Bayesian Linear Models

.title[
# Bayesian Linear Models
]
.subtitle[
## 03 - Interactions
]
.author[
### Stefano Coretta
]
.institute[
### University of Edinburgh
]
.date[
### 2023/07/07
]

---

## Many predictors

---

---

```r
my_seed <- 9899

rt_bm_3 <- brm(
  RT ~ IsWord + PhonLev,
  data = mald,
  backend = "cmdstanr",
  cores = 4,
  file = "./data/cache/rt_bm_3",
  seed = my_seed
)
```

---

```
## Population-Level Effects: 
##             Estimate Est.Error l-89% CI u-89% CI Rhat Bulk_ESS Tail_ESS
## Intercept     806.93     34.54   752.38   861.65 1.00     3876     3233
## IsWordFALSE   131.14     12.47   111.01   151.30 1.00     4793     3214
## PhonLev        24.71      4.73    17.09    32.19 1.00     3914     3289
```

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
- The `Intercept` is the mean RT when [IsWord=TRUE] and [PhonLev=0].

- `IsWordFALSE` is the **difference** between the mean RT when [IsWord=FALSE] and the mean RT when [IsWord=TRUE].

- `PhonLev` is the **difference** between the mean RT when [PhonLev=1] and [PhoneLev=0].
]

---

---

---

---

## We need an interaction

---

```r
rt_bm_4 <- brm(
  RT ~ IsWord + PhonLev + IsWord:PhonLev,
  data = mald,
  backend = "cmdstanr",
  cores = 4,
  file = "./data/cache/rt_bm_4",
  seed = my_seed
)
```

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
- `isWord:PhonLev` is called an **interaction** term.
- `isWord:PhonLev` allows the model to capture differences in the effect of `isWord` depending on the value of `PhonLev` (and vice versa).
]

---

```
## Population-Level Effects: 
##                     Estimate Est.Error l-89% CI u-89% CI Rhat Bulk_ESS Tail_ESS
## Intercept             756.94     47.75   679.47   833.45 1.00     2008     2259
## IsWordFALSE           229.58     68.39   121.26   341.01 1.00     1590     1633
## PhonLev                31.75      6.65    21.03    42.54 1.00     1965     2330
## IsWordFALSE:PhonLev   -13.85      9.48   -29.32     1.10 1.00     1614     1700
```

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
- The `Intercept` is the mean RT when [IsWord=TRUE] and [PhonLev=0].

- `IsWordFALSE` is the **difference** between the mean RT when [IsWord=FALSE] and the mean RT when [IsWord=TRUE], when [PhonLev=0].

- `PhonLev` is the **difference** between the mean RT when [PhonLev=1] and [PhoneLev=0], when [IsWord=TRUE].

- `IsWordFALSE:PhonLev` is the adjustment to the difference in mean RT when [IsWord=FALSE] and [PhonLev=1].
]

---