Interactive Dashboard • millsratio

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library(millsratio)
library(tidyverse)
library(plotly)
library(DT)

Overview

The millsratio package includes two interactive dashboards for exploring Mills ratios and their relationship to distribution tail thickness:

Simple Dashboard (launch_dashboard_simple()) - Focused on core concepts
Full Dashboard (launch_dashboard()) - Comprehensive 12-page analysis

Running the Dashboard

The dashboards are interactive Shiny applications that run locally on your computer. They are not available on the website. To use them:

Install the millsratio package

Run library(millsratio) in R

Execute launch_dashboard() or launch_dashboard_simple()

Your browser will open with the interactive dashboard

The dashboards require a local R installation and cannot be embedded in static documentation.

Dashboard Features

Simple Dashboard

The simplified dashboard focuses on essential functionality:

Analysis Page: Compare Mills ratios across distributions
Theory Page: Mathematical foundations and asymptotic behavior
Hazard Function Page: Explore the h(x) = 1/m(x) relationship
R Code Page: Ready-to-use tidyverse examples

Launch it with:

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launch_dashboard_simple()

Full Dashboard (12 Pages)

The comprehensive dashboard is organized into three main sections:

Analysis Section

Dashboard Overview: Introduction and navigation guide
Distribution Comparison: Side-by-side analysis of multiple distributions
The t(30) Paradox: Deep dive into why t(30) appears normal but isn’t
Tail Thickness Analysis: Quantitative measures of tail behavior

Theory Section

Mathematical Foundation: Definitions and derivations
Asymptotic Behavior: Long-run properties of Mills ratios
Properties Comparison: Table of distribution characteristics
References: Academic sources and further reading

Playground Section

Interactive Explorer: Real-time parameter adjustment
Code Playground: Write and test custom Mills ratio code
Simulation Lab: Monte Carlo verification tools
Export Results: Download analyses and visualizations

Launch it with:

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launch_dashboard()

Key Interactive Features

Real-time Visualization

Both dashboards provide instant feedback as you adjust parameters:

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# Example of what the dashboard visualizes
x <- seq(0.1, 5, by = 0.1)
data <- tibble(x = x) %>%
  mutate(
    Normal = mills_ratio_normal(x),
    `t(30)` = mills_ratio_t(x, df = 30),
    `t(3)` = mills_ratio_t(x, df = 3),
    Exponential = mills_ratio_exp(x)
  ) %>%
  pivot_longer(-x, names_to = "distribution", values_to = "mills_ratio")

p <- ggplot(data, aes(x, mills_ratio, color = distribution)) +
  geom_line(size = 1.2) +
  scale_y_log10() +
  labs(
    title = "Mills Ratio Comparison",
    subtitle = "Note the divergence in tail behavior",
    x = "x",
    y = "Mills Ratio m(x) (log scale)"
  ) +
  theme_minimal() +
  theme(legend.position = "bottom")

Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
ℹ Please use `linewidth` instead.

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ggplotly(p)

Distribution Comparison Table

The dashboards generate comparison tables like this:

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# Compare Mills ratios at key points
x_points <- c(1, 2, 3, 4, 5)
comparison <- tibble(
  x = x_points,
  Normal = mills_ratio_normal(x_points),
  `t(30)` = mills_ratio_t(x_points, df = 30),
  `t(3)` = mills_ratio_t(x_points, df = 3),
  Exponential = mills_ratio_exp(x_points)
) %>%
  mutate(
    `t(30)/Normal` = `t(30)` / Normal,
    `t(3)/Normal` = `t(3)` / Normal
  )

comparison %>%
  DT::datatable(
    caption = "Mills Ratio Values and Ratios",
    options = list(pageLength = 5, dom = 't')
  ) %>%
  formatRound(2:7, digits = 4)

The t(30) Paradox Visualization

A key feature demonstrates why t(30) is dangerous to treat as normal:

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# Analyze the paradox
paradox_data <- tibble(x = seq(0, 5, by = 0.1)) %>%
  mutate(
    normal = mills_ratio_normal(x),
    t30 = mills_ratio_t(x, df = 30),
    ratio = t30 / normal,
    percent_diff = (ratio - 1) * 100
  )

# Plot the divergence
p_paradox <- ggplot(paradox_data, aes(x, percent_diff)) +
  geom_line(color = "#e74c3c", size = 1.2) +
  geom_hline(yintercept = 0, linetype = "dashed", alpha = 0.5) +
  labs(
    title = "The t(30) Paradox: Percentage Difference from Normal",
    subtitle = "t(30) Mills ratio increasingly exceeds normal in the tails",
    x = "x",
    y = "% Difference from Normal"
  ) +
  theme_minimal() +
  annotate(
    "text", x = 4, y = 40,
    label = "At x=4: t(30) is 54% higher",
    color = "#e74c3c", fontface = "bold"
  )

ggplotly(p_paradox)

Hazard Function Connection

The dashboard explores the reciprocal relationship between Mills ratio and hazard function:

Show code

# Show both functions together
hazard_data <- tibble(x = seq(0.5, 5, by = 0.1)) %>%
  mutate(
    mills_normal = mills_ratio_normal(x),
    hazard_normal = 1 / mills_normal,
    mills_t30 = mills_ratio_t(x, df = 30),
    hazard_t30 = 1 / mills_t30
  ) %>%
  pivot_longer(
    -x,
    names_to = c("func_type", "distribution"),
    names_sep = "_",
    values_to = "value"
  )

p_hazard <- hazard_data %>%
  ggplot(aes(x, value, color = interaction(func_type, distribution))) +
  geom_line(size = 1) +
  scale_y_log10() +
  labs(
    title = "Mills Ratio m(x) and Hazard Function h(x) = 1/m(x)",
    x = "x",
    y = "Value (log scale)"
  ) +
  theme_minimal() +
  scale_color_manual(
    values = c(
      "mills.normal" = "#2c3e50",
      "hazard.normal" = "#e74c3c",
      "mills.t30" = "#3498db",
      "hazard.t30" = "#e67e22"
    ),
    labels = c(
      "mills.normal" = "Mills Normal",
      "hazard.normal" = "Hazard Normal",
      "mills.t30" = "Mills t(30)",
      "hazard.t30" = "Hazard t(30)"
    )
  ) +
  theme(legend.position = "bottom", legend.title = element_blank())

ggplotly(p_hazard)

Code Examples in Dashboard

The dashboard includes ready-to-copy tidyverse code:

Basic Mills Ratio Calculation

library(millsratio)
library(tidyverse)

# Calculate and compare Mills ratios
df <- tibble(x = seq(0.5, 5, by = 0.5)) %>%
  mutate(
    normal = mills_ratio_normal(x),
    t30 = mills_ratio_t(x, df = 30),
    exponential = mills_ratio_exp(x),
    t30_normal_ratio = t30 / normal
  )

Hazard Function Analysis

# Hazard function from Mills ratio
df_hazard <- df %>%
  mutate(
    hazard_normal = 1 / normal,
    hazard_t30 = 1 / t30,
    hazard_exp = 1 / exponential
  )

# Check if hazard is increasing (IFR property)
df_hazard %>%
  arrange(x) %>%
  mutate(
    normal_increasing = hazard_normal > lag(hazard_normal),
    t30_increasing = hazard_t30 > lag(hazard_t30),
    exp_constant = abs(hazard_exp - lag(hazard_exp)) < 1e-10
  )

Visualization Pipeline

# Create comparison plot
df %>%
  select(x, normal, t30, exponential) %>%
  pivot_longer(-x, names_to = "distribution", values_to = "mills_ratio") %>%
  ggplot(aes(x, mills_ratio, color = distribution)) +
  geom_line(linewidth = 1) +
  scale_y_log10() +
  labs(
    title = "Mills Ratio Comparison",
    y = "Mills Ratio m(x) (log scale)"
  ) +
  theme_minimal()

Dashboard Architecture

The dashboard is built with:

UI: bslib for Bootstrap 5 theming
Server: Reactive Shiny server with efficient caching
Plots: plotly for interactive visualizations
Tables: DT for sortable, searchable tables
Math: MathJax for rendering mathematical notation

File Structure

inst/shiny/
├── ui_simple.R         # Simplified UI (4 pages)
├── server_simple.R     # Simplified server logic
├── ui.R               # Full UI (12 pages)
├── server.R           # Full server logic
└── www/              # Static assets
    └── custom.css    # Custom styling

Extending the Dashboard

To add new features to the dashboard:

Add new distributions: Implement in R/mills_ratio.R
Add visualization: Update server.R reactive functions
Add theory content: Edit ui.R nav_panel sections
Add examples: Update the Code Playground section

Performance Considerations

The dashboards are optimized for performance:

Reactive caching prevents redundant calculations
Efficient vectorized operations throughout
Lazy evaluation for expensive computations
Debounced inputs to reduce update frequency

Troubleshooting

Common issues and solutions:

Dashboard won’t launch: Ensure all dependencies are installed
```
remotes::install_github("JohnGavin/millsratio")
```
Plots not rendering: Check that plotly is installed
```
install.packages("plotly")
```
Math notation not displaying: MathJax requires internet connection
Slow performance: Reduce the number of points in x-range

Next Steps

After exploring the dashboard: