BST430 Lecture 15

.title[
# BST430 Lecture 15
]
.subtitle[
## Linear Models
]
.author[
### Tanzy Love, based on the course by Andrew McDavid
]
.institute[
### U of Rochester
]
.date[
### 2021-10-28 (updated: 2024-10-29 by TL)
]

---

# What is a model?

Here's the [R code in this lecture](l15/l15-linear-models.R)

and the [auction dataset](l15/data/paris-paintings.csv)

---

## Modelling

- Use models to explain the relationship between variables and to make predictions
- For now we will focus on **linear** models (but remember there are *many* *many* other types of models too!)

.pull-left[
<img src="l15-linear-models_files/figure-html/unnamed-chunk-2-1.png" width="100%" style="display: block; margin: auto;" />
]
.pull-right[
<img src="l15-linear-models_files/figure-html/unnamed-chunk-3-1.png" width="100%" style="display: block; margin: auto;" />
]

---
class: middle

# Data: Paris Paintings

---

## Paris Paintings

``` r
pp = read_csv("l15/data/paris-paintings.csv", na = c("n/a", "", "NA"))
```

- Source: Printed catalogs of 28 auction sales in Paris, 1764 - 1780
- Data curators Sandra van Ginhoven and Hilary Coe Cronheim (who were PhD students in the Duke Art, Law, and Markets Initiative at the time of putting together this dataset) translated and tabulated the catalogs
- 3393 paintings, their prices, and descriptive details from sales catalogs over 60 variables

---

## Auctions today

.center[
<iframe width="840" height="473" src="https://www.youtube.com/embed/apaE1Q7r4so" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
]

---

## Auctions back in the day

.footnote[
.small[
Pierre-Antoine de Machy, Public Sale at the Hôtel Bullion, Musée Carnavalet, Paris (18th century)
]
]

---

## Paris auction market

---

## Départ pour la chasse

---

## Auction catalog text

.pull-left[
<img src="l15/img/auction-catalogue.png" width="60%" style="display: block; margin: auto;" />
]
.pull-right[
.small[
Two paintings very rich in composition, of a beautiful execution, and whose merit is very remarkable, each 17 inches 3 lines high, 23 inches wide; the first, painted on wood, comes from the Cabinet of Madame la Comtesse de Verrue; it represents a departure for the hunt: it shows in the front a child on a white horse, a man who gives the horn to gather the dogs, a falconer and other figures nicely distributed across the width of the painting; two horses drinking from a fountain; on the right in the corner a lovely country house topped by a terrace, on which people are at the table, others who play instruments; trees and fabriques pleasantly enrich the background.
]
]

---

---
class: code40

``` r
pp %>%
  filter(name == "R1777-89a") %>%
  glimpse()
```

```
## Rows: 1
## Columns: 61
## $ name              <chr> "R1777-89a"
## $ sale              <chr> "R1777"
## $ lot               <chr> "89"
## $ position          <dbl> 0.3755274
## $ dealer            <chr> "R"
## $ year              <dbl> 1777
## $ origin_author     <chr> "D/FL"
## $ origin_cat        <chr> "D/FL"
## $ school_pntg       <chr> "D/FL"
## $ diff_origin       <dbl> 0
## $ logprice          <dbl> 8.575462
## $ price             <dbl> 5300
## $ count             <dbl> 1
## $ subject           <chr> "D\x8epart pour la chasse"
## $ authorstandard    <chr> "Wouwerman, Philips"
## $ artistliving      <dbl> 0
## $ authorstyle       <chr> NA
## $ author            <chr> "Philippe Wouwermans"
## $ winningbidder     <chr> "Langlier, Jacques for Poullain, Anto…
## $ winningbiddertype <chr> "DC"
## $ endbuyer          <chr> "C"
...
```
]
.pull-right[

```
...
## $ Interm            <dbl> 1
## $ type_intermed     <chr> "D"
## $ Height_in         <dbl> 17.25
## $ Width_in          <dbl> 23
## $ Surface_Rect      <dbl> 396.75
## $ Diam_in           <dbl> NA
## $ Surface_Rnd       <dbl> NA
## $ Shape             <chr> "squ_rect"
## $ Surface           <dbl> 396.75
## $ material          <chr> "bois"
## $ mat               <chr> "b"
## $ materialCat       <chr> "wood"
## $ quantity          <dbl> 1
## $ nfigures          <dbl> 0
## $ engraved          <dbl> 0
## $ original          <dbl> 0
## $ prevcoll          <dbl> 1
## $ othartist         <dbl> 0
## $ paired            <dbl> 1
## $ figures           <dbl> 0
## $ finished          <dbl> 0
...
```
]

---

# Modeling the relationship between variables

---

## Heights

``` r
ggplot(data = pp, aes(x = Height_in)) +
  geom_histogram(binwidth = 5) +
  labs(x = "Height, in inches", y = NULL)
```

<img src="l15-linear-models_files/figure-html/height-dist-1.png" width="60%" style="display: block; margin: auto;" />
]

---

## Widths

``` r
ggplot(data = pp, aes(x = Width_in)) +
  geom_histogram(binwidth = 5) +
  labs(x = "Width, in inches", y = NULL)
```

<img src="l15-linear-models_files/figure-html/width-dist-1.png" width="60%" style="display: block; margin: auto;" />
]

---

## Height as a function of width

.panelset[
.panel[.panel-name[Plot]
<img src="l15-linear-models_files/figure-html/unnamed-chunk-13-1.png" width="60%" style="display: block; margin: auto;" />
]
.panel[.panel-name[Code]

``` r
ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
  geom_smooth(method = "lm") +
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )
```
]
]

---

## ... without showing uncertainty

.panelset[
.panel[.panel-name[Plot]
<img src="l15-linear-models_files/figure-html/unnamed-chunk-14-1.png" width="60%" style="display: block; margin: auto;" />
]
.panel[.panel-name[Code]

``` r
ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
  geom_smooth(method = "lm", 
*             se = FALSE) +
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )
```
]
]

---

## ... with different cosmetic choices

.panelset[
.panel[.panel-name[Plot]
<img src="l15-linear-models_files/figure-html/unnamed-chunk-15-1.png" width="60%" style="display: block; margin: auto;" />
]
.panel[.panel-name[Code]

``` r
ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE,
*             color = "#8E2C90", linetype = "dashed", size = 3) +
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )
```
]
]

---

## Other smoothing methods: gam

.panelset[
.panel[.panel-name[Plot]
<img src="l15-linear-models_files/figure-html/unnamed-chunk-16-1.png" width="60%" style="display: block; margin: auto;" />
]
.panel[.panel-name[Code]

``` r
ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
* geom_smooth(method = "gam",
              se = FALSE, color = "#8E2C90") + 
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )
```
]
]

---

## Other smoothing methods: loess

.panelset[
.panel[.panel-name[Plot]
<img src="l15-linear-models_files/figure-html/unnamed-chunk-17-1.png" width="60%" style="display: block; margin: auto;" />
]
.panel[.panel-name[Code]

``` r
ggplot(data = pp, aes(x = Width_in, y = Height_in)) +
  geom_point() +
* geom_smooth(method = "loess",
              se = FALSE, color = "#8E2C90") + 
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  )
```
]
]

---

## Vocabulary

- **Response variable:** Variable whose behavior or variation you are trying to understand, on the y-axis

--
- **Explanatory variables:** Other variables that you want to use to explain the variation in the response, on the x-axis

--
- **Predicted value:** Output of the **model function**
  - The model function gives the typical (expected) value of the response variable *conditioning* on the explanatory variables
  
--
- **Residuals:** A measure of how far each case is from its predicted value (based on a particular model)
  - Residual = Observed value - Predicted value
  - Tells how far above/below the expected value each case is

---

## Residuals

.panelset[
.panel[.panel-name[Plot]
<img src="l15-linear-models_files/figure-html/unnamed-chunk-18-1.png" width="60%" style="display: block; margin: auto;" />
]
.panel[.panel-name[Code]
.code60[

``` r
ht_wt_fit = linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ Width_in, data = pp)

ht_wt_fit_tidy = tidy(ht_wt_fit$fit) 
ht_wt_fit_aug  = augment(ht_wt_fit$fit) %>%
  mutate(res_cat = ifelse(.resid > 0, TRUE, FALSE))

ggplot(data = ht_wt_fit_aug) +
  geom_point(aes(x = Width_in, y = Height_in, color = res_cat)) +
  geom_line(aes(x = Width_in, y = .fitted), size = 0.75, color = "#8E2C90") + 
  labs(
    title = "Height vs. width of paintings",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)"
  ) +
  guides(color = 'none') +
  scale_color_manual(values = c("#260b27", "#e6b0e7")) +
  geom_text(aes(x = 0, y = 150), label = "Positive residual", color = "#e6b0e7", hjust = 0, size = 8) +
  geom_text(aes(x = 150, y = 25), label = "Negative residual", color = "#260b27", hjust = 0, size = 8)
```
]
]
]

---

.question[
The plot below displays the relationship between height and width of paintings. The only difference from the previous plots is that it uses a smaller alpha value, making the points somewhat transparent. What feature is apparent in this plot that was not (as) apparent in the previous plots? What might be the reason for this feature?
]

---

## Landscape paintings

- Landscape painting is the depiction in art of landscapes – natural scenery such as mountains, valleys, trees, rivers, and forests, especially where the main subject is a wide view – with its elements arranged into a coherent
composition.<sup>1</sup>
  - Landscape paintings tend to be wider than they are long.
- Portrait painting is a genre in painting, where the intent is to depict a human subject.<sup>2</sup>
  - Portrait paintings tend to be longer than they are wide.

.footnote[
[1] Source: Wikipedia, [Landscape painting](https://en.wikipedia.org/wiki/Landscape_painting)

[2] Source: Wikipedia, [Portait painting](https://en.wikipedia.org/wiki/Portrait_painting)
]

---

## Multiple explanatory variables

.panelset[
.panel[.panel-name[Plot]
.pull-left-narrow[
.question[
How, if at all, does the relationship between width and height of paintings vary by whether or not they have any landscape elements?
]
]
.pull-right-wide[
<img src="l15-linear-models_files/figure-html/unnamed-chunk-19-1.png" width="80%" style="display: block; margin: auto;" />
]
]
.panel[.panel-name[Code]

``` r
ggplot(data = pp, aes(x = Width_in, y = Height_in, color = factor(landsALL))) +
  geom_point(alpha = 0.4) +
  geom_smooth(method = "lm", se = FALSE) +
  labs(
    title = "Height vs. width of paintings, by landscape features",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)",
    color = "landscape"
  ) +
  scale_color_manual(values = c("#E48957", "#071381"))
```
]
]

---

## Extending regression lines

.panelset[
.panel[.panel-name[Plot]
<img src="l15-linear-models_files/figure-html/unnamed-chunk-20-1.png" width="65%" style="display: block; margin: auto;" />
]
.panel[.panel-name[Code]

``` r
ggplot(data = pp, aes(x = Width_in, y = Height_in, color = factor(landsALL))) +
  geom_point(alpha = 0.4) +
  geom_smooth(method = "lm", se = FALSE,
*             fullrange = TRUE) +
  labs(
    title = "Height vs. width of paintings, by landscape features",
    subtitle = "Paris auctions, 1764 - 1780",
    x = "Width (inches)",
    y = "Height (inches)",
    color = "landscape"
  ) +
  scale_color_manual(values = c("#E48957", "#071381"))
```
]
]

---

## Models - upsides and downsides

- Models can sometimes reveal patterns that are not evident in a graph of the data. This is a great advantage of modeling over simple visual inspection of data. 
- There is a real risk, however, that a model is imposing structure that is not really there on the scatter of data, just as people imagine animal shapes in the stars. A skeptical approach is always warranted.

---

## Variation around the model...

is just as important as the model, if not more!

*Statistics is the explanation of variation in the context of what remains unexplained.*

- The scatter suggests that there might be other factors that account for large parts of painting-to-painting variability, or perhaps just that randomness plays a big role.
- Adding more explanatory variables to a model can sometimes usefully reduce the size of the scatter around the model.

---

## How do we use models?

- Explanation (and inference!): Characterize the relationship between `$y$` and `$x$` via *slopes* for numerical explanatory variables or *differences* for categorical explanatory variables
- Inference: based on our sample, what conclusions can we make about the population, and with how much certainty?
- Prediction: Plug in `$x$`, get the predicted `$y$`

---

# Models with numerical explanatory variables

---

## Goal: Predict painting height from width

`$$\widehat{
\text{height}}_{i} = \beta_0 + \beta_1 \times 
\text{width}_{i}$$`

---

---

## Step 1: Specify model

``` r
linear_reg()
```

```
## Linear Regression Model Specification (regression)
## 
## Computational engine: lm
```

[More models...](https://www.tidymodels.org/find/parsnip/)

---

## Step 2: Set model fitting *engine*

``` r
linear_reg() %>%
  set_engine("lm") # lm: linear model
```

```
## Linear Regression Model Specification (regression)
## 
## Computational engine: lm
```

[More engines](https://parsnip.tidymodels.org/reference/linear_reg.html)
---

## Step 3: Fit model & estimate parameters

... using **formula syntax**

``` r
linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ Width_in, data = pp)
```

```
## parsnip model object
## 
## 
## Call:
## stats::lm(formula = Height_in ~ Width_in, data = data)
## 
## Coefficients:
## (Intercept)     Width_in  
##      3.6214       0.7808
```

---

## A closer look at model output

```
## parsnip model object
## 
## 
## Call:
## stats::lm(formula = Height_in ~ Width_in, data = data)
## 
## Coefficients:
## (Intercept)     Width_in  
##      3.6214       0.7808
```

---

## A tidy look at model output

``` r
linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ Width_in, data = pp) %>%
  tidy()
```

```
## # A tibble: 2 × 5
##   term        estimate std.error statistic  p.value
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)    3.62    0.254        14.3 8.82e-45
## 2 Width_in       0.781   0.00950      82.1 0
```

[More classes to tidy](https://broom.tidymodels.org/reference/index.html)

---

## A glance at model summaries

``` r
linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ Width_in, data = pp) %>%
  glance()
```

```
## # A tibble: 1 × 12
##   r.squared adj.r.squared sigma statistic p.value    df  logLik
##       <dbl>         <dbl> <dbl>     <dbl>   <dbl> <dbl>   <dbl>
## 1     0.683         0.683  8.30     6749.       0     1 -11083.
## # ℹ 5 more variables: AIC <dbl>, BIC <dbl>, deviance <dbl>,
## #   df.residual <int>, nobs <int>
```

More of this coming

---

## Slope and intercept

- **Slope:** For each additional inch the painting is wider, the height is expected to be higher, on average, by 0.781 inches.

--
- **Intercept:** Paintings that are 0 inches wide are expected to be 3.62 inches high, on average. (Does this make sense?)

---

## Correlation does not imply causation

Remember this when interpreting model coefficients

---

# Parameter estimation

---

## Linear model with a single predictor

- We're interested in `$\beta_0$` (population parameter for the intercept) and `$\beta_1$` (population parameter for the slope) in the following model:

`$$\hat{y}_{i} = \beta_0 + \beta_1~x_{i}$$`

--
- Tough luck, you can't have them...

--
- So we make an estimate (e.g. `$\mathbf{b} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}$`) from a sample instead:

`$$\hat{y}_{i} = b_0 + b_1~x_{i}$$`
---

## Visualizing residuals

---

# Models with categorical explanatory variables

---

## Categorical predictor with 2 levels

```
## # A tibble: 3,393 × 3
##    name      Height_in landsALL
##    <chr>         <dbl>    <dbl>
##  1 L1764-2          37        0
##  2 L1764-3          18        0
##  3 L1764-4          13        1
##  4 L1764-5a         14        1
##  5 L1764-5b         14        1
##  6 L1764-6           7        0
##  7 L1764-7a          6        0
##  8 L1764-7b          6        0
##  9 L1764-8          15        0
## 10 L1764-9a          9        0
## 11 L1764-9b          9        0
## 12 L1764-10a        16        1
## 13 L1764-10b        16        1
## 14 L1764-10c        16        1
## 15 L1764-11         20        0
## 16 L1764-12a        14        1
## 17 L1764-12b        14        1
## 18 L1764-13a        15        1
## 19 L1764-13b        15        1
## 20 L1764-14         37        0
## # ℹ 3,373 more rows
```
]
]
.pull-right-wide[
- `landsALL = 0`: No landscape features
- `landsALL = 1`: Some landscape features
]

---

## Height & landscape features

``` r
linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ factor(landsALL), data = pp) %>%
  tidy()
```

```
## # A tibble: 2 × 5
##   term              estimate std.error statistic  p.value
##   <chr>                <dbl>     <dbl>     <dbl>    <dbl>
## 1 (Intercept)          22.7      0.328      69.1 0       
## 2 factor(landsALL)1    -5.65     0.532     -10.6 7.97e-26
```

---

## Height & landscape features

`$$\widehat{height_{in}} = 22.7 - 5.645~landsALL$$`

- **Slope:** Paintings with landscape features are expected, on average, to be 5.645 inches shorter than paintings that without landscape features
  - Compares baseline level (`landsALL = 0`) to the other level (`landsALL = 1`)
    
- **Intercept:** Paintings that don't have landscape features are expected, on average, to be 22.7 inches tall

---

## Relationship between height and school

``` r
linear_reg() %>%
  set_engine("lm") %>%
  fit(Height_in ~ school_pntg, data = pp) %>%
  tidy()
```

---

## Dummy variables

- When the categorical explanatory variable has many levels, they're encoded to **dummy variables**
- Each coefficient describes the expected difference between heights in that particular school compared to the baseline level

---

## Categorical predictor with 3+ levels

.pull-left-wide[
<table class="table" style="margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> school_pntg </th>
   <th style="text-align:center;"> D_FL </th>
   <th style="text-align:center;"> F </th>
   <th style="text-align:center;"> G </th>
   <th style="text-align:center;"> I </th>
   <th style="text-align:center;"> S </th>
   <th style="text-align:center;"> X </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> A </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> D/FL </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(122, 209, 81, 255) !important;"> 1 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> F </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(122, 209, 81, 255) !important;"> 1 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> G </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(122, 209, 81, 255) !important;"> 1 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> I </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(122, 209, 81, 255) !important;"> 1 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> S </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(122, 209, 81, 255) !important;"> 1 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> X </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(68, 1, 84, 255) !important;"> 0 </td>
   <td style="text-align:center;width: 10em; color: white !important;background-color: rgba(122, 209, 81, 255) !important;"> 1 </td>
  </tr>
</tbody>
</table>
]
.pull-right-narrow[
.code50[

```
## # A tibble: 3,393 × 3
##    name      Height_in school_pntg
##    <chr>         <dbl> <chr>      
##  1 L1764-2          37 F          
##  2 L1764-3          18 I          
##  3 L1764-4          13 D/FL       
##  4 L1764-5a         14 F          
##  5 L1764-5b         14 F          
##  6 L1764-6           7 I          
##  7 L1764-7a          6 F          
##  8 L1764-7b          6 F          
##  9 L1764-8          15 I          
## 10 L1764-9a          9 D/FL       
## 11 L1764-9b          9 D/FL       
## 12 L1764-10a        16 X          
## 13 L1764-10b        16 X          
## 14 L1764-10c        16 X          
## 15 L1764-11         20 D/FL       
## 16 L1764-12a        14 D/FL       
## 17 L1764-12b        14 D/FL       
## 18 L1764-13a        15 D/FL       
## 19 L1764-13b        15 D/FL       
## 20 L1764-14         37 F          
## # ℹ 3,373 more rows
```
]
]

---

## Relationship between height and school

```
## # A tibble: 7 × 5
##   term            estimate std.error statistic p.value
##   <chr>              <dbl>     <dbl>     <dbl>   <dbl>
## 1 (Intercept)        14.0       10.0     1.40  0.162  
## 2 school_pntgD/FL     2.33      10.0     0.232 0.816  
## 3 school_pntgF       10.2       10.0     1.02  0.309  
## 4 school_pntgG        1.65      11.9     0.139 0.889  
## 5 school_pntgI       10.3       10.0     1.02  0.306  
## 6 school_pntgS       30.4       11.4     2.68  0.00744
## 7 school_pntgX        2.87      10.3     0.279 0.780
```
]
.font70[
- **Austrian school (A)** paintings are expected, on average, to be **14 inches** tall.
- **Dutch/Flemish school (D/FL)** paintings are expected, on average, to be **2.33 inches taller** than *Austrian school* paintings.
- **French school (F)** paintings are expected, on average, to be **10.2 inches taller** than *Austrian school* paintings.
- **German school (G)** paintings are expected, on average, to be **1.65 inches taller** than *Austrian school* paintings.
- **Italian school (I)** paintings are expected, on average, to be **10.3 inches taller** than *Austrian school* paintings.
- **Spanish school (S)** paintings are expected, on average, to be **30.4 inches taller** than *Austrian school* paintings.
- Paintings whose school is **unknown (X)** are expected, on average, to be **2.87 inches taller** than *Austrian school* paintings.
]

---

## Non-tidy models

I will confess, I don't use tidymodels yet in my typical workflow, though I think it is beneficial to learn because it makes resampling procedures (e.g. cross-validation) much easier.

IMO, it's important to have a feel for the underpinnings of base R models, because their interface appears in many different packages you will use.

---

### Non-tidy models cheatsheet

*  `fit = lm(formula, data)`: fit `formula` to data frame `data`

*  `fit = glm(formula, data, family = binomial)`: fit a logistic regression using `formula` to data frame `data`

*  `coef(fit)` extract coefficients as `numeric` vector

*  `vcov(fit)` extract variance-covariance as `numeric` matrix

*  `residuals(fit, type = ...)` get residuals. `type = ...` is an important argument (standardized, studentized, pearson, etc.)

*  `predict(fit, newdata = ..., type = ...)` get predictions.  If `newdata` is set, it must be a data frame containing the same variable names as `data`. `type = "response"` vs `type = "link"` is important for glm's.

*  `broom::tidy` is still very very useful!

---

# Acknowledgments

Data science in a box [u4-d01](https://rstudio-education.github.io/datascience-box/course-materials/slides/u4-d01-language-of-models/u4-d01-language-of-models.html#1)