Inference for Categorical Variables

background-image: url("img/DAW.png")
background-position: left
background-size: 50%
class: middle, center,

## .base-blue[Inference]

## .base-blue[for]

## .base-blue[Categorical Variables]

<br>

### .purple[Kelly McConville]

#### .purple[ Stat 100 | Week 14 | Fall 2022]

]

---

### One more reminder to RSVP to the `ggparty`:

### [bit.ly/stat100-ggparty](https://bit.ly/stat100-ggparty)

---

### Announcements

* Extra Credit Lecture Quiz due Dec 1st.
* P-Set 9 (the LAST p-set) is due Th, Dec 1st!
* Project Assignment 3 is due Tu, Dec 6th.
    + Also provide [feedback on work distribution](https://forms.gle/EJ1zgcC3c5rNrn9f7).
* Let's discuss the review sheet.

****************************

### Goals for Today

* Learn about Chi-Squared Random Variables.

* Consider inference for categorical variables with **more than** 2 categories!

]

* Quick wrap-up!

]

---

### Inference for Categorical Variables

Consider the situation where:

* Response variable: categorical

* Explanatory variable: categorical

* Parameter of interest: `$p_1 - p_2$`

This parameter of interest only makes sense if **both** variables only have two categories.

It is time to learn how to study the relationship between two categorical variables when **at least one has more than two categories.**

---

### Hypotheses

Consider the situation where:

* Response variable: categorical

* Explanatory variable: categorical

`$H_o$`: The two variables are independent.

`$H_a$`: The two variables are dependent.

---

### Example

Near-sightedness typically develops during the childhood years. Quinn, Shin, Maguire, and Stone (1999) explored whether there is a relationship between the type of light children were exposed to and their eye health based on questionnaires filled out by the children's parents at a university pediatric ophthalmology clinic.

```r
library(tidyverse)
library(infer)

# Import data
eye_data <- read_csv("~/shared_data/stat100/data/eye_lighting.csv")

# Contingency table
eye_data %>%
  count(Lighting, Eye)
```

```
## # A tibble: 9 × 3
##   Lighting Eye        n
##   <chr>    <chr>  <int>
## 1 dark     Far       40
## 2 dark     Near      18
## 3 dark     Normal   114
## 4 night    Far       39
## 5 night    Near      78
## 6 night    Normal   115
## 7 room     Far       12
## 8 room     Near      41
## 9 room     Normal    22
```

---

### Eyesight Example

Does there appear to be a relationship/dependence?

```r
ggplot(data = eye_data, 
       mapping = aes(x = Lighting,
                     fill = Eye)) + 
  geom_bar(position = "fill")
```

]

]

---

### Eyesight Example

Need a test statistic!

* Won't be a single sample statistic.

* Needs to measure the discrepancy between the observed sample and the sample we'd expect to see if `$H_o$` (no relationship) were true.

* Would be nice if its null distribution could be approximated by a known probability model.

---

#### Table of Observed Results

```r
table(eye_data$Eye, eye_data$Lighting) %>%
  addmargins() %>%
  kable(format = "html")
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> dark </th>
   <th style="text-align:right;"> night </th>
   <th style="text-align:right;"> room </th>
   <th style="text-align:right;"> Sum </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Far </td>
   <td style="text-align:right;"> 40 </td>
   <td style="text-align:right;"> 39 </td>
   <td style="text-align:right;"> 12 </td>
   <td style="text-align:right;"> 91 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Near </td>
   <td style="text-align:right;"> 18 </td>
   <td style="text-align:right;"> 78 </td>
   <td style="text-align:right;"> 41 </td>
   <td style="text-align:right;"> 137 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Normal </td>
   <td style="text-align:right;"> 114 </td>
   <td style="text-align:right;"> 115 </td>
   <td style="text-align:right;"> 22 </td>
   <td style="text-align:right;"> 251 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sum </td>
   <td style="text-align:right;"> 172 </td>
   <td style="text-align:right;"> 232 </td>
   <td style="text-align:right;"> 75 </td>
   <td style="text-align:right;"> 479 </td>
  </tr>
</tbody>
</table>

**Question**: If `$H_o$` were correct, is this the table that we'd expect to see?

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> Far </th>
   <th style="text-align:right;"> Near </th>
   <th style="text-align:right;"> Normal </th>
   <th style="text-align:right;"> Sum </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Far </td>
   <td style="text-align:right;"> 53 </td>
   <td style="text-align:right;"> 53 </td>
   <td style="text-align:right;"> 53 </td>
   <td style="text-align:right;"> 159 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Near </td>
   <td style="text-align:right;"> 53 </td>
   <td style="text-align:right;"> 53 </td>
   <td style="text-align:right;"> 53 </td>
   <td style="text-align:right;"> 159 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Normal </td>
   <td style="text-align:right;"> 53 </td>
   <td style="text-align:right;"> 53 </td>
   <td style="text-align:right;"> 53 </td>
   <td style="text-align:right;"> 159 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sum </td>
   <td style="text-align:right;"> 159 </td>
   <td style="text-align:right;"> 159 </td>
   <td style="text-align:right;"> 159 </td>
   <td style="text-align:right;"> 477 </td>
  </tr>
</tbody>
</table>

---

#### Table of Observed Results

**Question**: If `$H_o$` were correct, what table would we expect to see?

Want a `$H_o$` table that respects the eye condition proportions:

`$$\hat{p}_{far} = 91/479$$`

`$$\hat{p}_{nor} = 251/479$$`

`$$\hat{p}_{nea} = 137/479$$`

---

#### Table of Observed Results

**Question**: If `$H_o$` were correct, what table would we expect to see?

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> dark </th>
   <th style="text-align:right;"> night </th>
   <th style="text-align:right;"> room </th>
   <th style="text-align:right;"> Sum </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Far </td>
   <td style="text-align:right;"> (91/479)172 </td>
   <td style="text-align:right;"> (91/479)232 </td>
   <td style="text-align:right;"> (91/479)75 </td>
   <td style="text-align:right;"> 91 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Near </td>
   <td style="text-align:right;"> (137/479)172 </td>
   <td style="text-align:right;"> (137/479)232 </td>
   <td style="text-align:right;"> (137/479)75 </td>
   <td style="text-align:right;"> 137 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Normal </td>
   <td style="text-align:right;"> (251/479)172 </td>
   <td style="text-align:right;"> (251/479)232 </td>
   <td style="text-align:right;"> (251/479)75 </td>
   <td style="text-align:right;"> 251 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sum </td>
   <td style="text-align:right;"> 172 </td>
   <td style="text-align:right;"> 232 </td>
   <td style="text-align:right;"> 75 </td>
   <td style="text-align:right;"> 479 </td>
  </tr>
</tbody>
</table>

* Still have the same totals but distributed the values differently within the table

---

#### Table of Observed Results

**Question**: If `$H_o$` were correct, what table would we expect to see?

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> dark </th>
   <th style="text-align:right;"> night </th>
   <th style="text-align:right;"> room </th>
   <th style="text-align:right;"> Sum </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Far </td>
   <td style="text-align:right;"> 32.67641 </td>
   <td style="text-align:right;"> 44.07516 </td>
   <td style="text-align:right;"> 14.24843 </td>
   <td style="text-align:right;"> 91 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Near </td>
   <td style="text-align:right;"> 49.19415 </td>
   <td style="text-align:right;"> 66.35491 </td>
   <td style="text-align:right;"> 21.45094 </td>
   <td style="text-align:right;"> 137 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Normal </td>
   <td style="text-align:right;"> 90.12944 </td>
   <td style="text-align:right;"> 121.56994 </td>
   <td style="text-align:right;"> 39.30063 </td>
   <td style="text-align:right;"> 251 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sum </td>
   <td style="text-align:right;"> 172.00000 </td>
   <td style="text-align:right;"> 232.00000 </td>
   <td style="text-align:right;"> 75.00000 </td>
   <td style="text-align:right;"> 479 </td>
  </tr>
</tbody>
</table>

---

### Expected Table

* How does this table represent `$H_o$`?

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> dark </th>
   <th style="text-align:right;"> night </th>
   <th style="text-align:right;"> room </th>
   <th style="text-align:right;"> Sum </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Far </td>
   <td style="text-align:right;"> 32.67641 </td>
   <td style="text-align:right;"> 44.07516 </td>
   <td style="text-align:right;"> 14.24843 </td>
   <td style="text-align:right;"> 91 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Near </td>
   <td style="text-align:right;"> 49.19415 </td>
   <td style="text-align:right;"> 66.35491 </td>
   <td style="text-align:right;"> 21.45094 </td>
   <td style="text-align:right;"> 137 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Normal </td>
   <td style="text-align:right;"> 90.12944 </td>
   <td style="text-align:right;"> 121.56994 </td>
   <td style="text-align:right;"> 39.30063 </td>
   <td style="text-align:right;"> 251 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sum </td>
   <td style="text-align:right;"> 172.00000 </td>
   <td style="text-align:right;"> 232.00000 </td>
   <td style="text-align:right;"> 75.00000 </td>
   <td style="text-align:right;"> 479 </td>
  </tr>
</tbody>
</table>

]

]

---

### Test Statistic

Want the test statistic to quantify the difference between the observed table and the expected table.

<table style="display: inline-block;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> dark </th>
   <th style="text-align:right;"> night </th>
   <th style="text-align:right;"> room </th>
   <th style="text-align:right;"> Sum </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Far </td>
   <td style="text-align:right;"> 40 </td>
   <td style="text-align:right;"> 39 </td>
   <td style="text-align:right;"> 12 </td>
   <td style="text-align:right;"> 91 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Near </td>
   <td style="text-align:right;"> 18 </td>
   <td style="text-align:right;"> 78 </td>
   <td style="text-align:right;"> 41 </td>
   <td style="text-align:right;"> 137 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Normal </td>
   <td style="text-align:right;"> 114 </td>
   <td style="text-align:right;"> 115 </td>
   <td style="text-align:right;"> 22 </td>
   <td style="text-align:right;"> 251 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sum </td>
   <td style="text-align:right;"> 172 </td>
   <td style="text-align:right;"> 232 </td>
   <td style="text-align:right;"> 75 </td>
   <td style="text-align:right;"> 479 </td>
  </tr>
</tbody>
</table>

<table style="display: inline-block;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> dark </th>
   <th style="text-align:right;"> night </th>
   <th style="text-align:right;"> room </th>
   <th style="text-align:right;"> Sum </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Far </td>
   <td style="text-align:right;"> 32.68 </td>
   <td style="text-align:right;"> 44.08 </td>
   <td style="text-align:right;"> 14.25 </td>
   <td style="text-align:right;"> 91 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Near </td>
   <td style="text-align:right;"> 49.19 </td>
   <td style="text-align:right;"> 66.35 </td>
   <td style="text-align:right;"> 21.45 </td>
   <td style="text-align:right;"> 137 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Normal </td>
   <td style="text-align:right;"> 90.13 </td>
   <td style="text-align:right;"> 121.57 </td>
   <td style="text-align:right;"> 39.30 </td>
   <td style="text-align:right;"> 251 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sum </td>
   <td style="text-align:right;"> 172.00 </td>
   <td style="text-align:right;"> 232.00 </td>
   <td style="text-align:right;"> 75.00 </td>
   <td style="text-align:right;"> 479 </td>
  </tr>
</tbody>
</table>

For each cell: Compute a **Z-score**!

`\begin{align*}
\mbox{Z-score} &= \frac{\mbox{stat - mean}}{\mbox{SE}} \\
& = \frac{\mbox{observed - expected}}{\sqrt{\mbox{expected}}}
\end{align*}`

---

### Test Statistic

Want the test statistic to quantify the difference between the observed table and the expected table.

<table style="display: inline-block;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> dark </th>
   <th style="text-align:right;"> night </th>
   <th style="text-align:right;"> room </th>
   <th style="text-align:right;"> Sum </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Far </td>
   <td style="text-align:right;"> 32.68 </td>
   <td style="text-align:right;"> 44.08 </td>
   <td style="text-align:right;"> 14.25 </td>
   <td style="text-align:right;"> 91 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Near </td>
   <td style="text-align:right;"> 49.19 </td>
   <td style="text-align:right;"> 66.35 </td>
   <td style="text-align:right;"> 21.45 </td>
   <td style="text-align:right;"> 137 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Normal </td>
   <td style="text-align:right;"> 90.13 </td>
   <td style="text-align:right;"> 121.57 </td>
   <td style="text-align:right;"> 39.30 </td>
   <td style="text-align:right;"> 251 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sum </td>
   <td style="text-align:right;"> 172.00 </td>
   <td style="text-align:right;"> 232.00 </td>
   <td style="text-align:right;"> 75.00 </td>
   <td style="text-align:right;"> 479 </td>
  </tr>
</tbody>
</table>

**Test Statistic:**

`\begin{align*}
\chi^2 = \sum \left(\frac{\mbox{observed - expected}}{\sqrt{\mbox{expected}}} \right)^2
\end{align*}`

&rarr; Large test statistics signify that results are unusual if `$H_o$` is true.

---

<table style="display: inline-block;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> dark </th>
   <th style="text-align:right;"> night </th>
   <th style="text-align:right;"> room </th>
   <th style="text-align:right;"> Sum </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Far </td>
   <td style="text-align:right;"> 32.68 </td>
   <td style="text-align:right;"> 44.08 </td>
   <td style="text-align:right;"> 14.25 </td>
   <td style="text-align:right;"> 91 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Near </td>
   <td style="text-align:right;"> 49.19 </td>
   <td style="text-align:right;"> 66.35 </td>
   <td style="text-align:right;"> 21.45 </td>
   <td style="text-align:right;"> 137 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Normal </td>
   <td style="text-align:right;"> 90.13 </td>
   <td style="text-align:right;"> 121.57 </td>
   <td style="text-align:right;"> 39.30 </td>
   <td style="text-align:right;"> 251 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sum </td>
   <td style="text-align:right;"> 172.00 </td>
   <td style="text-align:right;"> 232.00 </td>
   <td style="text-align:right;"> 75.00 </td>
   <td style="text-align:right;"> 479 </td>
  </tr>
</tbody>
</table>

```r
library(infer)
#Compute Chi-square test stat
test_stat <- eye_data %>%
  specify(Eye ~ Lighting) %>%
  calculate(stat = "Chisq") 
test_stat
```

```
## Response: Eye (factor)
## Explanatory: Lighting (factor)
## # A tibble: 1 × 1
##    stat
##   <dbl>
## 1  56.5
```

Is 56.5 **large**?  Is 56.5 **unusual** under `$H_o$`?

---

### Generating the Null Distribution

#### Steps:

1. Shuffle lighting.
2. Compute the new observed table.  
3. Compute the test statistic.
4. Repeat 1 - 3 many times.

```
## # A tibble: 10 × 2
##    Eye    Lighting
##    <chr>  <chr>   
##  1 Far    night   
##  2 Far    dark    
##  3 Near   room    
##  4 Far    dark    
##  5 Normal room    
##  6 Near   room    
##  7 Normal night   
##  8 Normal dark    
##  9 Far    night   
## 10 Near   room
```

---

### Generating the Null Distribution

```r
# Construct null distribution
null_dist <- eye_data %>%
  specify(Eye ~ Lighting) %>%
  hypothesize(null = "independence") %>%
  generate(reps = 1000, type = "permute") %>%
  calculate(stat = "Chisq")

visualize(null_dist)
```

]

]

---

### The Null Distribution

**Key Observations about the distribution**:

* Smallest possible value?

* Shape?

]

]

---

### The Null Distribution

**Key Observations about the distribution**:

* Smallest possible value?

* Shape?

* Is our observed test statistic of 56.5 unusual?

]

]

---

### The P-value

```r
# Compute p-value
null_dist %>%
  get_pvalue(obs_stat = test_stat, direction = "greater")
```

```
## # A tibble: 1 × 1
##   p_value
##     <dbl>
## 1       0
```

---

### Approximating the Null Distribution

If there are at least 5 observations in each cell, then

$$
\mbox{test statistic} \sim \chi^2(df = (\mbox{# of rows} - 1)(\mbox{# of columns} - 1))
$$

The `$df$` controls the center and spread of the distribution.

]

]

---

### The Chi-Squared Test

```r
chisq.test(table(eye_data$Eye, eye_data$Lighting))
```

```
## 
## 	Pearson's Chi-squared test
## 
## data:  table(eye_data$Eye, eye_data$Lighting)
## X-squared = 56.513, df = 4, p-value = 1.565e-11
```

Conclusions?

* Causation?

--
* Decisions, decisions...

---

background-image: url("img/DAW.png")
background-position: left
background-size: 50%
class: middle

## Before talking about what else there is to study...

]

## Let's acknowledge that we actually learned a lot this semester!

]

---

background-image: url("img/DAW.png")
background-position: left
background-size: 50%
class: middle

####  (Some of the) Course Learning Objectives

]

* Learn how to **analyze** data in `R`.
 
]

* Master **creating** graphs with `ggplot2`.

]

* Apply data wrangling operations with `dplyr`.
]

* Translate a research problem into a set of relevant questions that can be answered with data.

]

* Reflect on how **sample design structures** impact potential conclusions.

]

* Appropriately apply and draw inferences from a statistical model, including **quantifying and interpreting the uncertainty** in model estimates.

]

* Develop a **reproducible** workflow using `R` Markdown documents.

]

---

## What if I want to learn more stats and data science?

#### For more modeling: Stat 139: Linear Models

#### For more theory and inference: Stat 110: Introduction to Probability Theory and Stat 111:  	Introduction to Statistical Inference

#### For more coding: Stat 108: Introduction to Statistical Computing in R

---

# Thanks for a wonderful semester!