Data Science for Linguists

class: center, middle, inverse, title-slide

.title[
# Data Science for Linguists
]
.subtitle[
## Getting the most out of your model
]
.author[
### Joseph V. Casillas, PhD
]
.date[
### Rutgers University</br>Spring 2023</br>Last update: 2023-04-02
]

---

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background-image: url(https://www.jvcasillas.com/media/rstats/memes/lm_interactions.png)
background-size: 700px

---
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# Fine-tuning and understanding your models

---
layout: true

# Fine-tuning and understanding your models

---

### Overview

### 1. Interactions

### 2. Coding - categorical variables

- Dummy coding
- Sum coding
- ~~Deviation coding~~

### 3. Transformations - continuous variables

- Centering
- Standardizing
- ~~Log transformation~~

---
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# Interactions

---
layout: true

# Interactions

---

### What is an interaction?

.Large[

.content-box-blue[
When the effect of one of the predictor on the criterion changes at different 
levels/values of one of the other predictors. 
]

- Interactions are possible between continuous variables, continuous *and* 
discontinuous variables, as well as *only* discontinuous variables.

- It is often the case that a significant interaction will make main effects 
uninterpretable.

]

---
background-color: black

### Continuous variables

background-image: url(./assets/img/continuous_interaction1.png)
background-position: 70% 50%
background-size: 900px

background-image: url(./assets/img/continuous_interaction2.png)
background-position: 70% 50%
background-size: 900px

---

### Continuous and</br>categorical variables

---

### Categorical variables

#### Interaction or no?

---

### Categorical variables

#### Interaction or no?

---

### Categorical variables

#### Interaction or no?

---

### Categorical variables

#### Interaction or no?

---

### Categorical variables

#### Interaction or no?

---

### Categorical variables

#### Interaction or no?

---

### Categorical variables

#### Interaction or no?

---

### Categorical variables

#### Interaction or no?

---

### Interpreting main effects

.large[
.pull-left[
- Imagine you fit a model looking at  
`val ~ condition * group`

- The model yields a main effect of *condition*, *group*, and a *condition* 
`$\times$` *group* interaction.

- Can you interpret the main effects? What does a main effect of *condition* 
mean? What about a main effect of *group*?
]
]

---

### Main effect of condition

.large[
.pull-left[
- Collapsing over *group*, there is a difference between the *control* and the 
*treatment* conditions, i.e., the difference in means is not 0.

- Thus, one might interpret the main effect of *condition* as something like 
"Overall, the control group had higher scores than the treatment group".

- Technically this is "true", but careless, as it does not take into account the 
fact that the treatment had no effect on the `Late` group. 
]
]

---

### Main effect of group

.large[
.pull-left[
- Collapsing over *condition*, there is a difference between `Early` and `Late`, i.e., the difference in means is not 0.

- Thus, one might interpret the main effect as something like 
"Overall, the late group had higher average scores than the early group".

- Technically this is "true", but careless, as it does not take into account the 
fact that the treatment is only effective on the `Early` group. 
]
]

---

### Three-way interactions 😅

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background-image: url(https://media.giphy.com/media/3ofSB8NFxyfY6jw2Jy/giphy.gif)
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---

# Interactions

### Three-way interactions 😑

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background-image: url(https://i.imgflip.com/26x5rs.jpg)
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class: title-slide-section-red, middle

# Coding categorical variables

---
layout: true

# Coding categorical variables

---

### Overview

.pull-left[

- One of the benefits of 'doing ANOVA' with MRC is that you can include 
different types of predictors in your model, i.e., **categorical** and 
.blue[continuous].

- Neither classical ANOVA nor classical MRC can handle combinations of these 
two predictors

- This is possible because of dummy coding

- But there are other methods of coding categorical variables
]

.pull-right[

- Treatment (dummy) coding

- Sum coding

- Deviation coding

- ...and others

]

---
class: title-slide-section-grey, middle

## .RUred[Treatment coding]

---

### Treatment (dummy) coding

#### You get j-1 dummies

.Large[

| Groups | d<sub>1</sub> | d<sub>2</sub> | d<sub>3</sub> |
| :----- | :-----------: | :-----------: | :-----------: |
| A      | 0             | 0             | 0             |
| B      | 1             | 0             | 0             |
| C      | 0             | 1             | 0             |
| D      | 0             | 0             | 1             |

]

---

### Treatment (dummy) coding

#### One level is taken as the *reference* or *baseline*. This is the intercept.

.Large[

| Groups | d<sub>1</sub> | d<sub>2</sub> | d<sub>3</sub> |                  |
| :----- | :-----------: | :-----------: | :-----------: | :--------------- |
| **A**  | **0**         | **0**         | **0**         | ⬅︎ **Intercept** |
| B      | 1             | 0             | 0             |                  |
| C      | 0             | 1             | 0             |                  |
| D      | 0             | 0             | 1             |                  |

]

---

### Treatment (dummy) coding

#### j-1 dummies = j-1 comparisons

.Large[

| Groups   | d<sub>1</sub> | d<sub>2</sub> | d<sub>3</sub> |                  |
| :------- | :-----------: | :-----------: | :-----------: | :--------------- |
| **A**    | **0**         | **0**         | **0**         | ⬅︎ **Intercept** |
| .blue[B] | .blue[1]      | 0             | 0             |                  |
| C        | 0             | 1             | 0             |                  |
| D        | 0             | 0             | 1             |                  |
|          | ⬆︎</br>**A**.blue[B]|         |               |                  |
]

---

### Treatment (dummy) coding

#### j-1 dummies = j-1 comparisons

.Large[

| Groups   | d<sub>1</sub> | d<sub>2</sub> | d<sub>3</sub> |                  |
| :------- | :-----------: | :-----------: | :-----------: | :--------------- |
| **A**    | **0**         | **0**         | **0**         | ⬅︎ **Intercept** |
| B        | 1             | 0             | 0             |                  |
| .blue[C] | 0             | .blue[1]      | 0             |                  |
| D        | 0             | 0             | 1             |                  |
|          |               | ⬆︎</br>**A**.blue[C]|         |                  |
]

---

### Treatment (dummy) coding

#### j-1 dummies = j-1 comparisons

.Large[

| Groups   | d<sub>1</sub> | d<sub>2</sub> | d<sub>3</sub> |                  |
| :------- | :-----------: | :-----------: | :-----------: | :--------------- |
| **A**    | **0**         | **0**         | **0**         | ⬅︎ **Intercept** |
| B        | 1             | 0             | 0             |                  |
| C        | 0             | 1             | 0             |                  |
| .blue[D] | 0             | 0             | .blue[1]      |                  |
|          |               |               | ⬆︎</br>**A**.blue[D] |           |
]

---

### Review - Spanish spirantization

- We are going to look at L2 acquisition of Spanish voiced stops

- Spanish voiced stops: /b, d, g/
  - Stop/approximant alternation: /b, d, g/ → [β, ð, ɣ]
  - Believed to be underlyingly stops (but this is contested)
  - Surface as stops in utterance initial position, after nasals, and after /l/ 
  in the case of /d/
  - Approximants in all other contexts

- We have data from 30 participants (15 natives, 15 L2 learners) who produced 
/b/ in utterance initial and intervocalic position

- We are interested in measuring degree of spirantization to see if learners 
can acquire a new phonetic category, [β]

- Spirantization can be quantified by measuring the relative difference in the 
intensity (dB) of the target segment and that of the following vowel

---

### Treatment (dummy) coding - Spanish spirantization

.pull-left[

- We will fit a model that analyzes the `intensity difference` for the target 
segment ([β]) as a function group (native, learner)

<table class="table" style="font-size: 17px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> group </th>
   <th style="text-align:right;"> mean_int_diff </th>
   <th style="text-align:right;"> sd_int_diff </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> 17.003 </td>
   <td style="text-align:right;"> 3.279 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 11.001 </td>
   <td style="text-align:right;"> 2.488 </td>
  </tr>
</tbody>
</table>

```r
lm(intensity_diff ~ group, data = spir_tidy)
```

<table class="table" style="font-size: 17px; width: auto !important; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 17.003 </td>
   <td style="text-align:right;"> 0.556 </td>
   <td style="text-align:right;"> 30.593 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupnative </td>
   <td style="text-align:right;"> -6.002 </td>
   <td style="text-align:right;"> 0.963 </td>
   <td style="text-align:right;"> -6.235 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>

]

---

### Treatment (dummy) coding - Spanish spirantization

.pull-left[

- If the categorical predictor includes more than one level

```r
lm(intensity_diff ~ prof, data = spir_tidy)
```

<table class="table" style="font-size: 17px; width: auto !important; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 11.001 </td>
   <td style="text-align:right;"> 0.725 </td>
   <td style="text-align:right;"> 15.170 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> profadv </td>
   <td style="text-align:right;"> 4.506 </td>
   <td style="text-align:right;"> 1.026 </td>
   <td style="text-align:right;"> 4.393 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> profbeg </td>
   <td style="text-align:right;"> 7.498 </td>
   <td style="text-align:right;"> 1.026 </td>
   <td style="text-align:right;"> 7.311 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>

]

---

### Treatment (dummy) coding - Spanish spirantization

- If we want to test other comparisons we have to `relevel()` the factor.

```r
mod_spir3 <- spir_tidy %>% 
* mutate(prof = fct_relevel(prof, "beg")) %>%
  lm(intensity_diff ~ prof, data = .)
mod_spir3 %>% 
  tidy() %>% 
  kable(format = 'html', digits = 3) %>% 
  kable_styling(., full_width = FALSE, font_size = 22)
```

<table class="table" style="font-size: 22px; width: auto !important; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 18.499 </td>
   <td style="text-align:right;"> 0.725 </td>
   <td style="text-align:right;"> 25.510 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> profadv </td>
   <td style="text-align:right;"> -2.993 </td>
   <td style="text-align:right;"> 1.026 </td>
   <td style="text-align:right;"> -2.918 </td>
   <td style="text-align:right;"> 0.006 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> profnat </td>
   <td style="text-align:right;"> -7.498 </td>
   <td style="text-align:right;"> 1.026 </td>
   <td style="text-align:right;"> -7.311 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
</tbody>
</table>

---

### Treatment (dummy) coding - Summary

#### Advantages

- Easy to do (default in R and most stats programs)

- Easy to interpret if you know reference level and model is not too complicated

#### Disadvantages

- May give meaningless comparisons without releveling factor

- May give meaningless comparisons if model includes continuous predictor

---
class: title-slide-section-grey, middle

## .RUred[Sum coding]

---

### Sum coding

- As we have seen, treatment coded dummy variables are discontinuous predictors 
coded as 0/1

- We can also use other values to get different model outputs

- **Sum coding** (or "effect" coding) is often used when one wants the default 
model output to use an overall average value as the reference (rather than a 
possibly arbitrarily chosen group)

- The values of the coded vectors must *sum to 0*

- We still get j-1 estimates

---

### Sum coding

#### How do we do it?

- If our discontinuous variable only has two levels, we can add a new column to 
the dataframe that includes -1/1

- We can do this easily using the `mutate()` function

```r
spir_tidy %>% 
* mutate(groupSum = if_else(group == "native", true = 1, false = -1))
```

<table class="table" style="font-size: 18px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> prof </th>
   <th style="text-align:left;"> id </th>
   <th style="text-align:left;"> position </th>
   <th style="text-align:right;"> cIntensity </th>
   <th style="text-align:right;"> vIntensity </th>
   <th style="text-align:left;"> group </th>
   <th style="text-align:right;"> groupSum </th>
   <th style="text-align:right;"> intensity_diff </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> beg </td>
   <td style="text-align:left;"> 01 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 58.58501 </td>
   <td style="text-align:right;"> 76.92710 </td>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> -1 </td>
   <td style="text-align:right;"> 18.34209 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> beg </td>
   <td style="text-align:left;"> 02 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 60.72916 </td>
   <td style="text-align:right;"> 79.39805 </td>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> -1 </td>
   <td style="text-align:right;"> 18.66888 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:left;"> beg </td>
   <td style="text-align:left;"> 03 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 61.53707 </td>
   <td style="text-align:right;"> 78.94344 </td>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> -1 </td>
   <td style="text-align:right;"> 17.40637 </td>
  </tr>
</tbody>
</table>

<table class="table" style="font-size: 18px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> prof </th>
   <th style="text-align:left;"> id </th>
   <th style="text-align:left;"> position </th>
   <th style="text-align:right;"> cIntensity </th>
   <th style="text-align:right;"> vIntensity </th>
   <th style="text-align:left;"> group </th>
   <th style="text-align:right;"> groupSum </th>
   <th style="text-align:right;"> intensity_diff </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 43 </td>
   <td style="text-align:left;"> nat </td>
   <td style="text-align:left;"> 13 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 69.39163 </td>
   <td style="text-align:right;"> 84.15049 </td>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 14.75886 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 44 </td>
   <td style="text-align:left;"> nat </td>
   <td style="text-align:left;"> 14 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 70.74004 </td>
   <td style="text-align:right;"> 82.05478 </td>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 11.31475 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 45 </td>
   <td style="text-align:left;"> nat </td>
   <td style="text-align:left;"> 15 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 70.53420 </td>
   <td style="text-align:right;"> 82.41582 </td>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 11.88162 </td>
  </tr>
</tbody>
</table>

---
background-image: url(./index_files/figure-html/spirantization-p1-1.png)
background-position: 98% 55%
background-size: 550px

### Sum coding - Spanish spirantization

.pull-left[

```r
lm(intensity_diff ~ groupSum, data = spir_tidy)
```

<table class="table" style="font-size: 18px; width: auto !important; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 14.002 </td>
   <td style="text-align:right;"> 0.481 </td>
   <td style="text-align:right;"> 29.091 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum </td>
   <td style="text-align:right;"> -3.001 </td>
   <td style="text-align:right;"> 0.481 </td>
   <td style="text-align:right;"> -6.235 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>

]

.footnote[

<table class="table" style="font-size: 18px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> group </th>
   <th style="text-align:right;"> mean_int_diff </th>
   <th style="text-align:right;"> sd_int_diff </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> 17.003 </td>
   <td style="text-align:right;"> 3.279 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 11.001 </td>
   <td style="text-align:right;"> 2.488 </td>
  </tr>
</tbody>
</table>

- What is the intercept? The mean of group means.

- What is `groupSum`? The effect of group on the intercept.

- Calculate the intercept +/- the `groupSum` effect = group means.

]

---
background-image: url(./index_files/figure-html/spirantization-p2-1.png)
background-position: 90% 35%
background-size: 500px

### Sum coding - Spanish spirantization

.pull-left[

- What about with a 3-level factor?
- This requires more tweeking.

```r
spir_tidy$prof <- as.factor(spir_tidy$prof)
*spir_tidy$profSum <- C(spir_tidy$prof, sum)
contrasts(spir_tidy$profSum)
```

```
##     [,1] [,2]
## adv    1    0
## beg    0    1
## nat   -1   -1
```

```r
lm(intensity_diff ~ profSum, data = spir_tidy) 
```

<table class="table" style="font-size: 18px; width: auto !important; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 15.002 </td>
   <td style="text-align:right;"> 0.419 </td>
   <td style="text-align:right;"> 35.832 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> profSum1 </td>
   <td style="text-align:right;"> 0.504 </td>
   <td style="text-align:right;"> 0.592 </td>
   <td style="text-align:right;"> 0.852 </td>
   <td style="text-align:right;"> 0.399 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> profSum2 </td>
   <td style="text-align:right;"> 3.497 </td>
   <td style="text-align:right;"> 0.592 </td>
   <td style="text-align:right;"> 5.906 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
</tbody>
</table>

]

.pull-right[
.footnote[
- Is this useful? Why would we do this?
]
]

---

### Sum coding - Summary

#### Advantages

- With two level factor, sum (effect) coding gives omnibus assessment of main 
effects

- With more than two levels, sum (effect) coding assess the "uniqueness" of a 
given level compared to the overall average

- Avoids (possibly) arbitrary selection of reference group

- Makes MRC even more like ANOVA (advantage for researchers accustomed to ANOVA)

#### Disadvantages

- Involves releveling, adding columns to dataframe or resetting factor

- May not always provide useful information?

---
class: title-slide-section-grey, middle

## .RUred[Deviation coding]

---
layout: true

# Coding categorical variables

---

### Deviation coding

- Dummy variables of discontinuous predictors can be coded as 0/1 (treatment 
coding) or -1/1 (sum coding)

- There is another *sum to 0* possibility that is often used with 2-level 
factors: deviation coding

- **Deviation coding** is often used when one wants the default 
model output to provide an overall assessment of effect size.

- The values of the coded vectors still must *sum to 0*

- We still get j-1 estimates

---

### Deviation coding

#### How do we do it?

- We can add a new column to the dataframe that includes -0.5/0.5

- We can do this easily using the `mutate()` function

<table class="table" style="font-size: 18px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> prof </th>
   <th style="text-align:left;"> id </th>
   <th style="text-align:left;"> position </th>
   <th style="text-align:right;"> cIntensity </th>
   <th style="text-align:right;"> vIntensity </th>
   <th style="text-align:left;"> group </th>
   <th style="text-align:right;"> groupDev </th>
   <th style="text-align:right;"> intensity_diff </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 1 </td>
   <td style="text-align:left;"> beg </td>
   <td style="text-align:left;"> 01 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 58.58501 </td>
   <td style="text-align:right;"> 76.92710 </td>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> -0.5 </td>
   <td style="text-align:right;"> 18.34209 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 2 </td>
   <td style="text-align:left;"> beg </td>
   <td style="text-align:left;"> 02 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 60.72916 </td>
   <td style="text-align:right;"> 79.39805 </td>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> -0.5 </td>
   <td style="text-align:right;"> 18.66888 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 3 </td>
   <td style="text-align:left;"> beg </td>
   <td style="text-align:left;"> 03 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 61.53707 </td>
   <td style="text-align:right;"> 78.94344 </td>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> -0.5 </td>
   <td style="text-align:right;"> 17.40637 </td>
  </tr>
</tbody>
</table>

<table class="table" style="font-size: 18px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> prof </th>
   <th style="text-align:left;"> id </th>
   <th style="text-align:left;"> position </th>
   <th style="text-align:right;"> cIntensity </th>
   <th style="text-align:right;"> vIntensity </th>
   <th style="text-align:left;"> group </th>
   <th style="text-align:right;"> groupDev </th>
   <th style="text-align:right;"> intensity_diff </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 43 </td>
   <td style="text-align:left;"> nat </td>
   <td style="text-align:left;"> 13 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 69.39163 </td>
   <td style="text-align:right;"> 84.15049 </td>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 0.5 </td>
   <td style="text-align:right;"> 14.75886 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 44 </td>
   <td style="text-align:left;"> nat </td>
   <td style="text-align:left;"> 14 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 70.74004 </td>
   <td style="text-align:right;"> 82.05478 </td>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 0.5 </td>
   <td style="text-align:right;"> 11.31475 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 45 </td>
   <td style="text-align:left;"> nat </td>
   <td style="text-align:left;"> 15 </td>
   <td style="text-align:left;"> vcv </td>
   <td style="text-align:right;"> 70.53420 </td>
   <td style="text-align:right;"> 82.41582 </td>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 0.5 </td>
   <td style="text-align:right;"> 11.88162 </td>
  </tr>
</tbody>
</table>

---
background-image: url(./index_files/figure-html/spirantization-p1-1.png)
background-position: 98% 55%
background-size: 550px

### Deviation coding - Spanish spirantization

.pull-left[

```r
lm(intensity_diff ~ groupDev, data = spir_tidy)
```

<table class="table" style="font-size: 18px; width: auto !important; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 14.002 </td>
   <td style="text-align:right;"> 0.481 </td>
   <td style="text-align:right;"> 29.091 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupDev </td>
   <td style="text-align:right;"> -6.002 </td>
   <td style="text-align:right;"> 0.963 </td>
   <td style="text-align:right;"> -6.235 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>

]

.footnote[

- What is the intercept? The mean of group means.

- What is `groupDev`? The overall effect of group.

- Calculates the difference between group means

]

---

### Deviation coding - Summary

#### Advantages

- Deviation coding gives omnibus assessment of main effects

- Avoids (possibly) arbitrary selection of reference group

- Makes MRC even more like ANOVA (advantage for researchers accustomed to ANOVA)

#### Disadvantages

- Involves releveling, adding columns to dataframe or resetting factor

- May not always provide useful information?

- Doesn't take advantage of the intercept

---
layout: false
background-color: lightblue

# A note about ANOVA

.pull-left[
#### MRC

```

Call:
lm(formula = intensity_diff ~ groupSum, data = spir_tidy)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.1762 -2.1004 -0.5187  1.6664  7.9063

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  14.0016     0.4813  29.091  < 2e-16 ***
groupSum     -3.0009     0.4813  -6.235 1.66e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.044 on 43 degrees of freedom
Multiple R-squared:  0.4748,	Adjusted R-squared:  0.4626 
F-statistic: 38.87 on 1 and 43 DF,  p-value: 1.665e-07
```

]

.pull-right[

#### ANOVA

</br>

```
aov(intensity_diff ~ groupSum, data = spir_tidy)
```

```
            Df Sum Sq Mean Sq F value   Pr(>F)    
groupSum     1  360.2   360.2   38.87 1.66e-07 ***
Residuals   43  398.5     9.3                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

.footnote[
.content-box-red[
Notice: the F-ratio and DFs
]
]
]

---
background-color: lightblue

# A note about ANOVA

### We can make them exactly the same...

.pull-left[
#### MRC

```
Analysis of Variance Table

Response: intensity_diff
          Df Sum Sq Mean Sq F value    Pr(>F)    
groupSum   1 360.22  360.22  38.873 1.665e-07 ***
Residuals 43 398.46    9.27                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

]

.pull-right[

#### ANOVA

]

---
background-color: lightblue

# A note about ANOVA

.pull-left[
#### T-test

```

Welch Two Sample t-test

data:  intensity_diff by group
t = 6.8355, df = 35.836, p-value = 5.528e-08
alternative hypothesis: true difference in means between group learner and group native is not equal to 0
95 percent confidence interval:
 4.220795 7.782855
sample estimates:
mean in group learner  mean in group native 
             17.00251              11.00068 
```
]

.pull-right[

#### MRC

```
             Estimate Std. Error   t value     Pr(>|t|)
(Intercept) 14.001595  0.4813115 29.090506 6.433300e-30
groupSum    -3.000913  0.4813115 -6.234865 1.664552e-07
```

</br>

#### ANOVA

]

---
background-color: lightblue

# A note about ANOVA

### What it means

.Large[

- With a 1-way between-subjects design, MRC, ANOVA and the t-test essentially 
do the exact same thing
- This is because they are all part of the general linear model (remember when 
I said 'everything is a linear model'?)
- ANOVA and t-tests **require** categorical factors
- T-test can only compare two groups (1 factor, 2 levels)
- ANOVA can include many categorical factors with numerous levels
- Neither the t-test nor ANOVA can model continuous predictors
- MRC includes continuous and discontinuous predictors and gives us more 
information...
  - Parameter estimates
  - Comparisons
  - R<sup>2</sup>

]

---
class: title-slide-section-grey, middle

# Coding categorical predictors

## .RUred[Contrast coding]

---
layout: true

# Coding categorical predictors

---

### Contrast coding

- Ideally we want to avoid tests that are not theoretically motivated
  1. Because they are not interesting
  2. To avoid capitalizing on chance

- One way to avoid atheoretical comparisons is to use orthogonal contrast 
coding

- This mean the researcher specifies the contrasts to test by hand-coding the 
dummy vector

- For example, imagine we want to test the `adv` group with the native controls, 
as well as the `beg` and `adv` learner groups together with the natives.

.pull-left[

.center[**Adv. vs. Natives**]

<table class="table" style="font-size: 18px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> prof </th>
   <th style="text-align:right;"> mean_int_diff </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> adv </td>
   <td style="text-align:right;"> 15.506 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> nat </td>
   <td style="text-align:right;"> 11.001 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

.center[**Learners vs. Natives**]

<table class="table" style="font-size: 18px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> group </th>
   <th style="text-align:right;"> mean_int_diff </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> 17.003 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 11.001 </td>
  </tr>
</tbody>
</table>
]

---

### Contrast coding

.left-column[

1. Manually code contrasts
1. Put them in a matrix
1. Calculate general inverse of matrix
1. Fit model using `contrasts` argument

<table class="table" style="font-size: 18px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> group </th>
   <th style="text-align:right;"> mean_int_diff </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> learner </td>
   <td style="text-align:right;"> 17.003 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> native </td>
   <td style="text-align:right;"> 11.001 </td>
  </tr>
</tbody>
</table>

]

.right-column[

```r
contrast1 <- c(1,     0, -1) # adv       vs. native
contrast2 <- c(1/2, 1/2, -1) # adv + beg vs. native
prof_contrasts <- matrix(c(contrast1, contrast2), ncol = 2)
prof_contrasts <- t(MASS::ginv(prof_contrasts))
show(prof_contrasts)
```

```
##               [,1]       [,2]
## [1,]  1.000000e+00 -0.6666667
## [2,] -1.000000e+00  1.3333333
## [3,]  1.110223e-16 -0.6666667
```

```
lm(intensity_diff ~ prof, data = spir_tidy, contrasts = list(prof = prof_contrasts))
```

<table class="table" style="font-size: 17px; width: auto !important; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 15.0019 </td>
   <td style="text-align:right;"> 0.4187 </td>
   <td style="text-align:right;"> 35.8322 </td>
   <td style="text-align:right;"> 0e+00 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> prof1 </td>
   <td style="text-align:right;"> 4.5055 </td>
   <td style="text-align:right;"> 1.0255 </td>
   <td style="text-align:right;"> 4.3934 </td>
   <td style="text-align:right;"> 1e-04 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> prof2 </td>
   <td style="text-align:right;"> 6.0018 </td>
   <td style="text-align:right;"> 0.8881 </td>
   <td style="text-align:right;"> 6.7578 </td>
   <td style="text-align:right;"> 0e+00 </td>
  </tr>
</tbody>
</table>
]

---

### Contrast coding - Summary

#### Advantages

- Allows you to test only the comparisons you want

- Test only theoretically motivated comparisons

#### Disadvantages

- Requires thought, coding by hand

- What is the benefit of estimating the intercept?

---
layout: false
class: title-slide-section-red, middle

# Transforming continuous predictors

---
layout: true

# Transforming continuous predictors

---

### Overview

.large[
- What is a transformation?  
The process of converting data from one format to another

- We can use linear and non-linear transformations to make our models more 
interpretable (especially in conjunction with coding discontinuous predictors)

- Transformations do not affect the model fit, predictions, R<sup>2</sup>, etc.

- The goal is to make our model easier to understand
]

<p></p>

.large[
- Examples
  - Log transformation
  - Centering
  - Standardizing
]

---
class: title-slide-section-grey, middle

## .RUred[Log transformations]

---

### Log transformations

#### What?

- A log transformation converts the predictor to a non-linear logarithmic scale

- A log scale is one that is based on orders of magnitude

- The value of each 'tick' on the scale is the value of the previous 'tick' 
multiplied by a base value

<p></p>

- Some naturally occurring phenomena are better understood in a logarithmic 
scale 
  - Seismic activity (Richter scale)
  - Sound (loudness, dB)
  - Odds (logits)

#### Why?

- A log transformation can make a predictor variable more normal (i.e., correct 
for skewness)

- Log scales show percent change in an easily interpretable manner

---
layout: false
class: middle

.pull-left[

<table class="table" style="font-size: 18px; ">
 <thead>
  <tr>
   <th style="text-align:left;"> Typical experience </th>
   <th style="text-align:right;"> Pressure (μPa) </th>
   <th style="text-align:right;"> Decibel level </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Absolute threshold </td>
   <td style="text-align:right;"> 20 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Faint whisper </td>
   <td style="text-align:right;"> 200 </td>
   <td style="text-align:right;"> 20 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Quiet office </td>
   <td style="text-align:right;"> 2000 </td>
   <td style="text-align:right;"> 40 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Conversation </td>
   <td style="text-align:right;"> 20000 </td>
   <td style="text-align:right;"> 60 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> City bus </td>
   <td style="text-align:right;"> 200000 </td>
   <td style="text-align:right;"> 80 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Subway train </td>
   <td style="text-align:right;"> 2000000 </td>
   <td style="text-align:right;"> 100 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Loud thunder </td>
   <td style="text-align:right;"> 20000000 </td>
   <td style="text-align:right;"> 120 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Pain and damage </td>
   <td style="text-align:right;"> 200000000 </td>
   <td style="text-align:right;"> 140 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

]

---
layout: true

# Transforming continuous predictors

---

.pull-left[

### Log transformations

- It is most common to use the natural log (log<sub>e</sub>)<sup>1</sup>

- The values must be positive in order to be log-transformed

- We can do it in R with the `log()` function

- Consider the `language_diversity` data

```r
tidy_lang_div <- language_diversity %>% 
  dplyr::filter(Continent != 'Americas') %>% 
  pivot_wider(names_from = "Measurement", values_from = "Value") %>% 
  select(Country, MGS, Langs) %>% 
* mutate(log_langs = log(Langs))
```

]

.pull-right[

]

.footnote[.scriptsize[<sup>1</sup> A mathematical constant, irrational and transcendental number equal to ~ 2.718281828459]]

---

#### Log transformations

.pull-left[

<table class="table" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> -16.67491 </td>
   <td style="text-align:right;"> 45.328814 </td>
   <td style="text-align:right;"> -0.3678655 </td>
   <td style="text-align:right;"> 0.7143855 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> MGS </td>
   <td style="text-align:right;"> 18.08001 </td>
   <td style="text-align:right;"> 6.264528 </td>
   <td style="text-align:right;"> 2.8860932 </td>
   <td style="text-align:right;"> 0.0055630 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

<table class="table" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 2.4704391 </td>
   <td style="text-align:right;"> 0.3094346 </td>
   <td style="text-align:right;"> 7.983719 </td>
   <td style="text-align:right;"> 0.0e+00 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> MGS </td>
   <td style="text-align:right;"> 0.2151393 </td>
   <td style="text-align:right;"> 0.0427644 </td>
   <td style="text-align:right;"> 5.030798 </td>
   <td style="text-align:right;"> 5.5e-06 </td>
  </tr>
</tbody>
</table>

.tiny[
- A 1 month change in MGS is associated with a 22% increase in the number of 
languages.
]
]

---

### Log transformations - Summary

#### Advantages

- Can correct for skewness (note: this is not a guaranteed fix for non-normal 
data)

<p></p>

- Fairly easy to interpret
  - Coefficients with low values can be interpreted directly as percents
  - Coefficients with higher values should be exponentiated (`exp()`)

<p></p>

- If the criterion *and* the predictors are log transformed, parameter estimates 
interpreted as "for each 1% change in x, the predicted difference in y is \_\_%"

---
class: title-slide-section-grey, middle

## .RUred[Centering]

---
exclude: true

---

### Centering

.pull-left[

- As mentioned previously, the intercepts in many of our models are not 
meaningful
  - This is particularly true when the average of a continuous predictor is far 
  from 0
  - Or when having a value of 0 makes no sense, i.e., IQ, vocab size, etc.

<p></p>

- An easy way to fix this is to center the predictor

- A centered predictor has a mean of 0

- The standard deviation remains the same

- To center a predictor you subtract the mean from each value

]

.pull-right[

```r
mtcars %>% 
* mutate(disp_c = disp - mean(disp)) %>%
  summarize(
    mean_disp = mean(disp) %>% round(2), 
    sd_disp = sd(disp) %>% round(2), 
    mean_disp_c = mean(disp_c) %>% round(2), 
    sd_disp_c = sd(disp_c) %>% round(2)) %>% 
  t(.) 
```

```
              [,1]
mean_disp   230.72
sd_disp     123.94
mean_disp_c   0.00
sd_disp_c   123.94
```
]

---

### Centering - Ex. A

.center[
<table class="table" style="font-size: 18px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 15.971 </td>
   <td style="text-align:right;"> 3.905 </td>
   <td style="text-align:right;"> 4.090 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupb </td>
   <td style="text-align:right;"> -12.700 </td>
   <td style="text-align:right;"> 5.523 </td>
   <td style="text-align:right;"> -2.300 </td>
   <td style="text-align:right;"> 0.024 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x </td>
   <td style="text-align:right;"> 0.246 </td>
   <td style="text-align:right;"> 0.322 </td>
   <td style="text-align:right;"> 0.765 </td>
   <td style="text-align:right;"> 0.446 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupb:x </td>
   <td style="text-align:right;"> 1.068 </td>
   <td style="text-align:right;"> 0.455 </td>
   <td style="text-align:right;"> 2.347 </td>
   <td style="text-align:right;"> 0.022 </td>
  </tr>
</tbody>
</table>
]

---
layout: false
background-image: url(./index_files/figure-html/centered-plot-a-1.png)
background-position: 50% 8%
background-size: 1025px

.pull-left[
<table class="table" style="font-size: 15px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 18.949 </td>
   <td style="text-align:right;"> 0.324 </td>
   <td style="text-align:right;"> 58.505 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupb </td>
   <td style="text-align:right;"> 0.218 </td>
   <td style="text-align:right;"> 0.458 </td>
   <td style="text-align:right;"> 0.476 </td>
   <td style="text-align:right;"> 0.635 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x_c </td>
   <td style="text-align:right;"> 0.246 </td>
   <td style="text-align:right;"> 0.322 </td>
   <td style="text-align:right;"> 0.765 </td>
   <td style="text-align:right;"> 0.446 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupb:x_c </td>
   <td style="text-align:right;"> 1.068 </td>
   <td style="text-align:right;"> 0.455 </td>
   <td style="text-align:right;"> 2.347 </td>
   <td style="text-align:right;"> 0.022 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

<table class="table" style="font-size: 15px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 19.058 </td>
   <td style="text-align:right;"> 0.229 </td>
   <td style="text-align:right;"> 83.215 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum </td>
   <td style="text-align:right;"> -0.109 </td>
   <td style="text-align:right;"> 0.229 </td>
   <td style="text-align:right;"> -0.476 </td>
   <td style="text-align:right;"> 0.635 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x_c </td>
   <td style="text-align:right;"> 0.780 </td>
   <td style="text-align:right;"> 0.228 </td>
   <td style="text-align:right;"> 3.429 </td>
   <td style="text-align:right;"> 0.001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum:x_c </td>
   <td style="text-align:right;"> -0.534 </td>
   <td style="text-align:right;"> 0.228 </td>
   <td style="text-align:right;"> -2.347 </td>
   <td style="text-align:right;"> 0.022 </td>
  </tr>
</tbody>
</table>
]

---
layout: true

# Transforming continuous predictors

---

### Centering - Ex. B

.center[
<table class="table" style="font-size: 18px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 5.592 </td>
   <td style="text-align:right;"> 3.481 </td>
   <td style="text-align:right;"> 1.606 </td>
   <td style="text-align:right;"> 0.112 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupd </td>
   <td style="text-align:right;"> -3.090 </td>
   <td style="text-align:right;"> 4.924 </td>
   <td style="text-align:right;"> -0.628 </td>
   <td style="text-align:right;"> 0.532 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x </td>
   <td style="text-align:right;"> 0.261 </td>
   <td style="text-align:right;"> 0.287 </td>
   <td style="text-align:right;"> 0.910 </td>
   <td style="text-align:right;"> 0.366 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupd:x </td>
   <td style="text-align:right;"> 1.699 </td>
   <td style="text-align:right;"> 0.406 </td>
   <td style="text-align:right;"> 4.189 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
</tbody>
</table>
]

---
layout: false
background-image: url(./index_files/figure-html/centered-plot-b-1.png)
background-position: 50% 8%
background-size: 1025px

.pull-left[
<table class="table" style="font-size: 15px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 8.750 </td>
   <td style="text-align:right;"> 0.289 </td>
   <td style="text-align:right;"> 30.303 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupd </td>
   <td style="text-align:right;"> 17.465 </td>
   <td style="text-align:right;"> 0.408 </td>
   <td style="text-align:right;"> 42.770 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x_c </td>
   <td style="text-align:right;"> 0.261 </td>
   <td style="text-align:right;"> 0.287 </td>
   <td style="text-align:right;"> 0.910 </td>
   <td style="text-align:right;"> 0.366 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupd:x_c </td>
   <td style="text-align:right;"> 1.699 </td>
   <td style="text-align:right;"> 0.406 </td>
   <td style="text-align:right;"> 4.189 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

<table class="table" style="font-size: 15px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 17.482 </td>
   <td style="text-align:right;"> 0.204 </td>
   <td style="text-align:right;"> 85.625 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum </td>
   <td style="text-align:right;"> -8.732 </td>
   <td style="text-align:right;"> 0.204 </td>
   <td style="text-align:right;"> -42.770 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x_c </td>
   <td style="text-align:right;"> 1.111 </td>
   <td style="text-align:right;"> 0.203 </td>
   <td style="text-align:right;"> 5.476 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum:x_c </td>
   <td style="text-align:right;"> -0.850 </td>
   <td style="text-align:right;"> 0.203 </td>
   <td style="text-align:right;"> -4.189 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>
]

---
layout: true

# Transforming continuous predictors

---
background-color: black
class: middle

## .white[Spanish spirantization (revisited)]

---

### Sum coding - Spanish spirantization (revisited)

.pull-left[

- We get j-1 contrasts
- `lm()` conducts tests of independence: H<sub>0</sub> `$\neq$` 0
- If we center the criterion, the mean = 0
- We can remove the intercept and test whether each level is = 0

```r
lm(int_diff_c ~ -1 + profSum, data = spir_tidy) 
```

<table class="table" style="font-size: 18px; width: auto !important; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> profSumadv </td>
   <td style="text-align:right;"> 0.504 </td>
   <td style="text-align:right;"> 0.725 </td>
   <td style="text-align:right;"> 0.695 </td>
   <td style="text-align:right;"> 0.491 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> profSumbeg </td>
   <td style="text-align:right;"> 3.497 </td>
   <td style="text-align:right;"> 0.725 </td>
   <td style="text-align:right;"> 4.822 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> profSumnat </td>
   <td style="text-align:right;"> -4.001 </td>
   <td style="text-align:right;"> 0.725 </td>
   <td style="text-align:right;"> -5.518 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
</tbody>
</table>

]

---

### Centering - Summary

#### Advantages

- We can make the intercept relevant

- Combined with sum/deviation coding, interactions are much easier to interpret

- Centering uncorrelates interaction terms

#### Disadvantages

- ???

<p></p>

- In some cases it might not help (but it doesn't hurt)
  - When 0 is meaningful
  - When there are no interaction terms

---
class: title-slide-section-grey, middle

## .RUred[Standardizing]

---

### Standardizing

.pull-left[
- Sometimes we are interested in comparing effect sizes within/between 
models

- Typically this is not possible because the parameter estimates are on different 
scales

- We know we can calculate z-scores (standardize) variables when we 
want to "put them on the same scale"

- When the criterion and the predictors are standardized the model coefficients 
are beta weights (assess effect size and are comparable between models)

- A standardized variable has a mean of 0 and a standard deviation of 1
]

.pull-right[

```r
mtcars %>% 
* mutate(disp_std = (disp - mean(disp)) / sd(disp)) %>%
  summarize(
    mean_disp = mean(disp), 
    sd_disp = sd(disp), 
    mean_disp_std = mean(disp_std), 
    sd_disp_std = sd(disp_std)
  ) %>% 
  pivot_longer(cols = everything(), names_to = "metric", values_to = "val") %>% 
  mutate(val = round(val)) %>% 
  knitr::kable()
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> metric </th>
   <th style="text-align:right;"> val </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> mean_disp </td>
   <td style="text-align:right;"> 231 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> sd_disp </td>
   <td style="text-align:right;"> 124 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> mean_disp_std </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> sd_disp_std </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
</tbody>
</table>
]

---
layout: false
background-image: url(./index_files/figure-html/std-plot-a-1.png)
background-position: 50% 8%
background-size: 1025px

.pull-left[
<table class="table" style="font-size: 15px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 19.058 </td>
   <td style="text-align:right;"> 0.229 </td>
   <td style="text-align:right;"> 83.215 </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum </td>
   <td style="text-align:right;"> -0.109 </td>
   <td style="text-align:right;"> 0.229 </td>
   <td style="text-align:right;"> -0.476 </td>
   <td style="text-align:right;"> 0.635 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x_c </td>
   <td style="text-align:right;"> 0.780 </td>
   <td style="text-align:right;"> 0.228 </td>
   <td style="text-align:right;"> 3.429 </td>
   <td style="text-align:right;"> 0.001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum:x_c </td>
   <td style="text-align:right;"> -0.534 </td>
   <td style="text-align:right;"> 0.228 </td>
   <td style="text-align:right;"> -2.347 </td>
   <td style="text-align:right;"> 0.022 </td>
  </tr>
</tbody>
</table>
]

.pull-right[
<table class="table" style="font-size: 15px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 0.000 </td>
   <td style="text-align:right;"> 0.103 </td>
   <td style="text-align:right;"> 0.000 </td>
   <td style="text-align:right;"> 1.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum </td>
   <td style="text-align:right;"> -0.049 </td>
   <td style="text-align:right;"> 0.103 </td>
   <td style="text-align:right;"> -0.476 </td>
   <td style="text-align:right;"> 0.635 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x_std </td>
   <td style="text-align:right;"> 0.355 </td>
   <td style="text-align:right;"> 0.103 </td>
   <td style="text-align:right;"> 3.429 </td>
   <td style="text-align:right;"> 0.001 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum:x_std </td>
   <td style="text-align:right;"> -0.243 </td>
   <td style="text-align:right;"> 0.103 </td>
   <td style="text-align:right;"> -2.347 </td>
   <td style="text-align:right;"> 0.022 </td>
  </tr>
</tbody>
</table>
]

---
background-image: url(./index_files/figure-html/std-plot-b-1.png)
background-position: 50% 8%
background-size: 1025px

.pull-left[
<table class="table" style="font-size: 15px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 17.482 </td>
   <td style="text-align:right;"> 0.204 </td>
   <td style="text-align:right;"> 85.625 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum </td>
   <td style="text-align:right;"> -8.732 </td>
   <td style="text-align:right;"> 0.204 </td>
   <td style="text-align:right;"> -42.770 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x_c </td>
   <td style="text-align:right;"> 1.111 </td>
   <td style="text-align:right;"> 0.203 </td>
   <td style="text-align:right;"> 5.476 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum:x_c </td>
   <td style="text-align:right;"> -0.850 </td>
   <td style="text-align:right;"> 0.203 </td>
   <td style="text-align:right;"> -4.189 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>
]

.pull-right[
<table class="table" style="font-size: 15px; margin-left: auto; margin-right: auto;">
 <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>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:right;"> 0.000 </td>
   <td style="text-align:right;"> 0.022 </td>
   <td style="text-align:right;"> 0.000 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum </td>
   <td style="text-align:right;"> -0.962 </td>
   <td style="text-align:right;"> 0.022 </td>
   <td style="text-align:right;"> -42.770 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> x_std </td>
   <td style="text-align:right;"> 0.124 </td>
   <td style="text-align:right;"> 0.023 </td>
   <td style="text-align:right;"> 5.476 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> groupSum:x_std </td>
   <td style="text-align:right;"> -0.095 </td>
   <td style="text-align:right;"> 0.023 </td>
   <td style="text-align:right;"> -4.189 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table>
]

---

### Standardizing - Summary

#### Advantages

- Makes intercept meaningful

- Model output provides beta weights (effect size)

- Beta weights can be compared with other variables (even in other models)

#### Disadvantages

- Interpreting predictions is more difficult (but we don't usually do that)

---

### Practice

[model tuning practice](./assets/tuning_walkthrough/tuning.zip)

---
exclude: true

(Wickham and Grolemund, 2016; Hardy, 1993b; Hardy, 1993c; Hardy, 1993d)

---
layout: false
class: title-slide-final, left

# References

[1] M. Hardy. "Assessing Group Differences in Effects". In: _Regression
with Dummy Variables_. Ed. by M. Hardy. Sage University Paper Series on
Quantitative Applications in the Social Sciences - 93. Newbury Park,
CA: Sage, 1993, pp. 29-63. ISBN: 9780803951280.

[2] M. Hardy. "Creating Dummy Variables". In: _Regression with Dummy
Variables_. Ed. by M. Hardy. Sage University Paper Series on
Quantitative Applications in the Social Sciences - 93. Newbury Park,
CA: Sage, 1993, pp. 7-17. ISBN: 9780803951280.

[3] M. Hardy. "Using Dummy Variables as Regressors". In: _Regression
with Dummy Variables_. Ed. by M. Hardy. Sage University Paper Series on
Quantitative Applications in the Social Sciences - 93. Newbury Park,
CA: Sage, 1993, pp. 18-28. ISBN: 9780803951280.

[4] H. Wickham and G. Grolemund. _R for Data Science: Import, Tidy,
Transform, Visualize, and Model Data_. O'Reilly Media, 2016.

Figueredo, A. J. (2013). Continuous and Categorical Predictors. *Statistical Methods in Psychological Research*.

Figueredo, A. J. (2013). The General Linear Model: ANOVA. *Statistical Methods in Psychological Research*.