NRI 7350 - Lab 4

class: title-slide, nobar

![:img right:0, bottom: 0, 65%, , ](./R Figs/title-plot-regression-1.png)
![:img right: 15%, top: 5%, 30%, ,](https://raw.githubusercontent.com/allisonhorst/palmerpenguins/master/man/figures/lter_penguins.png)

.footnote[Artwork by [@allison_horst](https://github.com/allisonhorst/stats-illustrations)]

## NRI 7350
# Linear Models
Regressions,    
ANOVAs, and   
Model assumptions

---
class: section

# Getting started (again)

Open RStudio  
Open your NRI project  
Open a **new** script for today:

File > New File > R Script

Make sure to load packages at the top: 
`library(tidyverse)` 
`library(palmerpenguins)`

---
# How Are we Doing?

![:img center, bottom: 0, 75%, ,](https://raw.githubusercontent.com/allisonhorst/stats-illustrations/master/other-stats-artwork/debugging.jpg)

.footnote[Artwork by [@allison_horst](https://github.com/allisonhorst/stats-illustrations)]

---
class: section
# Linear Models

---
# Linear Models

### Running models in R

- **y** is the **response** variable (**dependent**)
- **x** are the **explanatory** variables (**independent**, **predictor**)

Here we're assuming a **continuous** **y**

### Different types of models

- If we only have one **x** which is continuous, this is a **simple linear regression**
- If both **x** are continuous, this is a **multiple linear regression**
- If both **x** are categorical, this is an **ANOVA**
- If **x1** is continuous and **x2** is categorical, this is an **ANCOVA**

![:box 50%, 85%, 30%](R will figure it out for you)

---
# Regressions

### Real example

- Is penguin body mass a function of skeletal size?   
- Can it be predicted by flipper length and bill length?

---
# Regressions

### Real example

<code class ='r hljs remark-code'>lm(body_mass_g ~ flipper_length_mm + bill_length_mm, data = penguins)</code>

```

## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
##     data = penguins)
## 
## Coefficients:
##       (Intercept)  flipper_length_mm     bill_length_mm  
##         -5736.897             48.145              6.047

```

.footnote[As we have **two continuous** predictors, this is technically a *multiple* regression]

---
# Regressions

### Real example

```

## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
##     data = penguins)
## 
## Coefficients:
*##       (Intercept)  flipper_length_mm     bill_length_mm  
*##         -5736.897             48.145              6.047

```

![:box 50%, 70%, 55%](Hmm, not a lot of detail... Only <code>Intercept</code> and Slopes (<code>flipper_length_mm</code> and <code>bill_length_mm</code>&rpar;)

.footnote[As we have **two** predictors, this is technically a *multiple* regression]

---
# Regressions

### Assign model to `m` .small[(or any other name you want to give it)]

`m` is a model object

```r
class(m)
```

```
## [1] "lm"
```

This contains all the information about the model

---
class: split-25
# Regressions

Use `summary()` to show summary table:

.columnl[

```r
summary(m)
```
]

.columnr[
.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]
]

![:box 50%, 40%, 50%](Your turn! Create a model with your response variable by two of your continuous predictors. Look at the output of <code>summary(&rpar;</code>)

![:box 50%, 75%, 40%](Wait! Shouldn't interpret until we know the model is solid)

---
# Model Diagnostics

### Model Assumptions

- Normality (of residuals)
- Constant Variance (no heteroscedasticity)

### Other cautions

- Influential observations (Cook's D)
- Multiple collinearity (with more than one `x` or explanatory variables)

.footnote[There are other assumptions (independence, etc.) but they reflect experimental design, not patterns in the data]

---
class: split-45
# Model Diagnostics

First let's get our relevant variables into a diagnostic data frame:

- **`residuals`** (regular and standardized)
- **`fitted` values**
- **`cooks` distance**
- **`obs` number**

.footnote[Remember! `n()` only works *inside* `summarize()`/`mutate()`]

.columnl[.small[

```r
d <- data.frame(residuals = residuals(m), 
 std_residuals = rstudent(m), 
 fitted = fitted(m), 
 cooks = cooks.distance(m))

d <- mutate(d, obs = 1:n())
```
]]

--
.columnr[
.small[

```r
head(d)
```

```
##     residuals std_residuals   fitted        cooks obs
## 1  536.220898   1.368529806 3213.779 5.539153e-03   1
## 2  343.077607   0.873050231 3456.922 1.609402e-03   2
## 3 -645.064115  -1.644516798 3895.064 3.797384e-03   3
## 5 -327.003441  -0.833002736 3777.003 1.992863e-03   4
## 6    1.707668   0.004338503 3648.292 3.338060e-08   5
## 7  412.430396   1.051400111 3212.570 3.272886e-03   6
```
]]

---
# Side Note: `tidyverse` functions

- From `dplyr` package (part of `tidyverse`)

### `mutate()`
- `tidyverse` functions always start with the **data**, followed by **other arguments**
- `mutate()` adds **new columns** to your data
- Also note: `1:5` is the same as `c(1,2,3,4,5)`

.footnote[Remember! `n()` only works *inside* `summarize()`/`mutate()`]

---
class: split-50
# Normality

.columnl[
### Histogram of residuals

```r
ggplot(data = d, aes(x = std_residuals)) +
  geom_histogram(bins = 20)
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-14-1.png" width="90%" style="display: block; margin: auto;" />
]

.columnr[
### QQ Normality plot of residuals

```r
ggplot(data = d, aes(sample = std_residuals)) +
  stat_qq() +
  stat_qq_line()
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-15-1.png" width="90%" style="display: block; margin: auto;" />
]

---
class: split-45
# Variance and Influence

.columnl[
### Check heteroscedasticity
.small[

```r
ggplot(d, aes(x = fitted, y = std_residuals)) +
  geom_point() +
  geom_hline(yintercept = 0)
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-16-1.png" width="90%" style="display: block; margin: auto;" />
]]

.columnr[
### Cook's D
.small[

```r
ggplot(d, aes(x = obs, y = cooks)) +
  geom_point() +
* geom_hline(yintercept = 1, linetype = "dotted") +
  geom_hline(yintercept = 4/nrow(penguins), 
             linetype = "dashed")
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-17-1.png" width="90%" style="display: block; margin: auto;" />
]]

---
class: split-45
# Variance and Influence

.columnl[
### Check heteroscedasticity
.small[

```r
ggplot(d, aes(x = fitted, y = std_residuals)) +
  geom_point() +
  geom_hline(yintercept = 0)
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-18-1.png" width="90%" style="display: block; margin: auto;" />
]]

.columnr[
### Cook's D
.small[

```r
ggplot(d, aes(x = obs, y = cooks)) +
  geom_point() +
  geom_hline(yintercept = 4/nrow(penguins), 
             linetype = "dashed")
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-19-1.png" width="90%" style="display: block; margin: auto;" />
]]

.box-b[Pretty good!]

---
class: split-50
# What is a 'Good' Normality Plot?

### Problematic
.columnl[
<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-21-1.png" width="90%" style="display: block; margin: auto;" />
]

.columnr[
<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-22-1.png" width="90%" style="display: block; margin: auto;" />
]

---
class: split-50
# What is a 'Good' Normality Plot?

### Good
.columnl[
<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-23-1.png" width="90%" style="display: block; margin: auto;" />
]

.columnr[

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-24-1.png" width="90%" style="display: block; margin: auto;" />
]

![:box 50%, 50%, 40%](No clear cutoff!)

---
class: split-50
# What is a 'Good' Heteroscedasticity Plot?

## Problematic
.columnl[
<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-25-1.png" width="90%" style="display: block; margin: auto;" />
]

.columnr[

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-26-1.png" width="90%" style="display: block; margin: auto;" />
]

---
class: split-50
# What is a 'Good' Heteroscedasticity Plot?

## Good
.columnl[
<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-27-1.png" width="90%" style="display: block; margin: auto;" />
]

.columnr[
<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-28-1.png" width="90%" style="display: block; margin: auto;" />
]

![:box 50%, 50%, 40%](No clear cutoff!)

---
# Multicollinearity (collinearity)

- Only relevant with **more than one explanatory variable**
- If explanatory variables too correlated, can interfere with model interpretation

### Look at our two explanatory variables

```r
ggplot(data = penguins, aes(x = flipper_length_mm, y = bill_length_mm)) +
  geom_point() +
  stat_smooth(method = "lm")
```

.box-b[Correlated, but not necessarily a problem]

---
# Multicollinearity (collinearity)

- Only relevant with **more than one explanatory variable**
- If explanatory variables too correlated, can interfere with model interpretation
- Correlations between variables *might* be problematic (but not necessarily)

### Use `vif()` function from `car` package .small[(vif = variance inflation factor*)]

```r
library(car)
vif(m)
```

```
## flipper_length_mm    bill_length_mm 
##          1.756154          1.756154
```

![:spacer 10px]()
Hmm, that's pretty good (looking for < 10)

.footnote[\* Can be interpreted as how much influence the variable has on the model]

---
layout: true
class: split-40
# Interpreting Regressions

```r
m <- lm(body_mass_g ~ flipper_length_mm + bill_length_mm, data = penguins)
```

.columnl[

```r
summary(m)
```
]

---
.columnr[
.small[

---
.columnr[.small[

```
## 
## Call:
*## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
*## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### Model
]

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
*## Estimate Std. Error t value Pr(>|t|) 
*## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
*## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
*## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### Effects
]

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
*## (2 observations deleted due to missingness)
## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### Missing observations
]

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
## (2 observations deleted due to missingness)
*## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### R2 and adjusted R2

- Adjusted for the number of parameters
]

---
.columnr[
.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
*## Estimate Std. Error t value Pr(>|t|) 
*## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
*## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
*## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]
]

![:spacer 20px]()

.columnl[
### Specific Details

.medium[
- `Estimate`
    - Slope of the effect
    
- `Std. Error` 
    - Variability in the estimates
    
- `t value`
   - Test statistic
   - Think of it as a holistic combination of estimate and variability

- `Pr(>|t|)` 
    - **P-value**, significance of the results
    - Probability of getting `t-value` by chance
]]

---
.columnr[
.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
*## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]
]

![:spacer 20px]()
.columnl[
### Specific Details

![:spacer 15px]()

**Intercept**

.medium[
- Significant (P < 2e-16\*)
- Penguins with a flipper length of 0 mm are predicted to have a body mass of -5736.9g
 - Not useful!
]]

.columnl[
.footnote[.small[\* 2e-16 = 0.0000000000000002, R uses this as the smallest number]]
]

---
.columnr[
.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
*## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]
]

![:spacer 20px]()
.columnl[
### Specific Details

![:spacer 15px]()

Effect of **Flipper Length**

.medium[
- Significant (P < 2e-16*)
- For each 1 mm increase in flipper length, body mass is predicted to increase by 48.14g
]]

.columnl[
.footnote[.small[\* 2e-16 = 0.0000000000000002, R uses this as the smallest number]]
]

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
*## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### Specific Details

![:spacer 15px]()

Effect of **Flipper Length**

.medium[
- Significant (P < 2e-16*)
- For each 1 mm increase in flipper length, body mass is predicted to increase by 48.14g
]

Effect of **Bill Length**

.medium[
- Non-significant (P = 0.244, i.e. P < 0.05)
- Therefore no effect (in this model) 
 (and no interpretation of estimate)
]]

.columnl[
.footnote[.small[\* 2e-16 = 0.0000000000000002, R uses this as the smallest number]]
]

![:box 50%, 20%, 70%](Therefore There is a significant relationship between flipper length and body mass But not between bill length and body mass (when including flipper length&rpar;)

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ flipper_length_mm + bill_length_mm, 
## data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1090.5 -285.7 -32.1 244.2 1287.5 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
*## (Intercept) -5736.897 307.959 -18.629 <2e-16 ***
*## flipper_length_mm 48.145 2.011 23.939 <2e-16 ***
*## bill_length_mm 6.047 5.180 1.168 0.244 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 394.1 on 339 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.76,	Adjusted R-squared: 0.7585 
## F-statistic: 536.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### Specific Details

![:spacer 15px]()

Effect of **Flipper Length**

.medium[
- Significant (P < 2e-16*)
- For each 1 mm increase in flipper length, body mass is predicted to increase by 48.14g
]

Effect of **Bill Length**

.medium[
- Non-significant (P = 0.244, i.e. P < 0.05)
- Therefore no effect (and no interpretation of estimate)
]]

.columnl[
.footnote[.small[\* 2e-16 = 0.0000000000000002, R uses this as the smallest number]]
]

![:box 30%, 40%, 30%](y = mx + b)

![:box 30%, 40%, 30%](y = m1x1 + m2x2 + b)

![:box 30%, 40%, 30%](y = 48.14x1 + 6.05x2 + (-5736.9&rpar;)

---
layout: false
class: split-40

# Extra

### Why no effect of Bill Length?

.columnl[
![:spacer 35px]()
<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-42-1.png" width="100%" style="display: block; margin: auto;" />
]

.columnr[
.small[

```r
m <- lm(body_mass_g ~ bill_length_mm, data = penguins)
summary(m)
```

```
## 
## Call:
## lm(formula = body_mass_g ~ bill_length_mm, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1762.08 -446.98 32.59 462.31 1636.86 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 362.307 283.345 1.279 0.202 
*## bill_length_mm 87.415 6.402 13.654 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 645.4 on 340 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.3542,	Adjusted R-squared: 0.3523 
## F-statistic: 186.4 on 1 and 340 DF, p-value: < 2.2e-16
```
]]

---
class: split-40, space-list

# Extra

### Why no effect of Bill Length?

.columnl[

- Hypothesis of *causation* but really just correlation
- Flipper length is the 'better' predictor of body mass
- When flipper length in the model, no extra variation explained by bill length
- When flipper length *not* in the model, some variation left to be explained
]

.columnr[
.small[

```r
m <- lm(body_mass_g ~ bill_length_mm, data = penguins)
summary(m)
```

---
# Homework (Practice)*

Consider **bill depth** your response and **bill length** your predictor

1. Plot the relationship
2. Create a linear regression model
3. Check your model diagnostics
    - Normality
    - Heteroscedasticity
    - Influential variables (i.e. Cook's distance)
4. Interpret the results of your model

.footnote[\* Not to be handed in, answers posted in these slides next week]

---
exclude: TRUE

# Homework (Practice) Answers

1\. Plot the relationship

```r
ggplot(data = penguins, aes(x = bill_length_mm, y = bill_depth_mm)) + 
  geom_point() + 
  stat_smooth(method = "lm")
```

> Looks like there's definitely a relationship

---
exclude: TRUE
# Homework (Practice) Answers

2\. Create a linear regression model

```r
m <- lm(bill_depth_mm ~ bill_length_mm, data = penguins)
```

---
exclude: TRUE
# Homework (Practice) Answers

3\. Check your model diagnostics

```r
d <- data.frame(residuals = residuals(m), 
 std_residuals = rstudent(m), 
 fitted = fitted(m), 
 cooks = cooks.distance(m))

d <- mutate(d, obs = 1:n())
d
```

```
##         residuals std_residuals   fitted        cooks obs
## 1    1.138864e+00  5.934367e-01 17.56114 9.242306e-04   1
## 2   -1.271277e-01 -6.619739e-02 17.52713 1.070470e-05   2
## 3    5.408893e-01  2.815911e-01 17.45911 1.682589e-04   3
## 5    1.534813e+00  8.012445e-01 17.76519 2.609482e-03   4
## 6    3.055868e+00  1.597348e+00 17.54413 6.414686e-03   5
## 7    2.218595e-01  1.155596e-01 17.57814 3.639240e-05   6
## 8    2.047366e+00  1.068025e+00 17.55263 2.932579e-03   7
## 9    1.137574e-01  5.946167e-02 17.98624 2.228931e-05   8
## 10   2.885425e+00  1.506316e+00 17.31457 3.727910e-03   9
## 11  -5.716639e-01 -2.979755e-01 17.67166 2.962510e-04  10
## 12  -3.716639e-01 -1.937124e-01 17.67166 1.252215e-04  11
## 13   2.089063e-01  1.087197e-01 17.39109 2.205657e-05  12
## 14   3.596353e+00  1.883252e+00 17.60365 1.010901e-02  13
## 15   3.156268e+00  1.655660e+00 17.94373 1.582700e-02  14
## 16   2.631056e-02  1.372336e-02 17.77369 7.806848e-07  15
## 17   1.404855e+00  7.323832e-01 17.59514 1.514240e-03  16
## 18   3.427936e+00  1.791849e+00 17.27206 4.996598e-03  17
## 19   4.392637e-01  2.295562e-01 17.96074 3.166945e-04  18
## 20   4.525511e+00  2.374213e+00 16.97449 9.342805e-03  19
## 21   6.283361e-01  3.275244e-01 17.67166 3.579005e-04  20
## 22   1.019834e+00  5.317653e-01 17.68017 9.603925e-04  21
## 23   1.366796e+00  7.138244e-01 17.83320 2.383424e-03  22
## 24   4.623446e-01  2.409262e-01 17.63766 1.799497e-04  23
## 25  -3.866426e-01 -2.014081e-01 17.58664 1.125918e-04  24
## 26   1.015783e+00  5.305890e-01 17.88422 1.459065e-03  25
## 27   1.166396e+00  6.074312e-01 17.43360 7.440885e-04  26
## 28   4.578936e-01  2.383584e-01 17.44211 1.165862e-04  27
## 29   9.368382e-01  4.883983e-01 17.66316 7.810571e-04  28
## 30   1.457894e+00  7.594942e-01 17.44211 1.181870e-03  29
## 31  -8.271277e-01 -4.308134e-01 17.52713 4.531471e-04  30
## 32   3.773233e-01  1.967375e-01 17.72268 1.440852e-04  31
## 33   2.728723e-01  1.420922e-01 17.52713 4.931884e-05  32
## 34   1.491902e+00  7.771431e-01 17.40810 1.160061e-03  33
## 35  -7.906937e-01 -4.125831e-01 17.79069 7.306193e-04  34
## 36   3.547366e+00  1.856778e+00 17.55263 8.803811e-03  35
## 37   2.413357e+00  1.260020e+00 17.58664 4.386626e-03  36
## 38   1.202430e+00  6.259686e-01 17.29757 6.331778e-04  37
## 39   1.611332e+00  8.407634e-01 17.68867 2.442060e-03  38
## 40   1.598379e+00  8.330413e-01 17.50162 1.603367e-03  39
## 41   2.178084e-01  1.136174e-01 17.78219 5.446392e-05  40
## 42   9.833999e-01  5.120182e-01 17.41660 5.120677e-04  41
## 43   6.752978e-01  3.524538e-01 17.82470 5.717729e-04  42
## 44   2.563970e+00  1.337318e+00 17.13603 2.619048e-03  43
## 45  -8.396809e-01 -4.379702e-01 17.73968 7.399384e-04  44
## 46   1.281374e+00  6.676370e-01 17.51863 1.068058e-03  45
## 47   1.608906e+00  8.381648e-01 17.39109 1.308267e-03  46
## 48   1.202830e+00  6.273646e-01 17.69717 1.386037e-03  47
## 49   7.529779e-02  3.929257e-02 17.82470 7.108817e-06  48
## 50   3.910932e+00  2.047309e+00 17.28907 6.629368e-03  49
## 51   1.813744e-01  9.444103e-02 17.51863 2.139907e-05  50
## 52   1.423885e+00  7.418546e-01 17.47611 1.206913e-03  51
## 53  -9.723491e-03 -5.078237e-03 17.90972 1.406049e-07  52
## 54   2.185425e+00  1.139261e+00 17.31457 2.138542e-03  53
## 55   1.477659e-01  7.720881e-02 17.95223 3.526030e-05  54
## 56   1.234413e+00  6.427463e-01 17.36559 7.371185e-04  55
## 57  -6.963837e-02 -3.627000e-02 17.56964 3.519928e-06  56
## 58   1.366396e+00  7.117307e-01 17.43360 1.021141e-03  57
## 59  -1.182192e+00 -6.170119e-01 17.78219 1.604483e-03  58
## 60   1.411332e+00  7.362283e-01 17.68867 1.873461e-03  59
## 61  -9.502086e-01 -4.961446e-01 17.85021 1.192498e-03  60
## 62   3.725911e+00  1.949754e+00 17.37409 6.811981e-03  61
## 63  -6.886682e-01 -3.590286e-01 17.68867 4.460735e-04  62
## 64   8.089063e-01  4.210764e-01 17.39109 3.306977e-04  63
## 65  -6.906937e-01 -3.603818e-01 17.79069 5.575009e-04  64
## 66   6.514170e-01  3.390210e-01 17.34858 1.997244e-04  65
## 67  -1.667213e+00 -8.713253e-01 17.86721 3.799229e-03  66
## 68   1.708906e+00  8.903784e-01 17.39109 1.475949e-03  67
## 69  -1.233204e+00 -6.439649e-01 17.83320 1.940280e-03  68
## 70   2.068421e+00  1.078095e+00 17.33158 1.962365e-03  69
## 71   9.627446e-01  5.037231e-01 18.03726 1.754376e-03  70
## 72   8.898765e-01  4.634768e-01 17.51012 5.059342e-04  71
## 73  -3.186256e-01 -1.659118e-01 17.51863 6.603957e-05  72
## 74   1.908506e+00  9.944436e-01 16.99149 1.622733e-03  73
## 75  -3.672129e-01 -1.917101e-01 17.86721 1.843101e-04  74
## 76   1.253442e+00  6.525028e-01 17.24656 6.519034e-04  75
## 77  -6.080979e-01 -3.165278e-01 17.40810 1.927291e-04  76
## 78   1.677323e+00  8.754996e-01 17.72268 2.847250e-03  77
## 79  -1.707698e+00 -8.920311e-01 17.80770 3.530510e-03  78
## 80   1.793928e+00  9.345730e-01 17.30607 1.424721e-03  79
## 81  -7.437320e-01 -3.886526e-01 17.94373 8.787873e-04  80
## 82   3.619446e-01  1.883066e-01 17.23806 5.397894e-05  81
## 83   1.034813e+00  5.399422e-01 17.76519 1.186225e-03  82
## 84   1.498779e+00  7.833984e-01 17.90122 3.285269e-03  83
## 85   8.582545e-02  4.474427e-02 17.71417 7.319861e-06  84
## 86   2.925911e+00  1.527837e+00 17.37409 4.200787e-03  85
## 87   1.700804e+00  8.883539e-01 17.79920 3.440873e-03  86
## 88   8.518169e-01  4.443347e-01 17.74818 7.753733e-04  87
## 89   1.570847e+00  8.192567e-01 17.62915 2.039058e-03  88
## 90   1.221860e+00  6.367960e-01 17.57814 1.103817e-03  89
## 91   1.497914e-01  7.818484e-02 17.85021 2.963422e-05  90
## 92   7.089063e-01  3.689990e-01 17.39109 2.539876e-04  91
## 93  -8.947448e-01 -4.678832e-01 17.99474 1.401036e-03  92
## 94   5.813744e-01  3.027566e-01 17.51863 2.198643e-04  93
## 95  -5.076980e-01 -2.649172e-01 17.80770 3.120514e-04  94
## 96   1.483400e+00  7.727300e-01 17.41660 1.165154e-03  95
## 97   9.538425e-01  4.972111e-01 17.64616 7.802351e-04  96
## 98   1.040889e+00  5.420662e-01 17.45911 6.231183e-04  97
## 99  -1.971264e+00 -1.033070e+00 18.07126 7.820177e-03  98
## 100  1.287451e+00  6.702053e-01 17.21255 6.712863e-04  99
## 101 -9.723491e-03 -5.078237e-03 17.90972 1.406049e-07 100
## 102  2.600404e+00  1.356996e+00 17.39960 3.470048e-03 101
## 103 -1.680166e+00 -8.767050e-01 17.68017 2.606719e-03 102
## 104  2.328336e+00  1.216113e+00 17.67166 4.914390e-03 103
## 105  9.368382e-01  4.883983e-01 17.66316 7.810571e-04 104
## 106  1.389877e+00  7.242235e-01 17.51012 1.234204e-03 105
## 107 -4.036469e-01 -2.102888e-01 17.60365 1.273465e-04 106
## 108  2.362345e+00  1.233664e+00 17.63766 4.697925e-03 107
## 109 -6.461575e-01 -3.367572e-01 17.64616 3.580544e-04 108
## 110  1.787451e+00  9.310614e-01 17.21255 1.293942e-03 109
## 111 -1.146158e+00 -5.975563e-01 17.64616 1.126577e-03 110
## 112  3.291502e+00  1.719965e+00 17.00850 4.722759e-03 111
## 113  1.898765e-01  9.886401e-02 17.51012 2.303441e-05 112
## 114  2.202430e+00  1.148119e+00 17.29757 2.124273e-03 113
## 115  3.181374e+00  1.663254e+00 17.51863 6.583737e-03 114
## 116  1.044940e+00  5.438653e-01 17.25506 4.565108e-04 115
## 117 -6.036469e-01 -3.145085e-01 17.60365 2.848064e-04 116
## 118  2.785825e+00  1.456898e+00 17.71417 7.712203e-03 117
## 119 -8.502086e-01 -4.438982e-01 17.85021 9.547084e-04 118
## 120  1.208906e+00  6.294996e-01 17.39109 7.386183e-04 119
## 121 -6.076980e-01 -3.171115e-01 17.80770 4.470858e-04 120
## 122  2.119834e+00  1.106865e+00 17.68017 4.149478e-03 121
## 123 -4.676128e-01 -2.434444e-01 17.46761 1.279321e-04 122
## 124  1.134413e+00  5.906214e-01 17.36559 6.225277e-04 123
## 125 -1.992719e+00 -1.042210e+00 17.89272 5.710757e-03 124
## 126  1.566396e+00  8.160987e-01 17.43360 1.341948e-03 125
## 127  1.335738e-02  6.957648e-03 17.58664 1.343784e-07 126
## 128  9.429148e-01  4.908297e-01 17.35709 4.241573e-04 127
## 129 -4.696384e-01 -2.446245e-01 17.56964 1.600899e-04 128
## 130  8.639702e-01  4.495803e-01 17.13603 2.973822e-04 129
## 131  2.878510e-01  1.499655e-01 17.61215 6.597464e-05 130
## 132  1.978949e+00  1.031116e+00 17.22105 1.594191e-03 131
## 133  7.433148e-01  3.877368e-01 17.75669 6.011595e-04 132
## 134  8.028297e-01  4.186003e-01 17.69717 6.174664e-04 133
## 135 -4.615752e-02 -2.405186e-02 17.64616 1.827078e-06 134
## 136  1.089063e-01  5.667666e-02 17.39109 5.994334e-06 135
## 137 -3.587107e-01 -1.872553e-01 17.85871 1.728908e-04 136
## 138  2.632387e+00  1.374141e+00 17.46761 4.054208e-03 137
## 139 -1.239681e+00 -6.468227e-01 17.73968 1.612823e-03 138
## 140  3.898765e-01  2.030085e-01 17.51012 9.711553e-05 139
## 141 -3.676128e-01 -1.913769e-01 17.46761 7.906565e-05 140
## 142 -2.336043e-01 -1.215921e-01 17.43360 2.984658e-05 141
## 143 -2.656285e+00 -1.395444e+00 18.15629 1.646374e-02 142
## 144 -4.251022e-01 -2.212716e-01 17.42510 9.723744e-05 143
## 145 -9.141745e-01 -4.767544e-01 17.71417 8.304794e-04 144
## 146  1.130362e+00  5.890303e-01 17.56964 9.274094e-04 145
## 147  1.047366e+00  5.456897e-01 17.55263 7.674597e-04 146
## 148  6.263106e-01  3.267294e-01 17.77369 4.423796e-04 147
## 149 -2.470221e-02 -1.289031e-02 17.82470 7.650759e-07 148
## 150  4.283361e-01  2.232541e-01 17.67166 1.663212e-04 149
## 151 -7.247022e-01 -3.782496e-01 17.82470 6.584943e-04 150
## 152  1.142915e+00  5.950380e-01 17.35709 6.231747e-04 151
## 153 -3.765987e+00 -1.970755e+00 16.96599 6.551384e-03 152
## 154 -3.344043e-01 -1.742861e-01 16.63440 1.005550e-04 153
## 155 -2.644932e+00 -1.381343e+00 16.74493 4.944971e-03 154
## 156 -1.434404e+00 -7.481717e-01 16.63440 1.850135e-03 155
## 157 -2.338455e+00 -1.219969e+00 16.83846 3.175337e-03 156
## 158 -3.431979e+00 -1.794396e+00 16.93198 5.743320e-03 157
## 159 -2.425502e+00 -1.264880e+00 17.02550 2.514480e-03 158
## 160 -1.614975e+00 -8.413224e-01 16.91497 1.309499e-03 159
## 161 -3.804047e+00 -1.990470e+00 17.20405 5.834348e-03 160
## 162 -1.506472e+00 -7.847137e-01 16.90647 1.156796e-03 161
## 163 -3.708098e+00 -1.940547e+00 17.40810 7.166427e-03 162
## 164 -6.194256e-01 -3.226915e-01 16.71943 2.866427e-04 163
## 165 -3.317000e+00 -1.733378e+00 17.01700 4.747908e-03 164
## 166 -2.170438e+00 -1.132337e+00 16.77044 3.152199e-03 165
## 167 -2.391494e+00 -1.247146e+00 16.99149 2.547993e-03 166
## 168 -9.939192e-01 -5.179906e-01 16.69392 7.802058e-04 167
## 169 -3.814575e+00 -1.996373e+00 17.31457 6.515353e-03 168
## 170 -1.502421e+00 -7.833586e-01 16.70242 1.749987e-03 169
## 171 -2.457485e+00 -1.281830e+00 16.95749 2.826017e-03 170
## 172 -1.644932e+00 -8.576038e-01 16.74493 1.912628e-03 171
## 173 -2.317400e+00 -1.210493e+00 16.61740 5.010523e-03 172
## 174 -2.551009e+00 -1.330611e+00 17.05101 2.711538e-03 173
## 175 -2.431979e+00 -1.268554e+00 16.93198 2.883988e-03 174
## 176 -1.148983e+00 -5.981944e-01 16.94898 6.260576e-04 175
## 177 -4.138055e+00 -2.167635e+00 17.23806 7.055579e-03 176
## 178 -1.865987e+00 -9.722847e-01 16.96599 1.608395e-03 177
## 179 -2.802021e+00 -1.462252e+00 17.10202 3.159942e-03 178
## 180 -1.821451e+00 -9.495004e-01 16.82145 1.994403e-03 179
## 181 -2.487443e+00 -1.298378e+00 16.78744 3.993179e-03 180
## 182 -1.334404e+00 -6.959353e-01 16.63440 1.601161e-03 181
## 183 -1.563962e+00 -8.148433e-01 16.86396 1.350228e-03 182
## 184 -3.046558e+00 -1.590852e+00 17.24656 3.851174e-03 183
## 185 -2.551009e+00 -1.330611e+00 17.05101 2.711538e-03 184
## 186  1.181800e+00  6.227336e-01 15.81820 5.412211e-03 185
## 187 -1.910923e+00 -9.968563e-01 16.71092 2.779001e-03 186
## 188 -4.704383e-01 -2.449907e-01 16.77044 1.480894e-04 187
## 189 -3.563562e+00 -1.863430e+00 17.26356 5.352831e-03 188
## 190  1.894765e-01  9.857006e-02 17.11052 1.439823e-05 189
## 191 -3.544532e+00 -1.853217e+00 17.14453 5.001025e-03 190
## 192 -1.044932e+00 -5.444349e-01 16.74493 7.718106e-04 191
## 193 -3.555060e+00 -1.858915e+00 17.25506 5.283988e-03 192
## 194 -6.684128e-01 -3.483315e-01 16.66841 3.730573e-04 193
## 195 -3.334004e+00 -1.742293e+00 17.03400 4.708117e-03 194
## 196 -1.668413e+00 -8.702789e-01 16.66841 2.324305e-03 195
## 197 -6.918936e-01 -3.607699e-01 16.59189 4.719199e-04 196
## 198 -3.278540e+00 -1.712902e+00 17.17854 4.292729e-03 197
## 199 -3.117000e+00 -1.628020e+00 17.01700 4.192614e-03 198
## 200 -6.918936e-01 -3.607699e-01 16.59189 4.719199e-04 199
## 201 -3.768013e+00 -1.971454e+00 17.06801 5.833339e-03 200
## 202 -1.242506e+00 -6.468147e-01 17.04251 6.483743e-04 201
## 203 -2.723477e+00 -1.421498e+00 16.92348 3.669427e-03 202
## 204 -2.661936e+00 -1.390145e+00 16.76194 4.828616e-03 203
## 205 -2.651009e+00 -1.383060e+00 17.05101 2.928290e-03 204
## 206 -1.625902e+00 -8.483062e-01 16.62590 2.421400e-03 205
## 207 -2.531979e+00 -1.320979e+00 16.93198 3.126036e-03 206
## 208 -1.659511e+00 -8.642937e-01 17.05951 1.139125e-03 207
## 209 -3.261536e+00 -1.703934e+00 17.16154 4.235596e-03 208
## 210 -2.017000e+00 -1.051100e+00 17.01700 1.755590e-03 209
## 211 -2.712549e+00 -1.415293e+00 17.21255 2.979899e-03 210
## 212 -1.300396e+00 -6.783433e-01 16.60040 1.636778e-03 211
## 213 -3.234004e+00 -1.689587e+00 17.03400 4.429923e-03 212
## 214 -2.057485e+00 -1.072412e+00 16.95749 1.980917e-03 213
## 215 -3.099996e+00 -1.619124e+00 17.00000 4.233975e-03 214
## 216 -5.688128e-01 -2.975257e-01 16.26881 6.082565e-04 215
## 217 -2.791494e+00 -1.456955e+00 16.99149 3.471627e-03 216
## 218  1.485915e-01  7.743151e-02 16.65141 1.913276e-05 217
## 219 -2.557485e+00 -1.334259e+00 16.95749 3.060689e-03 218
## 220 -4.769149e-01 -2.485001e-01 16.67691 1.864270e-04 219
## 221 -2.987043e+00 -1.559466e+00 17.18704 3.572264e-03 220
## 222 -1.574889e+00 -8.219539e-01 16.57489 2.535885e-03 221
## 223 -1.829953e+00 -9.539081e-01 16.82995 1.978332e-03 222
## 224 -1.340481e+00 -6.980436e-01 16.94048 8.639154e-04 223
## 225 -1.187443e+00 -6.186265e-01 16.78744 9.099927e-04 224
## 226 -2.131979e+00 -1.111459e+00 16.93198 2.216356e-03 225
## 227 -1.940481e+00 -1.011285e+00 16.94048 1.810376e-03 226
## 228 -7.534341e-01 -3.924567e-01 16.75343 3.939652e-04 227
## 229 -2.646957e+00 -1.381718e+00 16.84696 3.999585e-03 228
## 230 -2.408809e-01 -1.256312e-01 16.54088 6.377210e-05 229
## 231 -3.242506e+00 -1.694044e+00 17.04251 4.415605e-03 230
## 232 -6.425064e-01 -3.343201e-01 17.04251 1.733738e-04 231
## 233 -2.210923e+00 -1.153928e+00 16.71092 3.720056e-03 232
## 234 -8.218511e-01 -4.292121e-01 16.42185 9.480558e-04 233
## 235 -2.255460e+00 -1.176407e+00 16.85546 2.855377e-03 234
## 236 -7.344043e-01 -3.828252e-01 16.63440 4.849875e-04 235
## 237 -3.268013e+00 -1.707427e+00 17.06801 4.387933e-03 236
## 238  7.336127e-01  3.826089e-01 16.56639 5.603371e-04 237
## 239 -2.795545e+00 -1.458847e+00 17.19554 3.138850e-03 238
## 240 -2.323877e+00 -1.214798e+00 16.52388 6.151666e-03 239
## 241 -2.846957e+00 -1.486775e+00 16.84696 4.626823e-03 240
## 242  5.441404e-01  2.840396e-01 16.45586 3.881127e-04 241
## 243 -1.846957e+00 -9.627260e-01 16.84696 1.947308e-03 242
## 244  6.526425e-01  3.407230e-01 16.44736 5.680213e-04 243
## 245 -2.517000e+00 -1.312853e+00 17.01700 2.733869e-03 244
## 246 -5.769149e-01 -3.006186e-01 16.67691 2.728039e-04 245
## 247 -2.402021e+00 -1.252463e+00 17.10202 2.322148e-03 246
## 248 -8.663873e-01 -4.518947e-01 16.56639 7.815194e-04 247
## 249 -8.854170e-01 -4.614308e-01 16.68542 6.307573e-04 248
## 250 -2.297970e+00 -1.198481e+00 16.89797 2.733474e-03 249
## 251 -2.370438e+00 -1.237130e+00 16.77044 3.759897e-03 250
## 252 -4.088087e-02 -2.132091e-02 16.54088 1.836819e-06 251
## 253 -1.761936e+00 -9.186669e-01 16.76194 2.115474e-03 252
## 254  8.672213e-01  4.545030e-01 16.13278 1.794658e-03 253
## 255 -1.372464e+00 -7.148861e-01 16.87246 1.022855e-03 254
## 256 -1.710923e+00 -8.922645e-01 16.71092 2.227734e-03 255
## 257 -3.063962e+00 -1.600817e+00 16.86396 5.182289e-03 256
## 258 -8.064724e-01 -4.198154e-01 16.90647 3.315229e-04 257
## 259 -2.640081e+00 -1.377570e+00 17.34008 3.237810e-03 258
## 260 -5.453319e-01 -2.849868e-01 16.34533 4.846441e-04 259
## 261 -3.204047e+00 -1.673680e+00 17.20405 4.139028e-03 260
## 262 -1.695945e+00 -8.840249e-01 16.79594 1.823257e-03 261
## 263 -1.391894e+00 -7.261915e-01 16.59189 1.909861e-03 262
## 264 -7.514085e-01 -3.916466e-01 16.65141 4.892627e-04 263
## 265 -1.987043e+00 -1.035321e+00 17.18704 1.580793e-03 264
## 266 -2.068724e-01 -1.079251e-01 16.50687 5.051238e-05 265
## 267 -2.857485e+00 -1.491744e+00 16.95749 3.820858e-03 266
## 268 -2.007957e-01 -1.051075e-01 16.20080 8.560371e-05 267
## 269 -1.402021e+00 -7.299300e-01 17.10202 7.911256e-04 268
## 270 -5.364298e-01 -2.794163e-01 16.73643 2.071817e-04 269
## 271 -3.172464e+00 -1.657903e+00 16.87246 5.465195e-03 270
## 273 -2.606472e+00 -1.360163e+00 16.90647 3.462907e-03 271
## 274 -9.003958e-01 -4.695200e-01 16.60040 7.847033e-04 272
## 275 -2.242506e+00 -1.169017e+00 17.04251 2.112010e-03 273
## 276 -5.429064e-01 -2.829579e-01 16.64291 2.601838e-04 274
## 277  9.680212e-01  5.039279e-01 16.93198 4.569236e-04 275
## 278  2.865596e+00  1.498372e+00 16.63440 7.383972e-03 276
## 279  2.676123e+00  1.399928e+00 16.52388 8.157909e-03 277
## 280  1.674498e+00  8.721607e-01 17.02550 1.198434e-03 278
## 281  3.395153e+00  1.781222e+00 16.40485 1.673476e-02 279
## 282  7.574936e-01  3.941775e-01 17.04251 2.409830e-04 280
## 283  1.234013e+00  6.424865e-01 16.96599 7.034198e-04 281
## 284  1.676123e+00  8.752742e-01 16.52388 3.200211e-03 282
## 285  1.925511e+00  1.003369e+00 16.97449 1.691352e-03 283
## 286  3.376123e+00  1.769139e+00 16.52388 1.298384e-02 284
## 287  8.765234e-01  4.562774e-01 16.92348 3.800826e-04 285
## 288  3.810132e+00  1.999701e+00 16.48987 1.774923e-02 286
## 289  4.105319e-01  2.136769e-01 16.88947 8.862061e-05 287
## 290  1.635638e+00  8.545464e-01 16.46436 3.446669e-03 288
## 291  1.170085e-01  6.088102e-02 16.98299 6.170717e-06 289
## 292  3.008106e+00  1.573917e+00 16.59189 8.920230e-03 290
## 293  3.391102e+00  1.775838e+00 16.60890 1.092824e-02 291
## 294  1.845766e+00  9.710618e-01 15.95423 1.081632e-02 292
## 295  1.659519e+00  8.645110e-01 16.94048 1.324081e-03 293
## 296  1.497579e+00  7.808291e-01 16.70242 1.738724e-03 294
## 297  1.943399e-02  1.011090e-02 17.28057 1.620941e-07 295
## 298  7.380638e-01  3.844293e-01 16.76194 3.712061e-04 296
## 299 -6.125490e-01 -3.187098e-01 17.21255 1.519596e-04 297
## 300  2.816608e+00  1.473154e+00 16.58339 7.964772e-03 298
## 301  9.850255e-01  5.128138e-01 16.91497 4.871567e-04 299
## 302  2.535638e+00  1.326762e+00 16.46436 8.283229e-03 300
## 303  1.808106e+00  9.438466e-01 16.59189 3.222832e-03 301
## 304  2.323085e+00  1.212974e+00 16.67691 4.423411e-03 302
## 305  8.595191e-01  4.473977e-01 16.94048 3.551903e-04 303
## 306  3.603655e+00  1.891898e+00 16.39634 1.917179e-02 304
## 307 -8.080979e-01 -4.206797e-01 17.40810 3.403520e-04 305
## 308  4.522685e+00  2.384861e+00 16.27731 3.786033e-02 306
## 309 -5.720639e-01 -2.976617e-01 17.27206 1.391548e-04 307
## 310  2.250617e+00  1.176089e+00 16.54938 5.467870e-03 308
## 311  1.940089e+00  1.012447e+00 16.65991 3.201729e-03 309
## 312 -4.695748e-02 -2.444320e-02 16.84696 1.258724e-06 310
## 313  1.461545e+00  7.614680e-01 16.83846 1.240383e-03 311
## 314  4.235638e+00  2.226655e+00 16.46436 2.311334e-02 312
## 315 -2.979703e-01 -1.550804e-01 16.89797 4.595914e-05 313
## 316  3.563170e+00  1.871638e+00 16.33683 2.102843e-02 314
## 317  2.780574e+00  1.452827e+00 16.71943 5.776060e-03 315
## 318  5.425149e-01  2.823269e-01 16.95749 1.377261e-04 316
## 319  2.542115e+00  1.329070e+00 16.55789 6.851224e-03 317
## 320 -1.700004e-02 -8.844675e-03 17.01700 1.247129e-07 318
## 321  1.342115e+00  7.003707e-01 16.55789 1.909665e-03 319
## 322  1.933613e+00  1.009755e+00 16.56639 3.892733e-03 320
## 323  1.274098e+00  6.644820e-01 16.62590 1.486905e-03 321
## 324  2.880574e+00  1.505420e+00 16.71943 6.198989e-03 322
## 325  2.193128e+00  1.146348e+00 16.50687 5.677024e-03 323
## 326  6.485915e-01  3.380371e-01 16.65141 3.645292e-04 324
## 327 -3.959447e-01 -2.061648e-01 16.79594 9.937875e-05 325
## 328  2.484626e+00  1.299330e+00 16.51537 7.158405e-03 326
## 329  3.000042e-01  1.560951e-01 17.00000 3.965343e-05 327
## 330  3.125111e+00  1.635829e+00 16.57489 9.985280e-03 328
## 331  2.793611e-02  1.453409e-02 17.27206 3.318494e-07 329
## 332  2.352643e+00  1.230767e+00 16.44736 7.381179e-03 330
## 333 -4.425064e-01 -2.302327e-01 17.04251 8.223708e-05 331
## 334  3.206081e+00  1.677134e+00 16.69392 8.118129e-03 332
## 335  2.182600e+00  1.139802e+00 16.61740 4.444566e-03 333
## 336  2.391502e+00  1.247106e+00 17.00850 2.493153e-03 334
## 337  3.027136e+00  1.585561e+00 16.47286 1.160234e-02 335
## 338 -4.064724e-01 -2.115513e-01 16.90647 8.421640e-05 336
## 339  4.212984e-06  2.191977e-06 17.00000 7.819982e-15 337
## 340  3.658719e+00  1.927160e+00 16.14128 3.148828e-02 338
## 341  9.129574e-01  4.750915e-01 17.18704 3.337042e-04 339
## 342  1.531587e+00  7.987675e-01 16.66841 1.958707e-03 340
## 343  2.433613e+00  1.271979e+00 16.56639 6.166214e-03 341
## 344  2.082600e+00  1.087393e+00 16.61740 4.046623e-03 342
```

---
exclude: TRUE
# Homework (Practice) Answers

3\. Check your model diagnostics - Normality

```r
ggplot(data = d, aes(sample = std_residuals)) +
  stat_qq() +
  stat_qq_line()
```

![:box 25%, 75%, 30%](Not bad, could be better, but good enough)

---
exclude: TRUE
# Homework (Practice) Answers

3\. Check your model diagnostics - Heteroscedasticity

```r
ggplot(d, aes(x = fitted, y = std_residuals)) +
  geom_point() +
  geom_hline(yintercept = 0)
```

![:box 25%, 75%, 30%](Again, not bad, could be better)

---
exclude: TRUE
# Homework (Practice) Answers

3\. Check your model diagnostics - Influence (Cook's d)

```r
ggplot(d, aes(x = obs, y = cooks)) +
  geom_point() +
  geom_hline(yintercept = 4/nrow(penguins), 
             linetype = "dashed")
```

![:box 15%, 75%, 25%](Some interesting observations, but nothing crazy)

---
exclude: TRUE
# Homework (Practice) Answers

4\. Interpret the results of your model

.small[

```r
summary(m)
```

```
## 
## Call:
## lm(formula = bill_depth_mm ~ bill_length_mm, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -4.1381 -1.4263 0.0164 1.3841 4.5255 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 20.88547 0.84388 24.749 < 2e-16 ***
*## bill_length_mm -0.08502 0.01907 -4.459 1.12e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.922 on 340 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.05525,	Adjusted R-squared: 0.05247 
## F-statistic: 19.88 on 1 and 340 DF, p-value: 1.12e-05
```
]

![:box 25%, 50%, 40%](Significant (p < 0.0001&rpar; relationship between bill length and bill depth. For every 1 mm increase in bill length, bill depth decreases by 0.085 mm Is this biologically meaningfull?)

---
class: section
layout: false

# ANOVAs

---
# Linear Models

### Running models in R

Here we're assuming a **continuous** **y**

### Different types of models

![:box 50%, 85%, 30%](R will figure it out for you)

---
# ANOVAs

### Real example

- Are male penguins larger than female penguins? 
- Are different species different sizes?
- Can body mass be predicted by species and sex?

---
# ANOVAs

### Real example

As we have two **categorical** predictors, this is an ANOVA

![:box 50%, 45%, 50%](Your turn! Create a model with your response variable by your one categorical predictor. Look at the output of <code>summary(&rpar;</code> and <code>anova(&rpar;</code>)

![:box 50%, 75%, 40%](Wait! Shouldn't interpret until we know the model is solid)

---
class: split-45
# Model Diagnostics

Same as before! Let's get our relevant variables into a diagnostic data frame:

- **`residuals`** (regular and standardized)
- **`fitted` values**
- **`cooks` distance**
- **`obs` number**

.footnote[Remember! `n()` only works *inside* `summarize()`/`mutate()`]

.columnl[.small[

```r
d <- data.frame(residuals = residuals(m), 
 std_residuals = rstudent(m), 
 fitted = fitted(m), 
 cooks = cooks.distance(m))

d <- mutate(d, obs = 1:n())
```
]]

--
.columnr[
.small[

```r
head(d)
```

```
##    residuals std_residuals   fitted        cooks obs
## 1 -289.94196    -0.9201062 4039.942 0.0021071103   1
## 2  427.61319     1.3590567 3372.387 0.0045831843   2
## 3 -122.38681    -0.3879729 3372.387 0.0003754346   3
## 5   77.61319     0.2460043 3372.387 0.0001509858   4
## 6 -389.94196    -1.2387413 4039.942 0.0038112292   5
## 7  252.61319     0.8013974 3372.387 0.0015994740   6
```
]]

---
class: split-50
# Normality

.columnl[
#### Histogram of residuals
.small[

```r
ggplot(data = d, aes(x = std_residuals)) +
  geom_histogram(bins = 20)
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-59-1.png" width="90%" style="display: block; margin: auto;" />
]]

.columnr[
#### QQ Normality plot of residuals
.small[

```r
ggplot(data = d, aes(sample = std_residuals)) +
  stat_qq() +
  stat_qq_line()
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-60-1.png" width="90%" style="display: block; margin: auto;" />
]]

---
class: split-50
# Variance and Influence

.columnl[
#### Check heteroscedasticity
.small[

```r
ggplot(d, aes(x = fitted, y = std_residuals)) +
  geom_point() +
  geom_hline(yintercept = 0)
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-61-1.png" width="90%" style="display: block; margin: auto;" />
]]

.columnr[
#### Cook's D

.small[

```r
ggplot(d, aes(x = obs, y = cooks)) +
  geom_point() +
* geom_hline(yintercept = 1, linetype = "dotted") +
  geom_hline(yintercept = 4/nrow(penguins), 
             linetype = "dashed")
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-62-1.png" width="90%" style="display: block; margin: auto;" />
]]

---
class: split-50
# Variance and Influence

.columnl[
#### Check heteroscedasticity
.small[

```r
ggplot(d, aes(x = fitted, y = std_residuals)) +
  geom_point() +
  geom_hline(yintercept = 0)
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-63-1.png" width="90%" style="display: block; margin: auto;" />
]]

.columnr[
#### Cook's D

.small[

```r
ggplot(d, aes(x = obs, y = cooks)) +
  geom_point() +
  geom_hline(yintercept = 4/nrow(penguins), 
             linetype = "dashed")
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-64-1.png" width="90%" style="display: block; margin: auto;" />
]]

--
![:box 50%, 80%, 20%](Overall not too bad)

---
# Multicollinearity (collinearity)

### `vif()` function from `car` package

```r
library(car)
vif(m)
```

```
##             GVIF Df GVIF^(1/(2*Df))
## species 1.000146  2        1.000036
## sex     1.000146  1        1.000073
```

Here we consider the `GVIF^(1/2*Df))` value*

Looks good!

.footnote[\* See `?vif` and the reference therein: Fox, J. and Monette, G. (1992) Generalized collinearity diagnostics. JASA, 87, 178–183.]

---
layout: true
class: split-40
# Interpreting ANOVA Summaries

```r
m <- lm(body_mass_g ~ species + sex, data = penguins)
```

.columnl[

```r
summary(m)
```
]

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ species + sex, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -816.87 -217.80 -16.87 227.61 882.20 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 3372.39 31.43 107.308 <2e-16 ***
## speciesChinstrap 26.92 46.48 0.579 0.563 
## speciesGentoo 1377.86 39.10 35.236 <2e-16 ***
## sexmale 667.56 34.70 19.236 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 316.6 on 329 degrees of freedom
## (11 observations deleted due to missingness)
## Multiple R-squared: 0.8468,	Adjusted R-squared: 0.8454 
## F-statistic: 606.1 on 3 and 329 DF, p-value: < 2.2e-16
```
]]

---
.columnr[.small[

```
## 
## Call:
*## lm(formula = body_mass_g ~ species + sex, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -816.87 -217.80 -16.87 227.61 882.20 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 3372.39 31.43 107.308 <2e-16 ***
## speciesChinstrap 26.92 46.48 0.579 0.563 
## speciesGentoo 1377.86 39.10 35.236 <2e-16 ***
## sexmale 667.56 34.70 19.236 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 316.6 on 329 degrees of freedom
## (11 observations deleted due to missingness)
## Multiple R-squared: 0.8468,	Adjusted R-squared: 0.8454 
## F-statistic: 606.1 on 3 and 329 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### Model
]

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ species + sex, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -816.87 -217.80 -16.87 227.61 882.20 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
*## (Intercept) 3372.39 31.43 107.308 <2e-16 ***
*## speciesChinstrap 26.92 46.48 0.579 0.563 
*## speciesGentoo 1377.86 39.10 35.236 <2e-16 ***
*## sexmale 667.56 34.70 19.236 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 316.6 on 329 degrees of freedom
## (11 observations deleted due to missingness)
## Multiple R-squared: 0.8468,	Adjusted R-squared: 0.8454 
## F-statistic: 606.1 on 3 and 329 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### Effects
]

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ species + sex, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -816.87 -217.80 -16.87 227.61 882.20 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 3372.39 31.43 107.308 <2e-16 ***
## speciesChinstrap 26.92 46.48 0.579 0.563 
## speciesGentoo 1377.86 39.10 35.236 <2e-16 ***
## sexmale 667.56 34.70 19.236 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 316.6 on 329 degrees of freedom
*## (11 observations deleted due to missingness)
## Multiple R-squared: 0.8468,	Adjusted R-squared: 0.8454 
## F-statistic: 606.1 on 3 and 329 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### Missing observations
]

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ species + sex, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -816.87 -217.80 -16.87 227.61 882.20 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
## (Intercept) 3372.39 31.43 107.308 <2e-16 ***
## speciesChinstrap 26.92 46.48 0.579 0.563 
## speciesGentoo 1377.86 39.10 35.236 <2e-16 ***
## sexmale 667.56 34.70 19.236 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 316.6 on 329 degrees of freedom
## (11 observations deleted due to missingness)
*## Multiple R-squared: 0.8468,	Adjusted R-squared: 0.8454 
## F-statistic: 606.1 on 3 and 329 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()
.columnl[
### R2 and adjusted R2

- Adjusted for the number of parameters
]

---
.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ species + sex, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -816.87 -217.80 -16.87 227.61 882.20 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
*## (Intercept) 3372.39 31.43 107.308 <2e-16 ***
*## speciesChinstrap 26.92 46.48 0.579 0.563 
*## speciesGentoo 1377.86 39.10 35.236 <2e-16 ***
*## sexmale 667.56 34.70 19.236 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 316.6 on 329 degrees of freedom
## (11 observations deleted due to missingness)
## Multiple R-squared: 0.8468,	Adjusted R-squared: 0.8454 
## F-statistic: 606.1 on 3 and 329 DF, p-value: < 2.2e-16
```
]]

![:spacer 20px]()

.columnl[
### Specific Details

.medium[
- `Estimate`
    - Treatment contrasts
    - Average *differences* among categories compared to the base category
    
- `Std. Error` 
    - Variability in the estimates
    
- `t value`
   - Test statistic

- `Pr(>|t|)` 
    - **P-value**, significance of the *differences*
    - Probability of getting `t-value` by chance
]]

![:box 50%, 50%, 60%](Easier to interpret estimates if we consider a simpler model)

---
layout: false
class: split-40, space-list

# Interpreting ANOVA Summaries

```r
m <- lm(body_mass_g ~ species, data = penguins)
```

.columnl[

```r
summary(m)
```
]

.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ species, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -1126.02 -333.09 -33.09 316.91 1223.98 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
*## (Intercept) 3700.66 37.62 98.37 <2e-16 ***
*## speciesChinstrap 32.43 67.51 0.48 0.631 
*## speciesGentoo 1375.35 56.15 24.50 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 462.3 on 339 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.6697,	Adjusted R-squared: 0.6677 
## F-statistic: 343.6 on 2 and 339 DF, p-value: < 2.2e-16
```
]]

.columnl[
Effect of **Species**
.medium[
- `(Intercept)` represents base category (i.e. Adelie penguins)
- Adelie have mean body mass of 3700.66 g 
- On average, Chinstrap penguins are 32.43 g heavier than Adelie penguins
- On average, Gentoo penguins are 1375.35 g heavier than Adelie penguins
]]

![:img right: 10%, top: 0%, 30%, ,](https://raw.githubusercontent.com/allisonhorst/palmerpenguins/master/man/figures/lter_penguins.png)

![:box 50%, 50%, 30%](Back to original model)

---
class: split-40, space-list
# Interpreting ANOVA Summaries

```r
m <- lm(body_mass_g ~ species + sex, data = penguins)
```

.columnl[

```r
summary(m)
```
]

.columnr[.small[

```
## 
## Call:
## lm(formula = body_mass_g ~ species + sex, data = penguins)
## 
## Residuals:
## Min 1Q Median 3Q Max 
## -816.87 -217.80 -16.87 227.61 882.20 
## 
## Coefficients:
## Estimate Std. Error t value Pr(>|t|) 
*## (Intercept) 3372.39 31.43 107.308 <2e-16 ***
*## speciesChinstrap 26.92 46.48 0.579 0.563 
*## speciesGentoo 1377.86 39.10 35.236 <2e-16 ***
*## sexmale 667.56 34.70 19.236 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 316.6 on 329 degrees of freedom
## (11 observations deleted due to missingness)
## Multiple R-squared: 0.8468,	Adjusted R-squared: 0.8454 
## F-statistic: 606.1 on 3 and 329 DF, p-value: < 2.2e-16
```
]]

.columnl[
Effect of **Species** and **Sex**

.medium[
- `(Intercept)` represents base category but is a combination of factors
- Much more complicated to interpret
- Comparisons are often not of interest anyway (unless you've set up contrasts, which are advanced stats but awesome!)
]]

![:box 80%, 80%, 25%](So let's look at ANOVA tables instead)

---
class: split-50
layout: true
# Interpreting ANOVA Tables

### Type I

```r
m <- lm(body_mass_g ~ species + sex, data = penguins)
```

.columnl[

```r
anova(m)
```
]

---
.columnr[.small[

```
## Analysis of Variance Table
## 
## Response: body_mass_g
## Df Sum Sq Mean Sq F value Pr(>F) 
## species 2 145190219 72595110 724.21 < 2.2e-16 ***
## sex 1 37090262 37090262 370.01 < 2.2e-16 ***
## Residuals 329 32979185 100241 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]]

---
.columnr[.small[

```
## Analysis of Variance Table
## 
## Response: body_mass_g
## Df Sum Sq Mean Sq F value Pr(>F) 
*## species 2 145190219 72595110 724.21 < 2.2e-16 ***
*## sex 1 37090262 37090262 370.01 < 2.2e-16 ***
## Residuals 329 32979185 100241 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]]

.columnl[
Overall effects of **Species** and **Sex**

.medium[
- Yes there are differences among **Species** (P < 2.2e-16)
- Yes there are differences between **Sexes** (P < 2.2e-16)
]]

![:box 75%, 75%, 45%](Not a whole lot of information... Stay tuned for Post-Hoc tests next week!)

---
class: split-50, space-list
layout: false
# Interpreting ANOVA Tables

### Type I

.columnl[
.small[

```r
*m1 <- lm(body_mass_g ~ species + sex, data = penguins)
anova(m1)
```

.columnr[
.small[

```r
*m2 <- lm(body_mass_g ~ sex + species, data = penguins)
anova(m2)
```

```
## Analysis of Variance Table
## 
## Response: body_mass_g
## Df Sum Sq Mean Sq F value Pr(>F) 
*## sex 1 38878897 38878897 387.86 < 2.2e-16 ***
*## species 2 143401584 71700792 715.29 < 2.2e-16 ***
## Residuals 329 32979185 100241 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]]

- For Type I ANOVAs, order matters with unbalanced samples
    - See that `Sum sq`, `Mean Sq` and `F value` all differ between the models
- Here, pretty minor, but important to remember with greater unbalances

---
layout: true
class: split-50

# Interpreting ANOVA Tables

### Type III

```r
m <- lm(body_mass_g ~ species + sex, data = penguins)
```

.footnote[Type II ANOVAs do exist as well, but generally we use Type III in natural sciences]

.columnl[

```r
library(car)
Anova(m, type = "3")
```
]

---
.columnr[.small[

```
## Anova Table (Type III tests)
## 
## Response: body_mass_g
## Sum Sq Df F value Pr(>F) 
## (Intercept) 1154266972 1 11514.96 < 2.2e-16 ***
## species 143401584 2 715.29 < 2.2e-16 ***
## sex 37090262 1 370.01 < 2.2e-16 ***
## Residuals 32979185 329 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]]

---
layout: false
class: split-50

# Interpreting ANOVA Tables

### Type III

.columnl[
.small[

```r
*m1 <- lm(body_mass_g ~ species + sex, data = penguins)
Anova(m1, type = "3")
```

```
## Anova Table (Type III tests)
## 
## Response: body_mass_g
## Sum Sq Df F value Pr(>F) 
## (Intercept) 1154266972 1 11514.96 < 2.2e-16 ***
*## species 143401584 2 715.29 < 2.2e-16 ***
*## sex 37090262 1 370.01 < 2.2e-16 ***
## Residuals 32979185 329 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]]

.columnr[
.small[

```r
*m2 <- lm(body_mass_g ~ sex + species, data = penguins)
Anova(m2, type = "3")
```

```
## Anova Table (Type III tests)
## 
## Response: body_mass_g
## Sum Sq Df F value Pr(>F) 
## (Intercept) 1154266972 1 11514.96 < 2.2e-16 ***
*## sex 37090262 1 370.01 < 2.2e-16 ***
*## species 143401584 2 715.29 < 2.2e-16 ***
## Residuals 32979185 329 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]]

- Type III and unbalanced samples: Not dependent on variable order

---
# Homework (Practice)*

Consider flipper length your response variable and species and sex your predictor variables

1. Plot the relationship between flipper length and species and between flipper length and sex

2. Create an ANOVA model of flipper length and species

3. Check diagnostics

4. Interpret the **summary table**

5. Interpret the **ANOVA Table**

6. Create an ANOVA model of flipper length and species and sex

7. Check diagnostics

8. Interpret the **ANOVA Table**

.footnote[\* Not to be handed in, answers posted in these slides next week]

---
exclude: TRUE
class: split-50
# Homework (Practice) Answers

1\. Plot the relationship between flipper length and species and between flipper length and sex

.footnote[Use `drop_na()` to omit missing `sex`]

.columnl[
.small[

```r
ggplot(data = penguins, 
       aes(x = species, y = flipper_length_mm)) +
  geom_boxplot()
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-86-1.png" width="90%" style="display: block; margin: auto;" />
]]

.columnr[
.small[

```r
ggplot(data = drop_na(penguins), 
       aes(x = sex, y = flipper_length_mm)) +
  geom_boxplot()
```

<img src="4 Regressions and ANOVAs_files/figure-html/unnamed-chunk-87-1.png" width="90%" style="display: block; margin: auto;" />
]]

---
exclude: TRUE
# Homework (Practice) Answers

2\. Create an ANOVA model of flipper length and species

```r
m <- lm(flipper_length_mm ~ species, data = penguins)
```

---
exclude: TRUE
# Homework (Practice) Answers

3\. Check your model diagnostics

```r
d <- data.frame(residuals = residuals(m), 
 std_residuals = rstudent(m), 
 fitted = fitted(m), 
 cooks = cooks.distance(m))

d <- mutate(d, obs = 1:n())
d
```

```
##        residuals std_residuals   fitted        cooks obs
## 1    -8.95364238  -1.354279014 189.9536 4.065712e-03   1
## 2    -3.95364238  -0.596704490 189.9536 7.927420e-04   2
## 3     5.04635762   0.761875329 189.9536 1.291496e-03   3
## 5     3.04635762   0.459673943 189.9536 4.706508e-04   4
## 6     0.04635762   0.006992853 189.9536 1.089881e-07   5
## 7    -8.95364238  -1.354279014 189.9536 4.065712e-03   6
## 8     5.04635762   0.761875329 189.9536 1.291496e-03   7
## 9     3.04635762   0.459673943 189.9536 4.706508e-04   8
## 10    0.04635762   0.006992853 189.9536 1.089881e-07   9
## 11   -3.95364238  -0.596704490 189.9536 7.927420e-04  10
## 12   -9.95364238  -1.506497837 189.9536 5.024596e-03  11
## 13   -7.95364238  -1.202336705 189.9536 3.208257e-03  12
## 14    1.04635762   0.157844495 189.9536 5.552612e-05  13
## 15    8.04635762   1.216413288 189.9536 3.283490e-03  14
## 16   -4.95364238  -0.747854240 189.9536 1.244476e-03  15
## 17    5.04635762   0.761875329 189.9536 1.291496e-03  16
## 18    7.04635762   1.064694521 189.9536 2.518062e-03  17
## 19   -5.95364238  -0.899155542 189.9536 1.797639e-03  18
## 20    4.04635762   0.610712824 189.9536 8.303584e-04  19
## 21  -15.95364238  -2.427426485 189.9536 1.290794e-02  20
## 22   -9.95364238  -1.506497837 189.9536 5.024596e-03  21
## 23   -0.95364238  -0.143857376 189.9536 4.612200e-05  22
## 24   -4.95364238  -0.747854240 189.9536 1.244476e-03  23
## 25   -9.95364238  -1.506497837 189.9536 5.024596e-03  24
## 26   -2.95364238  -0.445675516 189.9536 4.424385e-04  25
## 27   -6.95364238  -1.050639325 189.9536 2.452233e-03  26
## 28   -2.95364238  -0.445675516 189.9536 4.424385e-04  27
## 29  -17.95364238  -2.738103033 189.9536 1.634716e-02  28
## 30   -9.95364238  -1.506497837 189.9536 5.024596e-03  29
## 31  -11.95364238  -1.811892817 189.9536 7.246655e-03  30
## 32  -11.95364238  -1.811892817 189.9536 7.246655e-03  31
## 33   -1.95364238  -0.294736665 189.9536 1.935652e-04  32
## 34   -5.95364238  -0.899155542 189.9536 1.797639e-03  33
## 35    5.04635762   0.761875329 189.9536 1.291496e-03  34
## 36    6.04635762   0.913192246 189.9536 1.854064e-03  35
## 37    0.04635762   0.006992853 189.9536 1.089881e-07  36
## 38   -9.95364238  -1.506497837 189.9536 5.024596e-03  37
## 39   -8.95364238  -1.354279014 189.9536 4.065712e-03  38
## 40   -5.95364238  -0.899155542 189.9536 1.797639e-03  39
## 41   -7.95364238  -1.202336705 189.9536 3.208257e-03  40
## 42    5.04635762   0.761875329 189.9536 1.291496e-03  41
## 43   -3.95364238  -0.596704490 189.9536 7.927420e-04  42
## 44    6.04635762   0.913192246 189.9536 1.854064e-03  43
## 45   -4.95364238  -0.747854240 189.9536 1.244476e-03  44
## 46    0.04635762   0.006992853 189.9536 1.089881e-07  45
## 47   -7.95364238  -1.202336705 189.9536 3.208257e-03  46
## 48  -10.95364238  -1.659025042 189.9536 6.084910e-03  47
## 49    0.04635762   0.006992853 189.9536 1.089881e-07  48
## 50    1.04635762   0.157844495 189.9536 5.552612e-05  49
## 51   -3.95364238  -0.596704490 189.9536 7.927420e-04  50
## 52   -1.95364238  -0.294736665 189.9536 1.935652e-04  51
## 53    0.04635762   0.006992853 189.9536 1.089881e-07  52
## 54   10.04635762   1.520625971 189.9536 5.118637e-03  53
## 55   -2.95364238  -0.445675516 189.9536 4.424385e-04  54
## 56    1.04635762   0.157844495 189.9536 5.552612e-05  55
## 57   -3.95364238  -0.596704490 189.9536 7.927420e-04  56
## 58    3.04635762   0.459673943 189.9536 4.706508e-04  57
## 59   -8.95364238  -1.354279014 189.9536 4.065712e-03  58
## 60    4.04635762   0.610712824 189.9536 8.303584e-04  59
## 61   -4.95364238  -0.747854240 189.9536 1.244476e-03  60
## 62    5.04635762   0.761875329 189.9536 1.291496e-03  61
## 63   -4.95364238  -0.747854240 189.9536 1.244476e-03  62
## 64    2.04635762   0.308728023 189.9536 2.123734e-04  63
## 65   -5.95364238  -0.899155542 189.9536 1.797639e-03  64
## 66    2.04635762   0.308728023 189.9536 2.123734e-04  65
## 67    5.04635762   0.761875329 189.9536 1.291496e-03  66
## 68   -1.95364238  -0.294736665 189.9536 1.935652e-04  67
## 69    0.04635762   0.006992853 189.9536 1.089881e-07  68
## 70    8.04635762   1.216413288 189.9536 3.283490e-03  69
## 71    0.04635762   0.006992853 189.9536 1.089881e-07  70
## 72    0.04635762   0.006992853 189.9536 1.089881e-07  71
## 73    6.04635762   0.913192246 189.9536 1.854064e-03  72
## 74    7.04635762   1.064694521 189.9536 2.518062e-03  73
## 75    0.04635762   0.006992853 189.9536 1.089881e-07  74
## 76    5.04635762   0.761875329 189.9536 1.291496e-03  75
## 77    1.04635762   0.157844495 189.9536 5.552612e-05  76
## 78   -5.95364238  -0.899155542 189.9536 1.797639e-03  77
## 79   -2.95364238  -0.445675516 189.9536 4.424385e-04  78
## 80    5.04635762   0.761875329 189.9536 1.291496e-03  79
## 81   -0.95364238  -0.143857376 189.9536 4.612200e-05  80
## 82    6.04635762   0.913192246 189.9536 1.854064e-03  81
## 83   -2.95364238  -0.445675516 189.9536 4.424385e-04  82
## 84    3.04635762   0.459673943 189.9536 4.706508e-04  83
## 85    1.04635762   0.157844495 189.9536 5.552612e-05  84
## 86    4.04635762   0.610712824 189.9536 8.303584e-04  85
## 87    0.04635762   0.006992853 189.9536 1.089881e-07  86
## 88   -0.95364238  -0.143857376 189.9536 4.612200e-05  87
## 89   -0.95364238  -0.143857376 189.9536 4.612200e-05  88
## 90    0.04635762   0.006992853 189.9536 1.089881e-07  89
## 91   12.04635762   1.826084387 189.9536 7.359505e-03  90
## 92   15.04635762   2.287176547 189.9536 1.148153e-02  91
## 93   -4.95364238  -0.747854240 189.9536 1.244476e-03  92
## 94   -3.95364238  -0.596704490 189.9536 7.927420e-04  93
## 95   -2.95364238  -0.445675516 189.9536 4.424385e-04  94
## 96   18.04635762   2.752559125 189.9536 1.651643e-02  95
## 97    0.04635762   0.006992853 189.9536 1.089881e-07  96
## 98    6.04635762   0.913192246 189.9536 1.854064e-03  97
## 99  -11.95364238  -1.811892817 189.9536 7.246655e-03  98
## 100   2.04635762   0.308728023 189.9536 2.123734e-04  99
## 101   2.04635762   0.308728023 189.9536 2.123734e-04 100
## 102  13.04635762   1.979361527 189.9536 8.632083e-03 101
## 103  -6.95364238  -1.050639325 189.9536 2.452233e-03 102
## 104   0.04635762   0.006992853 189.9536 1.089881e-07 103
## 105   3.04635762   0.459673943 189.9536 4.706508e-04 104
## 106  -5.95364238  -0.899155542 189.9536 1.797639e-03 105
## 107   9.04635762   1.368379901 189.9536 4.150349e-03 106
## 108   0.04635762   0.006992853 189.9536 1.089881e-07 107
## 109  -8.95364238  -1.354279014 189.9536 4.065712e-03 108
## 110   7.04635762   1.064694521 189.9536 2.518062e-03 109
## 111   8.04635762   1.216413288 189.9536 3.283490e-03 110
## 112   1.04635762   0.157844495 189.9536 5.552612e-05 111
## 113   3.04635762   0.459673943 189.9536 4.706508e-04 112
## 114   7.04635762   1.064694521 189.9536 2.518062e-03 113
## 115   1.04635762   0.157844495 189.9536 5.552612e-05 114
## 116   6.04635762   0.913192246 189.9536 1.854064e-03 115
## 117  -1.95364238  -0.294736665 189.9536 1.935652e-04 116
## 118   9.04635762   1.368379901 189.9536 4.150349e-03 117
## 119  -0.95364238  -0.143857376 189.9536 4.612200e-05 118
## 120  -0.95364238  -0.143857376 189.9536 4.612200e-05 119
## 121  -2.95364238  -0.445675516 189.9536 4.424385e-04 120
## 122   8.04635762   1.216413288 189.9536 3.283490e-03 121
## 123 -13.95364238  -2.118780628 189.9536 9.874435e-03 122
## 124  12.04635762   1.826084387 189.9536 7.359505e-03 123
## 125  -3.95364238  -0.596704490 189.9536 7.927420e-04 124
## 126   9.04635762   1.368379901 189.9536 4.150349e-03 125
## 127   1.04635762   0.157844495 189.9536 5.552612e-05 126
## 128   5.04635762   0.761875329 189.9536 1.291496e-03 127
## 129   1.04635762   0.157844495 189.9536 5.552612e-05 128
## 130  20.04635762   3.065661869 189.9536 2.038018e-02 129
## 131   0.04635762   0.006992853 189.9536 1.089881e-07 130
## 132   7.04635762   1.064694521 189.9536 2.518062e-03 131
## 133   3.04635762   0.459673943 189.9536 4.706508e-04 132
## 134   9.04635762   1.368379901 189.9536 4.150349e-03 133
## 135  -2.95364238  -0.445675516 189.9536 4.424385e-04 134
## 136   0.04635762   0.006992853 189.9536 1.089881e-07 135
## 137   1.04635762   0.157844495 189.9536 5.552612e-05 136
## 138  10.04635762   1.520625971 189.9536 5.118637e-03 137
## 139  -4.95364238  -0.747854240 189.9536 1.244476e-03 138
## 140   3.04635762   0.459673943 189.9536 4.706508e-04 139
## 141   3.04635762   0.459673943 189.9536 4.706508e-04 140
## 142  -2.95364238  -0.445675516 189.9536 4.424385e-04 141
## 143  -1.95364238  -0.294736665 189.9536 1.935652e-04 142
## 144   0.04635762   0.006992853 189.9536 1.089881e-07 143
## 145   2.04635762   0.308728023 189.9536 2.123734e-04 144
## 146  -4.95364238  -0.747854240 189.9536 1.244476e-03 145
## 147   0.04635762   0.006992853 189.9536 1.089881e-07 146
## 148  -5.95364238  -0.899155542 189.9536 1.797639e-03 147
## 149   5.04635762   0.761875329 189.9536 1.291496e-03 148
## 150   3.04635762   0.459673943 189.9536 4.706508e-04 149
## 151  -2.95364238  -0.445675516 189.9536 4.424385e-04 150
## 152  11.04635762   1.673183393 189.9536 6.188356e-03 151
## 153  -6.18699187  -0.935198481 217.1870 2.390491e-03 152
## 154  12.81300813   1.945051998 217.1870 1.025251e-02 153
## 155  -7.18699187  -1.086845476 217.1870 3.225688e-03 154
## 156   0.81300813   0.122734782 217.1870 4.127793e-05 155
## 157  -2.18699187  -0.330202454 217.1870 2.986912e-04 156
## 158  -7.18699187  -1.086845476 217.1870 3.225688e-03 157
## 159  -6.18699187  -0.935198481 217.1870 2.390491e-03 158
## 160   1.81300813   0.273722799 217.1870 2.052710e-04 159
## 161  -8.18699187  -1.238713865 217.1870 4.185784e-03 160
## 162  -2.18699187  -0.330202454 217.1870 2.986912e-04 161
## 163  -3.18699187  -0.481274446 217.1870 6.342931e-04 162
## 164  -1.18699187  -0.179197301 217.1870 8.798803e-05 163
## 165  -3.18699187  -0.481274446 217.1870 6.342931e-04 164
## 166  -4.18699187  -0.632443927 217.1870 1.094794e-03 165
## 167  -7.18699187  -1.086845476 217.1870 3.225688e-03 166
## 168  -0.18699187  -0.028228404 217.1870 2.183602e-06 167
## 169  -7.18699187  -1.086845476 217.1870 3.225688e-03 168
## 170   3.81300813   0.575895638 217.1870 9.079534e-04 169
## 171  -8.18699187  -1.238713865 217.1870 4.185784e-03 170
## 172   4.81300813   0.727141791 217.1870 1.446643e-03 171
## 173   0.81300813   0.122734782 217.1870 4.127793e-05 172
## 174  -2.18699187  -0.330202454 217.1870 2.986912e-04 173
## 175  -4.18699187  -0.632443927 217.1870 1.094794e-03 174
## 176  -2.18699187  -0.330202454 217.1870 2.986912e-04 175
## 177  -2.18699187  -0.330202454 217.1870 2.986912e-04 176
## 178  -2.18699187  -0.330202454 217.1870 2.986912e-04 177
## 179  -1.18699187  -0.179197301 217.1870 8.798803e-05 178
## 180  -2.18699187  -0.330202454 217.1870 2.986912e-04 179
## 181  -7.18699187  -1.086845476 217.1870 3.225688e-03 180
## 182   2.81300813   0.424766211 217.1870 4.941628e-04 181
## 183   4.81300813   0.727141791 217.1870 1.446643e-03 182
## 184  -8.18699187  -1.238713865 217.1870 4.185784e-03 183
## 185 -10.18699187  -1.543240931 217.1870 6.480672e-03 184
## 186  12.81300813   1.945051998 217.1870 1.025251e-02 185
## 187   2.81300813   0.424766211 217.1870 4.941628e-04 186
## 188   2.81300813   0.424766211 217.1870 4.941628e-04 187
## 189  -4.18699187  -0.632443927 217.1870 1.094794e-03 188
## 190   1.81300813   0.273722799 217.1870 2.052710e-04 189
## 191  -9.18699187  -1.390835109 217.1870 5.270778e-03 190
## 192  -9.18699187  -1.390835109 217.1870 5.270778e-03 191
## 193  -9.18699187  -1.390835109 217.1870 5.270778e-03 192
## 194   7.81300813   1.181889650 217.1870 3.812103e-03 193
## 195  -7.18699187  -1.086845476 217.1870 3.225688e-03 194
## 196  -1.18699187  -0.179197301 217.1870 8.798803e-05 195
## 197   4.81300813   0.727141791 217.1870 1.446643e-03 196
## 198  -0.18699187  -0.028228404 217.1870 2.183602e-06 197
## 199  -7.18699187  -1.086845476 217.1870 3.225688e-03 198
## 200   7.81300813   1.181889650 217.1870 3.812103e-03 199
## 201  -4.18699187  -0.632443927 217.1870 1.094794e-03 200
## 202  -2.18699187  -0.330202454 217.1870 2.986912e-04 201
## 203  -7.18699187  -1.086845476 217.1870 3.225688e-03 202
## 204   2.81300813   0.424766211 217.1870 4.941628e-04 203
## 205  -7.18699187  -1.086845476 217.1870 3.225688e-03 204
## 206   7.81300813   1.181889650 217.1870 3.812103e-03 205
## 207  -0.18699187  -0.028228404 217.1870 2.183602e-06 206
## 208   2.81300813   0.424766211 217.1870 4.941628e-04 207
## 209  -9.18699187  -1.390835109 217.1870 5.270778e-03 208
## 210   2.81300813   0.424766211 217.1870 4.941628e-04 209
## 211  -9.18699187  -1.390835109 217.1870 5.270778e-03 210
## 212   6.81300813   1.030107729 217.1870 2.898718e-03 211
## 213  -9.18699187  -1.390835109 217.1870 5.270778e-03 212
## 214   3.81300813   0.575895638 217.1870 9.079534e-04 213
## 215  -3.18699187  -0.481274446 217.1870 6.342931e-04 214
## 216  13.81300813   2.098760725 217.1870 1.191529e-02 215
## 217   1.81300813   0.273722799 217.1870 2.052710e-04 216
## 218  12.81300813   1.945051998 217.1870 1.025251e-02 217
## 219  -3.18699187  -0.481274446 217.1870 6.342931e-04 218
## 220  11.81300813   1.791745587 217.1870 8.714633e-03 219
## 221   2.81300813   0.424766211 217.1870 4.941628e-04 220
## 222   5.81300813   0.878535496 217.1870 2.110231e-03 221
## 223  -1.18699187  -0.179197301 217.1870 8.798803e-05 222
## 224   3.81300813   0.575895638 217.1870 9.079534e-04 223
## 225   3.81300813   0.575895638 217.1870 9.079534e-04 224
## 226  -0.18699187  -0.028228404 217.1870 2.183602e-06 225
## 227  -1.18699187  -0.179197301 217.1870 8.798803e-05 226
## 228  12.81300813   1.945051998 217.1870 1.025251e-02 227
## 229  -8.18699187  -1.238713865 217.1870 4.185784e-03 228
## 230   2.81300813   0.424766211 217.1870 4.941628e-04 229
## 231  -2.18699187  -0.330202454 217.1870 2.986912e-04 230
## 232   5.81300813   0.878535496 217.1870 2.110231e-03 231
## 233  -5.18699187  -0.783741645 217.1870 1.680193e-03 232
## 234   3.81300813   0.575895638 217.1870 9.079534e-04 233
## 235  -5.18699187  -0.783741645 217.1870 1.680193e-03 234
## 236   6.81300813   1.030107729 217.1870 2.898718e-03 235
## 237  -5.18699187  -0.783741645 217.1870 1.680193e-03 236
## 238  10.81300813   1.638808541 217.1870 7.301652e-03 237
## 239   0.81300813   0.122734782 217.1870 4.127793e-05 238
## 240   0.81300813   0.122734782 217.1870 4.127793e-05 239
## 241  -5.18699187  -0.783741645 217.1870 1.680193e-03 240
## 242  12.81300813   1.945051998 217.1870 1.025251e-02 241
## 243   0.81300813   0.122734782 217.1870 4.127793e-05 242
## 244  10.81300813   1.638808541 217.1870 7.301652e-03 243
## 245  -5.18699187  -0.783741645 217.1870 1.680193e-03 244
## 246   6.81300813   1.030107729 217.1870 2.898718e-03 245
## 247  -3.18699187  -0.481274446 217.1870 6.342931e-04 246
## 248   8.81300813   1.333912631 217.1870 4.850388e-03 247
## 249  -1.18699187  -0.179197301 217.1870 8.798803e-05 248
## 250   4.81300813   0.727141791 217.1870 1.446643e-03 249
## 251 -14.18699187  -2.156355425 217.1870 1.256923e-02 250
## 252   7.81300813   1.181889650 217.1870 3.812103e-03 251
## 253   1.81300813   0.273722799 217.1870 2.052710e-04 252
## 254  10.81300813   1.638808541 217.1870 7.301652e-03 253
## 255  -2.18699187  -0.330202454 217.1870 2.986912e-04 254
## 256  10.81300813   1.638808541 217.1870 7.301652e-03 255
## 257  -1.18699187  -0.179197301 217.1870 8.798803e-05 256
## 258  -2.18699187  -0.330202454 217.1870 2.986912e-04 257
## 259  -7.18699187  -1.086845476 217.1870 3.225688e-03 258
## 260   1.81300813   0.273722799 217.1870 2.052710e-04 259
## 261  -9.18699187  -1.390835109 217.1870 5.270778e-03 260
## 262  -8.18699187  -1.238713865 217.1870 4.185784e-03 261
## 263  -1.18699187  -0.179197301 217.1870 8.798803e-05 262
## 264  11.81300813   1.791745587 217.1870 8.714633e-03 263
## 265  -4.18699187  -0.632443927 217.1870 1.094794e-03 264
## 266  12.81300813   1.945051998 217.1870 1.025251e-02 265
## 267  -0.18699187  -0.028228404 217.1870 2.183602e-06 266
## 268  12.81300813   1.945051998 217.1870 1.025251e-02 267
## 269  -0.18699187  -0.028228404 217.1870 2.183602e-06 268
## 270   4.81300813   0.727141791 217.1870 1.446643e-03 269
## 271  -3.18699187  -0.481274446 217.1870 6.342931e-04 270
## 273  -2.18699187  -0.330202454 217.1870 2.986912e-04 271
## 274   4.81300813   0.727141791 217.1870 1.446643e-03 272
## 275  -5.18699187  -0.783741645 217.1870 1.680193e-03 273
## 276  -4.18699187  -0.632443927 217.1870 1.094794e-03 274
## 277  -3.82352941  -0.579412040 195.8235 1.673520e-03 275
## 278   0.17647059   0.026728852 195.8235 3.564894e-06 276
## 279  -2.82352941  -0.427776924 195.8235 9.126129e-04 277
## 280  -7.82352941  -1.187446937 195.8235 7.006601e-03 278
## 281   1.17647059   0.178200525 195.8235 1.584397e-04 279
## 282   2.17647059   0.329708494 195.8235 5.422600e-04 280
## 283 -17.82352941  -2.729194710 195.8235 3.636548e-02 281
## 284   1.17647059   0.178200525 195.8235 1.584397e-04 282
## 285  -0.82352941  -0.124737379 195.8235 7.763547e-05 283
## 286   2.17647059   0.329708494 195.8235 5.422600e-04 284
## 287  -2.82352941  -0.427776924 195.8235 9.126129e-04 285
## 288  -1.82352941  -0.276229014 195.8235 3.806515e-04 286
## 289 -10.82352941  -1.645924475 195.8235 1.341034e-02 287
## 290   5.17647059   0.784759436 195.8235 3.067393e-03 288
## 291  -5.82352941  -0.883068084 195.8235 3.882170e-03 289
## 292   5.17647059   0.784759436 195.8235 3.067393e-03 290
## 293   1.17647059   0.178200525 195.8235 1.584397e-04 291
## 294 -14.82352941  -2.262153692 195.8235 2.515389e-02 292
## 295  -5.82352941  -0.883068084 195.8235 3.882170e-03 293
## 296  -0.82352941  -0.124737379 195.8235 7.763547e-05 294
## 297 -14.82352941  -2.262153692 195.8235 2.515389e-02 295
## 298  -4.82352941  -0.731165380 195.8235 2.663372e-03 296
## 299  -8.82352941  -1.339986260 195.8235 8.912235e-03 297
## 300  -2.82352941  -0.427776924 195.8235 9.126129e-04 298
## 301  -0.82352941  -0.124737379 195.8235 7.763547e-05 299
## 302   1.17647059   0.178200525 195.8235 1.584397e-04 300
## 303   4.17647059   0.632956942 195.8235 1.996737e-03 301
## 304   4.17647059   0.632956942 195.8235 1.996737e-03 302
## 305  -4.82352941  -0.731165380 195.8235 2.663372e-03 303
## 306   9.17647059   1.393887858 195.8235 9.639473e-03 304
## 307  -8.82352941  -1.339986260 195.8235 8.912235e-03 305
## 308   5.17647059   0.784759436 195.8235 3.067393e-03 306
## 309  -8.82352941  -1.339986260 195.8235 8.912235e-03 307
## 310   7.17647059   1.088876944 195.8235 5.895542e-03 308
## 311  -0.82352941  -0.124737379 195.8235 7.763547e-05 309
## 312   3.17647059   0.481283647 195.8235 1.155026e-03 310
## 313  -0.82352941  -0.124737379 195.8235 7.763547e-05 311
## 314  14.17647059   2.162011704 195.8235 2.300585e-02 312
## 315  -3.82352941  -0.579412040 195.8235 1.673520e-03 313
## 316   9.17647059   1.393887858 195.8235 9.639473e-03 314
## 317  14.17647059   2.162011704 195.8235 2.300585e-02 315
## 318  -8.82352941  -1.339986260 195.8235 8.912235e-03 316
## 319   0.17647059   0.026728852 195.8235 3.564894e-06 317
## 320   0.17647059   0.026728852 195.8235 3.564894e-06 318
## 321   0.17647059   0.026728852 195.8235 3.564894e-06 319
## 322   5.17647059   0.784759436 195.8235 3.067393e-03 320
## 323  -5.82352941  -0.883068084 195.8235 3.882170e-03 321
## 324  16.17647059   2.472194735 195.8235 2.995501e-02 322
## 325  -8.82352941  -1.339986260 195.8235 8.912235e-03 323
## 326   2.17647059   0.329708494 195.8235 5.422600e-04 324
## 327   3.17647059   0.481283647 195.8235 1.155026e-03 325
## 328   5.17647059   0.784759436 195.8235 3.067393e-03 326
## 329  -2.82352941  -0.427776924 195.8235 9.126129e-04 327
## 330   7.17647059   1.088876944 195.8235 5.895542e-03 328
## 331  -8.82352941  -1.339986260 195.8235 8.912235e-03 329
## 332   1.17647059   0.178200525 195.8235 1.584397e-04 330
## 333  -4.82352941  -0.731165380 195.8235 2.663372e-03 331
## 334   7.17647059   1.088876944 195.8235 5.895542e-03 332
## 335   6.17647059   0.936722318 195.8235 4.366995e-03 333
## 336  -1.82352941  -0.276229014 195.8235 3.806515e-04 334
## 337  10.17647059   1.546807973 195.8235 1.185486e-02 335
## 338  -6.82352941  -1.035151443 195.8235 5.329913e-03 336
## 339  -0.82352941  -0.124737379 195.8235 7.763547e-05 337
## 340  11.17647059   1.700047551 195.8235 1.429919e-02 338
## 341   6.17647059   0.936722318 195.8235 4.366995e-03 339
## 342  -2.82352941  -0.427776924 195.8235 9.126129e-04 340
## 343  14.17647059   2.162011704 195.8235 2.300585e-02 341
## 344   2.17647059   0.329708494 195.8235 5.422600e-04 342
```

---
exclude: TRUE
# Homework (Practice) Answers

3\. Check your model diagnostics - Normality

```r
ggplot(data = d, aes(sample = std_residuals)) +
  stat_qq() +
  stat_qq_line()
```

![:box 25%, 75%, 30%](Great)

---
exclude: TRUE
# Homework (Practice) Answers

3\. Check your model diagnostics - Heteroscedasticity

```r
ggplot(d, aes(x = fitted, y = std_residuals)) +
  geom_point() +
  geom_hline(yintercept = 0)
```

![:box 25%, 75%, 30%](Not bad)

---
exclude: TRUE
# Homework (Practice) Answers

3\. Check your model diagnostics - Influence (Cook's d)

```r
ggplot(d, aes(x = obs, y = cooks)) +
  geom_point() +
  geom_hline(yintercept = 4/nrow(penguins), 
             linetype = "dashed")
```

![:box 15%, 75%, 25%](Some interesting observations, but nothing crazy)

---
exclude: TRUE
# Homework (Practice) Answers

4\. Interpret the **summary table**

.small[

```r
summary(m)
```

```
## 
## 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
```
]

![:box 25%, 20%, 40%](Adelie penguins have an average flipper length of 189.95mm)

![:box 25%, 40%, 40%](Chinstrap flippers are on average 5.87mm longer than Adelie penguins (P < 0.0001&rpar;)

![:box 25%, 60%, 40%](Gentoo flippers are on average 27.23mm longer than Adelie penguins (P < 0.0001&rpar;)

---
exclude: TRUE
# Homework (Practice) Answers

5\. Interpret the **ANOVA Table**

.small[

```r
anova(m)
```

```
## Analysis of Variance Table
## 
## Response: flipper_length_mm
## Df Sum Sq Mean Sq F value Pr(>F) 
*## species 2 52473 26236.6 594.8 < 2.2e-16 ***
## Residuals 339 14953 44.1 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]

![:box 50%, 75%, 60%](Flipper length differs significantly (P < 0.0001&rpar; among species But we don't know exactly how)

---
exclude: TRUE
# Homework (Practice) Answers
6\. Create an ANOVA model of flipper length and species and sex

```r
m <- lm(flipper_length_mm ~ species + sex, data = penguins)
```

---
exclude: TRUE
# Homework (Practice) Answers

7\. Check your model diagnostics

```r
d <- data.frame(residuals = residuals(m), 
 std_residuals = rstudent(m), 
 fitted = fitted(m), 
 cooks = cooks.distance(m))

d <- mutate(d, obs = 1:n())
d
```

```
##       residuals std_residuals   fitted        cooks obs
## 1   -12.5278243   -2.21168802 193.5278 1.202679e-02   1
## 2    -0.6776552   -0.11875498 186.6777 3.518972e-05   2
## 3     8.3223448    1.46316112 186.6777 5.307497e-03   3
## 5     6.3223448    1.11000797 186.6777 3.063054e-03   4
## 6    -3.5278243   -0.61857713 193.5278 9.537010e-04   5
## 7    -5.6776552   -0.99645826 186.6777 2.470224e-03   6
## 8     1.4721757    0.25801055 193.5278 1.660801e-04   7
## 13   -4.6776552   -0.82055414 186.6777 1.676698e-03   8
## 14   -2.5278243   -0.44310900 193.5278 4.896572e-04   9
## 15    4.4721757    0.78443976 193.5278 1.532626e-03  10
## 16   -1.6776552   -0.29403137 186.6777 2.156770e-04  11
## 17    8.3223448    1.46316112 186.6777 5.307497e-03  12
## 18    3.4721757    0.60880846 193.5278 9.238506e-04  13
## 19   -2.6776552   -0.46939040 186.6777 5.494240e-04  14
## 20    0.4721757    0.08274503 193.5278 1.708463e-05  15
## 21  -12.6776552   -2.23854117 186.6777 1.231619e-02  16
## 22  -13.5278243   -2.39119205 193.5278 1.402343e-02  17
## 23    2.3223448    0.40707096 186.6777 4.132870e-04  18
## 24   -8.5278243   -1.49953135 193.5278 5.572818e-03  19
## 25  -13.5278243   -2.39119205 193.5278 1.402343e-02  20
## 26    0.3223448    0.05648805 186.6777 7.962323e-06  21
## 27  -10.5278243   -1.85454541 193.5278 8.493283e-03  22
## 28    0.3223448    0.05648805 186.6777 7.962323e-06  23
## 29  -14.6776552   -2.59845420 186.6777 1.650866e-02  24
## 30  -13.5278243   -2.39119205 193.5278 1.402343e-02  25
## 31   -8.6776552   -1.52606303 186.6777 5.770363e-03  26
## 32  -15.5278243   -2.75234251 193.5278 1.847649e-02  27
## 33    1.3223448    0.23174694 186.6777 1.339948e-04  28
## 34   -9.5278243   -1.67679881 193.5278 6.956421e-03  29
## 35    8.3223448    1.46316112 186.6777 5.307497e-03  30
## 36    2.4721757    0.43334857 193.5278 4.683355e-04  31
## 37   -3.5278243   -0.61857713 193.5278 9.537010e-04  32
## 38   -6.6776552   -1.17264400 186.6777 3.417011e-03  33
## 39   -5.6776552   -0.99645826 186.6777 2.470224e-03  34
## 40   -9.5278243   -1.67679881 193.5278 6.956421e-03  35
## 41   -4.6776552   -0.82055414 186.6777 1.676698e-03  36
## 42    1.4721757    0.25801055 193.5278 1.660801e-04  37
## 43   -0.6776552   -0.11875498 186.6777 3.518972e-05  38
## 44    2.4721757    0.43334857 193.5278 4.683355e-04  39
## 45   -1.6776552   -0.29403137 186.6777 2.156770e-04  40
## 46   -3.5278243   -0.61857713 193.5278 9.537010e-04  41
## 47  -11.5278243   -2.03282396 193.5278 1.018341e-02  42
## 49    3.3223448    0.58250944 186.6777 8.458391e-04  43
## 50   -2.5278243   -0.44310900 193.5278 4.896572e-04  44
## 51   -0.6776552   -0.11875498 186.6777 3.518972e-05  45
## 52   -5.5278243   -0.97008568 193.5278 2.341568e-03  46
## 53    3.3223448    0.58250944 186.6777 8.458391e-04  47
## 54    6.4721757    1.13641600 193.5278 3.209955e-03  48
## 55    0.3223448    0.05648805 186.6777 7.962323e-06  49
## 56   -2.5278243   -0.44310900 193.5278 4.896572e-04  50
## 57   -0.6776552   -0.11875498 186.6777 3.518972e-05  51
## 58   -0.5278243   -0.09249724 193.5278 2.134898e-05  52
## 59   -5.6776552   -0.99645826 186.6777 2.470224e-03  53
## 60    0.4721757    0.08274503 193.5278 1.708463e-05  54
## 61   -1.6776552   -0.29403137 186.6777 2.156770e-04  55
## 62    1.4721757    0.25801055 193.5278 1.660801e-04  56
## 63   -1.6776552   -0.29403137 186.6777 2.156770e-04  57
## 64   -1.5278243   -0.26776549 193.5278 1.788732e-04  58
## 65   -2.6776552   -0.46939040 186.6777 5.494240e-04  59
## 66   -1.5278243   -0.26776549 193.5278 1.788732e-04  60
## 67    8.3223448    1.46316112 186.6777 5.307497e-03  61
## 68   -5.5278243   -0.97008568 193.5278 2.341568e-03  62
## 69    3.3223448    0.58250944 186.6777 8.458391e-04  63
## 70    4.4721757    0.78443976 193.5278 1.532626e-03  64
## 71    3.3223448    0.58250944 186.6777 8.458391e-04  65
## 72   -3.5278243   -0.61857713 193.5278 9.537010e-04  66
## 73    9.3223448    1.64033660 186.6777 6.659608e-03  67
## 74    3.4721757    0.60880846 193.5278 9.238506e-04  68
## 75    3.3223448    0.58250944 186.6777 8.458391e-04  69
## 76    1.4721757    0.25801055 193.5278 1.660801e-04  70
## 77    4.3223448    0.75811190 186.6777 1.431651e-03  71
## 78   -9.5278243   -1.67679881 193.5278 6.956421e-03  72
## 79    0.3223448    0.05648805 186.6777 7.962323e-06  73
## 80    1.4721757    0.25801055 193.5278 1.660801e-04  74
## 81    2.3223448    0.40707096 186.6777 4.132870e-04  75
## 82    2.4721757    0.43334857 193.5278 4.683355e-04  76
## 83    0.3223448    0.05648805 186.6777 7.962323e-06  77
## 84   -0.5278243   -0.09249724 193.5278 2.134898e-05  78
## 85    4.3223448    0.75811190 186.6777 1.431651e-03  79
## 86    0.4721757    0.08274503 193.5278 1.708463e-05  80
## 87   -3.5278243   -0.61857713 193.5278 9.537010e-04  81
## 88    2.3223448    0.40707096 186.6777 4.132870e-04  82
## 89   -4.5278243   -0.79421942 193.5278 1.571005e-03  83
## 90    3.3223448    0.58250944 186.6777 8.458391e-04  84
## 91   15.3223448    2.71509662 186.6777 1.799073e-02  85
## 92   11.4721757    2.02288813 193.5278 1.008533e-02  86
## 93   -1.6776552   -0.29403137 186.6777 2.156770e-04  87
## 94   -7.5278243   -1.32269092 193.5278 4.342475e-03  88
## 95    0.3223448    0.05648805 186.6777 7.962323e-06  89
## 96   14.4721757    2.56134434 193.5278 1.604967e-02  90
## 97    3.3223448    0.58250944 186.6777 8.458391e-04  91
## 98    2.4721757    0.43334857 193.5278 4.683355e-04  92
## 99   -8.6776552   -1.52606303 186.6777 5.770363e-03  93
## 100  -1.5278243   -0.26776549 193.5278 1.788732e-04  94
## 101   5.3223448    0.93392806 186.6777 2.170723e-03  95
## 102   9.4721757    1.66692203 193.5278 6.875398e-03  96
## 103  -3.6776552   -0.64488146 186.6777 1.036431e-03  97
## 104  -3.5278243   -0.61857713 193.5278 9.537010e-04  98
## 105   6.3223448    1.11000797 186.6777 3.063054e-03  99
## 106  -9.5278243   -1.67679881 193.5278 6.956421e-03 100
## 107  12.3223448    2.17488469 186.6777 1.163550e-02 101
## 108  -3.5278243   -0.61857713 193.5278 9.537010e-04 102
## 109  -5.6776552   -0.99645826 186.6777 2.470224e-03 103
## 110   3.4721757    0.60880846 193.5278 9.238506e-04 104
## 111  11.3223448    1.99614536 186.6777 9.823610e-03 105
## 112  -2.5278243   -0.44310900 193.5278 4.896572e-04 106
## 113   6.3223448    1.11000797 186.6777 3.063054e-03 107
## 114   3.4721757    0.60880846 193.5278 9.238506e-04 108
## 115   4.3223448    0.75811190 186.6777 1.431651e-03 109
## 116   2.4721757    0.43334857 193.5278 4.683355e-04 110
## 117   1.3223448    0.23174694 186.6777 1.339948e-04 111
## 118   5.4721757    0.96029224 193.5278 2.294660e-03 112
## 119   2.3223448    0.40707096 186.6777 4.132870e-04 113
## 120  -4.5278243   -0.79421942 193.5278 1.571005e-03 114
## 121   0.3223448    0.05648805 186.6777 7.962323e-06 115
## 122   4.4721757    0.78443976 193.5278 1.532626e-03 116
## 123 -10.6776552   -1.88122187 186.6777 8.736755e-03 117
## 124   8.4721757    1.48967969 193.5278 5.500324e-03 118
## 125  -0.6776552   -0.11875498 186.6777 3.518972e-05 119
## 126   5.4721757    0.96029224 193.5278 2.294660e-03 120
## 127   4.3223448    0.75811190 186.6777 1.431651e-03 121
## 128   1.4721757    0.25801055 193.5278 1.660801e-04 122
## 129   4.3223448    0.75811190 186.6777 1.431651e-03 123
## 130  16.4721757    2.92396594 193.5278 2.079219e-02 124
## 131   3.3223448    0.58250944 186.6777 8.458391e-04 125
## 132   3.4721757    0.60880846 193.5278 9.238506e-04 126
## 133   6.3223448    1.11000797 186.6777 3.063054e-03 127
## 134   5.4721757    0.96029224 193.5278 2.294660e-03 128
## 135   0.3223448    0.05648805 186.6777 7.962323e-06 129
## 136  -3.5278243   -0.61857713 193.5278 9.537010e-04 130
## 137   4.3223448    0.75811190 186.6777 1.431651e-03 131
## 138   6.4721757    1.13641600 193.5278 3.209955e-03 132
## 139  -1.6776552   -0.29403137 186.6777 2.156770e-04 133
## 140  -0.5278243   -0.09249724 193.5278 2.134898e-05 134
## 141   6.3223448    1.11000797 186.6777 3.063054e-03 135
## 142  -6.5278243   -1.14622602 193.5278 3.265392e-03 136
## 143   1.3223448    0.23174694 186.6777 1.339948e-04 137
## 144  -3.5278243   -0.61857713 193.5278 9.537010e-04 138
## 145   5.3223448    0.93392806 186.6777 2.170723e-03 139
## 146  -8.5278243   -1.49953135 193.5278 5.572818e-03 140
## 147  -3.5278243   -0.61857713 193.5278 9.537010e-04 141
## 148  -2.6776552   -0.46939040 186.6777 5.494240e-04 142
## 149   8.3223448    1.46316112 186.6777 5.307497e-03 143
## 150  -0.5278243   -0.09249724 193.5278 2.134898e-05 144
## 151   0.3223448    0.05648805 186.6777 7.962323e-06 145
## 152   7.4721757    1.31286151 193.5278 4.278510e-03 146
## 153  -2.7238629   -0.47790867 213.7239 6.693794e-04 147
## 154   9.4259680    1.65990955 220.5740 7.801122e-03 148
## 155  -3.7238629   -0.65355898 213.7239 1.251092e-03 149
## 156  -2.5740320   -0.45153442 220.5740 5.817455e-04 150
## 157  -5.5740320   -0.97891486 220.5740 2.727998e-03 151
## 158  -3.7238629   -0.65355898 213.7239 1.251092e-03 152
## 159  -2.7238629   -0.47790867 213.7239 6.693794e-04 153
## 160  -1.5740320   -0.27606159 220.5740 2.175366e-04 154
## 161  -4.7238629   -0.82939369 213.7239 2.013244e-03 155
## 162  -5.5740320   -0.97891486 220.5740 2.727998e-03 156
## 163   0.2761371    0.04843227 213.7239 6.879414e-06 157
## 164  -4.5740320   -0.80291126 220.5740 1.836976e-03 158
## 165   0.2761371    0.04843227 213.7239 6.879414e-06 159
## 166  -7.5740320   -1.33180365 220.5740 5.036855e-03 160
## 167  -3.7238629   -0.65355898 213.7239 1.251092e-03 161
## 168  -3.5740320   -0.62713442 220.5740 1.121559e-03 162
## 169  -3.7238629   -0.65355898 213.7239 1.251092e-03 163
## 170   0.4259680    0.07470035 220.5740 1.593159e-05 164
## 171  -4.7238629   -0.82939369 213.7239 2.013244e-03 165
## 172   1.4259680    0.25008818 220.5740 1.785355e-04 166
## 173  -2.5740320   -0.45153442 220.5740 5.817455e-04 167
## 174   1.2761371    0.22384068 213.7239 1.469253e-04 168
## 175  -0.7238629   -0.12696251 213.7239 4.727315e-05 169
## 176  -5.5740320   -0.97891486 220.5740 2.727998e-03 170
## 177   1.2761371    0.22384068 213.7239 1.469253e-04 171
## 178  -5.5740320   -0.97891486 220.5740 2.727998e-03 172
## 180  -5.5740320   -0.97891486 220.5740 2.727998e-03 173
## 181  -3.7238629   -0.65355898 213.7239 1.251092e-03 174
## 182  -0.5740320   -0.10066646 220.5740 2.893193e-05 175
## 183   1.4259680    0.25008818 220.5740 1.785355e-04 176
## 184  -4.7238629   -0.82939369 213.7239 2.013244e-03 177
## 185  -6.7238629   -1.18181651 213.7239 4.078867e-03 178
## 186   9.4259680    1.65990955 220.5740 7.801122e-03 179
## 187   6.2761371    1.10281996 213.7239 3.553749e-03 180
## 188  -0.5740320   -0.10066646 220.5740 2.893193e-05 181
## 189  -0.7238629   -0.12696251 213.7239 4.727315e-05 182
## 190  -1.5740320   -0.27606159 220.5740 2.175366e-04 183
## 191  -5.7238629   -1.00546276 213.7239 2.955836e-03 184
## 192 -12.5740320   -2.22156785 220.5740 1.388207e-02 185
## 193  -5.7238629   -1.00546276 213.7239 2.955836e-03 186
## 194   4.4259680    0.77687195 220.5740 1.719973e-03 187
## 195  -3.7238629   -0.65355898 213.7239 1.251092e-03 188
## 196  -4.5740320   -0.80291126 220.5740 1.836976e-03 189
## 197   1.4259680    0.25008818 220.5740 1.785355e-04 190
## 198   3.2761371    0.57489598 213.7239 9.683359e-04 191
## 199  -3.7238629   -0.65355898 213.7239 1.251092e-03 192
## 200   4.4259680    0.77687195 220.5740 1.719973e-03 193
## 201  -0.7238629   -0.12696251 213.7239 4.727315e-05 194
## 202  -5.5740320   -0.97891486 220.5740 2.727998e-03 195
## 203  -3.7238629   -0.65355898 213.7239 1.251092e-03 196
## 204  -0.5740320   -0.10066646 220.5740 2.893193e-05 197
## 205  -3.7238629   -0.65355898 213.7239 1.251092e-03 198
## 206   4.4259680    0.77687195 220.5740 1.719973e-03 199
## 207   3.2761371    0.57489598 213.7239 9.683359e-04 200
## 208  -0.5740320   -0.10066646 220.5740 2.893193e-05 201
## 209  -5.7238629   -1.00546276 213.7239 2.955836e-03 202
## 210  -0.5740320   -0.10066646 220.5740 2.893193e-05 203
## 211  -5.7238629   -1.00546276 213.7239 2.955836e-03 204
## 212   3.4259680    0.60112443 220.5740 1.030556e-03 205
## 213  -5.7238629   -1.00546276 213.7239 2.955836e-03 206
## 214   0.4259680    0.07470035 220.5740 1.593159e-05 207
## 215   0.2761371    0.04843227 213.7239 6.879414e-06 208
## 216  10.4259680    1.83773464 220.5740 9.544164e-03 209
## 217   5.2761371    0.92659991 213.7239 2.511505e-03 210
## 218   9.4259680    1.65990955 220.5740 7.801122e-03 211
## 220   8.4259680    1.48255929 220.5740 6.233683e-03 212
## 221   6.2761371    1.10281996 213.7239 3.553749e-03 213
## 222   2.4259680    0.42554638 220.5740 5.167438e-04 214
## 223   2.2761371    0.39931210 213.7239 4.674108e-04 215
## 224   0.4259680    0.07470035 220.5740 1.593159e-05 216
## 225   0.4259680    0.07470035 220.5740 1.593159e-05 217
## 226   3.2761371    0.57489598 213.7239 9.683359e-04 218
## 227   2.2761371    0.39931210 213.7239 4.674108e-04 219
## 228   9.4259680    1.65990955 220.5740 7.801122e-03 220
## 229  -4.7238629   -0.82939369 213.7239 2.013244e-03 221
## 230  -0.5740320   -0.10066646 220.5740 2.893193e-05 222
## 231   1.2761371    0.22384068 213.7239 1.469253e-04 223
## 232   2.4259680    0.42554638 220.5740 5.167438e-04 224
## 233  -1.7238629   -0.30239302 213.7239 2.681065e-04 225
## 234   0.4259680    0.07470035 220.5740 1.593159e-05 226
## 235  -1.7238629   -0.30239302 213.7239 2.681065e-04 227
## 236   3.4259680    0.60112443 220.5740 1.030556e-03 228
## 237  -1.7238629   -0.30239302 213.7239 2.681065e-04 229
## 238   7.4259680    1.30563169 220.5740 4.841849e-03 230
## 239   4.2761371    0.75064197 213.7239 1.649701e-03 231
## 240  -2.5740320   -0.45153442 220.5740 5.817455e-04 232
## 241  -1.7238629   -0.30239302 213.7239 2.681065e-04 233
## 242   9.4259680    1.65990955 220.5740 7.801122e-03 234
## 243   4.2761371    0.75064197 213.7239 1.649701e-03 235
## 244   7.4259680    1.30563169 220.5740 4.841849e-03 236
## 245  -1.7238629   -0.30239302 213.7239 2.681065e-04 237
## 246   3.4259680    0.60112443 220.5740 1.030556e-03 238
## 247   0.2761371    0.04843227 213.7239 6.879414e-06 239
## 248   5.4259680    0.95283881 220.5740 2.584994e-03 240
## 249  -4.5740320   -0.80291126 220.5740 1.836976e-03 241
## 250   8.2761371    1.45624893 213.7239 6.179555e-03 242
## 251 -10.7238629   -1.89110027 213.7239 1.037539e-02 243
## 252   4.4259680    0.77687195 220.5740 1.719973e-03 244
## 253   5.2761371    0.92659991 213.7239 2.511505e-03 245
## 254   7.4259680    1.30563169 220.5740 4.841849e-03 246
## 255   1.2761371    0.22384068 213.7239 1.469253e-04 247
## 256   7.4259680    1.30563169 220.5740 4.841849e-03 248
## 258  -5.5740320   -0.97891486 220.5740 2.727998e-03 249
## 259  -3.7238629   -0.65355898 213.7239 1.251092e-03 250
## 260  -1.5740320   -0.27606159 220.5740 2.175366e-04 251
## 261  -5.7238629   -1.00546276 213.7239 2.955836e-03 252
## 262 -11.5740320   -2.04254313 220.5740 1.176182e-02 253
## 263   2.2761371    0.39931210 213.7239 4.674108e-04 254
## 264   8.4259680    1.48255929 220.5740 6.233683e-03 255
## 265  -0.7238629   -0.12696251 213.7239 4.727315e-05 256
## 266   9.4259680    1.65990955 220.5740 7.801122e-03 257
## 267   3.2761371    0.57489598 213.7239 9.683359e-04 258
## 268   9.4259680    1.65990955 220.5740 7.801122e-03 259
## 270   1.4259680    0.25008818 220.5740 1.785355e-04 260
## 271   0.2761371    0.04843227 213.7239 6.879414e-06 261
## 273   1.2761371    0.22384068 213.7239 1.469253e-04 262
## 274   1.4259680    0.25008818 220.5740 1.785355e-04 263
## 275  -1.7238629   -0.30239302 213.7239 2.681065e-04 264
## 276  -7.5740320   -1.33180365 220.5740 5.036855e-03 265
## 277  -0.3984449   -0.07010276 192.3984 2.221743e-05 266
## 278  -3.2486140   -0.57184471 199.2486 1.476908e-03 267
## 279  -6.2486140   -1.10140984 199.2486 5.464177e-03 268
## 280  -4.3984449   -0.77456809 192.3984 2.707420e-03 269
## 281  -2.2486140   -0.39571469 199.2486 7.075989e-04 270
## 282   5.6015551    0.98699786 192.3984 4.391113e-03 271
## 283 -14.3984449   -2.55840828 192.3984 2.901273e-02 272
## 284  -2.2486140   -0.39571469 199.2486 7.075989e-04 273
## 285   2.6015551    0.45786284 192.3984 9.471602e-04 274
## 286  -1.2486140   -0.21969682 199.2486 2.181794e-04 275
## 287   0.6015551    0.10583918 192.3984 5.064170e-05 276
## 288  -5.2486140   -0.92464182 199.2486 3.855197e-03 277
## 289  -7.3984449   -1.30505466 192.3984 7.660167e-03 278
## 290   1.7513860    0.30818278 199.2486 4.292607e-04 279
## 291  -2.3984449   -0.42209604 192.3984 8.050388e-04 280
## 292   1.7513860    0.30818278 199.2486 4.292607e-04 281
## 293  -2.2486140   -0.39571469 199.2486 7.075989e-04 282
## 294 -11.3984449   -2.01784702 192.3984 1.818229e-02 283
## 295  -2.3984449   -0.42209604 192.3984 8.050388e-04 284
## 296  -4.2486140   -0.74813698 199.2486 2.526108e-03 285
## 297 -11.3984449   -2.01784702 192.3984 1.818229e-02 286
## 298  -8.2486140   -1.45593987 199.2486 9.521805e-03 287
## 299  -5.3984449   -0.95110912 192.3984 4.078446e-03 288
## 300  -6.2486140   -1.10140984 199.2486 5.464177e-03 289
## 301   2.6015551    0.45786284 192.3984 9.471602e-04 290
## 302  -2.2486140   -0.39571469 199.2486 7.075989e-04 291
## 303   7.6015551    1.34107627 192.3984 8.086531e-03 292
## 304   0.7513860    0.13220209 199.2486 7.901028e-05 293
## 305  -1.3984449   -0.24606455 192.3984 2.736831e-04 294
## 306   5.7513860    1.01347976 199.2486 4.629162e-03 295
## 307  -5.3984449   -0.95110912 192.3984 4.078446e-03 296
## 308   1.7513860    0.30818278 199.2486 4.292607e-04 297
## 309  -5.3984449   -0.95110912 192.3984 4.078446e-03 298
## 310   3.7513860    0.66045612 199.2486 1.969431e-03 299
## 311  -4.2486140   -0.74813698 199.2486 2.526108e-03 300
## 312   6.6015551    1.16387092 192.3984 6.098877e-03 301
## 313   2.6015551    0.45786284 192.3984 9.471602e-04 302
## 314  10.7513860    1.90199748 199.2486 1.617656e-02 303
## 315  -0.3984449   -0.07010276 192.3984 2.221743e-05 304
## 316   5.7513860    1.01347976 199.2486 4.629162e-03 305
## 317  10.7513860    1.90199748 199.2486 1.617656e-02 306
## 318  -5.3984449   -0.95110912 192.3984 4.078446e-03 307
## 319  -3.2486140   -0.57184471 199.2486 1.476908e-03 308
## 320   3.6015551    0.63404440 192.3984 1.815254e-03 309
## 321   3.6015551    0.63404440 192.3984 1.815254e-03 310
## 322   1.7513860    0.30818278 199.2486 4.292607e-04 311
## 323  -2.3984449   -0.42209604 192.3984 8.050388e-04 312
## 324  12.7513860    2.26088764 199.2486 2.275475e-02 313
## 325 -12.2486140   -2.17043654 199.2486 2.099574e-02 314
## 326   5.6015551    0.98699786 192.3984 4.391113e-03 315
## 327   6.6015551    1.16387092 192.3984 6.098877e-03 316
## 328   1.7513860    0.30818278 199.2486 4.292607e-04 317
## 329   0.6015551    0.10583918 192.3984 5.064170e-05 318
## 330   3.7513860    0.66045612 199.2486 1.969431e-03 319
## 331  -5.3984449   -0.95110912 192.3984 4.078446e-03 320
## 332  -2.2486140   -0.39571469 199.2486 7.075989e-04 321
## 333  -1.3984449   -0.24606455 192.3984 2.736831e-04 322
## 334   3.7513860    0.66045612 199.2486 1.969431e-03 323
## 335   2.7513860    0.48425079 199.2486 1.059401e-03 324
## 336   1.6015551    0.28181110 192.3984 3.589559e-04 325
## 337   6.7513860    1.19039934 199.2486 6.378862e-03 326
## 338  -3.3984449   -0.59824718 192.3984 1.616284e-03 327
## 339   2.6015551    0.45786284 192.3984 9.471602e-04 328
## 340   7.7513860    1.36765891 199.2486 8.408453e-03 329
## 341   9.6015551    1.69669088 192.3984 1.290151e-02 330
## 342  -6.2486140   -1.10140984 199.2486 5.464177e-03 331
## 343  10.7513860    1.90199748 199.2486 1.617656e-02 332
## 344   5.6015551    0.98699786 192.3984 4.391113e-03 333
```

---
exclude: TRUE
# Homework (Practice) Answers

7\. Check your model diagnostics - Normality

```r
ggplot(data = d, aes(sample = std_residuals)) +
  stat_qq() +
  stat_qq_line()
```

![:box 25%, 75%, 30%](No problem)

---
exclude: TRUE
# Homework (Practice) Answers

7\. Check your model diagnostics - Heteroscedasticity

```r
ggplot(d, aes(x = fitted, y = std_residuals)) +
  geom_point() +
  geom_hline(yintercept = 0)
```

![:box 25%, 25%, 30%](Not bad, perhaps a smidge of funnelling but I wouldn't worry about it)

---
exclude: TRUE
# Homework (Practice) Answers

7\. Check your model diagnostics - Influence (Cook's d)

```r
ggplot(d, aes(x = obs, y = cooks)) +
  geom_point() +
  geom_hline(yintercept = 4/nrow(penguins), 
             linetype = "dashed")
```

![:box 15%, 75%, 25%](Some interesting observations, but nothing crazy)

---
exclude: TRUE
# Homework (Practice) Answers

8\. Interpret the **ANOVA Table**

.small[

```r
anova(m)
```

```
## Analysis of Variance Table
## 
## Response: flipper_length_mm
## Df Sum Sq Mean Sq F value Pr(>F) 
*## species 2 50526 25262.9 770.50 < 2.2e-16 ***
*## sex 1 3906 3905.6 119.12 < 2.2e-16 ***
## Residuals 329 10787 32.8 
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]

![:box 50%, 75%, 90%](Flipper length differs significantly among species (P < 0.0001&rpar; and between sexes (P < 0.0001&rpar; We don't know exactly how they differ among species, but looking at our boxplots, we know that male penguins have longer flippers than female penguins)