black",
                         binwidth = (x_axis_limits[2]-x_axis_limits[1])/50) +
          scale_y_continuous(expand = c(0.01, 0.01))
      }
      if (input$compute_pvalue && !is.na(pvalue()[["approx"]]) && !is.na(observed_stat())){
        if (input$statistic == "n_runs"){
          shade_min = -Inf
          shade_max = observed_stat()
        }else{
          shade_min = observed_stat() 
          shade_max = Inf
        }
        pvalue_fill_scale <- scale_fill_manual(
          name = "pvalue_color", 
          values = as.character(c("FALSE"="white", "TRUE"=pvalue_color)),
          limits = c("FALSE","TRUE")
        )
        null_plot = null_plot +
          pvalue_fill_scale +
          annotate("rect", xmin = shade_min, xmax = shade_max, ymin = -Inf, ymax = 0, fill=pvalue_color, color=pvalue_color) + 
          geom_vline(xintercept = observed_stat(), color = observed_color, linetype="dashed", size = 1.5)
      }else{
        null_plot = null_plot + scale_fill_manual(values=c("white","white"))
      }
      null_plot = null_plot +
        annotate("text",  x=Inf, y = Inf,
                 label = paste("Median = ", round(median(plot_values()),3),
                               "\n Mean = ", round(mean(plot_values()),3),
                               "\n SD = ", round(sd(plot_values()),3)),
                 vjust=1, hjust=1) +
        coord_cartesian(xlim=c(x_axis_limits)) + 
        ggtitle(paste("Based on ",length(plot_values())," simulated values
                      \n (resulting from  ", total_nrep$current," repetitions of the simulation)")) +
        theme(plot.title = element_text(hjust = 0.5)) +
        labs(x="Streak statistic",y="Frequency") +
        theme(legend.position = "none",
              panel.grid.major = element_blank(),
              panel.grid.minor = element_blank(),
              panel.background = element_blank(),
              axis.line.x = element_line(color="black", size = 0.5),
              axis.line.y = element_line(color="black", size = 0.5),
              plot.title = element_text(face="bold", size=14),
              axis.title = element_text(face="bold", size=14))
      print(null_plot)
    }
  })
  
  # Output value of observed statistic for plot
  output$observed_stat <- renderText({
    if (input$compute_pvalue){ 
      if (is.na(observed_stat()) & input$input_type == 'input_data'){
        paste("Warning: observed value of the streak statistic cannot be computed
              because there are no trials following a streak of length ",
              streak_length(), " in the observed data")
      }else if (is.na(observed_stat()) & input$input_type == 'input_stat'){
        paste("Please enter the observed value of the streak statistic")
      }else{ ####
        paste("Observed value of streak statistic = ", round(observed_stat(),4))
      }
    }
  })
  
  # Output the p-value calculation
  output$pvalue <- renderText({
    if (input$compute_pvalue){ 
      if (is.na(observed_stat())){
        paste("The p-value cannot be computed because the observed statistic cannot be computed")
      }else if (pvalue()[["nrep"]] == 0){
        paste("There must be at least one simulated value of the streak statistic in order to compute an approximate p-value")
      }else{
        paste("Simulated p-value = ", pvalue()[["count"]], "/",
              pvalue()[["nrep"]], " = ",
              pvalue()[["approx"]])
      }
    }
  })
}

shinyApp(ui = ui, server = server)

DESCRIPTION

Title: Hot Hand Phenomenon
Author: Kevin Ross
AuthorUrl: https://statistics.calpoly.edu
License: MIT
DisplayMode: Normal
Tags: Simulation-based inference, Hot hand in basketball, Hot hand fallacy
Type: Shiny

instructions.html

<h3>Background</h3>
<p>Many basketball players and fans alike believe in the "hot hand" phenomemon: the idea that making several shots in a row increases a player's chance of making the next shot.   Is there really a hot hand in basketball?  More generally, in success/failure trials is there a tendency for trials following streaks of successes to be more likely to result in success?  This app can be used to perform a statistical test for "hot hand" type behavior in sequences of success/failure trials, such as the shot attempts of a basketball player.</p>
<h3>Data</h3>
<p>The data consist of a fixed number of success/failure trials with the outcomes recorded in sequence. (The probability that a trial results in success is assumed to be the same for all trials.) The method to input the observed data is controlled by the radio buttions:</p>
<ol>
<li><strong>Input results sequence:</strong> Input the observed sequence of trial outcomes itself, with&nbsp;1 representing success and 0 failure, separated by commas. For example, an entry of 0,1,1,1,0 corresponds to failure on the first trial, success on the second, third, and fourth trials, and failure on the fifth trial. (Click on the References tab for information about <a href="nba3point.csv" target="_blank">some sample data</a> that can be copied into the app.)</li>
<li><strong>Input summary statistics:</strong> Rather than entering the results of each individual trial, the user can enter three summary statistics.
<ul>
<li>Number of trials</li>
<li>Total number of successes</li>
<li>Observed value of the streak statistic (see below)</li>
</ul>
</li>
</ol>
<p>After entering the results sequence or the summary statistics (number of trials and number of successes), as well as the streak length (see below), click the button to <strong>Accept inputs and initialization simulation</strong>.  The initialization process might take a minute; when the "please wait" message disappers, the app is ready to perform repetitions of the simulation (see <em>Permutation test</em> below).  Note that the Accept inputs button should be pushed whenever these inputs change.</p>
<h3>A "streak statistic" that measures the hot hand?</h3>
<p>While there is no consensus definition of what constitutes the "hot hand", the term generally refers to a tendency for trials following streaks of successes to be more likely to result in success. Several statistics are used in practice; those included in the app are described below.</p>
<p>For statistics 1 through 4 below, the user must <strong>define the streak length</strong>: How many successes must be observed in a row in order to consider it a hot streak? The minimum possible streak length is 1; the maximum allowed streak length in the app is 7. Note: in applications concerning the hot hand in basketball, 3 is commonly used for the streak length.</p>
<p>In the app, the user can <strong>choose a streak statistic</strong> from the following.&nbsp;</p>
<ol>
<li><em>Proportion of S after streaks of S</em>. The proportion of success on those trials that are immediately preceded by a streak of successes. For example, if the streak length is 3 and the sequence is 0,1,1,1,<strong>1</strong>,<strong>0</strong>,1,1,1,<strong>1</strong>, trials 5, 6, and 10 are preceded by a streak of 3 successes, and the proportion of success on these trials is 2/3 = 0.6667.&nbsp;</li>
<li><em>Difference in proportion of S (after streaks of S - other trials).</em> The difference between the statistic in item 1 and the proportion of success on the remanining trials. In the previous example,&nbsp;<span style="text-decoration: underline;">0</span>,<span style="text-decoration: underline;">1</span>,<span style="text-decoration: underline;">1</span>,<span style="text-decoration: underline;">1</span>,<strong>1</strong>,<strong>0</strong>,<span style="text-decoration: underline;">1</span>,<span style="text-decoration: underline;">1</span>,<span style="text-decoration: underline;">1</span>,<strong>1</strong>, the value of the statistic is 2/3 - 6/7 = -0.1905.</li>
<li><em>Difference in proportion of S (after streaks of S - after streaks of F).</em> The difference between the statistic in item 1 and the proportion of success on those trials that are immediately preceded by a streak of failures. In the previous example, this statistic cannot be computed since there are no trials which follow a streak of 3 failures. In the sequence 1,0,0,0,<span style="text-decoration: underline;">0</span>,<span style="text-decoration: underline;">1</span>,1,1,<strong>1</strong>,<strong>1</strong>, with streak length 3, the value of the statistic is 2/2 - 1/2 = 0.5.&nbsp;</li>
<li><em>Frequency of S streaks.</em>&nbsp;The proportion of trials that are immediately preceded by a streak of successes. In the example&nbsp;0,1,1,1,<strong>1</strong>,<strong>0</strong>,1,1,1,<strong>1</strong>, the value of the statistic is 3/7 = 0.4286. Note, with a streak length of 3 the first 3 trials are not counted in determining the frequency of trials preceded by a streak of successes, and similarly for other values of streak length.&nbsp;</li>
<li><em>Longest run of S.</em> The largest number of successes in a row in the observed sequence. In the example&nbsp;0,1,1,1,1,0,1,1,1,1, the value of the statistic is 4.</li>
<li><em>Total number of runs.</em> The total number of runs, of any length, of both success and failure. In the example&nbsp;0,1,1,1,1,0,1,1,1,1, the value of the statistic is 4.  (This statistic is equivalent to the total number of "switches" or "alterations" between S and F if the first trial is counted as the first switch.)</li>
</ol>
<h3>Permutation test</h3>
<p>The app can be used to perform a randomization-based hypothesis test of the hot hand phenomenon. The null hypothesis is that there is no hot hand; that is, that the trials are independent. Under the null hypothesis (together with the assumption of constant probability of success on all trials), given the number of successes any possible <em>ordering</em> of the successes and failures in the sequence is equally likely. Therefore, to simulate one hypothetical value of the streak statistic under the null hypothesis of no hot hand:
<ul>
<li>shuffle the observed successes and failures,</li>
<li>deal them out in sequence (first trial, second trial, and so on),</li>
<li>and compute the value of the streak statistic for this hypothetical sequence.</li>
</ul>
The null distribution of the streak statistic can be simulated by repeating this process many times, by pressing the <strong>Simulate shuffles button</strong> in the app and changing the desired <strong>number of shuffles to simulate</strong>. Checking the box for <strong>show most recent shuffle</strong> will illustrate the process. Note: the observed value of the streak statistic is not necessary to simulate the null distribution.</p>
<p>An approximate p-value can be computed by comparing the observed value of the streak statistic relative to its null distribution, by <strong>checking the box for Compute p-value</strong>. Since we are interested if there is evidence of presence of the hot hand, one-sided p-values are computed.</p>
<ul>
<li>If total number of runs is selected as the streak statistic, smaller values are stronger evidence to reject the null hypothesis of no hot hand. (In streaky behavior we would expect longer, but fewer, runs of success.)</li>
<li>For all other streak statistics, larger values of the statistic are stronger evidence to reject the null hypothesis of no hot hand.&nbsp;</li>
</ul>
<p> As illustrated by the example for streak statistic 3 above, for streak statistics 1 through 4 the value of the statistic cannot be computed for a permutation in which there are no streaks of the specified length.  Thus the app distinguishes between the number of repetitions performed and the number of simulated values of the statistic.  The latter count is the denominator of the simulated p-value.
</p>
<p>Note: The <strong>clear plot</strong> button will simply clear the plot.  To rerun the simulation the Accept inputs and initialize simulation button must be pressed first.</p>
<h3>A note on analyzing the "cold hand" phenomenon</h3>
<p>The hot hand refers to a tendency for successes to cluster together. The "cold" hand refers to a similar tendency for failures. The app can be used to analyze the cold hand by simply interchanging the roles of success and failure. For example, to assess evidence of the cold hand based on the observed sequence 0,0,0,1,0,0,1, enter the data as 1,1,1,0,1,1,0 and perform a "hot hand" analysis. KEVIN ROSS</p>

LICENSE

The MIT License (MIT)
 
Copyright (c) 2017 Kevin Ross
 
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.

references.html

<p>The following paper accompanies this app and provides a discussion of the methods upon which the app is based, illustrations of hot hand analyses that can be performed using the app, and suggested activities.</p>
<ul>
<li> Ross, K. (2017).  <a href="hot_hand_paper_final.pdf">&ldquo;Classroom Investigations of Recent Research Concerning the Hot Hand Phenomenon.&rdquo;</a>  <em>Journal of Statistics Education.</em> 25(3).</li> 
</ul>
<p><a href="nba3point.csv" target="_blank">This data set</a> accompanies the paper. It contains the results from the NBA Three-Point Contest for 2013-2017.  For each year, player, and round, the results of all the player's attempts in the round are recorded in sequence, with 1 indicating a made field goal (success) and 0 indicating a miss (failure).  (In most cases there are 25 attempts in a round.)  The results sequence for a particular player can be copied and pasted into the <strong>Enter observed results sequence</strong> box.</p>
<p>The hot hand analysis which the app performs is based on the methods in a series of papers by <a href="http://didattica.unibocconi.eu/mypage/index.php?IdUte=111643&amp;cognome=MILLER&amp;nome=JOSHUA%20BENJAMIN&amp;urlBackMy=">Joshua B. Miller</a>&nbsp;and&nbsp;<a href="https://sites.google.com/site/adamangelsanjurjo/home">Adam Sanjurjo</a>&nbsp;which have&nbsp;been a subject of much <a href="http://andrewgelman.com/2015/07/09/hey-guess-what-there-really-is-a-hot-hand/">discussion</a>&nbsp;and <a href="http://didattica.unibocconi.eu/myigier/index.php?IdUte=111643&amp;idr=26734&amp;lingua=eng&amp;comando=Apri">media</a>&nbsp;<a href="http://fae.ua.es/FAEX/the-hot-hand-is-back/">coverage</a>.</p>
<ul>
<li>Miller, J. B. and Sanjurjo, A. (2014). <a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2450479">&ldquo;A Cold Shower for the Hot Hand Fallacy&rdquo;</a>. Working paper (Dec 2014).</li>
<li>&mdash; (2015). <a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2611987">&ldquo;Is it a Fallacy to Believe in the Hot Hand in the NBA Three-Point Contest?&rdquo;</a>. Working paper (Jun 2015).</li>
<li>&mdash; (2016). <a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2627354">&ldquo;Surprised by the Gambler&rsquo;s and Hot Hand Fallacies? A Truth in the Law of Small Numbers&rdquo;</a>. Working paper (Aug 2016).</li>
<li>&mdash; (2016). <a href="http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2728151">&ldquo;A Primer and Frequently Asked Questions for &lsquo;Surprised by the Gambler's and Hot Hand Fallacies? A Truth in the Law of Small Numbers&rsquo; (Miller and Sanjurjo 2015)&rdquo;</a>. Working paper (Feb 2016).</li>

</ul>
<p>The paper accompanying the app also uses data from the following seminal study on the hot hand in basketball, which is referenced numerous times in the Miller and Sanjurjo papers and related literature.</p>
<ul>
<li>Gilovich, T., Vallone, R., and Tversky, A. (1985). <a href="http://www.sciencedirect.com/science/article/pii/0010028585900106">&ldquo;The Hot Hand in Basketball: On the Misperception of Random Sequences&rdquo;</a>. <em>Cognitive Psychology</em> 17, 295-314.</li>
</ul>

streak_stats.r

# function that returns basic streak statistics given a sequence
streak_stats = function(results_sequence, streak_length){
  n_trials = length(results_sequence)
  n_success = sum(results_sequence)
  
  run_lengths = rle(results_sequence)$lengths
  if_run_success = rle(results_sequence)$values
  n_runs = length(run_lengths)
  
  # runs of successes
  longest_run_success = max(run_lengths*if_run_success)
  n_runs_success = sum(if_run_success)
  
  # successes after streaks of Successes
  run_lengths_adj = run_lengths - streak_length + 1
  run_values_adj_S =
    if_run_success *
    (!((if_run_success == 1) &
         (run_lengths < streak_length)))
  n_after_Sstreak = sum(run_lengths_adj * run_values_adj_S) -
    run_values_adj_S[length(run_values_adj_S)]
  n_success_after_Sstreak = sum((run_lengths - streak_length) * run_values_adj_S)
  
  # runs of failures
  longest_run_failure = max(run_lengths*(1-if_run_success))
  n_runs_failure = sum(1-if_run_success)
  
  # successes after streaks of Failures
  run_values_adj_F =
    (1-if_run_success) *
    (!((if_run_success == 0) &
         (run_lengths < streak_length)))
  n_after_Fstreak = sum(run_lengths_adj * run_values_adj_F) -
    run_values_adj_F[length(run_values_adj_F)]
  n_success_after_Fstreak = n_after_Fstreak -
    sum((run_lengths - streak_length) * run_values_adj_F)
  
  # Proportion of hits on trials after streaks
  phat_after_Sstreak = n_success_after_Sstreak / n_after_Sstreak
  
  # Difference in proportion of hits (trials after streaks - all other trials)
  phat_after_no_Sstreak = (n_success - n_success_after_Sstreak)/
    (n_trials - n_after_Sstreak)
  phat_Sstreak_vs_others = phat_after_Sstreak - phat_after_no_Sstreak
  
  # Difference in proportion of hits (trials after hit streaks - trials after miss streaks)
  phat_after_Fstreak = n_success_after_Fstreak / n_after_Fstreak
  phat_Sstreak_vs_Fstreak = phat_after_Sstreak - phat_after_Fstreak
  
  # Hit streak frequency - proportion of trials that occur after hit streaks
  Sstreak_frequency  = n_after_Sstreak / (n_trials - streak_length)
  
  # collect the streak stats for the results sequence
  return(data.frame(phat_after_Sstreak,
                    phat_Sstreak_vs_others,
                    phat_Sstreak_vs_Fstreak,
                    Sstreak_frequency,
                    longest_run_success,
                    n_runs
                    #                     n_runs_success,
                    #                     phat_after_Fstreak,
                    #                     n_runs_failure,
                    #                     longest_run_failure,
  ))
}