Categorical outcomes: logistic regression

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# Categorical outcomes: logistic regression
## 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>
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<div>
  <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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  <span style="line-height:40px;">www.discovr.rocks</span>
</div>

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# Categorical outcomes: .red[Christmas] log~~istic~~ regression
## ~~Professor Andy Field~~ .red[Santa]

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???

Santa: long intro

[After Santa says about Krampus and head popping out of bucket ...]

Andy: Santa, I can help. Tell me about your operations.

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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)

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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.

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# A festive example

.pull-right[
Santa Claus wanted to test the effects of different types of treats on whether presents got delivered:

* Predictors

+ **treat**: Christmas pudding, Mulled wine

* Outcome

+ **delivered**: Did the presents get delivered?
]

???

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<\/thead>\n<\/table>","options":{"dom":"tp","columnDefs":[{"className":"dt-center","targets":[1,2,3]},{"orderable":false,"targets":0}],"pageLength":10,"order":[],"autoWidth":false,"orderClasses":false},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":[],"jsHooks":[]}</script>

---
class: inverse
background-image: url("media/as_snowy_trees_126530339.jpg")
background-size: cover
#  If it were a standard linear model

.center[.eq_lrge[.white[
$$
`\begin{aligned}
\text{delivered}_{i} &= \hat{b}_{0} + \hat{b}_{1}\text{treat}_{i} + e_{i}
\end{aligned}`
$$
]]]

</br>

.pull-left[
Assumption of linearity

* Violated with categorical outcomes
* We can’t fit this model
]

.pull-right[.whitebox9[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Group </th>
   <th style="text-align:center;"> Treat </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Christmas pudding </td>
   <td style="text-align:center;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Mulled wine </td>
   <td style="text-align:center;"> 1 </td>
  </tr>
</tbody>
</table>
]]

---
class: inverse
background-image: url("media/as_blu_globe_391811093.jpg")
background-size: cover
#  But it’s not a standard linear model

.pull-right[
We predict the probability of the outcome occurring

.eq_lrge[.white[
$$
`\begin{aligned}
P(Y) &= \frac{1}{1+ e^{-(\hat{b}_0 + \hat{b}_1X_i+e_i)}} \\
P(\text{delivery}) &= \frac{1}{1+ e^{-(\hat{b}_0 + \hat{b}_1\text{treat}_i+e_i)}} \\
\end{aligned}`
$$
]]

Note the equation contains the linear model
]

---
class: inverse
background-image: url("media/as_snowy_trees_126530339.jpg")
background-size: cover

#  Or …

.eq_lrge[.white[
$$
`\begin{aligned}
\ln\bigg(\frac{P(Y)}{1- P(Y)}\bigg) &= \hat{b}_0 + \hat{b}_1X_{i} + e_i \\
\ln\bigg(\frac{P(\text{delivery})}{1- P(\text{delivery})}\bigg) &= \hat{b}_0 + \hat{b}_1\text{treat}_{i} + e_i \\
\end{aligned}`
$$
]]

.pull-left[.blubox[
### Outcome

* We predict the log odds of the outcome occurring
]]

.pull-right[.blubox[
### `$\hat{b}_0$` and `$\hat{b}_1$`

* Note the the logistic regression equation is the same as the linear model, except we predict the log odds of the outcome

* `$\hat{b}_1$` is the change in the log odds of the outcome associated with a unit change in the predictor
]]

---
class: inverse
background-image: url("media/as_red_santa_globe_45715114.jpg")
background-size: cover

#  Logs and exponents

.pull-right[

![](media/log_exp.png)

]

---
background-image: url("media/as_snowman_wave_303329070.jpg")
background-size: cover

#  The odds ratio: exp(*b*)

.center[.eq_lrge[
$$
`\begin{aligned}
\exp(b) = \frac{\text{odds after a unit change in the predictor}}{\text{original odds}}
\end{aligned}`
$$
]]

### *b*<sub>0</sub>

* Log odds of outcome when the predictors are 0

* Easier to interpret exp(*b*<sub>0</sub>), the odds of outcome when predictor is 0

### *b*<sub>1</sub>

* Change in the log odds of outcome associated with a unit change in the predictor

* Easier to interpret exp (*b*<sub>1</sub>), the odds ratio associated with a unit change in the predictor

* OR > 1: Predictor `$\uparrow$`, probability of outcome occurring `$\uparrow$`

* OR < 1: Predictor `$\uparrow$`, probability of outcome occurring `$\downarrow$`

---
class: inverse
background-image: url("media/as_santa_vortex_94489562.jpg")
background-size: cover
exclude: ![:isSV]
---
class: inverse
background-image: url("media/as_santa_vortex_94489562.jpg")
background-size: cover

#  Classification table

.pull-left[.whitebox9[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> Delivered </th>
   <th style="text-align:center;"> Not delivered </th>
   <th style="text-align:center;"> Total </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Christmas pudding </td>
   <td style="text-align:center;"> 150 </td>
   <td style="text-align:center;"> 28 </td>
   <td style="text-align:center;color: #912D25 !important;"> 178 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Mulled wine </td>
   <td style="text-align:center;"> 100 </td>
   <td style="text-align:center;"> 122 </td>
   <td style="text-align:center;color: #912D25 !important;"> 222 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #912D25 !important;"> Total </td>
   <td style="text-align:center;color: #912D25 !important;"> 250 </td>
   <td style="text-align:center;color: #912D25 !important;"> 150 </td>
   <td style="text-align:center;color: #912D25 !important;color: #912D25 !important;"> 400 </td>
  </tr>
</tbody>
</table>
]]

.pull-right[.whitebox9[
$$
`\begin{aligned}
\text{odds}_\text{delivery} &= \frac{\text{Number delivered}}{\text{Number not delivered}} \\
&= \frac{250}{150} \\
&= 1.67
\end{aligned}`
$$
]]
---
class: inverse
background-image: url("media/as_gingerbread_229820523.jpg")
background-size: cover
exclude: ![:isSV]

---
class: inverse
background-image: url("media/as_gingerbread_229820523.jpg")
background-size: cover

#  The odds ratio

.whitebox9[
$$
`\begin{aligned}
\text{odds}_\text{delivered after pudding} &= \frac{\text{Number delivered after pudding}}{\text{Number not delivered after pudding}} \\
&= \frac{150}{28} \\
&= 5.36
\end{aligned}`
$$
]

</br>

.whitebox9[
$$
`\begin{aligned}
\text{odds}_\text{delivered after wine} &= \frac{\text{Number delivered after wine}}{\text{Number not delivered after wine}} \\
&= \frac{100}{122} \\
&= 0.82
\end{aligned}`
$$
]

---
class: inverse
background-image: url("media/as_gingerbread_229820523.jpg")
background-size: cover

#  The odds ratio

.whitebox9[.eq_lrge[
$$
`\begin{aligned}
\text{odds ratio} &= \frac{\text{odds}_\text{delivered after wine}}{\text{odds}_\text{delivered after pudding}} \\
&= \frac{0.82}{5.36} \\
&= 0.15
\end{aligned}`
$$
]]

---
class: inverse
background-image: url("media/as_gingerbread_229820523.jpg")
background-size: cover

#  The odds ratio

.whitebox9[.eq_lrge[
$$
`\begin{aligned}
\text{odds ratio} &= \frac{\text{odds}_\text{delivered after pudding}}{\text{odds}_\text{delivered after wine}} \\
&= \frac{5.36}{0.82} \\
&= 6.54
\end{aligned}`
$$
]]

---
class: inverse
background-image: url("media/as_santa_over_village_71686107.jpg")
background-size: cover
exclude: ![:isSV]

---
class: inverse
background-image: url("media/as_santa_over_village_71686107.jpg")
background-size: cover

#  Fitting the model

```r
santa_mod <- glm(delivered ~ treat, data = santa_tib, family = binomial())

santa_mod |> 
  broom::tidy(conf.int = TRUE) |> 
  knitr::kable(digits = 2, align = 'lcccccc')
```

---
class: inverse
background-image: url("media/as_santa_over_village_71686107.jpg")
background-size: cover

# Odds ratios

```r
santa_mod |> 
  broom::tidy(conf.int = TRUE, `exponentiate = TRUE`) |> 
  knitr::kable(digits = 2, align = 'lcccccc')
```

---
background-image: none
background-color: #000000
class: no-scribble
exclude: ![:isSV]

---
class: inverse
background-image: url("media/as_blu_house_93514381.jpg")
background-size: cover

# A festive example

.pull-right[
Santa Claus wanted to test the effects of different types of treats and the quantity of them consumed on whether presents got delivered:

* Predictors

+ **treat**: Christmas pudding, Mulled wine
  
  + **quantity**: 0, 1, 2, 3, or 4

* Outcome

+ **delivered**: Did the presents get delivered?
]

---
class: inverse
background-image: url("media/as_santa_moon_36326866.jpg")
background-size: cover

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---
class: inverse
background-image: url("media/as_red_snowflake_228468210.jpg")
background-size: cover

# Extending the model

.center[.eq_med[.white[
$$
`\begin{aligned}
\text{delivered}_{i} &= \hat{b}_{0} + \hat{b}_{1}\text{treat}_{i} + \hat{b}_{2}\text{quantity}_{i} + \hat{b}_{3}\text{treat} \times \text{quantity}_{i} + e_{i}
\end{aligned}`
$$
]]]

</br>

.eq_med[.white[
$$
`\begin{aligned}
P(\text{delivery}) &= \frac{1}{1+e^{-(\hat{b}_0 + \hat{b}_1\text{treat}_{i} + \hat{b}_2\text{quantity}_{i} + \hat{b}_3\text{treat} \times \text{quantity}_{i} + e_i)}} \\
\end{aligned}`
$$
]]

.eq_med[.white[
$$
`\begin{aligned}
\ln\bigg(\frac{P(\text{delivery})}{1- P(\text{delivery})}\bigg) &= \hat{b}_0 + \hat{b}_1\text{treat}_{i} + \hat{b}_2\text{quantity}_{i} + \hat{b}_3\text{treat} \times \text{quantity}_{i} + e_i \\
\end{aligned}`
$$
]]

---
class: inverse
background-image: url("media/as_tree_train_93514522.jpg")
background-size: cover

#  Building the model

.pull-left[

* Forced entry: all variables entered simultaneously.

* Hierarchical: variables entered in blocks.

+ Blocks should be based on past research, or theory being tested.
  + Good method.

* Stepwise: variables entered on the basis of statistical criteria (i.e., Relative contribution to predicting outcome).

+ Should be used only for exploratory analysis.
]

---
class: inverse
background-image: url("media/as_blu_house_93514381.jpg")
background-size: cover

#  Things that can go wrong

.pull-right[
### Things we’ve met before

.eq_med[
* Linearity (of the logit)

* Spherical residuals
  + Independent errors

* Multicollinearity
]

### Unique problems

.eq_med[
* Incomplete information

* Complete separation
]
]

---
class: inverse
background-image: url("media/as_santa_moon_37337682.jpg")
background-size: cover

#  Incomplete information

.pull-left[.blubox[
### Empty cells

.eq_lrge[* We don’t know how many presents are delivered after two puddings or not delivered after 5 wines

* Problem quickly escalates with continuous predictors

* Inflates standard errors
]]]

.pull-right[.blubox[

<table>
 <thead>
  <tr>
   <th style="text-align:left;background-color: #15224c !important;font-size: 18px;">   </th>
   <th style="text-align:center;background-color: #15224c !important;font-size: 18px;"> Quantity </th>
   <th style="text-align:center;background-color: #15224c !important;font-size: 18px;"> Delivered </th>
   <th style="text-align:center;background-color: #15224c !important;font-size: 18px;"> Not delivered </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Pudding </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 0 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 27 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Pudding </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 1 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 36 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Pudding </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 2 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> - </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 5 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Pudding </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 3 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 37 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 6 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Pudding </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 4 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 12 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Mulled wine </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 0 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 27 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Mulled wine </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 1 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 32 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 11 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Mulled wine </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 2 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 24 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 36 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Mulled wine </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 3 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 14 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 48 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> Mulled wine </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 4 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> 3 </td>
   <td style="text-align:center;color: white !important;background-color: rgba(32, 21, 34, 0.3) !important;font-size: 18px;"> - </td>
  </tr>
</tbody>
</table>
]]

---
background-image: url("media/as_krampus_374161953.jpg")
background-size: cover

#  Complete separation

When the outcome variable can be perfectly predicted

* Predicting whether someone is a krampus based on weight

.pull-left[
### Santa vs. krampus

![](categorical_outcomes_files/figure-html/unnamed-chunk-9-1.png)

]

![:abs 10%, 15%, 60%](media/santa.png)
![:abs 9%, 35%, 60%](media/krampus.png)

.pull-right[
### Elves vs. krampus

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

]

![:abs 7%, 60%, 60%](media/elf.jpg)
![:abs 9%, 80%, 60%](media/krampus.png)

---
exclude: ![:isSV]
class: no-scribble

---
class: no-scribble
exclude: ![:isSV]

---
class: inverse
background-image: url("media/as_snowy_trees_white_126530339.jpg")
background-size: cover

# Fitting the model

```r
santa_full_mod <- glm(delivered ~ treat*quantity, data = santa_tib, family = binomial())

santa_full_mod |> 
  broom::tidy(conf.int = TRUE) |> 
  knitr::kable(digits = 2, align = 'lcccccc')
```

</br>

.blubox[
<table class="table" style="font-size: 28px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;color: #2C5577 !important;"> term </th>
   <th style="text-align:center;color: #2C5577 !important;"> estimate </th>
   <th style="text-align:center;color: #2C5577 !important;"> std.error </th>
   <th style="text-align:center;color: #2C5577 !important;"> statistic </th>
   <th style="text-align:center;color: #2C5577 !important;"> p.value </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.low </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 1.83 </td>
   <td style="text-align:center;"> 0.38 </td>
   <td style="text-align:center;"> 4.80 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 1.13 </td>
   <td style="text-align:center;"> 2.63 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #2C5577 !important;"> treatMulled wine </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.20 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.52 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.38 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.70 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -0.83 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 1.22 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> quantity </td>
   <td style="text-align:center;"> -0.08 </td>
   <td style="text-align:center;"> 0.17 </td>
   <td style="text-align:center;"> -0.48 </td>
   <td style="text-align:center;"> 0.63 </td>
   <td style="text-align:center;"> -0.41 </td>
   <td style="text-align:center;"> 0.25 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #2C5577 !important;"> treatMulled wine:quantity </td>
   <td style="text-align:center;color: #2C5577 !important;"> -1.03 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.23 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -4.45 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.00 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -1.49 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -0.58 </td>
  </tr>
</tbody>
</table>
]

---
class: inverse
background-image: url("media/as_snowy_trees_white_126530339.jpg")
background-size: cover

</br>

.pull-left[

```r
santa_tib |> 
  filter(treat == "Pudding") |> 
  glm(delivered ~ quantity, data = _, family = binomial()) 
```

.blubox[
<table class="table" style="font-size: 16px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;color: #2C5577 !important;"> term </th>
   <th style="text-align:center;color: #2C5577 !important;"> estimate </th>
   <th style="text-align:center;color: #2C5577 !important;"> statistic </th>
   <th style="text-align:center;color: #2C5577 !important;"> p.value </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.low </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 1.83 </td>
   <td style="text-align:center;"> 4.80 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 1.13 </td>
   <td style="text-align:center;"> 2.63 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #2C5577 !important;"> quantity </td>
   <td style="text-align:center;color: #2C5577 !important;"> -0.08 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -0.48 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.63 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -0.41 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.25 </td>
  </tr>
</tbody>
</table>
]]

![:abs 10%, 20%, 3%](media/pudding.png)

.pull-right[

```r
santa_tib |> 
  filter(treat == "Mulled wine") |> 
  glm(delivered ~ quantity, data = _, family = binomial()) 
```

.blubox[
<table class="table" style="font-size: 16px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;color: #2C5577 !important;"> term </th>
   <th style="text-align:center;color: #2C5577 !important;"> estimate </th>
   <th style="text-align:center;color: #2C5577 !important;"> statistic </th>
   <th style="text-align:center;color: #2C5577 !important;"> p.value </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.low </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 2.03 </td>
   <td style="text-align:center;"> 5.73 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1.37 </td>
   <td style="text-align:center;"> 2.76 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #2C5577 !important;"> quantity </td>
   <td style="text-align:center;color: #2C5577 !important;"> -1.11 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -6.99 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -1.44 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -0.81 </td>
  </tr>
</tbody>
</table>
]]

![:abs 6%, 70%, 3%](media/mulled_wine.png)

</br>

.center[.eq_lrge[.blu_dk[
$$
`\begin{aligned}
b_\text{wine} − b_\text{pudding} &= −1.11−(−0.08) \\
&= −1.03
\end{aligned}`
$$
]]]

---
class: inverse
background-image: url("media/as_snowy_trees_white_126530339.jpg")
background-size: cover

# Odds ratio

```r
santa_full_mod |> 
  broom::tidy(conf.int = TRUE, `exponentiate = TRUE`) |> 
  knitr::kable(digits = 2, align = 'lcccccc')
```

.blubox[
<table class="table" style="font-size: 28px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;color: #2C5577 !important;"> term </th>
   <th style="text-align:center;color: #2C5577 !important;"> estimate </th>
   <th style="text-align:center;color: #2C5577 !important;"> std.error </th>
   <th style="text-align:center;color: #2C5577 !important;"> statistic </th>
   <th style="text-align:center;color: #2C5577 !important;"> p.value </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.low </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 6.23 </td>
   <td style="text-align:center;"> 0.38 </td>
   <td style="text-align:center;"> 4.80 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 3.08 </td>
   <td style="text-align:center;"> 13.86 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #2C5577 !important;"> treatMulled wine </td>
   <td style="text-align:center;color: #2C5577 !important;"> 1.22 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.52 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.38 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.70 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.43 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 3.37 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> quantity </td>
   <td style="text-align:center;"> 0.92 </td>
   <td style="text-align:center;"> 0.17 </td>
   <td style="text-align:center;"> -0.48 </td>
   <td style="text-align:center;"> 0.63 </td>
   <td style="text-align:center;"> 0.66 </td>
   <td style="text-align:center;"> 1.28 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #2C5577 !important;"> treatMulled wine:quantity </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.36 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.23 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -4.45 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.00 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.23 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0.56 </td>
  </tr>
</tbody>
</table>
]

---
class: inverse
background-image: url("media/as_snowy_trees_white_126530339.jpg")
background-size: cover

</br>

.pull-left[

```r
santa_tib |> 
  filter(treat == "Pudding") |> 
  glm(delivered ~ quantity, data = _, family = binomial()) 
```

![:abs 10%, 20%, 3%](media/pudding.png)

.pull-right[

```r
santa_tib |> 
  filter(treat == "Mulled wine") |> 
  glm(delivered ~ quantity, data = _, family = binomial()) 
```

.blubox[
<table class="table" style="font-size: 16px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;color: #2C5577 !important;"> term </th>
   <th style="text-align:center;color: #2C5577 !important;"> estimate </th>
   <th style="text-align:center;color: #2C5577 !important;"> statistic </th>
   <th style="text-align:center;color: #2C5577 !important;"> p.value </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.low </th>
   <th style="text-align:center;color: #2C5577 !important;"> conf.high </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:center;"> 2.03 </td>
   <td style="text-align:center;"> 5.73 </td>
   <td style="text-align:center;"> 0 </td>
   <td style="text-align:center;"> 1.37 </td>
   <td style="text-align:center;"> 2.76 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #2C5577 !important;"> quantity </td>
   <td style="text-align:center;color: #2C5577 !important;"> -1.11 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -6.99 </td>
   <td style="text-align:center;color: #2C5577 !important;"> 0 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -1.44 </td>
   <td style="text-align:center;color: #2C5577 !important;"> -0.81 </td>
  </tr>
</tbody>
</table>
]]

![:abs 6%, 70%, 3%](media/mulled_wine.png)

</br>

.center[.eq_lrge[.blu_dk[
$$
`\begin{aligned}
b_\text{wine} − b_\text{pudding} &= −1.11−(−0.08) \\
&= −1.03
\end{aligned}`
$$
]]]

.center[.eq_lrge[.blu_dk[
$$
`\begin{aligned}
\exp(b)_\text{difference} &= e^{−1.03} \\
&= 0.36
\end{aligned}`
$$
]]]

---
class: inverse, center, middle
background-image: url("media/as_snowman_wave_303329070.jpg")
background-size: cover
exclude: ![:isSV]

---
class: inverse, center, middle
background-image: url("media/as_snowman_wave_303329070.jpg")
background-size: cover

![](categorical_outcomes_files/figure-html/unnamed-chunk-23-1.png)

![:abs 6%, 78%, 60%](media/mulled_wine.png)
![:abs 10%, 75%, 20%](media/pudding.png)

---
class: inverse, center, middle
background-image: url("media/as_snowman_wave_303329070.jpg")
background-size: cover
exclude: true

![](categorical_outcomes_files/figure-html/unnamed-chunk-24-1.png)

---
class: inverse
background-image: url("media/as_blu_snowman_389261412.jpg")
background-size: cover
exclude: true

#  Conclusions

.pull-left[
### It’s OK for elves to eat puddings

### It’s not OK for elves to drink more than 1 glass of mulled wine!
]

---
class: inverse
background-image: url("media/as_log_cabin_184043999.jpg")
background-size: cover
exclude: true

.left[.blubox[

You’d better watch out, you might want to cry </br>
You’d better beware, I’m telling you why </br>
Logistic regression is coming to town

Santa’s checking the output, he’s checking it twice </br>
To try to predict, who’s naughty or nice </br>
Logistic regression is coming to town

He’s got a binary outcome. Residuals aren’t normal </br>
But with a logit link function, everything is cool </br>

So, there’s nothing to fear. There’s no need to cry </br>
It’s like every other model we’ve tried </br>
Logistic regression is coming to town

The effect of predictors, is easy to see </br>
You just to need to check the exponent of *b* </br>
Logistic regression is coming to town

But design cells can be empty, that’s incomplete information </br>
And if prediction is perfect then you have complete separation </br>

But there’s nothing to fear. There’s no need to cry </br>
It’s like every other, model we’ve tried </br>
Logistic regression is coming to town
]]

---

background-image: none
background-color: #000000
class: no-scribble
exclude: ![:isSV]

???

---

background-image: none
background-color: #000000
class: no-scribble
exclude: ![:isSV]

---