Bootstrap Confidence Intervals

# Bootstrap Confidence Intervals
### Dr. D’Agostino McGowan

---

<div class="my-footer">
<span>
Dr. Lucy D'Agostino McGowan
</span>
</div>

---

## Simulation

* Generate `$\epsilon$` from the known error distribution
--

* Calculate `$\mathbf{y}=\mathbf{X}\beta+\epsilon$` from the known `$\beta$` and the observed data, `$\mathbf{X}$`
--

* Use this to compute `$\hat\beta$`
--

* Do this _many times_ to get a distribution of `$\hat\beta$` and estimate the uncertainty from that

---

## Simulation

* This sounds **great!** 
--

* but...we don't know the true error distribution and we don't know the true `$\beta$` (womp womp!)

---

## Bootstrap

* That is where the bootstrap comes in!
--

* The bootstrap tries to estimate the same procedure as the simulation method except instead of sampling from the **true** distribution, we sample from the **observed data**

---

## Bootstrap

_"To pull oneself up by one's bootstraps"_

The phrase likely came from an 18th century writing "The Surprising Adventures of Baron Munchausen" by Rudolph Erich Raspe

_"The Baron had fallen to the bottom of a deep lake. Just
when it looked like all was lost, he thought to **pick himself
up by his own bootstraps**."_
---

## Bootstrap

![](img/10/boot.png)

## Bootstrap

* Used to estimate _uncertainty_.
--

* For example, the standard error of the estimate or an approximate confidence interval

---

## Bootstrap

* Generate `$\epsilon^*$` from sampling with replacement from `$e_1,\dots,e_n$` (the residuals)
--

* Calculate `$\mathbf{y}^* =\mathbf{X}\hat\beta+\epsilon^*$`
--

* Compute `$\hat\beta^*$` from `$(\mathbf{X}, \mathbf{y}^*)$`

---

## Generate `$\epsilon^*$` from sampling with replacement from the residuals

|     e|
|-----:|
|  0.02|
|  0.02|
| -0.23|
|  0.42|
| -0.23|

---

## Generate `$\epsilon^*$` from sampling with replacement from the residuals

|     e|  e* 1|  e* 2| e* 3|  e* 4|  e* 5|
|-----:|-----:|-----:|----:|-----:|-----:|
|  0.02|  0.02| -0.23| 0.02| -0.23| -0.23|
|  0.02|  0.02| -0.23| 0.02|  0.42|  0.02|
| -0.23| -0.23|  0.42| 0.42| -0.23|  0.42|
|  0.42| -0.23|  0.42| 0.02|  0.02|  0.02|
| -0.23|  0.02|  0.02| 0.42|  0.42|  0.02|

---

## Let's do this in R!

* The `sample()` function allows you to resample with replacement.

```r
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 2 1 2
```

```r
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 3 2 2
```

```r
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 2 2 1
```

---

## Let's do this in R!

* The `sample()` function allows you to resample with replacement.

```r
set.seed(1)
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 1 2 2
```

```r
set.seed(1)
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 1 2 2
```

---

## Let's do this in R!

* The `sample()` function allows you to resample with replacement.

```r
set.seed(1)
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 1 2 2
```

```r
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 3 1 3
```

```r
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 3 2 2
```

]
]

```r
set.seed(1)
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 1 2 2
```

```r
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 3 1 3
```

```r
sample(c(1, 2, 3), replace = TRUE)
```

```
## [1] 3 2 2
```
]
]

* **Always set a seed when generating random processes so you can replicate your results**

---

## <svg style="height:0.8em;top:.04em;position:relative;" viewBox="0 0 640 512"><path d="M512 64v256H128V64h384m16-64H112C85.5 0 64 21.5 64 48v288c0 26.5 21.5 48 48 48h416c26.5 0 48-21.5 48-48V48c0-26.5-21.5-48-48-48zm100 416H389.5c-3 0-5.5 2.1-5.9 5.1C381.2 436.3 368 448 352 448h-64c-16 0-29.2-11.7-31.6-26.9-.5-2.9-3-5.1-5.9-5.1H12c-6.6 0-12 5.4-12 12v36c0 26.5 21.5 48 48 48h544c26.5 0 48-21.5 48-48v-36c0-6.6-5.4-12-12-12z"/></svg> `Application Exercise`

* Set a seed
* Use the `sample()` function to sample from the following vector with replacement: `c(1, 2, 3, 4, 5)`

---
## Let's do this in R

* Generate `$\epsilon^*$` from sampling with replacement from `$e_1,\dots,e_n$` (the residuals)

```r
e_star <- sample(e, replace = TRUE) 
```
--

* Calculate `$\mathbf{y}^* =\mathbf{X}\hat\beta+\epsilon^*$`

```r
y_star <- y_hat + e_star
```

* Compute `$\hat\beta^*$` from `$(\mathbf{X}, \mathbf{y}^*)$`

```r
new_mod <- update(model, y_star ~ ., data = data)
coef(new_mod)
```

---

## Let's do this in R

Let's build a function to do this **once**

```r
*bootstrap_residual <- function(e, y_hat, model, data) {
  e_star <- sample(e, replace = TRUE) 
  data$y_star <- y_hat + e_star
  new_mod <- update(model, y_star ~ ., data = data)
  return(coef(new_mod))
}
```
]

---

## Let's do this in R

Let's build a function to do this **once**

```r
bootstrap_residual <- function(e, y_hat, model, data) { 
* e_star <- sample(e, replace = TRUE)
  data$y_star <- y_hat + e_star
  new_mod <- update(model, y_star ~ ., data = data)
  return(coef(new_mod))
}
```

---

## Let's do this in R

Let's build a function to do this **once**

```r
bootstrap_residual <- function(e, y_hat, model, data) { 
  e_star <- sample(e, replace = TRUE) 
* data$y_star <- y_hat + e_star
  new_mod <- update(model, y_star ~ ., data = data)
  return(coef(new_mod))
}
```

---

## Let's do this in R

Let's build a function to do this **once**

```r
bootstrap_residual <- function(e, y_hat, model, data) { 
  e_star <- sample(e, replace = TRUE) 
  data$y_star <- y_hat + e_star
* new_mod <- update(model, y_star ~ ., data = data)
  return(coef(new_mod))
}
```

* The `update()` function will update a model with a new formula. Here we are swapping out `y` from our original `model` with `y_star`

---
## Let's do this in R

Let's build a function to do this **once**

```r
bootstrap_residual <- function(e, y_hat, model, data) { 
  e_star <- sample(e, replace = TRUE) 
  data$y_star <- y_hat + e_star
  new_mod <- update(model, y_star ~ ., data = data)
* return(coef(new_mod))
}
```

---

* Write a function like the one on the previous slide to perform a bootstrap.

<div class="countdown" id="timer_5f74ae70" style="right:0;bottom:0;" data-warnwhen="0">
<code class="countdown-time"><span class="countdown-digits minutes">02</span><span class="countdown-digits colon">:</span><span class="countdown-digits seconds">00</span></code>
</div>
---

## Let's do this in R

```r
model <- lm(mpg ~ horsepower, data = Auto)
y_hat <- fitted(model)
e <- residuals(model)
```

```r
set.seed(1)
e_star <- sample(e, replace = TRUE) 
Auto$y_star <- y_hat + e_star
new_mod <- update(model, y_star ~ ., data = Auto)
coef(new_mod)
```

```
## (Intercept)  horsepower 
##  38.9999721  -0.1506159
```
]
]

```r
set.seed(1)
bootstrap_residual(e, y_hat, model, Auto)
```

```
## (Intercept)  horsepower 
##  38.9999721  -0.1506159
```

]]

---

## Let's do this in R

```r
model <- lm(mpg ~ horsepower, data = Auto)
y_hat <- fitted(model)
e <- residuals(model)
```

```r
set.seed(1)
e_star <- sample(e, replace = TRUE) 
Auto$y_star <- y_hat + e_star
new_mod <- update(model, y_star ~ ., data = Auto)
coef(new_mod)
```

```
## (Intercept)  horsepower 
##  38.9999721  -0.1506159
```
]
]

```r
set.seed(1)
bootstrap_residual(e, y_hat, model, Auto)
```

```
## (Intercept)  horsepower 
##  38.9999721  -0.1506159
```

```r
bootstrap_residual(e, y_hat, model, Auto)
```

```
## (Intercept)  horsepower 
##  40.0583859  -0.1597723
```

]]

---

## Let's do this in R

```r
model <- lm(mpg ~ horsepower, data = Auto)
y_hat <- fitted(model)
e <- residuals(model)
```

```r
set.seed(1)
e_star <- sample(e, replace = TRUE) 
Auto$y_star <- y_hat + e_star
new_mod <- update(model, y_star ~ ., data = Auto)
coef(new_mod)
```

```
## (Intercept)  horsepower 
##  38.9999721  -0.1506159
```
]
]

```r
set.seed(1)
bootstrap_residual(e, y_hat, model, Auto)
```

```
## (Intercept)  horsepower 
##  38.9999721  -0.1506159
```

```r
bootstrap_residual(e, y_hat, model, Auto)
```

```
## (Intercept)  horsepower 
##  40.0583859  -0.1597723
```

```r
bootstrap_residual(e, y_hat, model, Auto)
```

```
## (Intercept)  horsepower 
##  39.0514849  -0.1513228
```

]]

---

## purrr

- **purrr** is a package for _iterating_ in R
- The functions we will use are all `map_xxx()` functions

---

## `map(.x, .f, ...)`

### for every element of `.x` do `.f`

---

## Example

```r
library(purrr)
map(1:5, ~ .x * 2)
```

```
## [[1]]
## [1] 2
## 
## [[2]]
## [1] 4
## 
## [[3]]
## [1] 6
## 
## [[4]]
## [1] 8
## 
## [[5]]
## [1] 10
```
]

* By default, the output will be a **list**
--

* You can dictate your desired output type by specifying `map_xxx()` for example to output a numeric (double) vector, use `map_dbl()`

---

## Example

```r
map_dbl(1:5, ~ .x * 2)
```

```
## [1]  2  4  6  8 10
```
]

---

## `map_xxx(.x, .f)`

* `map()`: list
* `map_dbl()`: double
* `map_int()`
* `map_lgl()`
* `map_chr()`
* `map_df()`

---

## `map_xxx(.x, .f)`

* `map()`: list
* `map_dbl()`: double
* `map_int()`: integer
* `map_lgl()`: logical
* `map_chr()`: character
* `map_df()`: data frame

---

## Back to our example!

```r
B <- 1000
map(1:B, ~ bootstrap_residual(e, y_hat, model, Auto))
```

```
## [[1]]
## (Intercept)  horsepower 
##  39.5577878  -0.1537403 
## 
## [[2]]
## (Intercept)  horsepower 
##  38.7517839  -0.1445242 
## 
## [[3]]
## (Intercept)  horsepower 
##   40.299449   -0.159421 
## 
## [[4]]
## (Intercept)  horsepower 
##  39.5579743  -0.1562991 
## 
## [[5]]
## (Intercept)  horsepower 
##  39.9851338  -0.1570814 
## 
## [[6]]
## (Intercept)  horsepower 
##  41.3035838  -0.1641114 
## 
## [[7]]
## (Intercept)  horsepower 
##  41.2766538  -0.1702292 
## 
## [[8]]
## (Intercept)  horsepower 
##  41.0797054  -0.1730696 
## 
## [[9]]
## (Intercept)  horsepower 
##  40.8505331  -0.1643151 
## 
## [[10]]
## (Intercept)  horsepower 
##  40.0168081  -0.1580494 
## 
## [[11]]
## (Intercept)  horsepower 
##  40.2916142  -0.1598027 
## 
## [[12]]
## (Intercept)  horsepower 
##  40.1242482  -0.1581572 
## 
## [[13]]
## (Intercept)  horsepower 
##  39.9658178  -0.1568639 
## 
## [[14]]
## (Intercept)  horsepower 
##  39.7701365  -0.1558859 
## 
## [[15]]
## (Intercept)  horsepower 
##  40.5372073  -0.1627034 
## 
## [[16]]
## (Intercept)  horsepower 
##  39.5850590  -0.1565891 
## 
## [[17]]
## (Intercept)  horsepower 
##  40.2910406  -0.1609778 
## 
## [[18]]
## (Intercept)  horsepower 
##  40.2798489  -0.1609537 
## 
## [[19]]
## (Intercept)  horsepower 
##  41.2150687  -0.1711352 
## 
## [[20]]
## (Intercept)  horsepower 
##  39.2629638  -0.1544094 
## 
## [[21]]
## (Intercept)  horsepower 
##  39.5349343  -0.1537218 
## 
## [[22]]
## (Intercept)  horsepower 
##  39.0080677  -0.1493604 
## 
## [[23]]
## (Intercept)  horsepower 
##  40.2583847  -0.1610556 
## 
## [[24]]
## (Intercept)  horsepower 
##  38.8722471  -0.1433313 
## 
## [[25]]
## (Intercept)  horsepower 
##  40.1509600  -0.1604275 
## 
## [[26]]
## (Intercept)  horsepower 
##  39.1729288  -0.1566467 
## 
## [[27]]
## (Intercept)  horsepower 
##  39.7792045  -0.1582528 
## 
## [[28]]
## (Intercept)  horsepower 
##   40.078146   -0.159936 
## 
## [[29]]
## (Intercept)  horsepower 
##  39.7678203  -0.1555682 
## 
## [[30]]
## (Intercept)  horsepower 
##  41.4134066  -0.1683716 
## 
## [[31]]
## (Intercept)  horsepower 
##  38.6201544  -0.1490515 
## 
## [[32]]
## (Intercept)  horsepower 
##  39.7114520  -0.1561636 
## 
## [[33]]
## (Intercept)  horsepower 
##  39.0898140  -0.1536236 
## 
## [[34]]
## (Intercept)  horsepower 
##  39.8639302  -0.1599347 
## 
## [[35]]
## (Intercept)  horsepower 
##  40.8588984  -0.1677346 
## 
## [[36]]
## (Intercept)  horsepower 
##  41.0818814  -0.1674714 
## 
## [[37]]
## (Intercept)  horsepower 
##  40.1840424  -0.1592952 
## 
## [[38]]
## (Intercept)  horsepower 
##   40.367334   -0.164715 
## 
## [[39]]
## (Intercept)  horsepower 
##  40.0898365  -0.1622775 
## 
## [[40]]
## (Intercept)  horsepower 
##  37.6499657  -0.1369237 
## 
## [[41]]
## (Intercept)  horsepower 
##   39.823175   -0.158254 
## 
## [[42]]
## (Intercept)  horsepower 
##  39.6551554  -0.1554174 
## 
## [[43]]
## (Intercept)  horsepower 
##  40.4102540  -0.1594586 
## 
## [[44]]
## (Intercept)  horsepower 
##  41.0654751  -0.1677579 
## 
## [[45]]
## (Intercept)  horsepower 
##  39.6167314  -0.1524366 
## 
## [[46]]
## (Intercept)  horsepower 
##  39.8079910  -0.1569379 
## 
## [[47]]
## (Intercept)  horsepower 
##  40.1026108  -0.1581709 
## 
## [[48]]
## (Intercept)  horsepower 
##  39.6178025  -0.1530939 
## 
## [[49]]
## (Intercept)  horsepower 
##  40.2222307  -0.1599964 
## 
## [[50]]
## (Intercept)  horsepower 
##  40.7234411  -0.1626942 
## 
## [[51]]
## (Intercept)  horsepower 
##  38.8285126  -0.1519009 
## 
## [[52]]
## (Intercept)  horsepower 
##   39.323665   -0.152499 
## 
## [[53]]
## (Intercept)  horsepower 
##  39.7705791  -0.1539415 
## 
## [[54]]
## (Intercept)  horsepower 
##  40.6498290  -0.1631471 
## 
## [[55]]
## (Intercept)  horsepower 
##  40.6031043  -0.1641427 
## 
## [[56]]
## (Intercept)  horsepower 
##  39.4958141  -0.1553134 
## 
## [[57]]
## (Intercept)  horsepower 
##  41.4847308  -0.1686765 
## 
## [[58]]
## (Intercept)  horsepower 
##  40.3336274  -0.1572135 
## 
## [[59]]
## (Intercept)  horsepower 
##  41.2418036  -0.1720063 
## 
## [[60]]
## (Intercept)  horsepower 
##  40.1031117  -0.1630496 
## 
## [[61]]
## (Intercept)  horsepower 
##  39.3444628  -0.1522149 
## 
## [[62]]
## (Intercept)  horsepower 
##  40.5606501  -0.1639916 
## 
## [[63]]
## (Intercept)  horsepower 
##  40.0301164  -0.1587286 
## 
## [[64]]
## (Intercept)  horsepower 
##  40.3977062  -0.1610259 
## 
## [[65]]
## (Intercept)  horsepower 
##  39.8567555  -0.1571448 
## 
## [[66]]
## (Intercept)  horsepower 
##  40.0225487  -0.1590777 
## 
## [[67]]
## (Intercept)  horsepower 
##  39.5250737  -0.1549172 
## 
## [[68]]
## (Intercept)  horsepower 
##  39.7936246  -0.1589207 
## 
## [[69]]
## (Intercept)  horsepower 
##  39.7257813  -0.1580421 
## 
## [[70]]
## (Intercept)  horsepower 
##   39.623322   -0.153161 
## 
## [[71]]
## (Intercept)  horsepower 
##  39.9780714  -0.1620281 
## 
## [[72]]
## (Intercept)  horsepower 
##  40.2904159  -0.1638038 
## 
## [[73]]
## (Intercept)  horsepower 
##  39.9442227  -0.1643489 
## 
## [[74]]
## (Intercept)  horsepower 
##  40.7109727  -0.1656063 
## 
## [[75]]
## (Intercept)  horsepower 
##  40.1146796  -0.1592135 
## 
## [[76]]
## (Intercept)  horsepower 
##  40.7509373  -0.1649399 
## 
## [[77]]
## (Intercept)  horsepower 
##  40.2625021  -0.1560143 
## 
## [[78]]
## (Intercept)  horsepower 
##  40.1475082  -0.1583246 
## 
## [[79]]
## (Intercept)  horsepower 
##  39.8926351  -0.1585041 
## 
## [[80]]
## (Intercept)  horsepower 
##  38.6615396  -0.1469887 
## 
## [[81]]
## (Intercept)  horsepower 
##  39.6107093  -0.1556742 
## 
## [[82]]
## (Intercept)  horsepower 
##  39.4734357  -0.1541891 
## 
## [[83]]
## (Intercept)  horsepower 
##   40.801377   -0.165111 
## 
## [[84]]
## (Intercept)  horsepower 
##  41.0306958  -0.1666446 
## 
## [[85]]
## (Intercept)  horsepower 
##  39.6439258  -0.1526369 
## 
## [[86]]
## (Intercept)  horsepower 
##  40.9855636  -0.1646064 
## 
## [[87]]
## (Intercept)  horsepower 
##  39.9900474  -0.1559717 
## 
## [[88]]
## (Intercept)  horsepower 
##  40.9847786  -0.1644254 
## 
## [[89]]
## (Intercept)  horsepower 
##  40.0109059  -0.1569243 
## 
## [[90]]
## (Intercept)  horsepower 
##  39.1407418  -0.1520801 
## 
## [[91]]
## (Intercept)  horsepower 
##  40.6240903  -0.1676859 
## 
## [[92]]
## (Intercept)  horsepower 
##   39.020788   -0.153221 
## 
## [[93]]
## (Intercept)  horsepower 
##  40.5863430  -0.1606872 
## 
## [[94]]
## (Intercept)  horsepower 
##  39.6809119  -0.1566499 
## 
## [[95]]
## (Intercept)  horsepower 
##  40.0267718  -0.1592652 
## 
## [[96]]
## (Intercept)  horsepower 
##  39.8662546  -0.1592607 
## 
## [[97]]
## (Intercept)  horsepower 
##  39.6652009  -0.1557058 
## 
## [[98]]
## (Intercept)  horsepower 
##  38.5325581  -0.1447881 
## 
## [[99]]
## (Intercept)  horsepower 
##  40.0227756  -0.1567542 
## 
## [[100]]
## (Intercept)  horsepower 
##  40.2906138  -0.1611927 
## 
## [[101]]
## (Intercept)  horsepower 
##  40.2475817  -0.1610279 
## 
## [[102]]
## (Intercept)  horsepower 
##  41.4177392  -0.1684197 
## 
## [[103]]
## (Intercept)  horsepower 
##  40.1412265  -0.1587343 
## 
## [[104]]
## (Intercept)  horsepower 
##   38.815354   -0.144407 
## 
## [[105]]
## (Intercept)  horsepower 
##  39.8134453  -0.1546786 
## 
## [[106]]
## (Intercept)  horsepower 
##  40.0463617  -0.1552516 
## 
## [[107]]
## (Intercept)  horsepower 
##  39.5956595  -0.1547384 
## 
## [[108]]
## (Intercept)  horsepower 
##  39.9132199  -0.1548267 
## 
## [[109]]
## (Intercept)  horsepower 
##  40.7247670  -0.1647174 
## 
## [[110]]
## (Intercept)  horsepower 
##  40.3719495  -0.1587214 
## 
## [[111]]
## (Intercept)  horsepower 
##  39.8990600  -0.1585696 
## 
## [[112]]
## (Intercept)  horsepower 
##  39.2929986  -0.1535196 
## 
## [[113]]
## (Intercept)  horsepower 
##  38.9802463  -0.1484273 
## 
## [[114]]
## (Intercept)  horsepower 
##  41.0569632  -0.1655651 
## 
## [[115]]
## (Intercept)  horsepower 
##  38.5552634  -0.1427499 
## 
## [[116]]
## (Intercept)  horsepower 
##  39.3985976  -0.1559174 
## 
## [[117]]
## (Intercept)  horsepower 
##  40.4101332  -0.1598915 
## 
## [[118]]
## (Intercept)  horsepower 
##  40.1662186  -0.1595035 
## 
## [[119]]
## (Intercept)  horsepower 
##  40.7177105  -0.1638234 
## 
## [[120]]
## (Intercept)  horsepower 
##  39.7728086  -0.1553169 
## 
## [[121]]
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## 
## [[822]]
## (Intercept)  horsepower 
##  40.7453981  -0.1641011 
## 
## [[823]]
## (Intercept)  horsepower 
##   39.450741   -0.152538 
## 
## [[824]]
## (Intercept)  horsepower 
##  40.5839070  -0.1638367 
## 
## [[825]]
## (Intercept)  horsepower 
##   40.362568   -0.161486 
## 
## [[826]]
## (Intercept)  horsepower 
##  39.8320856  -0.1589011 
## 
## [[827]]
## (Intercept)  horsepower 
##  40.9775815  -0.1678421 
## 
## [[828]]
## (Intercept)  horsepower 
##  39.1749287  -0.1521002 
## 
## [[829]]
## (Intercept)  horsepower 
##   38.459457   -0.140866 
## 
## [[830]]
## (Intercept)  horsepower 
##  41.6444912  -0.1766301 
## 
## [[831]]
## (Intercept)  horsepower 
##  39.4293392  -0.1535648 
## 
## [[832]]
## (Intercept)  horsepower 
##  39.7343139  -0.1565904 
## 
## [[833]]
## (Intercept)  horsepower 
##  39.4283872  -0.1533811 
## 
## [[834]]
## (Intercept)  horsepower 
##   39.014854   -0.149972 
## 
## [[835]]
## (Intercept)  horsepower 
##  39.7859277  -0.1556053 
## 
## [[836]]
## (Intercept)  horsepower 
##  40.0978772  -0.1639104 
## 
## [[837]]
## (Intercept)  horsepower 
##  39.0880224  -0.1482984 
## 
## [[838]]
## (Intercept)  horsepower 
##  38.8017574  -0.1528747 
## 
## [[839]]
## (Intercept)  horsepower 
##  40.6073056  -0.1613577 
## 
## [[840]]
## (Intercept)  horsepower 
##  39.7329117  -0.1516594 
## 
## [[841]]
## (Intercept)  horsepower 
##  39.9323679  -0.1581045 
## 
## [[842]]
## (Intercept)  horsepower 
##  40.7947251  -0.1643277 
## 
## [[843]]
## (Intercept)  horsepower 
##  39.6968178  -0.1591358 
## 
## [[844]]
## (Intercept)  horsepower 
##  40.7671139  -0.1647306 
## 
## [[845]]
## (Intercept)  horsepower 
##  39.3687445  -0.1524165 
## 
## [[846]]
## (Intercept)  horsepower 
##  40.7347456  -0.1621298 
## 
## [[847]]
## (Intercept)  horsepower 
##  41.0065441  -0.1702513 
## 
## [[848]]
## (Intercept)  horsepower 
##  39.1202891  -0.1499724 
## 
## [[849]]
## (Intercept)  horsepower 
##  40.3298037  -0.1627495 
## 
## [[850]]
## (Intercept)  horsepower 
##  37.9122330  -0.1412086 
## 
## [[851]]
## (Intercept)  horsepower 
##  40.5853504  -0.1638791 
## 
## [[852]]
## (Intercept)  horsepower 
##  40.0443605  -0.1599826 
## 
## [[853]]
## (Intercept)  horsepower 
##  40.2486937  -0.1586009 
## 
## [[854]]
## (Intercept)  horsepower 
##   39.330655   -0.155513 
## 
## [[855]]
## (Intercept)  horsepower 
##  41.0670879  -0.1678794 
## 
## [[856]]
## (Intercept)  horsepower 
##  38.5968985  -0.1461197 
## 
## [[857]]
## (Intercept)  horsepower 
##  39.3678340  -0.1540749 
## 
## [[858]]
## (Intercept)  horsepower 
##  38.7168643  -0.1484614 
## 
## [[859]]
## (Intercept)  horsepower 
##  40.9312260  -0.1632645 
## 
## [[860]]
## (Intercept)  horsepower 
##  39.8932726  -0.1596032 
## 
## [[861]]
## (Intercept)  horsepower 
##  38.7208992  -0.1480864 
## 
## [[862]]
## (Intercept)  horsepower 
##  39.7101885  -0.1553889 
## 
## [[863]]
## (Intercept)  horsepower 
##  39.2426820  -0.1512771 
## 
## [[864]]
## (Intercept)  horsepower 
##  40.0137156  -0.1567188 
## 
## [[865]]
## (Intercept)  horsepower 
##  37.9622670  -0.1431672 
## 
## [[866]]
## (Intercept)  horsepower 
##  39.3866860  -0.1550671 
## 
## [[867]]
## (Intercept)  horsepower 
##  38.8735300  -0.1487488 
## 
## [[868]]
## (Intercept)  horsepower 
##  40.7069622  -0.1621565 
## 
## [[869]]
## (Intercept)  horsepower 
##  40.3735518  -0.1640552 
## 
## [[870]]
## (Intercept)  horsepower 
##   41.085591   -0.166701 
## 
## [[871]]
## (Intercept)  horsepower 
##  39.8576160  -0.1543065 
## 
## [[872]]
## (Intercept)  horsepower 
##  39.6240808  -0.1556993 
## 
## [[873]]
## (Intercept)  horsepower 
##  39.3946900  -0.1503367 
## 
## [[874]]
## (Intercept)  horsepower 
##  39.5260850  -0.1561312 
## 
## [[875]]
## (Intercept)  horsepower 
##  39.7629896  -0.1553082 
## 
## [[876]]
## (Intercept)  horsepower 
##  41.0836253  -0.1682755 
## 
## [[877]]
## (Intercept)  horsepower 
##  41.2663210  -0.1682109 
## 
## [[878]]
## (Intercept)  horsepower 
##  39.8209050  -0.1552811 
## 
## [[879]]
## (Intercept)  horsepower 
##  39.1517491  -0.1498476 
## 
## [[880]]
## (Intercept)  horsepower 
##   40.343630   -0.162037 
## 
## [[881]]
## (Intercept)  horsepower 
##  41.2989182  -0.1689022 
## 
## [[882]]
## (Intercept)  horsepower 
##  39.5738972  -0.1524727 
## 
## [[883]]
## (Intercept)  horsepower 
##  39.9793011  -0.1585686 
## 
## [[884]]
## (Intercept)  horsepower 
##  40.2429128  -0.1625834 
## 
## [[885]]
## (Intercept)  horsepower 
##  40.1028545  -0.1580146 
## 
## [[886]]
## (Intercept)  horsepower 
##  38.3195628  -0.1451329 
## 
## [[887]]
## (Intercept)  horsepower 
##  40.5850698  -0.1595182 
## 
## [[888]]
## (Intercept)  horsepower 
##  41.5881272  -0.1707614 
## 
## [[889]]
## (Intercept)  horsepower 
##   40.925603   -0.167252 
## 
## [[890]]
## (Intercept)  horsepower 
##  40.4697539  -0.1634783 
## 
## [[891]]
## (Intercept)  horsepower 
##  40.2145210  -0.1604052 
## 
## [[892]]
## (Intercept)  horsepower 
##  41.0290760  -0.1660027 
## 
## [[893]]
## (Intercept)  horsepower 
##  40.6987911  -0.1594369 
## 
## [[894]]
## (Intercept)  horsepower 
##  39.9398971  -0.1563004 
## 
## [[895]]
## (Intercept)  horsepower 
##  39.4969895  -0.1542574 
## 
## [[896]]
## (Intercept)  horsepower 
##  40.4092498  -0.1600834 
## 
## [[897]]
## (Intercept)  horsepower 
##  39.3444405  -0.1551739 
## 
## [[898]]
## (Intercept)  horsepower 
##  41.2351706  -0.1688288 
## 
## [[899]]
## (Intercept)  horsepower 
##  40.1269588  -0.1610919 
## 
## [[900]]
## (Intercept)  horsepower 
##  39.4841926  -0.1530155 
## 
## [[901]]
## (Intercept)  horsepower 
##  39.8824452  -0.1561253 
## 
## [[902]]
## (Intercept)  horsepower 
##  39.4839344  -0.1578613 
## 
## [[903]]
## (Intercept)  horsepower 
##  40.7325562  -0.1643893 
## 
## [[904]]
## (Intercept)  horsepower 
##  39.8739524  -0.1578806 
## 
## [[905]]
## (Intercept)  horsepower 
##  39.0486937  -0.1555495 
## 
## [[906]]
## (Intercept)  horsepower 
##  40.0218563  -0.1597314 
## 
## [[907]]
## (Intercept)  horsepower 
##  38.6249376  -0.1481092 
## 
## [[908]]
## (Intercept)  horsepower 
##  41.1513962  -0.1670602 
## 
## [[909]]
## (Intercept)  horsepower 
##  40.1679532  -0.1611735 
## 
## [[910]]
## (Intercept)  horsepower 
##  39.4328974  -0.1561585 
## 
## [[911]]
## (Intercept)  horsepower 
##  40.1092778  -0.1552901 
## 
## [[912]]
## (Intercept)  horsepower 
##  39.9705557  -0.1582378 
## 
## [[913]]
## (Intercept)  horsepower 
##  41.2011967  -0.1685721 
## 
## [[914]]
## (Intercept)  horsepower 
##  40.1347989  -0.1615469 
## 
## [[915]]
## (Intercept)  horsepower 
##  39.9382322  -0.1577842 
## 
## [[916]]
## (Intercept)  horsepower 
##  40.8277501  -0.1684406 
## 
## [[917]]
## (Intercept)  horsepower 
##   39.882815   -0.158755 
## 
## [[918]]
## (Intercept)  horsepower 
##  40.0000567  -0.1591823 
## 
## [[919]]
## (Intercept)  horsepower 
##  39.9477238  -0.1596909 
## 
## [[920]]
## (Intercept)  horsepower 
##  39.8306187  -0.1609955 
## 
## [[921]]
## (Intercept)  horsepower 
##   39.555435   -0.155202 
## 
## [[922]]
## (Intercept)  horsepower 
##  38.5891642  -0.1474612 
## 
## [[923]]
## (Intercept)  horsepower 
##  38.5892814  -0.1436393 
## 
## [[924]]
## (Intercept)  horsepower 
##  40.2487442  -0.1607852 
## 
## [[925]]
## (Intercept)  horsepower 
##  39.4633831  -0.1580345 
## 
## [[926]]
## (Intercept)  horsepower 
##  39.9829332  -0.1602359 
## 
## [[927]]
## (Intercept)  horsepower 
##  39.5063168  -0.1541429 
## 
## [[928]]
## (Intercept)  horsepower 
##  39.1798304  -0.1548976 
## 
## [[929]]
## (Intercept)  horsepower 
##  39.1968693  -0.1520828 
## 
## [[930]]
## (Intercept)  horsepower 
##  39.3421963  -0.1536696 
## 
## [[931]]
## (Intercept)  horsepower 
##  39.0620201  -0.1472987 
## 
## [[932]]
## (Intercept)  horsepower 
##  39.6743333  -0.1504059 
## 
## [[933]]
## (Intercept)  horsepower 
##  38.8316134  -0.1463643 
## 
## [[934]]
## (Intercept)  horsepower 
##  39.9230253  -0.1555708 
## 
## [[935]]
## (Intercept)  horsepower 
##  41.2815337  -0.1724662 
## 
## [[936]]
## (Intercept)  horsepower 
##   39.571080   -0.159058 
## 
## [[937]]
## (Intercept)  horsepower 
##  40.2498096  -0.1558503 
## 
## [[938]]
## (Intercept)  horsepower 
##  39.9039149  -0.1559185 
## 
## [[939]]
## (Intercept)  horsepower 
##  40.6018348  -0.1625995 
## 
## [[940]]
## (Intercept)  horsepower 
##    40.02401    -0.15590 
## 
## [[941]]
## (Intercept)  horsepower 
##  39.2628806  -0.1551312 
## 
## [[942]]
## (Intercept)  horsepower 
##  39.6572361  -0.1572104 
## 
## [[943]]
## (Intercept)  horsepower 
##  38.6154611  -0.1466608 
## 
## [[944]]
## (Intercept)  horsepower 
##  39.4962863  -0.1523947 
## 
## [[945]]
## (Intercept)  horsepower 
##  39.6893455  -0.1563132 
## 
## [[946]]
## (Intercept)  horsepower 
##  40.8672761  -0.1654187 
## 
## [[947]]
## (Intercept)  horsepower 
##  41.5917968  -0.1733266 
## 
## [[948]]
## (Intercept)  horsepower 
##  40.6229579  -0.1674325 
## 
## [[949]]
## (Intercept)  horsepower 
##  40.2141928  -0.1641932 
## 
## [[950]]
## (Intercept)  horsepower 
##   38.553069   -0.148483 
## 
## [[951]]
## (Intercept)  horsepower 
##  39.4802628  -0.1490459 
## 
## [[952]]
## (Intercept)  horsepower 
##  39.9501351  -0.1584735 
## 
## [[953]]
## (Intercept)  horsepower 
##  39.8317421  -0.1575276 
## 
## [[954]]
## (Intercept)  horsepower 
##  39.1135431  -0.1449971 
## 
## [[955]]
## (Intercept)  horsepower 
##  39.4594027  -0.1491685 
## 
## [[956]]
## (Intercept)  horsepower 
##  39.4189550  -0.1573291 
## 
## [[957]]
## (Intercept)  horsepower 
##  39.3321627  -0.1543486 
## 
## [[958]]
## (Intercept)  horsepower 
##  39.8528177  -0.1574034 
## 
## [[959]]
## (Intercept)  horsepower 
##  39.9475426  -0.1585826 
## 
## [[960]]
## (Intercept)  horsepower 
##  39.8427879  -0.1565519 
## 
## [[961]]
## (Intercept)  horsepower 
##   40.359369   -0.158876 
## 
## [[962]]
## (Intercept)  horsepower 
##  39.8323791  -0.1568189 
## 
## [[963]]
## (Intercept)  horsepower 
##  39.7776256  -0.1581397 
## 
## [[964]]
## (Intercept)  horsepower 
##  40.6175057  -0.1674063 
## 
## [[965]]
## (Intercept)  horsepower 
##  38.9046930  -0.1480393 
## 
## [[966]]
## (Intercept)  horsepower 
##  38.8969953  -0.1447938 
## 
## [[967]]
## (Intercept)  horsepower 
##  40.2522592  -0.1592828 
## 
## [[968]]
## (Intercept)  horsepower 
##  41.1272327  -0.1739835 
## 
## [[969]]
## (Intercept)  horsepower 
##  40.2862041  -0.1571654 
## 
## [[970]]
## (Intercept)  horsepower 
##  39.7370445  -0.1542555 
## 
## [[971]]
## (Intercept)  horsepower 
##  38.8141756  -0.1479584 
## 
## [[972]]
## (Intercept)  horsepower 
##  40.2582817  -0.1600649 
## 
## [[973]]
## (Intercept)  horsepower 
##  39.9391093  -0.1586584 
## 
## [[974]]
## (Intercept)  horsepower 
##  40.4329752  -0.1618723 
## 
## [[975]]
## (Intercept)  horsepower 
##  41.3109100  -0.1742695 
## 
## [[976]]
## (Intercept)  horsepower 
##  40.0561437  -0.1596104 
## 
## [[977]]
## (Intercept)  horsepower 
##  38.3533940  -0.1424435 
## 
## [[978]]
## (Intercept)  horsepower 
##  40.2019465  -0.1613147 
## 
## [[979]]
## (Intercept)  horsepower 
##  40.3854292  -0.1591208 
## 
## [[980]]
## (Intercept)  horsepower 
##  40.2281877  -0.1604024 
## 
## [[981]]
## (Intercept)  horsepower 
##  40.7873744  -0.1643383 
## 
## [[982]]
## (Intercept)  horsepower 
##  38.3800852  -0.1419252 
## 
## [[983]]
## (Intercept)  horsepower 
##  40.2015762  -0.1614201 
## 
## [[984]]
## (Intercept)  horsepower 
##   39.851291   -0.157331 
## 
## [[985]]
## (Intercept)  horsepower 
##  39.2264421  -0.1509098 
## 
## [[986]]
## (Intercept)  horsepower 
##  39.3224752  -0.1523126 
## 
## [[987]]
## (Intercept)  horsepower 
##  40.4872013  -0.1618712 
## 
## [[988]]
## (Intercept)  horsepower 
##  39.6132886  -0.1581214 
## 
## [[989]]
## (Intercept)  horsepower 
##  39.9354746  -0.1572458 
## 
## [[990]]
## (Intercept)  horsepower 
##  39.1977429  -0.1520091 
## 
## [[991]]
## (Intercept)  horsepower 
##  40.2363844  -0.1609104 
## 
## [[992]]
## (Intercept)  horsepower 
##  40.2655033  -0.1604819 
## 
## [[993]]
## (Intercept)  horsepower 
##  40.6224390  -0.1618327 
## 
## [[994]]
## (Intercept)  horsepower 
##  40.0776847  -0.1540276 
## 
## [[995]]
## (Intercept)  horsepower 
##  39.5230700  -0.1526811 
## 
## [[996]]
## (Intercept)  horsepower 
##  40.1546702  -0.1620206 
## 
## [[997]]
## (Intercept)  horsepower 
##  39.4325851  -0.1479894 
## 
## [[998]]
## (Intercept)  horsepower 
##  42.3139795  -0.1774157 
## 
## [[999]]
## (Intercept)  horsepower 
##  41.0829348  -0.1656016 
## 
## [[1000]]
## (Intercept)  horsepower 
##   39.092764   -0.149022
```
]
]

---

## Back to our example!

```r
B <- 1000
map_df(1:B, ~ bootstrap_residual(e, y_hat, model, Auto))
```

```
## # A tibble: 1,000 x 2
##    `(Intercept)` horsepower
##            <dbl>      <dbl>
##  1          40.3     -0.165
##  2          40.3     -0.162
##  3          39.3     -0.156
##  4          38.5     -0.145
##  5          38.9     -0.150
##  6          41.0     -0.168
##  7          40.4     -0.164
##  8          38.4     -0.144
##  9          39.2     -0.153
## 10          40.4     -0.163
## # … with 990 more rows
```
]
]

---

* Using the function created in the previous application exercise, run a bootstrap on the following model:

Using the `mtcars` data set fit a model predicting `mpg` from `wt` and `cyl`.

---
## Bootstrap

* Generate `$\epsilon^*$` from sampling with replacement from `$e_1,\dots,e_n$` (the residuals)
* Calculate `$\mathbf{y}^* =\mathbf{X}\hat\beta+\epsilon^*$`
* Compute `$\hat\beta^*$` from `$(\mathbf{X}, \mathbf{y}^*)$`

---

## Bootstrap

```r
boot_coefs <- map_df(1:B, ~ bootstrap_residual(e, y_hat, model, Auto))

ggplot(boot_coefs, aes(x = horsepower)) + 
  geom_histogram(bins = 25)
```

![](10-bootstrap-ci_files/figure-html/unnamed-chunk-37-1.png)

---

## Bootstrap

```r
ggplot(boot_coefs, aes(x = horsepower)) + 
  geom_histogram(bins = 25) + 
  geom_vline(xintercept = c(quantile(boot_coefs$horsepower, 0.025), quantile(boot_coefs$horsepower, 0.975)))
```

![](10-bootstrap-ci_files/figure-html/unnamed-chunk-38-1.png)

---

## Bootstrap Percentile Confidence Intervals

```r
quantile(boot_coefs$horsepower, 0.025)
```

```
##       2.5% 
## -0.1705285
```

```r
quantile(boot_coefs$horsepower, 0.975)
```

```
##      97.5% 
## -0.1449497
```

* The `quantile()` function calculates sample quantiles, here we are calculating the 2.5% and 97.5% of the `horsepower` bootstrapped coefficients.

---

## Bootstrap Percentile Confidence Intervals

* The bootstrap percentile confidence interval for the coefficient for horsepower is (-0.170, -0.145)

--
* Let's compare this to the confidence interval calculated using the usual method

```r
confint(lm(mpg ~ horsepower, data = Auto))
```

```
##                 2.5 %     97.5 %
## (Intercept) 38.525212 41.3465103
## horsepower  -0.170517 -0.1451725
```

---

## Bootstrap "standard error"

* We can also calculate the average coefficient and standard error

```r
(beta_avg <- mean(boot_coefs$horsepower))
```

```
## [1] -0.1579431
```

```r
(beta_se <- sd(boot_coefs$horsepower))
```

```
## [1] 0.006454474
```

* We can use the "standard error method" to calculate a slightly different bootstrap confidence interval

---

## Bootstrap standard error method

`$$\hat\beta_i \pm t^* \hat{SE}_{\beta_i}$$`

```r
t_star <- qt(0.975, nrow(Auto) - 2)
beta_avg - t_star * beta_se
```

```
## [1] -0.170633
```

```r
beta_avg + t_star * beta_se
```

```
## [1] -0.1452532
```

* Looks very similar!

---

## Bootstrap

```r
ggplot(boot_coefs, aes(x = horsepower)) + 
  geom_histogram(bins = 25) + 
  geom_vline(xintercept = c(quantile(boot_coefs$horsepower, 0.025), quantile(boot_coefs$horsepower, 0.975))) 
```

![](10-bootstrap-ci_files/figure-html/unnamed-chunk-43-1.png)
]
---

## Bootstrap

```r
ggplot(boot_coefs, aes(x = horsepower)) + 
  geom_histogram(bins = 25) + 
  geom_vline(xintercept = c(quantile(boot_coefs$horsepower, 0.025), quantile(boot_coefs$horsepower, 0.975))) + 
  geom_vline(xintercept = c(beta_avg - t_star * beta_se, beta_avg + t_star * beta_se), lty = 2) 
```

![](10-bootstrap-ci_files/figure-html/unnamed-chunk-44-1.png)

]
---

## Bootstrap

![](10-bootstrap-ci_files/figure-html/unnamed-chunk-45-1.png)
]

---

* Calculate a bootstrap percentile confidence interval for the coefficient associated with `wt`
* Calculate a bootstrap confidence interval using the standard error method for the coefficient associated with `wt`
* Compare these intervals to the interval computed from the `confint` function.