POL90: Statistics

class: center, middle, inverse, title-slide

# POL90: Statistics
## Multiple Comparisons with ANOVA
### Prof Wasow <br/> Assistant Professor, Politics <br/> Pomona College
### 2022-03-15

---

# Announcements

.large[
* Assignments

+ PS05 due <mark>Friday</mark>
    
    + Report 1

]

.large[
* Statistical Sleuth

+ Read Chapter 5
]

---
# Schedule

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> Week </th>
   <th style="text-align:left;"> Date </th>
   <th style="text-align:left;"> Day </th>
   <th style="text-align:left;"> Title </th>
   <th style="text-align:right;"> Chapter </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:left;"> Feb 9 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Confidence Intervals </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> Feb 14 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> A Closer Look at Assumptions </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> Feb 16 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> A Closer Look at Assumptions </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:left;"> Feb 21 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Alternatives to the t-Tools </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 6 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Feb 23 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Wed </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Comparison Among Several Samples </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 5 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> Feb 28 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Comparison Among Several Samples </td>
   <td style="text-align:right;"> 5 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> Mar 2 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Simple Linear Regression </td>
   <td style="text-align:right;"> 7 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:left;"> Mar 7 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Simple Linear Regression </td>
   <td style="text-align:right;"> 7 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:left;"> Mar 9 </td>
   <td style="text-align:left;"> Wed </td>
   <td style="text-align:left;"> Regression by Calculation </td>
   <td style="text-align:right;"> 7 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> Mar 14 </td>
   <td style="text-align:left;"> Mon </td>
   <td style="text-align:left;"> Spring Recess </td>
   <td style="text-align:right;"> - </td>
  </tr>
</tbody>
</table>

---
## Assignment schedule

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> Week </th>
   <th style="text-align:left;"> Date </th>
   <th style="text-align:left;"> Day </th>
   <th style="text-align:left;"> Assignment </th>
   <th style="text-align:right;"> Percent </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> Feb 18 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS04 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 6 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Feb 25 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Fri </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> PS05 </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 7 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Mar 4 </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Fri </td>
   <td style="text-align:left;color: black !important;background-color: yellow !important;"> Report1 </td>
   <td style="text-align:right;color: black !important;background-color: yellow !important;"> 6 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:left;"> Mar 11 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS06 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> Mar 18 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> Spring break </td>
   <td style="text-align:right;"> NA </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:left;"> Mar 25 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS07 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:left;"> Apr 1 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS08 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 12 </td>
   <td style="text-align:left;"> Apr 8 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> Report2 </td>
   <td style="text-align:right;"> 8 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 13 </td>
   <td style="text-align:left;"> Apr 15 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS09 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 14 </td>
   <td style="text-align:left;"> Apr 22 </td>
   <td style="text-align:left;"> Fri </td>
   <td style="text-align:left;"> PS10 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
</tbody>
</table>

---
## R Markdown & Knitr tips

* Useful chunk headers like:

- `echo = TRUE          # echo code in output`
  - `include = FALSE      # suppress all output`
  - `results = 'asis'     # output for HTML & Latex tables`
  - `results = 'hide'     # hide output `
  - `fig.height = 3       # set figure height to 3 inches`
  - `fig.width  = 4       # set figure width to 4 inches`
  - `fig.align = 'center' # center figure`
  - `out.width = '50%'    # shrink image 50%`

* Example: 
  - `{r some_chunk, echo = FALSE}`

* Chatty packages

- `suppressMessages(library(dplyr)) # suppress messages`

---
## Chunk tips
.large[
- When to create new chunks?

- Good to separate different parts of analysis 
  - SCRUBBING: load and clean data chunks
  - ANALYZING: run analyses chunks (e.g., statistical tests)
  - PRESENTING: display results chunks (e.g., tables and plots)
  
- Put tables and plots in separate chunks
  - `results = 'asis' # for tables`
  - `fig.height = 3 # for figures`
]

---
## Why `results = 'asis'` in Latex?

.large[
- What you write:
]

```r
t.test(mpg ~ am, data = mtcars) %>% 
  broom::tidy() %>% 
  select(-method, -alternative) %>% 
  gt() %>% 
  fmt_number(columns = everything(), decimals = 2) 
```

.large[
- What you see in your pdf:
]

---
## Why `results = 'asis'` in Latex?

.large[
- What you write:
]

```r
t.test(mpg ~ am, data = mtcars) %>% 
  broom::tidy() %>% 
  select(-method, -alternative) %>% 
  gt() %>% 
  fmt_number(columns = everything(), decimals = 2)
  as_latex()
```

.large[
- What actually creates a Latex table in the pdf:
]

```
[[1]]
[1] "\\captionsetup[table]{labelformat=empty,skip=1pt}"                                              
[2] "\\begin{longtable}{rrrrrrrr}"                                                                   
[3] "\\toprule"                                                                                      
[4] "estimate & estimate1 & estimate2 & statistic & p.value & parameter & conf.low & conf.high \\\\ "
[5] "\\midrule"                                                                                      
[6] "$-7.24$ & $17.15$ & $24.39$ & $-3.77$ & $0.00$ & $18.33$ & $-11.28$ & $-3.21$ \\\\ "            
[7] " \\bottomrule"                                                                                  
[8] "\\end{longtable}"                                                                               
[9] ""                                                                                               
```

---
## Why `results = 'asis'` in HTML?

.large[
- What you write:
]

```r
t.test(mpg ~ am, data = mtcars) %>% 
  broom::tidy() %>% 
  gt() %>% 
  fmt_number(columns = 1:8, decimals = 2)
```

<div id="nspzxzkxbv" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
<style>html {
  font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif;
}

#nspzxzkxbv .gt_table {
  display: table;
  border-collapse: collapse;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  font-weight: normal;
  font-style: normal;
  background-color: #FFFFFF;
  width: auto;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #A8A8A8;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
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}

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}

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  font-size: 125%;
  font-weight: initial;
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}

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}

#nspzxzkxbv .gt_col_heading {
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  border-left-color: #D3D3D3;
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}

#nspzxzkxbv .gt_column_spanner_outer {
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  padding-left: 4px;
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}

#nspzxzkxbv .gt_column_spanner_outer:first-child {
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}

#nspzxzkxbv .gt_column_spanner_outer:last-child {
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}

#nspzxzkxbv .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: bottom;
  padding-top: 5px;
  padding-bottom: 5px;
  overflow-x: hidden;
  display: inline-block;
  width: 100%;
}

#nspzxzkxbv .gt_group_heading {
  padding: 8px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
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  border-top-width: 2px;
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  border-bottom-width: 2px;
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  border-left-style: none;
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  border-left-color: #D3D3D3;
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  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
}

#nspzxzkxbv .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  vertical-align: middle;
}

#nspzxzkxbv .gt_from_md > :first-child {
  margin-top: 0;
}

#nspzxzkxbv .gt_from_md > :last-child {
  margin-bottom: 0;
}

#nspzxzkxbv .gt_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 1px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 1px;
  border-right-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#nspzxzkxbv .gt_stub {
  color: #333333;
  background-color: #FFFFFF;
  font-size: 100%;
  font-weight: initial;
  text-transform: inherit;
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

#nspzxzkxbv .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#nspzxzkxbv .gt_first_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#nspzxzkxbv .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  text-transform: inherit;
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
}

#nspzxzkxbv .gt_first_grand_summary_row {
  padding-top: 8px;
  padding-bottom: 8px;
  padding-left: 5px;
  padding-right: 5px;
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#nspzxzkxbv .gt_striped {
  background-color: rgba(128, 128, 128, 0.05);
}

#nspzxzkxbv .gt_table_body {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#nspzxzkxbv .gt_footnotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#nspzxzkxbv .gt_footnote {
  margin: 0px;
  font-size: 90%;
  padding: 4px;
}

#nspzxzkxbv .gt_sourcenotes {
  color: #333333;
  background-color: #FFFFFF;
  border-bottom-style: none;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  border-left-style: none;
  border-left-width: 2px;
  border-left-color: #D3D3D3;
  border-right-style: none;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
}

#nspzxzkxbv .gt_sourcenote {
  font-size: 90%;
  padding: 4px;
}

#nspzxzkxbv .gt_left {
  text-align: left;
}

#nspzxzkxbv .gt_center {
  text-align: center;
}

#nspzxzkxbv .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

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  font-weight: normal;
}

#nspzxzkxbv .gt_font_bold {
  font-weight: bold;
}

#nspzxzkxbv .gt_font_italic {
  font-style: italic;
}

#nspzxzkxbv .gt_super {
  font-size: 65%;
}

#nspzxzkxbv .gt_footnote_marks {
  font-style: italic;
  font-weight: normal;
  font-size: 65%;
}
</style>
<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">estimate</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">estimate1</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">estimate2</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">statistic</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">p.value</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">parameter</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">conf.low</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">conf.high</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">method</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">alternative</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">−7.24</td>
<td class="gt_row gt_right">17.15</td>
<td class="gt_row gt_right">24.39</td>
<td class="gt_row gt_right">−3.77</td>
<td class="gt_row gt_right">0.00</td>
<td class="gt_row gt_right">18.33</td>
<td class="gt_row gt_right">−11.28</td>
<td class="gt_row gt_right">−3.21</td>
<td class="gt_row gt_left">Welch Two Sample t-test</td>
<td class="gt_row gt_left">two.sided</td></tr>
  </tbody>
  
  
</table>
</div>

---
## Why `results = 'asis'` in HTML?

.large[
- What you write:
]

```r
t.test(mpg ~ am, data = mtcars) %>% 
  broom::tidy() %>% 
  gt() %>% 
  fmt_number(columns = 1:8, decimals = 2) 
```
.large[
- What actually creates an HTML table:
]

```
[[1]]
 [1] "<table style=\"font-family: -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen, Ubuntu, Cantarell, 'Helvetica Neue', 'Fira Sans', 'Droid Sans', Arial, sans-serif; display: table; border-collapse: collapse; margin-left: auto; margin-right: auto; color: #333333; font-size: 16px; font-weight: normal; font-style: normal; background-color: #FFFFFF; width: auto; border-top-style: solid; border-top-width: 2px; border-top-color: #A8A8A8; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #A8A8A8; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3;\">"
 [2] "  "                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
 [3] "  <thead style=\"border-top-style: solid; border-top-width: 2px; border-top-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3;\">"                                                                                                                                                                                                                                                                                                                                                                 
 [4] "    <tr>"                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
 [5] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\" rowspan=\"1\" colspan=\"1\">estimate</th>"                                                                                                                                                                                           
 [6] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\" rowspan=\"1\" colspan=\"1\">estimate1</th>"                                                                                                                                                                                          
 [7] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\" rowspan=\"1\" colspan=\"1\">estimate2</th>"                                                                                                                                                                                          
 [8] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\" rowspan=\"1\" colspan=\"1\">statistic</th>"                                                                                                                                                                                          
 [9] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\" rowspan=\"1\" colspan=\"1\">p.value</th>"                                                                                                                                                                                            
[10] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\" rowspan=\"1\" colspan=\"1\">parameter</th>"                                                                                                                                                                                          
[11] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\" rowspan=\"1\" colspan=\"1\">conf.low</th>"                                                                                                                                                                                           
[12] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\" rowspan=\"1\" colspan=\"1\">conf.high</th>"                                                                                                                                                                                          
[13] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: left;\" rowspan=\"1\" colspan=\"1\">method</th>"                                                                                                                                                                                                                                  
[14] "      <th style=\"color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; text-align: left;\" rowspan=\"1\" colspan=\"1\">alternative</th>"                                                                                                                                                                                                                             
[15] "    </tr>"                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       
[16] "  </thead>"                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
[17] "  <tbody style=\"border-top-style: solid; border-top-width: 2px; border-top-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3;\">"                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
[18] "    <tr><td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\">-7.244939</td>"                                                                                                                                                                                                                                     
[19] "<td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\">17.14737</td>"                                                                                                                                                                                                                                              
[20] "<td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\">24.39231</td>"                                                                                                                                                                                                                                              
[21] "<td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\">-3.767123</td>"                                                                                                                                                                                                                                             
[22] "<td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\">0.001373638</td>"                                                                                                                                                                                                                                           
[23] "<td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\">18.33225</td>"                                                                                                                                                                                                                                              
[24] "<td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\">-11.28019</td>"                                                                                                                                                                                                                                             
[25] "<td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: right; font-variant-numeric: tabular-nums;\">-3.209684</td>"                                                                                                                                                                                                                                             
[26] "<td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: left;\">Welch Two Sample t-test</td>"                                                                                                                                                                                                                                                                    
[27] "<td style=\"padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; text-align: left;\">two.sided</td></tr>"                                                                                                                                                                                                                                                                             
[28] "  </tbody>"                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      
[29] "  "                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
[30] "  "                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              
[31] "</table>"                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        
```

---

## What is `group_by` again? Split-Apply-Combine

---
class: center, middle, inverse

# Chapter 5: Multiple Comparisons with ANOVA

---

## Caloric restriction and longevity
<br>

.large[
* Question: 
  - Does reducing caloric intake increase longevity?

* Data: 
  - Female mice were randomly assigned to six diet groups and their lifetimes were measured

* Why do we care?
]

---