Unit 1

class: center, middle, inverse, title-slide

# Unit 1
### Jung-Jin Lee
### Last modified: Nov 03, 2019

---

## List of packages used

```r
library(tidyverse)
library(broom)
library(gt)
library(rstan)
# the following are recommended in startup messages
*options(mc.cores = parallel::detectCores())
*rstan_options(auto_write = TRUE)
```

Stan user's guide https://mc-stan.org/docs/2_18/stan-users-guide/index.html

---

## Installation of RStan

Follow instruction in https://github.com/stan-dev/rstan/wiki/RStan-Getting-Started:

- Remove existing `RStan`, if any:

```r
remove.packages("rstan")
if (file.exists(".RData")) file.remove(".RData")
```

- Restart `R` and install `rstan` package:

```r
install.packages("rstan", repos = "https://cloud.r-project.org/", 
                 dependencies = TRUE)
```

- Make sure you have the necessary C++ toolchain:

```r
pkgbuild::has_build_tools(debug = TRUE)
```

```
## [1] TRUE
```

---

## Example: simple linear regression

`mtcars` is one of built-in data sets in `R`.

```r
head(mtcars) %>% 
  rownames_to_column("name") %>%
  gt(rowname_col = vars(name))
```

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  <tr>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1">name</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">mpg</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">cyl</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">disp</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">hp</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">drat</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">wt</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">qsec</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">vs</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">am</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">gear</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">carb</th>
  </tr>
  <body class="gt_table_body">
    <tr>
      <td class="gt_row gt_left">Mazda RX4</td>
      <td class="gt_row gt_right">21.0</td>
      <td class="gt_row gt_right">6</td>
      <td class="gt_row gt_right">160</td>
      <td class="gt_row gt_right">110</td>
      <td class="gt_row gt_right">3.90</td>
      <td class="gt_row gt_right">2.620</td>
      <td class="gt_row gt_right">16.46</td>
      <td class="gt_row gt_right">0</td>
      <td class="gt_row gt_right">1</td>
      <td class="gt_row gt_right">4</td>
      <td class="gt_row gt_right">4</td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_striped">Mazda RX4 Wag</td>
      <td class="gt_row gt_right gt_striped">21.0</td>
      <td class="gt_row gt_right gt_striped">6</td>
      <td class="gt_row gt_right gt_striped">160</td>
      <td class="gt_row gt_right gt_striped">110</td>
      <td class="gt_row gt_right gt_striped">3.90</td>
      <td class="gt_row gt_right gt_striped">2.875</td>
      <td class="gt_row gt_right gt_striped">17.02</td>
      <td class="gt_row gt_right gt_striped">0</td>
      <td class="gt_row gt_right gt_striped">1</td>
      <td class="gt_row gt_right gt_striped">4</td>
      <td class="gt_row gt_right gt_striped">4</td>
    </tr>
    <tr>
      <td class="gt_row gt_left">Datsun 710</td>
      <td class="gt_row gt_right">22.8</td>
      <td class="gt_row gt_right">4</td>
      <td class="gt_row gt_right">108</td>
      <td class="gt_row gt_right">93</td>
      <td class="gt_row gt_right">3.85</td>
      <td class="gt_row gt_right">2.320</td>
      <td class="gt_row gt_right">18.61</td>
      <td class="gt_row gt_right">1</td>
      <td class="gt_row gt_right">1</td>
      <td class="gt_row gt_right">4</td>
      <td class="gt_row gt_right">1</td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_striped">Hornet 4 Drive</td>
      <td class="gt_row gt_right gt_striped">21.4</td>
      <td class="gt_row gt_right gt_striped">6</td>
      <td class="gt_row gt_right gt_striped">258</td>
      <td class="gt_row gt_right gt_striped">110</td>
      <td class="gt_row gt_right gt_striped">3.08</td>
      <td class="gt_row gt_right gt_striped">3.215</td>
      <td class="gt_row gt_right gt_striped">19.44</td>
      <td class="gt_row gt_right gt_striped">1</td>
      <td class="gt_row gt_right gt_striped">0</td>
      <td class="gt_row gt_right gt_striped">3</td>
      <td class="gt_row gt_right gt_striped">1</td>
    </tr>
    <tr>
      <td class="gt_row gt_left">Hornet Sportabout</td>
      <td class="gt_row gt_right">18.7</td>
      <td class="gt_row gt_right">8</td>
      <td class="gt_row gt_right">360</td>
      <td class="gt_row gt_right">175</td>
      <td class="gt_row gt_right">3.15</td>
      <td class="gt_row gt_right">3.440</td>
      <td class="gt_row gt_right">17.02</td>
      <td class="gt_row gt_right">0</td>
      <td class="gt_row gt_right">0</td>
      <td class="gt_row gt_right">3</td>
      <td class="gt_row gt_right">2</td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_striped">Valiant</td>
      <td class="gt_row gt_right gt_striped">18.1</td>
      <td class="gt_row gt_right gt_striped">6</td>
      <td class="gt_row gt_right gt_striped">225</td>
      <td class="gt_row gt_right gt_striped">105</td>
      <td class="gt_row gt_right gt_striped">2.76</td>
      <td class="gt_row gt_right gt_striped">3.460</td>
      <td class="gt_row gt_right gt_striped">20.22</td>
      <td class="gt_row gt_right gt_striped">1</td>
      <td class="gt_row gt_right gt_striped">0</td>
      <td class="gt_row gt_right gt_striped">3</td>
      <td class="gt_row gt_right gt_striped">1</td>
    </tr>
  </body>
  
  
</table></div>

---

## Example: simple linear regression

Scatter plot showing the relation between `wt` and `mpg`:

```r
mtcars %>% ggplot(aes(wt, mpg)) + 
  geom_point() +
  geom_smooth(method = "lm", se = FALSE)
```

---

## Example: simple linear regression

Simple linear regression using `lm()`:

```r
fit <- lm(mpg ~ wt, data = mtcars)
tidy(fit) %>% 
  gt(rowname_col = "term") %>%
  fmt_number(columns = vars(estimate, std.error, statistic), 
             decimals = 2) %>%
  fmt_scientific(columns = vars(p.value), decimals = 2)
```

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  border-top-width: 2px;
  /* footnotes.border.top.width */
  border-top-color: #D3D3D3;
  /* footnotes.border.top.color */
}

#ibhghmiynh .gt_footnote {
  font-size: 90%;
  /* footnote.font.size */
  margin: 0px;
  padding: 4px;
  /* footnote.padding */
}

#ibhghmiynh .gt_sourcenotes {
  border-top-style: solid;
  /* sourcenotes.border.top.style */
  border-top-width: 2px;
  /* sourcenotes.border.top.width */
  border-top-color: #D3D3D3;
  /* sourcenotes.border.top.color */
}

#ibhghmiynh .gt_sourcenote {
  font-size: 90%;
  /* sourcenote.font.size */
  padding: 4px;
  /* sourcenote.padding */
}

#ibhghmiynh .gt_center {
  text-align: center;
}

#ibhghmiynh .gt_left {
  text-align: left;
}

#ibhghmiynh .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#ibhghmiynh .gt_font_normal {
  font-weight: normal;
}

#ibhghmiynh .gt_font_bold {
  font-weight: bold;
}

#ibhghmiynh .gt_font_italic {
  font-style: italic;
}

#ibhghmiynh .gt_super {
  font-size: 65%;
}

#ibhghmiynh .gt_footnote_marks {
  font-style: italic;
  font-size: 65%;
}
</style>
<div id="ibhghmiynh" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <tr>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1"></th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1">estimate</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">std.error</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">statistic</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">p.value</th>
  </tr>
  <body class="gt_table_body">
    <tr>
      <td class="gt_row gt_left gt_stub">(Intercept)</td>
      <td class="gt_row gt_right">37.29</td>
      <td class="gt_row gt_right">1.88</td>
      <td class="gt_row gt_right">19.86</td>
      <td class="gt_row gt_right">8.24 × 10<sup class='gt_super'>−19</sup></td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_stub">wt</td>
      <td class="gt_row gt_right gt_striped">−5.34</td>
      <td class="gt_row gt_right gt_striped">0.56</td>
      <td class="gt_row gt_right gt_striped">−9.56</td>
      <td class="gt_row gt_right gt_striped">1.29 × 10<sup class='gt_super'>−10</sup></td>
    </tr>
  </body>
  
  
</table></div>

---

## Example: simple linear regression

In particular, the 95% confidence intervals of the coefficients are given below:

```r
confint(fit) %>%
  as.data.frame() %>%
  rownames_to_column("term") %>%
  gt(rowname_col = "term") %>%
  fmt_number(columns = everything(), 
             decimals = 2)
```

#mohdhsuimh .gt_table {
  display: table;
  border-collapse: collapse;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  background-color: #FFFFFF;
  /* table.background.color */
  width: auto;
  /* table.width */
  border-top-style: solid;
  /* table.border.top.style */
  border-top-width: 2px;
  /* table.border.top.width */
  border-top-color: #A8A8A8;
  /* table.border.top.color */
  border-bottom-style: solid;
  /* table.border.bottom.style */
  border-bottom-width: 2px;
  /* table.border.bottom.width */
  border-bottom-color: #A8A8A8;
  /* table.border.bottom.color */
}

#mohdhsuimh .gt_heading {
  background-color: #FFFFFF;
  /* heading.background.color */
  border-bottom-color: #FFFFFF;
}

#mohdhsuimh .gt_title {
  color: #333333;
  font-size: 125%;
  /* heading.title.font.size */
  padding-top: 4px;
  /* heading.top.padding - not yet used */
  padding-bottom: 4px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#mohdhsuimh .gt_subtitle {
  color: #333333;
  font-size: 85%;
  /* heading.subtitle.font.size */
  padding-top: 0;
  padding-bottom: 4px;
  /* heading.bottom.padding - not yet used */
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#mohdhsuimh .gt_bottom_border {
  border-bottom-style: solid;
  /* heading.border.bottom.style */
  border-bottom-width: 2px;
  /* heading.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* heading.border.bottom.color */
}

#mohdhsuimh .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  padding-top: 4px;
  padding-bottom: 4px;
}

#mohdhsuimh .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  /* column_labels.background.color */
  font-size: 16px;
  /* column_labels.font.size */
  font-weight: initial;
  /* column_labels.font.weight */
  vertical-align: middle;
  padding: 5px;
  margin: 10px;
  overflow-x: hidden;
}

#mohdhsuimh .gt_columns_top_border {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#mohdhsuimh .gt_columns_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#mohdhsuimh .gt_sep_right {
  border-right: 5px solid #FFFFFF;
}

#mohdhsuimh .gt_group_heading {
  padding: 8px;
  /* row_group.padding */
  color: #333333;
  background-color: #FFFFFF;
  /* row_group.background.color */
  font-size: 16px;
  /* row_group.font.size */
  font-weight: initial;
  /* row_group.font.weight */
  border-top-style: solid;
  /* row_group.border.top.style */
  border-top-width: 2px;
  /* row_group.border.top.width */
  border-top-color: #D3D3D3;
  /* row_group.border.top.color */
  border-bottom-style: solid;
  /* row_group.border.bottom.style */
  border-bottom-width: 2px;
  /* row_group.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* row_group.border.bottom.color */
  vertical-align: middle;
}

#mohdhsuimh .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  /* row_group.background.color */
  font-size: 16px;
  /* row_group.font.size */
  font-weight: initial;
  /* row_group.font.weight */
  border-top-style: solid;
  /* row_group.border.top.style */
  border-top-width: 2px;
  /* row_group.border.top.width */
  border-top-color: #D3D3D3;
  /* row_group.border.top.color */
  border-bottom-style: solid;
  /* row_group.border.bottom.style */
  border-bottom-width: 2px;
  /* row_group.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* row_group.border.bottom.color */
  vertical-align: middle;
}

#mohdhsuimh .gt_striped {
  background-color: #8080800D;
}

#mohdhsuimh .gt_from_md > :first-child {
  margin-top: 0;
}

#mohdhsuimh .gt_from_md > :last-child {
  margin-bottom: 0;
}

#mohdhsuimh .gt_row {
  padding: 8px;
  /* row.padding */
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#mohdhsuimh .gt_stub {
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

#mohdhsuimh .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  /* summary_row.background.color */
  padding: 8px;
  /* summary_row.padding */
  text-transform: inherit;
  /* summary_row.text_transform */
}

#mohdhsuimh .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  /* grand_summary_row.background.color */
  padding: 8px;
  /* grand_summary_row.padding */
  text-transform: inherit;
  /* grand_summary_row.text_transform */
}

#mohdhsuimh .gt_first_summary_row {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#mohdhsuimh .gt_first_grand_summary_row {
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#mohdhsuimh .gt_table_body {
  border-top-style: solid;
  /* table_body.border.top.style */
  border-top-width: 2px;
  /* table_body.border.top.width */
  border-top-color: #D3D3D3;
  /* table_body.border.top.color */
  border-bottom-style: solid;
  /* table_body.border.bottom.style */
  border-bottom-width: 2px;
  /* table_body.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* table_body.border.bottom.color */
}

#mohdhsuimh .gt_footnotes {
  border-top-style: solid;
  /* footnotes.border.top.style */
  border-top-width: 2px;
  /* footnotes.border.top.width */
  border-top-color: #D3D3D3;
  /* footnotes.border.top.color */
}

#mohdhsuimh .gt_footnote {
  font-size: 90%;
  /* footnote.font.size */
  margin: 0px;
  padding: 4px;
  /* footnote.padding */
}

#mohdhsuimh .gt_sourcenotes {
  border-top-style: solid;
  /* sourcenotes.border.top.style */
  border-top-width: 2px;
  /* sourcenotes.border.top.width */
  border-top-color: #D3D3D3;
  /* sourcenotes.border.top.color */
}

#mohdhsuimh .gt_sourcenote {
  font-size: 90%;
  /* sourcenote.font.size */
  padding: 4px;
  /* sourcenote.padding */
}

#mohdhsuimh .gt_center {
  text-align: center;
}

#mohdhsuimh .gt_left {
  text-align: left;
}

#mohdhsuimh .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#mohdhsuimh .gt_font_normal {
  font-weight: normal;
}

#mohdhsuimh .gt_font_bold {
  font-weight: bold;
}

#mohdhsuimh .gt_font_italic {
  font-style: italic;
}

#mohdhsuimh .gt_super {
  font-size: 65%;
}

#mohdhsuimh .gt_footnote_marks {
  font-style: italic;
  font-size: 65%;
}
</style>
<div id="mohdhsuimh" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <tr>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1"></th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1">2.5 %</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">97.5 %</th>
  </tr>
  <body class="gt_table_body">
    <tr>
      <td class="gt_row gt_left gt_stub">(Intercept)</td>
      <td class="gt_row gt_right">33.45</td>
      <td class="gt_row gt_right">41.12</td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_stub">wt</td>
      <td class="gt_row gt_right gt_striped">−6.49</td>
      <td class="gt_row gt_right gt_striped">−4.20</td>
    </tr>
  </body>
  
  
</table></div>

---

## Example: simple linear regression

The following model is saved as `simple_lm.stan`:

```bash
cat simple_lm.stan
```

```
data {
  int<lower=0> N;
  vector[N] x;
  vector[N] y;
}
parameters {
  real alpha;
  real beta;
  real<lower=0> sigma;
}
model {
  y ~ normal(alpha + beta * x, sigma);
}
```

---

## Example: simple linear regression

Data need to be provided:

```r
mtcarsData <- list(N = nrow(mtcars),
                   y = mtcars$mpg,
                   x = mtcars$wt)
fit_stan <- stan(file = 'simple_lm.stan', data = mtcarsData)
```

Recall that previously the following was obtained using `lm()`:

#pfcjpleufz .gt_table {
  display: table;
  border-collapse: collapse;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  background-color: #FFFFFF;
  /* table.background.color */
  width: auto;
  /* table.width */
  border-top-style: solid;
  /* table.border.top.style */
  border-top-width: 2px;
  /* table.border.top.width */
  border-top-color: #A8A8A8;
  /* table.border.top.color */
  border-bottom-style: solid;
  /* table.border.bottom.style */
  border-bottom-width: 2px;
  /* table.border.bottom.width */
  border-bottom-color: #A8A8A8;
  /* table.border.bottom.color */
}

#pfcjpleufz .gt_heading {
  background-color: #FFFFFF;
  /* heading.background.color */
  border-bottom-color: #FFFFFF;
}

#pfcjpleufz .gt_title {
  color: #333333;
  font-size: 125%;
  /* heading.title.font.size */
  padding-top: 4px;
  /* heading.top.padding - not yet used */
  padding-bottom: 4px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#pfcjpleufz .gt_subtitle {
  color: #333333;
  font-size: 85%;
  /* heading.subtitle.font.size */
  padding-top: 0;
  padding-bottom: 4px;
  /* heading.bottom.padding - not yet used */
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#pfcjpleufz .gt_bottom_border {
  border-bottom-style: solid;
  /* heading.border.bottom.style */
  border-bottom-width: 2px;
  /* heading.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* heading.border.bottom.color */
}

#pfcjpleufz .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  padding-top: 4px;
  padding-bottom: 4px;
}

#pfcjpleufz .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  /* column_labels.background.color */
  font-size: 16px;
  /* column_labels.font.size */
  font-weight: initial;
  /* column_labels.font.weight */
  vertical-align: middle;
  padding: 5px;
  margin: 10px;
  overflow-x: hidden;
}

#pfcjpleufz .gt_columns_top_border {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#pfcjpleufz .gt_columns_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#pfcjpleufz .gt_sep_right {
  border-right: 5px solid #FFFFFF;
}

#pfcjpleufz .gt_group_heading {
  padding: 8px;
  /* row_group.padding */
  color: #333333;
  background-color: #FFFFFF;
  /* row_group.background.color */
  font-size: 16px;
  /* row_group.font.size */
  font-weight: initial;
  /* row_group.font.weight */
  border-top-style: solid;
  /* row_group.border.top.style */
  border-top-width: 2px;
  /* row_group.border.top.width */
  border-top-color: #D3D3D3;
  /* row_group.border.top.color */
  border-bottom-style: solid;
  /* row_group.border.bottom.style */
  border-bottom-width: 2px;
  /* row_group.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* row_group.border.bottom.color */
  vertical-align: middle;
}

#pfcjpleufz .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  /* row_group.background.color */
  font-size: 16px;
  /* row_group.font.size */
  font-weight: initial;
  /* row_group.font.weight */
  border-top-style: solid;
  /* row_group.border.top.style */
  border-top-width: 2px;
  /* row_group.border.top.width */
  border-top-color: #D3D3D3;
  /* row_group.border.top.color */
  border-bottom-style: solid;
  /* row_group.border.bottom.style */
  border-bottom-width: 2px;
  /* row_group.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* row_group.border.bottom.color */
  vertical-align: middle;
}

#pfcjpleufz .gt_striped {
  background-color: #8080800D;
}

#pfcjpleufz .gt_from_md > :first-child {
  margin-top: 0;
}

#pfcjpleufz .gt_from_md > :last-child {
  margin-bottom: 0;
}

#pfcjpleufz .gt_row {
  padding: 8px;
  /* row.padding */
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#pfcjpleufz .gt_stub {
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

#pfcjpleufz .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  /* summary_row.background.color */
  padding: 8px;
  /* summary_row.padding */
  text-transform: inherit;
  /* summary_row.text_transform */
}

#pfcjpleufz .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  /* grand_summary_row.background.color */
  padding: 8px;
  /* grand_summary_row.padding */
  text-transform: inherit;
  /* grand_summary_row.text_transform */
}

#pfcjpleufz .gt_first_summary_row {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#pfcjpleufz .gt_first_grand_summary_row {
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#pfcjpleufz .gt_table_body {
  border-top-style: solid;
  /* table_body.border.top.style */
  border-top-width: 2px;
  /* table_body.border.top.width */
  border-top-color: #D3D3D3;
  /* table_body.border.top.color */
  border-bottom-style: solid;
  /* table_body.border.bottom.style */
  border-bottom-width: 2px;
  /* table_body.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* table_body.border.bottom.color */
}

#pfcjpleufz .gt_footnotes {
  border-top-style: solid;
  /* footnotes.border.top.style */
  border-top-width: 2px;
  /* footnotes.border.top.width */
  border-top-color: #D3D3D3;
  /* footnotes.border.top.color */
}

#pfcjpleufz .gt_footnote {
  font-size: 90%;
  /* footnote.font.size */
  margin: 0px;
  padding: 4px;
  /* footnote.padding */
}

#pfcjpleufz .gt_sourcenotes {
  border-top-style: solid;
  /* sourcenotes.border.top.style */
  border-top-width: 2px;
  /* sourcenotes.border.top.width */
  border-top-color: #D3D3D3;
  /* sourcenotes.border.top.color */
}

#pfcjpleufz .gt_sourcenote {
  font-size: 90%;
  /* sourcenote.font.size */
  padding: 4px;
  /* sourcenote.padding */
}

#pfcjpleufz .gt_center {
  text-align: center;
}

#pfcjpleufz .gt_left {
  text-align: left;
}

#pfcjpleufz .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#pfcjpleufz .gt_font_normal {
  font-weight: normal;
}

#pfcjpleufz .gt_font_bold {
  font-weight: bold;
}

#pfcjpleufz .gt_font_italic {
  font-style: italic;
}

#pfcjpleufz .gt_super {
  font-size: 65%;
}

#pfcjpleufz .gt_footnote_marks {
  font-style: italic;
  font-size: 65%;
}
</style>
<div id="pfcjpleufz" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <tr>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1"></th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1">estimate</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">std.error</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">statistic</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">p.value</th>
  </tr>
  <body class="gt_table_body">
    <tr>
      <td class="gt_row gt_left gt_stub">(Intercept)</td>
      <td class="gt_row gt_right">37.29</td>
      <td class="gt_row gt_right">1.88</td>
      <td class="gt_row gt_right">19.86</td>
      <td class="gt_row gt_right">8.24 × 10<sup class='gt_super'>−19</sup></td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_stub">wt</td>
      <td class="gt_row gt_right gt_striped">−5.34</td>
      <td class="gt_row gt_right gt_striped">0.56</td>
      <td class="gt_row gt_right gt_striped">−9.56</td>
      <td class="gt_row gt_right gt_striped">1.29 × 10<sup class='gt_super'>−10</sup></td>
    </tr>
  </body>
  
  
</table></div>

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<div id="svtyupdmci" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <tr>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1"></th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1">2.5 %</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">97.5 %</th>
  </tr>
  <body class="gt_table_body">
    <tr>
      <td class="gt_row gt_left gt_stub">(Intercept)</td>
      <td class="gt_row gt_right">33.45</td>
      <td class="gt_row gt_right">41.12</td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_stub">wt</td>
      <td class="gt_row gt_right gt_striped">−6.49</td>
      <td class="gt_row gt_right gt_striped">−4.20</td>
    </tr>
  </body>
  
  
</table></div>

---

## Example: simple linear regression

```r
print(fit_stan)
```

```
Inference for Stan model: simple_lm.
4 chains, each with iter=2000; warmup=1000; thin=1; 
post-warmup draws per chain=1000, total post-warmup draws=4000.

mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
alpha  37.23    0.05 1.94  33.41  35.98  37.18  38.49  41.09  1334 1.00
beta   -5.33    0.02 0.58  -6.48  -5.70  -5.32  -4.95  -4.19  1439 1.00
sigma   3.19    0.01 0.43   2.46   2.88   3.15   3.44   4.14  1315 1.00
lp__  -51.05    0.04 1.23 -54.25 -51.61 -50.72 -50.16 -49.64  1164 1.01

Samples were drawn using NUTS(diag_e) at Mon Oct 21 22:57:08 2019.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).
```

---

## Example: simple linear regression

Run `stan()` with options:

```r
fit_stan2 <- stan(file = 'simple_lm.stan', 
                 data = mtcarsData,
                 seed = 2019,
                 chains = 5,
                 iter = 10000,
                 warmup = 3000)
```

---

## Example: simple linear regression

```r
print(fit_stan2)
```

```
Inference for Stan model: simple_lm.
5 chains, each with iter=10000; warmup=3000; thin=1; 
post-warmup draws per chain=7000, total post-warmup draws=35000.

mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
alpha  37.28    0.02 1.99  33.30  35.98  37.29  38.60  41.23 11621    1
beta   -5.34    0.01 0.60  -6.52  -5.74  -5.35  -4.95  -4.16 11620    1
sigma   3.18    0.00 0.44   2.47   2.87   3.13   3.44   4.17 15359    1
lp__  -51.11    0.01 1.30 -54.47 -51.69 -50.78 -50.17 -49.63 10790    1

Samples were drawn using NUTS(diag_e) at Mon Oct 21 23:46:03 2019.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).
```

---

## Example: simple linear regression

```r
traceplot(fit_stan2)
```

---

## Example: simple linear regression

Output from `stan()` can be extracted:

```r
df <- as.data.frame(fit_stan2)
dim(df) # iter = 10000, warmup = 3000, chains = 5
```

```
## [1] 35000     4
```

```r
head(df)
```

```
##      alpha      beta    sigma      lp__
## 1 38.48587 -5.295838 3.269406 -52.50245
## 2 38.42898 -5.334596 3.259406 -51.80863
## 3 38.45064 -5.414004 3.401005 -51.21041
## 4 38.55132 -5.426852 3.555980 -51.60899
## 5 37.79401 -5.771271 2.736554 -51.74718
## 6 38.02972 -5.688517 2.731914 -50.31771
```

---

## Example: logistic regression

```r
# virginica or not
dIris <- iris %>%
  mutate(IsVirginica = ifelse(Species == "virginica", 1, 0))

fit <- glm(IsVirginica ~ Sepal.Length + Sepal.Width, 
           data = dIris,
           family = binomial(link = "logit"))

tidy(fit) %>% 
  gt(rowname_col = "term") %>%
  fmt_number(columns = vars(estimate, std.error, statistic), 
             decimals = 2) %>%
  fmt_scientific(columns = vars(p.value), decimals = 2)
```

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<div id="qrrpnqhkkl" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <tr>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1"></th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1">estimate</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">std.error</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">statistic</th>
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  </tr>
  <body class="gt_table_body">
    <tr>
      <td class="gt_row gt_left gt_stub">(Intercept)</td>
      <td class="gt_row gt_right">−14.18</td>
      <td class="gt_row gt_right">3.12</td>
      <td class="gt_row gt_right">−4.55</td>
      <td class="gt_row gt_right">5.36 × 10<sup class='gt_super'>−6</sup></td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_stub">Sepal.Length</td>
      <td class="gt_row gt_right gt_striped">2.60</td>
      <td class="gt_row gt_right gt_striped">0.44</td>
      <td class="gt_row gt_right gt_striped">5.91</td>
      <td class="gt_row gt_right gt_striped">3.48 × 10<sup class='gt_super'>−9</sup></td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_stub">Sepal.Width</td>
      <td class="gt_row gt_right">−0.75</td>
      <td class="gt_row gt_right">0.64</td>
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      <td class="gt_row gt_right">2.45 × 10<sup class='gt_super'>−1</sup></td>
    </tr>
  </body>
  
  
</table></div>

---

## Example: logistic regression

```r
confint(fit) %>%
  as.data.frame() %>%
  rownames_to_column("term") %>%
  gt(rowname_col = "term") %>%
  fmt_number(columns = everything(), 
             decimals = 2)
```

#nddyshqptf .gt_table {
  display: table;
  border-collapse: collapse;
  margin-left: auto;
  margin-right: auto;
  color: #333333;
  font-size: 16px;
  background-color: #FFFFFF;
  /* table.background.color */
  width: auto;
  /* table.width */
  border-top-style: solid;
  /* table.border.top.style */
  border-top-width: 2px;
  /* table.border.top.width */
  border-top-color: #A8A8A8;
  /* table.border.top.color */
  border-bottom-style: solid;
  /* table.border.bottom.style */
  border-bottom-width: 2px;
  /* table.border.bottom.width */
  border-bottom-color: #A8A8A8;
  /* table.border.bottom.color */
}

#nddyshqptf .gt_heading {
  background-color: #FFFFFF;
  /* heading.background.color */
  border-bottom-color: #FFFFFF;
}

#nddyshqptf .gt_title {
  color: #333333;
  font-size: 125%;
  /* heading.title.font.size */
  padding-top: 4px;
  /* heading.top.padding - not yet used */
  padding-bottom: 4px;
  border-bottom-color: #FFFFFF;
  border-bottom-width: 0;
}

#nddyshqptf .gt_subtitle {
  color: #333333;
  font-size: 85%;
  /* heading.subtitle.font.size */
  padding-top: 0;
  padding-bottom: 4px;
  /* heading.bottom.padding - not yet used */
  border-top-color: #FFFFFF;
  border-top-width: 0;
}

#nddyshqptf .gt_bottom_border {
  border-bottom-style: solid;
  /* heading.border.bottom.style */
  border-bottom-width: 2px;
  /* heading.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* heading.border.bottom.color */
}

#nddyshqptf .gt_column_spanner {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
  padding-top: 4px;
  padding-bottom: 4px;
}

#nddyshqptf .gt_col_heading {
  color: #333333;
  background-color: #FFFFFF;
  /* column_labels.background.color */
  font-size: 16px;
  /* column_labels.font.size */
  font-weight: initial;
  /* column_labels.font.weight */
  vertical-align: middle;
  padding: 5px;
  margin: 10px;
  overflow-x: hidden;
}

#nddyshqptf .gt_columns_top_border {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#nddyshqptf .gt_columns_bottom_border {
  border-bottom-style: solid;
  border-bottom-width: 2px;
  border-bottom-color: #D3D3D3;
}

#nddyshqptf .gt_sep_right {
  border-right: 5px solid #FFFFFF;
}

#nddyshqptf .gt_group_heading {
  padding: 8px;
  /* row_group.padding */
  color: #333333;
  background-color: #FFFFFF;
  /* row_group.background.color */
  font-size: 16px;
  /* row_group.font.size */
  font-weight: initial;
  /* row_group.font.weight */
  border-top-style: solid;
  /* row_group.border.top.style */
  border-top-width: 2px;
  /* row_group.border.top.width */
  border-top-color: #D3D3D3;
  /* row_group.border.top.color */
  border-bottom-style: solid;
  /* row_group.border.bottom.style */
  border-bottom-width: 2px;
  /* row_group.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* row_group.border.bottom.color */
  vertical-align: middle;
}

#nddyshqptf .gt_empty_group_heading {
  padding: 0.5px;
  color: #333333;
  background-color: #FFFFFF;
  /* row_group.background.color */
  font-size: 16px;
  /* row_group.font.size */
  font-weight: initial;
  /* row_group.font.weight */
  border-top-style: solid;
  /* row_group.border.top.style */
  border-top-width: 2px;
  /* row_group.border.top.width */
  border-top-color: #D3D3D3;
  /* row_group.border.top.color */
  border-bottom-style: solid;
  /* row_group.border.bottom.style */
  border-bottom-width: 2px;
  /* row_group.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* row_group.border.bottom.color */
  vertical-align: middle;
}

#nddyshqptf .gt_striped {
  background-color: #8080800D;
}

#nddyshqptf .gt_from_md > :first-child {
  margin-top: 0;
}

#nddyshqptf .gt_from_md > :last-child {
  margin-bottom: 0;
}

#nddyshqptf .gt_row {
  padding: 8px;
  /* row.padding */
  margin: 10px;
  border-top-style: solid;
  border-top-width: 1px;
  border-top-color: #D3D3D3;
  vertical-align: middle;
  overflow-x: hidden;
}

#nddyshqptf .gt_stub {
  border-right-style: solid;
  border-right-width: 2px;
  border-right-color: #D3D3D3;
  padding-left: 12px;
}

#nddyshqptf .gt_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  /* summary_row.background.color */
  padding: 8px;
  /* summary_row.padding */
  text-transform: inherit;
  /* summary_row.text_transform */
}

#nddyshqptf .gt_grand_summary_row {
  color: #333333;
  background-color: #FFFFFF;
  /* grand_summary_row.background.color */
  padding: 8px;
  /* grand_summary_row.padding */
  text-transform: inherit;
  /* grand_summary_row.text_transform */
}

#nddyshqptf .gt_first_summary_row {
  border-top-style: solid;
  border-top-width: 2px;
  border-top-color: #D3D3D3;
}

#nddyshqptf .gt_first_grand_summary_row {
  border-top-style: double;
  border-top-width: 6px;
  border-top-color: #D3D3D3;
}

#nddyshqptf .gt_table_body {
  border-top-style: solid;
  /* table_body.border.top.style */
  border-top-width: 2px;
  /* table_body.border.top.width */
  border-top-color: #D3D3D3;
  /* table_body.border.top.color */
  border-bottom-style: solid;
  /* table_body.border.bottom.style */
  border-bottom-width: 2px;
  /* table_body.border.bottom.width */
  border-bottom-color: #D3D3D3;
  /* table_body.border.bottom.color */
}

#nddyshqptf .gt_footnotes {
  border-top-style: solid;
  /* footnotes.border.top.style */
  border-top-width: 2px;
  /* footnotes.border.top.width */
  border-top-color: #D3D3D3;
  /* footnotes.border.top.color */
}

#nddyshqptf .gt_footnote {
  font-size: 90%;
  /* footnote.font.size */
  margin: 0px;
  padding: 4px;
  /* footnote.padding */
}

#nddyshqptf .gt_sourcenotes {
  border-top-style: solid;
  /* sourcenotes.border.top.style */
  border-top-width: 2px;
  /* sourcenotes.border.top.width */
  border-top-color: #D3D3D3;
  /* sourcenotes.border.top.color */
}

#nddyshqptf .gt_sourcenote {
  font-size: 90%;
  /* sourcenote.font.size */
  padding: 4px;
  /* sourcenote.padding */
}

#nddyshqptf .gt_center {
  text-align: center;
}

#nddyshqptf .gt_left {
  text-align: left;
}

#nddyshqptf .gt_right {
  text-align: right;
  font-variant-numeric: tabular-nums;
}

#nddyshqptf .gt_font_normal {
  font-weight: normal;
}

#nddyshqptf .gt_font_bold {
  font-weight: bold;
}

#nddyshqptf .gt_font_italic {
  font-style: italic;
}

#nddyshqptf .gt_super {
  font-size: 65%;
}

#nddyshqptf .gt_footnote_marks {
  font-style: italic;
  font-size: 65%;
}
</style>
<div id="nddyshqptf" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <tr>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1"></th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_left" rowspan="1" colspan="1">2.5 %</th>
    <th class="gt_col_heading gt_columns_bottom_border gt_columns_top_border gt_center" rowspan="1" colspan="1">97.5 %</th>
  </tr>
  <body class="gt_table_body">
    <tr>
      <td class="gt_row gt_left gt_stub">(Intercept)</td>
      <td class="gt_row gt_right">−20.85</td>
      <td class="gt_row gt_right">−8.56</td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_stub">Sepal.Length</td>
      <td class="gt_row gt_right gt_striped">1.82</td>
      <td class="gt_row gt_right gt_striped">3.57</td>
    </tr>
    <tr>
      <td class="gt_row gt_left gt_stub">Sepal.Width</td>
      <td class="gt_row gt_right">−2.09</td>
      <td class="gt_row gt_right">0.45</td>
    </tr>
  </body>
  
  
</table></div>

---

## Example: logistic regression

The following model is saved as `logistic_2.stan`:

```bash
cat logistic_2.stan
```

```
data {
  int<lower=0> N;
  vector[N] x1;
  vector[N] x2;
  int<lower=0, upper=1> y[N];
}
parameters {
  real alpha;
  real beta1;
  real beta2;
}
model {
  y ~ bernoulli_logit(alpha + beta1 * x1 + beta2 * x2);
}
```

---

## Example: logistic regression

```r
irisData <- list(N = nrow(dIris),
                 y = dIris$IsVirginica,
                 x1 = dIris$Sepal.Length,
                 x2 = dIris$Sepal.Width)
fit_logit <- stan(file = 'logistic_2.stan',
                  data = irisData,
                  seed = 2019,
                  chains = 5,
                  iter = 10000,
                  warmup = 3000)
```

---

## Example: logistic regression

```r
print(fit_logit)
```

```
Inference for Stan model: logistic_2.
5 chains, each with iter=10000; warmup=3000; thin=1; 
post-warmup draws per chain=7000, total post-warmup draws=35000.

mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
alpha -14.87    0.03 3.21 -21.57 -16.91 -14.71 -12.66  -9.06 10388    1
beta1   2.74    0.00 0.46   1.92   2.42   2.71   3.04   3.74 10766    1
beta2  -0.82    0.01 0.65  -2.14  -1.24  -0.79  -0.37   0.40 12745    1
lp__  -59.47    0.01 1.26 -62.74 -60.02 -59.14 -58.56 -58.06  9382    1

Samples were drawn using NUTS(diag_e) at Sun Nov  3 11:49:04 2019.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at 
convergence, Rhat=1).
```

---

## Example: logistic regression

```r
traceplot(fit_logit)
```