Categorical predictors: Comparing means

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# Categorical predictors: Comparing means
## Professor Andy Field

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  <span style="line-height:40px;">@profandyfield</span>
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  <img style="vertical-align:middle; width:60px" src="media/youtube.png">
  <span style="line-height:40px;">www.youtube.com/user/ProfAndyField/</span>
</div>
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  <img style="vertical-align:middle; width:30px; height:30px" src="media/ds_com_fav.png">
  <span style="line-height:40px;">www.discoveringstatistics.com</span>
</div>
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  <img style="vertical-align:middle; width:30px; height:30px" src="media/milton_grey_fav.png">
  <span style="line-height:40px;">www.milton-the-cat.rocks</span>
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  <img style="vertical-align:middle; width:30px; height:30px" src="media/discovr_fav.png">
  <span style="line-height:40px;">www.discovr.rocks</span>
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They don’t like Halloween because it’s the one night that they can’t scare humans. On Halloween everyone thinks that they are a small child in fancy dress.

So they gather underground and bemoan how scary small children are these days. Invariably they quarrel about who is the scariest.

The zombies argue that they are scariest because they look rotten, but the Werewolves believe that they are scariest because they have big teeth.

The zombies think the Werewolves look too cuddly.

Each year Azal, head of the Werewolves becomes so enraged that he eats all of the zombies to win the argument. They taste rotten too.

This year, the zombies are fighting back …]

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C: .... you look awful
AF: Well, it's been a long term etc.

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???

C: Is it true that you are the graetest statistician in the land?
AF: Well, on a good day I'm barely adequate. But I am cheap, and I am available.

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C: .... can you do it?
AF: yes, sure, leave it with me ... byE!

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C: Thank you

---
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![:scale 83%](media/spine_content_map.png)
???

We've seen this map of the process of fitting models before

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![:scale 83%](media/spine_lec_04.png)

???

Today we focus back on the model itself to look at the form of the model we're fitting. The faded stuff still applies though - we'll look at bias, robust models, and of course samples and estimation and so on. But they are the same as for other models, what is different is the form of the model we're fitting.

---

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{volume}_{i} + e_i
\end{aligned}`
$$
]

???

We've seen a few times already that when we enter categorical predictors into the GLM, their parameters represent differences between means. For example, we looked at how attending a concert affected how long people experienced ringing ears.
---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{musician}_{i} + e_i
\end{aligned}`
$$
]

![](categorical_predictors_files/figure-html/unnamed-chunk-46-1.png)

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{musician}_{i} + e_i
\end{aligned}`
$$
]

![](categorical_predictors_files/figure-html/unnamed-chunk-47-1.png)

---
class: center

.ong_dk[
$$
`\begin{aligned}
\text{ringing}_i &= \hat{b}_0 + \hat{b}_1\text{musician}_{i} + e_i
\end{aligned}`
$$
]

![](categorical_predictors_files/figure-html/unnamed-chunk-48-1.png)

???
b1 represents the differences between means
---

#  The GLM and experiments

* In experimental research, predictors in the linear model are defined by a manipulation
  + By manipulating a predictor variable can we cause (and therefore predict) a change in behaviour?
  
--
  
* The *F*-statistic
  + Still quantifies the fit of the model to the data
  + Still has an associated significance test
  + The 'fit' represents the experimental manipulation (which defines the predictor)
  + 'Significant' fit equates to a 'significant' effect of the experimental manipulation

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## A ghoulish example

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---

## Same ghoul

![](categorical_predictors_files/figure-html/unnamed-chunk-5-1.png)

]

---

## Same ghoul

![](categorical_predictors_files/figure-html/unnamed-chunk-6-1.png)

]

---

## Different ghouls

![](categorical_predictors_files/figure-html/unnamed-chunk-7-1.png)

]
---

<div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-84b887cab6bb147a8b38" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-84b887cab6bb147a8b38">{"x":{"filter":"none","vertical":false,"caption":"<caption>Table 1: Data for the zombie vs. werewolf costume experiment<\/caption>","data":[["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99","100"],["Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Zombie","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf","Werewolf"],[6,8,5,6,8,8,7,10,7,3,6,8,8,7,2,5,8,10,7,9,4,8,9,6,9,3,3,6,8,6,8,8,7,3,8,9,7,9,6,4,7,7,10,6,7,9,7,9,10,9,4,3,7,5,3,6,3,9,7,5,3,4,4,5,4,4,3,2,7,1,2,0,4,5,1,7,6,6,4,3,3,7,6,1,3,3,4,7,4,6,3,10,5,2,5,2,7,6,9,5],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th>ID<\/th>\n      <th>Costume<\/th>\n      <th>Fear<\/th>\n      <th>Numeric code<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"tp","columnDefs":[{"className":"dt-center","targets":[1,2,3]}],"pageLength":10,"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

---

.ong_dk[
$$
`\begin{aligned}
Y_i &= \hat{b}_0 + \hat{b}_1X_{i} + e_i \\
\text{Fear}_i &= \hat{b}_0 + \hat{b}_1\text{Costume}_{i} + e_i 
\end{aligned}`
$$
]]

.pull-right[
![](categorical_predictors_files/figure-html/unnamed-chunk-9-1.png)
]
---
.pull-left[
# The model
.ong_dk[
$$
`\begin{aligned}
Y_i &= \hat{b}_0 + \hat{b}_1X_{i} + e_i \\
\text{Fear}_i &= \hat{b}_0 + \hat{b}_1\text{Costume}_{i} + e_i 
\end{aligned}`
$$
]
]

![](categorical_predictors_files/figure-html/unnamed-chunk-10-1.png)
]

???

The intercept represents the mean of the control group

---
.pull-left[
# The model
.ong_dk[
$$
`\begin{aligned}
Y_i &= \hat{b}_0 + \hat{b}_1X_{i} + e_i \\
\text{Fear}_i &= \hat{b}_0 + \hat{b}_1\text{Costume}_{i} + e_i \\
\hat{\text{Fear}}_i &= \hat{b}_0 + \hat{b}_1\text{Costume}_{i} \\
\end{aligned}`
$$
]
]

![](categorical_predictors_files/figure-html/unnamed-chunk-11-1.png)
]

???

b_1 represents the difference between group means
---
# Dummy coding: *b*<sub>0</sub>

* Dummy Coding
  + Zombie = 0,
  Werewolf = 1

* When costume = zombie
  + Costume = 0
  + Predicted fear = mean of zombie group:

.center[
.eq_lrge[
.ong[
$$
`\begin{aligned}
\hat{\text{fear}}_i &= \hat{b}_0 + \hat{b}_1\text{Costume}_i \\
\bar{X}_\text{zombie} &= \hat{b}_0 + \hat{b}_1\times0 \\
\bar{X}_\text{zombie} &= \hat{b}_0
\end{aligned}`
$$
]
]
]

---
# Dummy coding: *b*<sub>1</sub>

* When costume = werewolf
  + Costume = 1
  + Predicted fear = mean of werewolf group:

.center[
.eq_lrge[
.ong[
$$
`\begin{aligned}
\hat{\text{fear}}_i &= \hat{b}_0 + \hat{b}_1\text{Costume}_i \\
\bar{X}_\text{werewolf} &= \hat{b}_0 + \hat{b}_1\times1 \\
\bar{X}_\text{werewolf} &= \hat{b}_0 + \hat{b}_1
\end{aligned}`
$$
]
]
]

.center[
.eq_lrge[
.ong[
$$
`\begin{aligned}
\bar{X}_\text{werewolf} &= \bar{X}_\text{zombie} + \hat{b}_1 \\
\hat{b}_1 &= \bar{X}_\text{werewolf} - \bar{X}_\text{zombie}
\end{aligned}`
$$
]
]
]

---
# The linear model

* We can fit a linear model with fear as the outcome and the type of costume (zombie or werewolf) as the predictor, note:
  + Intercept (*b*<sub>0</sub>) is the mean of ‘zero coded’ group
  + *b* for the dummy variable is the difference between the means of the two costume groups (4.5 − 7 = −2.5)

```r
zombie_lm <- lm(Fear ~ Costume, zombie_tib)
broom::tidy(zombie_lm, conf.int = TRUE)
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
   <th style="text-align:right;"> conf.low </th>
   <th style="text-align:right;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 7.0 </td>
   <td style="text-align:right;"> 0.299 </td>
   <td style="text-align:right;"> 23.400 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 6.406 </td>
   <td style="text-align:right;"> 7.594 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> CostumeWerewolf </td>
   <td style="text-align:right;"> -2.5 </td>
   <td style="text-align:right;"> 0.423 </td>
   <td style="text-align:right;"> -5.909 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> -3.340 </td>
   <td style="text-align:right;"> -1.660 </td>
  </tr>
</tbody>
</table>

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    <source src="media/cubit_facetime_live_lecture_05.mp4" type="video/mp4">
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C: Hello, it is I, Cubit Zenom, King of the zombies. 
AF: Hi.

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C: I took your results to Azal and he said you have done a foolish thing.
AF: Really?

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C: I need you to take another look.
AF: Ok, leave it with me.

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</video>
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C: Bye!
---
background-image: url("media/halloween_pumpkin_house.jpg")
background-size: cover
class: inverse

# Another ghoulish example

* Which is more scary?
  + Human
  + Zombie
  + Werewolf

* Design
  + Prank

* Outcome
  + Fear

![:abs 13%, 75%, 5%](media/oval_human.png)
![:abs 13%, 75%, 37%](media/oval_zombie.png)
![:abs 13%, 75%, 67%](media/oval_milton.png)
---
# The data

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<td headers="grand_summary_stub_1 Zombie" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row" style="background-color: rgba(202,62,52,0.8); color: #E8EAE5; text-align: right; font-weight: bold;">7.20</td>
<td headers="grand_summary_stub_1 Werewolf" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row" style="background-color: rgba(202,62,52,0.8); color: #E8EAE5; text-align: right; font-weight: bold;">3.80</td></tr>
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<td headers="grand_summary_stub_2 Zombie" class="gt_row gt_center gt_grand_summary_row">3.70</td>
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<td headers="grand_summary_stub_3 Zombie" class="gt_row gt_center gt_grand_summary_row gt_last_summary_row">1.92</td>
<td headers="grand_summary_stub_3 Werewolf" class="gt_row gt_center gt_grand_summary_row gt_last_summary_row">3.19</td></tr>
  </tbody>
  
  
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</div>
<br>
.center[
.eq_lrge[
.ong[
`$\text{Overall mean (} \bar{X}_\text{grand}\text{)} = 4.47$`
]
]
]

---
class: center

# The model

![](categorical_predictors_files/figure-html/unnamed-chunk-17-1.png)

---
class: center

# The model

![](categorical_predictors_files/figure-html/unnamed-chunk-18-1.png)

---
# Testing the model

## Overall fit (*F*-statistic)

* The ratio of how well the model fits to how much error it has
* In the case of experiments:
  + The model = differences between means
  + *F* is the ratio of the experimental effect to the background ‘error’
  + Tests whether group means differ **overall**
  
## Parameter estimates (and *t*-tests)

* Break down the overall fit
* Tell us, specifically, which means differ

---
# Total sum of squared error, SS<sub>T</sub>

---
# Total sum of squared error, SS<sub>T</sub>

]
---
# Residual sum of squared error, SS<sub>R</sub>

---
# Residual sum of squared error, SS<sub>R</sub>

]
---
# Model sum of squared error, SS<sub>M</sub>

---
# Model sum of squared error, SS<sub>M</sub>

---
# Model sum of squared error, SS<sub>M</sub>

]

---
# Dummy coding multiple categories

* You can code any categorical predictor into a series of dummy variables
  + Dummy variables must be entered in the same block
  + Choose a baseline category  - it is always coded as 0
    - (By default <svg style="height: 1em; top:.04em; position: relative; fill: #136CB9;" viewBox="0 0 581 512"><path d="M581 226.6C581 119.1 450.9 32 290.5 32S0 119.1 0 226.6C0 322.4 103.3 402 239.4 418.1V480h99.1v-61.5c24.3-2.7 47.6-7.4 69.4-13.9L448 480h112l-67.4-113.7c54.5-35.4 88.4-84.9 88.4-139.7zm-466.8 14.5c0-73.5 98.9-133 220.8-133s211.9 40.7 211.9 133c0 50.1-26.5 85-70.3 106.4-2.4-1.6-4.7-2.9-6.4-3.7-10.2-5.2-27.8-10.5-27.8-10.5s86.6-6.4 86.6-92.7-90.6-87.9-90.6-87.9h-199V361c-74.1-21.5-125.2-67.1-125.2-119.9zm225.1 38.3v-55.6c57.8 0 87.8-6.8 87.8 27.3 0 36.5-38.2 28.3-87.8 28.3zm-.9 72.5H365c10.8 0 18.9 11.7 24 19.2-16.1 1.9-33 2.8-50.6 2.9v-22.1z"/></svg> chooses the first level of the factor)
  + The *b* for each dummy variable will be the difference in means between each category and the baseline

<br>
--

.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Entity </th>
   <th style="text-align:center;"> Dummy 1 (Zombie vs. Human) </th>
   <th style="text-align:center;"> Dummy 2 (Werewolf vs. Human) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Human </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Zombie </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Werewolf </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
  </tr>
</tbody>
</table>

]

.center[.eq_lrge[.ong[
$$
`\begin{aligned}
\hat{\text{Fear}}_i &= \hat{b}_0 + \hat{b}_1\text{Zombie vs. Human}_i + \hat{b}_2\text{Werewolf vs. Human}_i
\end{aligned}`
$$
]]]

---
# Dummy coding: *b*<sub>0</sub>

* Predicted fear = mean of human group:

]

.pull-right[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Entity </th>
   <th style="text-align:center;"> Zombie vs. Human </th>
   <th style="text-align:center;"> Werewolf vs. Human </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Human </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Zombie </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Werewolf </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1 </td>
  </tr>
</tbody>
</table>
]

![:abs 13%, 80%, 60%](media/oval_human.png)

.eq_lrge[.ong[
$$
`\begin{aligned}
\hat{\text{Fear}}_i &= \hat{b}_0 + \hat{b}_1\text{Zombie vs. Human}_i + \hat{b}_2\text{Werewolf vs. Human}_i \\
\bar{X}_\text{human} &= \hat{b}_0 + \hat{b}_1\times 0 + \hat{b}_2\times 0 \\
\hat{b}_0 &= \bar{X}_\text{human}
\end{aligned}`
$$
]]

---
# Dummy coding: *b*<sub>1</sub>

* Predicted fear = mean of zombie group:

]

![:abs 13%, 80%, 60%](media/oval_zombie.png)

.eq_lrge[.ong[
$$
`\begin{aligned}
\hat{\text{Fear}}_i &= \hat{b}_0 + \hat{b}_1\text{Zombie vs. Human}_i + \hat{b}_2\text{Werewolf vs. Human}_i \\
\bar{X}_\text{zombie} &= \hat{b}_0 + \hat{b}_1\times 1 + \hat{b}_2\times 0 \\
\bar{X}_\text{zombie} &= \hat{b}_0 + \hat{b}_1 \\
\hat{b}_1 &= \bar{X}_\text{zombie}- \hat{b}_0 \\
 &= \bar{X}_\text{zombie}- \bar{X}_\text{human}
\end{aligned}`
$$
]]
---
# Dummy coding: *b*<sub>2</sub>

* Predicted fear = mean of werewolf group:

]

![:abs 13%, 80%, 60%](media/oval_milton.png)

.eq_lrge[.ong[
$$
`\begin{aligned}
\hat{\text{Fear}}_i &= \hat{b}_0 + \hat{b}_1\text{Zombie vs. Human}_i + \hat{b}_2\text{Werewolf vs. Human}_i \\
\bar{X}_\text{werewolf} &= \hat{b}_0 + \hat{b}_1\times 0 + \hat{b}_2\times 1 \\
\bar{X}_\text{werewolf} &= \hat{b}_0 + \hat{b}_2 \\
\hat{b}_2 &= \bar{X}_\text{werewolf}- \hat{b}_0 \\
 &= \bar{X}_\text{werewolf}- \bar{X}_\text{human}
\end{aligned}`
$$
]]
---
class: center

# The model

![](categorical_predictors_files/figure-html/unnamed-chunk-30-1.png)

---
class: center

# The model

![](categorical_predictors_files/figure-html/unnamed-chunk-31-1.png)
---
# The parameter values

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  <thead>
    
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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Zombie">Zombie</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="Werewolf">Werewolf</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
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<td headers="stub_1_1 Human" class="gt_row gt_center">2</td>
<td headers="stub_1_1 Zombie" class="gt_row gt_center">10</td>
<td headers="stub_1_1 Werewolf" class="gt_row gt_center">6</td></tr>
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<td headers="stub_1_2 Werewolf" class="gt_row gt_center">3</td></tr>
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<td headers="stub_1_3 Human" class="gt_row gt_center">1</td>
<td headers="stub_1_3 Zombie" class="gt_row gt_center">5</td>
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<td headers="stub_1_4 Human" class="gt_row gt_center">1</td>
<td headers="stub_1_4 Zombie" class="gt_row gt_center">7</td>
<td headers="stub_1_4 Werewolf" class="gt_row gt_center">2</td></tr>
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<td headers="stub_1_5 Human" class="gt_row gt_center">2</td>
<td headers="stub_1_5 Zombie" class="gt_row gt_center">6</td>
<td headers="stub_1_5 Werewolf" class="gt_row gt_center">8</td></tr>
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<td headers="grand_summary_stub_1 Human" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row" style="background-color: rgba(202,62,52,0.8); color: #E8EAE5; text-align: center; font-weight: bold;">2.4</td>
<td headers="grand_summary_stub_1 Zombie" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row" style="background-color: rgba(202,62,52,0.8); color: #E8EAE5; text-align: center; font-weight: bold;">7.2</td>
<td headers="grand_summary_stub_1 Werewolf" class="gt_row gt_center gt_grand_summary_row gt_first_grand_summary_row gt_last_summary_row" style="background-color: rgba(202,62,52,0.8); color: #E8EAE5; text-align: center; font-weight: bold;">3.8</td></tr>
  </tbody>
  
  
</table>
</div>

.eq_lrge[.ong[
`$\hat{b}_2 = \bar{X}_\text{werewolf} - \bar{X}_\text{human} = 3.80 - 2.40 = 1.40$`
]]
---
class: center

# The model

<br>

.eq_lrge[.ong[
$$
`\begin{aligned}
\hat{\text{Fear}}_i &= \hat{b}_0 + \hat{b}_1\text{Zombie vs. Human}_i + \hat{b}_2\text{Werewolf vs. Human}_i
\end{aligned}`
$$
]]

<br>

.eq_lrge[.ong[
$$
`\begin{aligned}
\hat{\text{Fear}}_i &= 2.4 + 4.8\text{Zombie vs. Human}_i + 1.4\text{Werewolf vs. Human}_i
\end{aligned}`
$$
]]

---
background-image: url("media/halloween_pumpkins_at_night.jpg")
background-size: cover

# .gry[Overall fit]

.center[
<table>
 <thead>
  <tr>
   <th style="text-align:right;"> r.squared </th>
   <th style="text-align:right;"> adj.r.squared </th>
   <th style="text-align:right;"> sigma </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
   <th style="text-align:right;"> df </th>
   <th style="text-align:right;"> logLik </th>
   <th style="text-align:right;"> AIC </th>
   <th style="text-align:right;"> BIC </th>
   <th style="text-align:right;"> deviance </th>
   <th style="text-align:right;"> df.residual </th>
   <th style="text-align:right;"> nobs </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 0.46 </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 0.36 </td>
   <td style="text-align:right;background-color: white !important;"> 2.46 </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 5.02 </td>
   <td style="text-align:right;background-color: white !important;"> 0.03 </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 2 </td>
   <td style="text-align:right;background-color: white !important;"> -33.13 </td>
   <td style="text-align:right;background-color: white !important;"> 74.26 </td>
   <td style="text-align:right;background-color: white !important;"> 77.1 </td>
   <td style="text-align:right;background-color: white !important;"> 72.8 </td>
   <td style="text-align:right;background-color: yellow !important;background-color: white !important;"> 12 </td>
   <td style="text-align:right;background-color: white !important;"> 15 </td>
  </tr>
</tbody>
</table>
]
<br>

.infobox[
<svg style="height: 1em; top:.04em; position: relative; fill: fill;" viewBox="0 0 512 512"><path d="M493.255 56.236l-37.49-37.49c-24.993-24.993-65.515-24.994-90.51 0L12.838 371.162.151 485.346c-1.698 15.286 11.22 28.203 26.504 26.504l114.184-12.687 352.417-352.417c24.992-24.994 24.992-65.517-.001-90.51zM164.686 347.313c6.249 6.249 16.379 6.248 22.627 0L368 166.627l30.059 30.059L174 420.745V386h-48v-48H91.255l224.059-224.059L345.373 144 164.686 324.687c-6.249 6.248-6.249 16.378 0 22.626zm-38.539 121.285l-58.995 6.555-30.305-30.305 6.555-58.995L63.255 366H98v48h48v34.745l-19.853 19.853zm344.48-344.48l-49.941 49.941-82.745-82.745 49.941-49.941c12.505-12.505 32.748-12.507 45.255 0l37.49 37.49c12.506 12.506 12.507 32.747 0 45.255z"/></svg> The type of monster in the prank had a significant effect on fear levels, *F*(2, 12) = 5.02, *p* = 0.026, *R*<sup>2</sup> = 0.46.
]

---
background-image: url("media/halloween_lots_of_pumpkins.jpg")
background-size: cover

# .gry[Parameter estimates]

<br>
.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> estimate </th>
   <th style="text-align:right;"> std.error </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
   <th style="text-align:right;"> conf.low </th>
   <th style="text-align:right;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;"> (Intercept) </td>
   <td style="text-align:right;background-color: white !important;"> 2.4 </td>
   <td style="text-align:right;background-color: white !important;"> 1.10 </td>
   <td style="text-align:right;background-color: white !important;"> 2.18 </td>
   <td style="text-align:right;background-color: white !important;"> 0.05 </td>
   <td style="text-align:right;background-color: white !important;"> 0.00 </td>
   <td style="text-align:right;background-color: white !important;"> 4.80 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;"> entityZombie </td>
   <td style="text-align:right;background-color: white !important;"> 4.8 </td>
   <td style="text-align:right;background-color: white !important;"> 1.56 </td>
   <td style="text-align:right;background-color: white !important;"> 3.08 </td>
   <td style="text-align:right;background-color: white !important;"> 0.01 </td>
   <td style="text-align:right;background-color: white !important;"> 1.41 </td>
   <td style="text-align:right;background-color: white !important;"> 8.19 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;"> entityWerewolf </td>
   <td style="text-align:right;background-color: white !important;"> 1.4 </td>
   <td style="text-align:right;background-color: white !important;"> 1.56 </td>
   <td style="text-align:right;background-color: white !important;"> 0.90 </td>
   <td style="text-align:right;background-color: white !important;"> 0.39 </td>
   <td style="text-align:right;background-color: white !important;"> -1.99 </td>
   <td style="text-align:right;background-color: white !important;"> 4.79 </td>
  </tr>
</tbody>
</table>
]

---
class: inverse
background-image: url("media/halloween_pumpkins_fire.jpg")
background-size: cover

# Testing assumptions

---
background-image: url("media/pumpkin_maths_background.png")
background-size: cover

# .gry[Robust *F*-statistic]

```r
stats::oneway.test(fear ~ entity, data = humans_tib)
```
]
.center[
.infobox[

```
## 
## 	One-way analysis of means (not assuming equal variances)
## 
## data:  fear and entity
## F = 6.8917, num df = 2.0000, denom df = 7.7365, p-value = 0.01911
```
]
]
--
<br>
.infobox[
<svg style="height: 1em; top:.04em; position: relative; fill: fill;" viewBox="0 0 512 512"><path d="M493.255 56.236l-37.49-37.49c-24.993-24.993-65.515-24.994-90.51 0L12.838 371.162.151 485.346c-1.698 15.286 11.22 28.203 26.504 26.504l114.184-12.687 352.417-352.417c24.992-24.994 24.992-65.517-.001-90.51zM164.686 347.313c6.249 6.249 16.379 6.248 22.627 0L368 166.627l30.059 30.059L174 420.745V386h-48v-48H91.255l224.059-224.059L345.373 144 164.686 324.687c-6.249 6.248-6.249 16.378 0 22.626zm-38.539 121.285l-58.995 6.555-30.305-30.305 6.555-58.995L63.255 366H98v48h48v34.745l-19.853 19.853zm344.48-344.48l-49.941 49.941-82.745-82.745 49.941-49.941c12.505-12.505 32.748-12.507 45.255 0l37.49 37.49c12.506 12.506 12.507 32.747 0 45.255z"/></svg> The type of monster in the prank had a significant effect on fear levels, *F*(2, 7.74) = 6.89, *p* = 0.019.
]

---
background-image: url("media/halloween_trick_or_treat.jpg")
background-size: cover

# .gry[Robust model using HC4 Standard errors]

```r
parameters::parameters(human_lm, vcov = "HC4")
```
]

.center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Parameter </th>
   <th style="text-align:right;"> Coefficient </th>
   <th style="text-align:right;"> SE </th>
   <th style="text-align:right;"> CI </th>
   <th style="text-align:right;"> CI_low </th>
   <th style="text-align:right;"> CI_high </th>
   <th style="text-align:right;"> t </th>
   <th style="text-align:right;"> df_error </th>
   <th style="text-align:right;"> p </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;background-color: white !important;"> (Intercept) </td>
   <td style="text-align:right;background-color: white !important;"> 2.4 </td>
   <td style="text-align:right;background-color: white !important;"> 0.927 </td>
   <td style="text-align:right;background-color: white !important;"> 0.95 </td>
   <td style="text-align:right;background-color: white !important;"> 0.379 </td>
   <td style="text-align:right;background-color: white !important;"> 4.421 </td>
   <td style="text-align:right;background-color: white !important;"> 2.588 </td>
   <td style="text-align:right;background-color: white !important;"> 12 </td>
   <td style="text-align:right;background-color: white !important;"> 0.024 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;"> entityZombie </td>
   <td style="text-align:right;background-color: white !important;"> 4.8 </td>
   <td style="text-align:right;background-color: white !important;"> 1.265 </td>
   <td style="text-align:right;background-color: white !important;"> 0.95 </td>
   <td style="text-align:right;background-color: white !important;"> 2.044 </td>
   <td style="text-align:right;background-color: white !important;"> 7.556 </td>
   <td style="text-align:right;background-color: white !important;"> 3.795 </td>
   <td style="text-align:right;background-color: white !important;"> 12 </td>
   <td style="text-align:right;background-color: white !important;"> 0.003 </td>
  </tr>
  <tr>
   <td style="text-align:left;background-color: white !important;"> entityWerewolf </td>
   <td style="text-align:right;background-color: white !important;"> 1.4 </td>
   <td style="text-align:right;background-color: white !important;"> 1.703 </td>
   <td style="text-align:right;background-color: white !important;"> 0.95 </td>
   <td style="text-align:right;background-color: white !important;"> -2.310 </td>
   <td style="text-align:right;background-color: white !important;"> 5.110 </td>
   <td style="text-align:right;background-color: white !important;"> 0.822 </td>
   <td style="text-align:right;background-color: white !important;"> 12 </td>
   <td style="text-align:right;background-color: white !important;"> 0.427 </td>
  </tr>
</tbody>
</table>
]

???

Using a robust model the results don't change.

---
background-image: none
background-color: #000000
class: no-scribble

<video width="100%" height="100%" controls id="my_video">
    <source src="media/cubit_azal_lecture_end_captioned.mp4" type="video/mp4">
</video>
---
background-image: none
background-color: #000000
class: no-scribble

---
background-image: none
background-color: #000000
class: no-scribble