Lec05: Linear Models

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# Lec05: Linear Models
## Stat41: Data Viz
### Prof Amanda Luby
### Swarthmore College

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# Today:

(1) Announcements

(2) Survey

(3) Regression Refresher

(4) IJALM

(5) Project 2

---
# Announcements

Project01/Labs/Milestones: if you don't think you'll be able to get them in by **Tuesday**, please get in touch with me so we can figure out a plan to catch up

If you're confident about catching up, I trust you.

If you DO start to worry, please let me know - I'm here to help!

This week: I've included a few options for integrating work from the labs or your final project into the mini-project, so there's an option to scale back the workload a bit

Milestone 02: we will talk about tomorrow

ggplot2 [cheat sheet](https://rstudio.com/resources/cheatsheets/)

---

# Survey

Please fill out this ["first week survey"](https://forms.gle/6Fi74BhDFvXMeyMi9) -- *some* groups will be switching around tomorrow, so I need this information!

---
# Regression Refresher

.pull-left[
Idea: draw a line through a scatterplot, but with math

Assumptions (LINE):

* **L**inear Relationship
* **I**ndependent Observations
* **N**ormal Residuals
* **E**qual variance in Residuals 
]

.pull-right[
![](Lec05_files/figure-html/unnamed-chunk-2-1.png)
]

---

# In groups:

(1) How do you check each of these assumptions?

(2) What do you do if they're violated?

(3) What are the necessary visualizations in a OLS regression analysis?

(4) Did any questions come up from reading?

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# Quick Check in

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**L**inear Relationship:

.pull-left[

```r
penguins %>% 
  filter(species == "Adelie") %>%
  ggplot(., aes(x = bill_length_mm, y = bill_depth_mm)) +
  geom_point() + 
  geom_smooth(se = FALSE, method = "lm") +
  theme_xaringan() + 
  labs(x = "Bill Length (mm)",
       y = "Bill Depth (mm)",
       title = "Adelie Penguins Bill Length/Depth",
       caption = "Source: palmerpenguins")
```
]

.pull-right[
![](Lec05_files/figure-html/adelie-linear-scatter-1.png)
]

---
**I**ndependent Observations

First, consider how data was collected: does order matter? Next, look at the *residuals* plotted against the *predictor* variable. We shouldn't be able to see any relationship

.pull-left[

```r
library(tidymodels)
adelie = penguins %>%
  filter(species == "Adelie")

lm_mod = lm(bill_depth_mm ~ bill_length_mm, data = adelie)
lm_res = augment(lm_mod)
ggplot(lm_res, aes(x = bill_length_mm, y = .resid)) +
  geom_point() +
  theme_xaringan()
```
]

.pull-right[
![](Lec05_files/figure-html/resid-by-x-1.png)
]

---
**N**ormal Residuals

If we look at the distribution of the residuals, it should be *symmetric*, *unimodal*, and roughly  *bell-shaped*

.pull-left[

```r
ggplot(lm_res, aes(x = .resid)) +
  geom_histogram(bins = 20, col = "white") +
  theme_xaringan()
```
]

.pull-right[
![](Lec05_files/figure-html/resid-hist-1.png)
]

---
**E**qual variance in the residuals

In the residual by fitted values scatterplot, there should be no relationship

.pull-left[

```r
ggplot(lm_res, aes(x = .fitted, y = .resid)) +
  geom_point() +
  theme_xaringan()
```
]

.pull-right[
![](Lec05_files/figure-html/resid-fitted-1.png)
]

---

# IJALM

(**I**t's **J**ust **A** **L**inear **M**odel)

If you can do a linear regression, you can do *all of the other* Stat011 methods.

We'll do quick examples of the following tests using the `penguins` data, showing that they're all equivalent to a linear model

* One-sample t-test

* Two-sample t-test

* ANOVA

If you want to read more, [this link](https://lindeloev.github.io/tests-as-linear/linear_tests_cheat_sheet.pdf) has further details and discusses additional methods.

---

# One-sample t-test

.pull-left[
`$$H_0: \mu = 40$$` 
`$$H_A: \mu \ne 40$$` 
]

.pull-right[
![](Lec05_files/figure-html/unnamed-chunk-3-1.png)
]

---

```r
t_res = t.test(adelie$bill_length_mm, mu = 40)
t_res
```

```
## 
## 	One Sample t-test
## 
## data:  adelie$bill_length_mm
## t = -5.5762, df = 150, p-value = 1.114e-07
## alternative hypothesis: true mean is not equal to 40
## 95 percent confidence interval:
##  38.36312 39.21966
## sample estimates:
## mean of x 
##  38.79139
```

```r
lm_res = lm((bill_length_mm - 40) ~ 1, data = adelie)
tidy(lm_res)
```

```
## # A tibble: 1 x 5
##   term        estimate std.error statistic     p.value
##   <chr>          <dbl>     <dbl>     <dbl>       <dbl>
## 1 (Intercept)    -1.21     0.217     -5.58 0.000000111
```

---

# Two-sample t-test

Test the mean of *Chinstrap* `bill_length` compared to *Gentoo* `bill_length`
.pull-left[
`$$H_0: \mu_1 = \mu_2$$` 
`$$H_A: \mu_1 \ne \mu_2$$` 
]

.pull-right[
![](Lec05_files/figure-html/unnamed-chunk-6-1.png)
]

---

```r
t_res = t.test(bill_length_mm ~ species, data = chinstrap_gentoo, var.equal = TRUE)
t_res
```

```
## 
## 	Two Sample t-test
## 
## data:  bill_length_mm by species
## t = 2.7694, df = 189, p-value = 0.006176
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.3823625 2.2755285
## sample estimates:
## mean in group Chinstrap    mean in group Gentoo 
##                48.83382                47.50488
```

```r
lm_res = lm(bill_length_mm ~ species + 1, data = chinstrap_gentoo)
tidy(lm_res)
```

```
## # A tibble: 2 x 5
##   term          estimate std.error statistic   p.value
##   <chr>            <dbl>     <dbl>     <dbl>     <dbl>
## 1 (Intercept)      48.8      0.385    127.   8.20e-185
## 2 speciesGentoo    -1.33     0.480     -2.77 6.18e-  3
```

---
# ANOVA

Test whether *all 3* species of penguins have the same *flipper length*

.pull-left[
`$$H_0: \mu_1 = \mu_2 = \mu_3$$` 
`$$H_A: \text{at least one} \ne$$` 
]

.pull-right[
![](Lec05_files/figure-html/unnamed-chunk-9-1.png)
]

---

```r
aov_res = aov(flipper_length_mm ~ species, data = penguins)
summary(aov_res)
```

```
##              Df Sum Sq Mean Sq F value Pr(>F)    
## species       2  52473   26237   594.8 <2e-16 ***
## Residuals   339  14953      44                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 2 observations deleted due to missingness
```

```r
lm_res = lm(flipper_length_mm ~ species, data = penguins)
summary(lm_res)
```

```
## 
## Call:
## lm(formula = flipper_length_mm ~ species, data = penguins)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.9536  -4.8235   0.0464   4.8130  20.0464 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      189.9536     0.5405 351.454  < 2e-16 ***
## speciesChinstrap   5.8699     0.9699   6.052 3.79e-09 ***
## speciesGentoo     27.2333     0.8067  33.760  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.642 on 339 degrees of freedom
##   (2 observations deleted due to missingness)
## Multiple R-squared:  0.7782,	Adjusted R-squared:  0.7769 
## F-statistic: 594.8 on 2 and 339 DF,  p-value: < 2.2e-16
```
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# Questions?

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# Project 2

The [project 2 prompt](https://aluby.domains.swarthmore.edu/stat041/Projects/proj-2.html) is posted