Quantitative Methods for LEL

.title[
# Quantitative Methods for LEL
]
.subtitle[
## Week 5
]
.author[
### Dr Stefano Coretta
]
.institute[
### University of Edinburgh
]
.date[
### 2023/10/17
]

---

---

## Summary from last week

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
- **Probability distributions**

- Continuous vs discrete distributions.
  - Describe distributions: density functions and parameters.
  
- **Modelling continuous variables**

- The Gaussian distribution has two parameters: mean `$\mu$` and SD `$\sigma$`.

- We can describe `$\mu$` and `$\sigma$` as probability distributions and estimate the (hyper-)parameters of those probability distributions.
  
- `brm()` from brms.
]

---

## Comparing groups

---

```r
polite
```

```
## # A tibble: 224 × 27
##    subject gender birthplace musicstudent months_ger scenario task  attitude total_duration
##    <chr>   <chr>  <chr>      <chr>             <dbl>    <dbl> <chr> <fct>             <dbl>
##  1 F1      F      seoul_area yes                  18        6 not   informal          55.2 
##  2 F1      F      seoul_area yes                  18        6 not   polite            28.5 
##  3 F1      F      seoul_area yes                  18        7 not   informal          60.3 
##  4 F1      F      seoul_area yes                  18        7 not   polite            40.8 
##  5 F1      F      seoul_area yes                  18        1 dct   polite            18.4 
##  6 F1      F      seoul_area yes                  18        1 dct   informal          13.6 
##  7 F1      F      seoul_area yes                  18        2 dct   polite             5.22
##  8 F1      F      seoul_area yes                  18        2 dct   informal           4.25
##  9 F1      F      seoul_area yes                  18        3 dct   polite             6.79
## 10 F1      F      seoul_area yes                  18        3 dct   informal           4.13
## # ℹ 214 more rows
## # ℹ 18 more variables: articulation_rate <dbl>, f0mn <dbl>, f0sd <dbl>, f0range <dbl>, inmn <dbl>,
## #   insd <dbl>, inrange <dbl>, shimmer <dbl>, jitter <dbl>, HNRmn <dbl>, H1H2 <dbl>,
## #   breath_count <dbl>, filler_count <dbl>, hiss_count <dbl>, nasal_count <dbl>, sil_count <dbl>,
## #   ya_count <dbl>, yey_count <dbl>
```

---

HNR is the ratio between periodicity and noise in the voice signal. Lower HNR values indicate less modal voice (i.e. creakier or breathier).

---

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
- We want to estimate the probability distribution of harmonics-to-noise ratio (HNR) in Korean speakers.

- Let's assume it is a Gaussian probability distribution.
  
  - So, `$\text{HNR} \sim Gaussian(\mu, \sigma)$`.
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
- But we also want to estimate the difference in the mean (`$\mu$`) in the informal vs polite attitude conditions!
  
  - In other words, we need to allow the model to estimate `$\mu$` based on attitude (informal vs polite).
]

---

--
.f3[

$$
`\begin{align}
\mu_{attitude=informal} & \sim Gaussian(\mu_1, \sigma_1) \\
\mu_{attitude=polite} & \sim Gaussian(\mu_2, \sigma_2) \\
\sigma & \sim TruncGaussian(\mu_3, \sigma_3)
\end{align}`
$$
]

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
That would be great, but alas these models are set up in a different way by default!
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
These models are called *linear models*, or *regression models*, or *linear regression models*. They are a generalisation of the formula of a line:

$$
y = a + b \cdot x
$$
]

---

## Linear models

---

---

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
.f3.center[
`brm(y ~ x)`
]

- The variable `y` to the left of `~` is called the **outcome variable** (aka response variable, dependent variable).

- The variable `x` to the right of `~` is called the **predictor variable** (aka independent variable).
]

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
There is a catch: linear models are meant to be used with numeric variables.

😱 But... `attitude` is not a numeric predictor! 
]

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt2[
Let's talk about coding of categorical variables!
]

---

## Coding categorical predictors

---

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
**NOTE**: What follows is for you how to understand how coding works, but remember that this is done automatically by R for you so you never have to do it by hand!!!
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
**Categorical predictors can be coded using numbers.**

There are two common types of coding systems:

- **Treatment** coding (this week's focus).

- **Sum** coding.
]

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt2[
As with anything else in stats, **naming of coding systems is not an established matter** and the same coding can have different names, and vice versa the same name could refer to different systems.

For an excellent overview, see <https://debruine.github.io/faux/articles/contrasts.html>.
]

---

**Treatment** coding uses `0` and `1` to code categorical predictors.

.f3[
|               | attitude_pol |
| ------------- | -------:     |
| informal      | 0            |
| polite        | 1            |
]

<br>

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
With **treatment coding**, the first level in the predictor is the **reference level** (in `attitude`, it is `informal`).
]

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
**Level order**

By default, the ordering of levels in a categorical predictor is based on alphabetical order. But you can specify the order manually using the `factor()` function. (You will see how in the tutorial).
]

---

```
## # A tibble: 224 × 3
##    HNRmn attitude attitude_pol
##    <dbl> <fct>           <dbl>
##  1  18.1 informal            0
##  2  17.8 polite              1
##  3  17   informal            0
##  4  17.1 polite              1
##  5  18.5 polite              1
##  6  18.8 informal            0
##  7  20.9 polite              1
##  8  14.6 informal            0
##  9  20.6 polite              1
## 10  16.1 informal            0
## # ℹ 214 more rows
```

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
**NOTE**: Remember that this is done automatically by R under the hood for you so you never have to do it by hand!!!

We will use `attitude` in the model.
]

---

## Comparing groups

---

.f4[
$$
`\begin{align}
\text{HNR} & \sim Gaussian(\mu, \sigma) \\
\mu & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
[\text{formula of line: } y & = a + b \cdot x]
\end{align}`
$$
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
**For [attitude = informal]**, `$attitude_{pol} = 0$`:

$$
`\begin{align}
\mu & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
    & = \beta_0 + \beta_1 \cdot 0 \\
    & = \beta_0
\end{align}`
$$

**For [attitude = polite]**, `$attitude_{pol} = 1$`:

$$
`\begin{align}
\mu & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
    & = \beta_0 + \beta_1 \cdot 1 \\
    & = \beta_0 + \beta_1
\end{align}`
$$
]

---

.f4[
$$
`\begin{align}
\text{HNR} & \sim Gaussian(\mu, \sigma) \\
\mu & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
[\text{formula of line: } y & = a + b \cdot x]
\end{align}`
$$
]

<br>

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[

- `$\beta_0$`: (`Intercept`) mean HNR when [attitude = informal], the reference level of `attitude`.

- `$\beta_1$`: **difference** of mean HNR between [attitude = polite] and [attitude = informal].
  
$$
`\begin{align}
\beta_1 & = \mu_{pol} - \mu_{inf} \\
        & = (\beta_0 + \beta_1) - (\beta_0) \\
        & = \beta_1
\end{align}`
$$
]

---

.f4[
$$
`\begin{align}
\text{HNR} & \sim Gaussian(\mu, \sigma) \\
\mu & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
[\text{formula of line: } y & = a + b \cdot x]
\end{align}`
$$
]

<br>

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
`$\beta_1$` is the **effect of [attitude = polite]** on mean HNR relative to the baseline mean HNR when [attitude = informal].
]

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
With **treatment coding**, the second level [polite] is compared against the reference level [informal].
]

---

---

## Comparing groups

---

.f4[
$$
`\begin{align}
\text{HNR} & \sim Gaussian(\mu, \sigma) \\
\mu & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
\beta_0 & \sim Gaussian(\mu_0, \sigma_0) \\
\beta_1 & \sim Gaussian(\mu_1, \sigma_1) \\
\sigma & \sim TruncGaussian(\mu_2, \sigma_2)
\end{align}`
$$
]

<br>

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
We need to estimate:

- The **probability distribution of `$\beta_0$`**, i.e. the mean HNR when [attitude = informal].

- The **probability distribution of `$\beta_1$`**, i.e. the difference in mean HNR between [attitude = polite] and [attitude = informal].

- The **probability distribution of `$\sigma$`**, i.e. the standard deviation. 
]

---

<br>

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
In other words, we need to estimate the following hyperparameters: `$\mu_0, \sigma_0, \mu_1, \sigma_1, \mu_2, \sigma_2$`.
]

---

## Modelling HNR by attitude

---

```r
# Attach the brms package
library(brms)

# Run a Bayesian model
vot_bm <- brm(
  # This is the formula of the model.
  HNRmn ~ 1 + attitude,
  # This is the probability distribution family.
  family = gaussian(),
  # And the data.
  data = polite
)
```

---

.f2.center[
`HNRmn ~ 1 + attitude`
]

**Read as**: Model HNR values (`HNRmn`) as a function of (`~`) the mean (`1`) and attitude (`attitude`).

.f2.center[
`HNRmn ~ attitude`
]

We can omit the `1` (it is implied by default when there are other terms in the formula)! **You can read as**: Model HNR values (`HNRmn`) as a function of (`~`) attitude (`attitude`).

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
- `HNRmn` is the **outcome variable** (aka response variable, dependent variable).

- `attitude` is the **predictor variable** (aka independent variables).
]

---

```r
vot_bm <- brm(
  # This is the formula of the model. We can omit `1`.
  HNRmn ~ attitude,
  # This is the probability distribution family.
  family = gaussian(),
  # And the data.
  data = polite
)
```

---

```
##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: HNRmn ~ attitude 
##    Data: polite (Number of observations: 224) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Population-Level Effects: 
##                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept         16.27      0.16    15.97    16.59 1.00     4136     3237
## attitudepolite     1.25      0.23     0.80     1.70 1.00     4396     2937
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     1.69      0.08     1.54     1.86 1.00     4065     2979
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

---

```
## Population-Level Effects: 
##                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept         16.27      0.16    15.97    16.59 1.00     4136     3237
## attitudepolite     1.25      0.23     0.80     1.70 1.00     4396     2937
```

<br>

.f4[
$$
`\begin{align}
\mu     & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
\beta_0 & \sim Gaussian(16.27, 0.16) \\
\beta_1 & \sim Gaussian(\mu_1, \sigma_1)
\end{align}`
$$
]

- Parameter `$\beta_0$` (`Intercept`): this is `$\mu$` when [attitude = informal].

- **Estimate**: `$\mu_0 = 16.27$` dB.

- **Est.Error**: `$\sigma_0 = 0.16$` dB.

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
When [attitude = informal], the probability distribution of `$\mu$` is the probability distribution of `$\beta_0$`, which is `$Gaussian(16.27, 0.16)$`.
]

---

---

There is a 95% probability that mean HNR when [attitude = informal] is between 15.97 and 16.59 dB.

---

<br>

.f4[
$$
`\begin{align}
\mu     & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
\beta_0 & \sim Gaussian(16.27, 0.16) \\
\beta_1 & \sim Gaussian(1.25, 0.23)
\end{align}`
$$
]

- Parameter: **`$\beta_1$`**: difference of `$\mu$` when [attitude = polite] and [attitude = informal].

- **Estimate**: `$\mu_1 = 1.25$` dB.

- **Est.Error**: `$\sigma_1 = 0.23$` dB.

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
The probability distribution of the difference between `$\mu_{\text{attitude = polite}}$` and `$\mu_{\text{attitude = informal}}$` is the probability distribution of `$\beta_1$`, which is `$Gaussian(1.25, 0.23)$`.
]

---

---

There is 95% probability that the difference in mean HNR between `$\mu_{\text{attitude = polite}}$` and `$\mu_{\text{attitude = informal}}$` is between 0.8 and 1.7 dB.

---

.f4[
$$
`\begin{align}
\mu     & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
\beta_0 & \sim Gaussian(16.27, 0.16) \\
\beta_1 & \sim Gaussian(1.25, 0.23)
\end{align}`
$$
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
What is the average `$\mu$` when [attitude = polite]?

$$
`\begin{align}
\mu & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
    & = \beta_0 + \beta_1 \cdot 1 \\
    & = \beta_0 + \beta_1 \\
    & = 16.27 + 1.25 \\
    & = 17.52
\end{align}`
$$
]

---

```
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     1.69      0.08     1.54     1.86 1.00     4065     2979
```

<br>

.f4[
$$
`\begin{align}
\text{HNR} & \sim Gaussian(\mu, \sigma) \\
\mu & = \beta_0 + \beta_1 \cdot attitude_{pol} \\
\beta_0 & \sim Gaussian(16.27, 0.16) \\
\beta_1 & \sim Gaussian(1.25, 0.23) \\
\sigma & \sim TruncGaussian(1.69, 0.08)
\end{align}`
$$
]

---

**Reporting**

> We fitted a Bayesian model with HNR as the outcome variable and attitude (informal vs polite) as the only predictor, using a Gaussian distribution as the distribution family of the outcome variable. The categorical predictor attitude was coded using the default treatment contrasts, with "informal" as the reference level.
>
> According to the model, the mean HNR for the informal attitude is between 16 and 16.5 db (`$\beta$` = 16.3, SD = 0.16), at 95% probability. When the attitude is polite, there is an increase in HNR between 0.8 and 1.7 db (`$\beta$` = 1.25, SD = 0.23).

---

---

There is a 95% probability that the standard deviation of the probability distribution of HNR values is between 1.54 and 1.86 dB.

---

## MID-TERM COURSE EVALUATION

## [bit.ly/45wgCEf](http://bit.ly/45wgCEf)

---

## Modelling VOT by stop

---

```r
alb_vot_vl
```

```
## # A tibble: 45 × 8
##    speaker file     label release voi_onset consonant   vot stop 
##    <chr>   <chr>    <chr>   <dbl>     <dbl> <chr>     <dbl> <chr>
##  1 s01     011-kati k       0.754     0.785 k         31.5  k    
##  2 s01     014-pata p       0.705     0.712 p          7.46 p    
##  3 s01     020-tapa t       0.825     0.833 t          8.00 t    
##  4 s01     052-kati k       0.772     0.807 k         35.0  k    
##  5 s01     055-pata p       0.823     0.838 p         15.0  p    
##  6 s01     061-tapa t       0.944     0.953 t          8.62 t    
##  7 s01     093-kati k       0.908     0.946 k         38.1  k    
##  8 s01     096-pata p       1.10      1.10  p          6.24 p    
##  9 s01     102-tapa t       0.951     0.964 t         13.1  t    
## 10 s02     011-tapa t       0.752     0.766 t         13.9  t    
## # ℹ 35 more rows
```

---

---

---

---

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
- We want to estimate the probability distribution of Voice Onset Time (VOT) in Albanian voiceless stops /k, p, t/.

- Let's assume it is a Gaussian probability distribution.
  
  - So, `$\text{vot} \sim Gaussian(\mu, \sigma)$`.
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
- But we also want to estimate the difference in the mean (`$\mu$`) in each stop /k, p, t/!
  
  - In other words, we need to allow the model to estimate `$\mu$` based on the stop.
]

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt2[
😱 But... `stop` is not a numeric predictor! 
]

---

**Treatment** coding uses `0` and `1` to code categorical predictors. **But** now we have three levels (`k, p, t`) so `0` and `1` are not enough.

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
In treatment coding, **N-1 dummy variables** are created where N is the number of levels.

- `attitude` has two levels (informal vs polite) so we need `$N-1 = 2-1 = 1$` dummy variable: `attitude_pol`.

- `stop` has three levels (/k, p, t/) so we need `$N-1 = 3-1 = 2$` dummy variables: `stop_p` and `stop_t`.
  - The reference level does not get a dummy variable!
]

<br>

|   | stop_p | stop_t |
|---|--------|--------|
| k | 0      | 0      |
| p | 1      | 0      |
| t | 0      | 1      |

---

```r
# Run a Bayesian model
vot_bm <- brm(
  # This is the formula of the model.
  vot ~ stop,
  # This is the probability distribution family.
  family = gaussian(),
  # And the data.
  data = alb_vot_vl
)
```

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
**Remember**: coding is done under the hood for you. You don't have to do it manually!
]

---

```
##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: vot ~ stop 
##    Data: alb_vot_vl (Number of observations: 45) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Population-Level Effects: 
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept    54.12      4.44    45.35    63.01 1.00     3233     2426
## stopp       -40.55      6.27   -53.23   -28.35 1.00     3669     2833
## stopt       -39.42      6.32   -51.81   -27.36 1.00     3259     2577
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    16.77      1.82    13.71    20.89 1.00     3141     2785
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

---

.f5[
$$
`\begin{align}
vot     & \sim Gaussian(\mu, \sigma) \\
\mu     & \sim \beta_0 + \beta_1 \cdot stop_p + \beta_2 \cdot stop_t \\
\sigma  & \sim TruncGaussian(\mu_3, \sigma_3)
\end{align}`
$$
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
**For [stop = k]**, `$stop_p = 0$` and `$stop_t = 0$`:

$$
`\begin{align}
\mu & = \beta_0 + \beta_1 \cdot stop_p + \beta_2 \cdot stop_t \\
    & = \beta_0 + \beta_1 \cdot 0 + \beta_2 \cdot 0 = \beta_0
\end{align}`
$$

**For [stop = p]**, `$stop_p = 1$` and `$stop_t = 0$`:

$$
`\begin{align}
\mu & = \beta_0 + \beta_1 \cdot stop_p + \beta_2 \cdot stop_t \\
    & = \beta_0 + \beta_1 \cdot 1 + \beta_2 \cdot 0 = \beta_0 + \beta_1
\end{align}`
$$

**For [stop = t]**, `$stop_p = 0$` and `$stop_t = 1$`:

$$
`\begin{align}
\mu & = \beta_0 + \beta_1 \cdot stop_p + \beta_2 \cdot stop_t \\
    & = \beta_0 + \beta_1 \cdot 0 + \beta_2 \cdot 1 = \beta_0 + \beta_2
\end{align}`
$$
]

---

.f5[
$$
`\begin{align}
vot     & \sim Gaussian(\mu, \sigma) \\
\mu     & \sim \beta_0 + \beta_1 \cdot stop_p + \beta_2 \cdot stop_t \\
\sigma  & \sim TruncGaussian(\mu_3, \sigma_3)
\end{align}`
$$
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
**For [stop = k]**, `$stop_p = 0$` and `$stop_t = 0$`:

$$
`\begin{align}
\mu & = 54.12 + (-40.55) \cdot stop_p + (-39.42) \cdot stop_t \\
    & = 54.12 - 40.55 \cdot 0 - 39.42 \cdot 0 = 54.12
\end{align}`
$$

**For [stop = p]**, `$stop_p = 1$` and `$stop_t = 0$`:

$$
`\begin{align}
\mu & = 54.12 + (-40.55) \cdot stop_p + (-39.42) \cdot stop_t \\
    & = 54.12 - 40.55 \cdot 1 - 39.42 \cdot 0 = 54.12 - 40.55 = 13.57
\end{align}`
$$

**For [stop = t]**, `$stop_p = 0$` and `$stop_t = 1$`:

$$
`\begin{align}
\mu & = 54.12 + (-40.55) \cdot stop_p + (-39.42) \cdot stop_t \\
    & = 54.12 - 40.55 \cdot 1 - 39.42 \cdot 1 = 54.12 - 39.42 = 14.7
\end{align}`
$$
]

---

.f4[
$$
`\begin{align}
vot     & \sim Gaussian(\mu, \sigma) \\
\mu     & \sim \beta_0 + \beta_1 \cdot stop_p + \beta_2 \cdot stop_t \\
\beta_0 & \sim Gaussian(54.12, 4.44) \\
\beta_1 & \sim Gaussian(-40.55, 6.27) \\
\beta_2 & \sim Gaussian(-39.42, 6.32) \\
\sigma  & \sim TruncGaussian(16.77, 1.82)
\end{align}`
$$
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
**IMPORTANT**

- While `$\beta_0$` is the probability distribution of mean VOT when [stop = k]

- `$\beta_1$` and `$\beta_2$` are the probability distributions of the **DIFFERENCE** between mean VOT when [stop = p] and when [stop = k], and when [stop = t] and when [stop = k], respectively.

- `$\beta_1$` and `$\beta_2$` are **NOT** the probability distributions of the mean VOT when [stop = p] and when [stop = t]!
]

---

## Summary

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
- **Comparing groups** with `brm()`

- `outcome ~ predictor`.
  - Categorical predictor with 2 and 3 levels.

- **Treatment coding** of categorical predictors.

- N-1 **dummy variables**, where N is number of levels in the predictor.
  - Level ordering is *alphabetical* but you can specify your own.
  - **NOTE**: you don't have to apply treatment coding yourself! It's done under the hood by R.

- **Remember**

- The **Intercept** `$\beta_0$` is the mean of the reference level.
  - The other `$\beta$`'s are the **difference** of the other levels relative to the reference level.
]