• Simple Linear Regression (SLR) is a type of Linear Regression model that uses one independent variable (input) to predict a target value (dependent variable) while assuming a staight-line relationship between the two.
• This presentation will be using the built-in base R data set “cars” and “penguins” as examples.

• First we’ll demonstrate SLR with the data ser ‘cars’.
• Data given for the speed of cars and the distances taken to stop.
• Note that the data were recorded in the 1920s.

##   speed dist
## 1     4    2
## 2     4   10
## 3     7    4
## 4     7   22
## 5     8   16
## 6     9   10

• Here is the scatter plot for the data set ‘cars’
• As you can see, you can tell there’s a direct relationship between car speed and breaking distance right away.

• \(y\) is the dependent variable (output).
• \(x\) is the independent variable (input).
• \(m\) is the slope of the line.
• \(b\) is the intercept (when x is zero). \[ \begin{aligned} y = m x + b \end{aligned} \]

• \(y\) is the dependent variable (predicted value).
• \(\beta_0\) is the \(y\)-intercept when \(x\) is zero.
• \(\beta_1\) is slope
  • Positive value indicates a direct relationship.
  • Negative value indicates an inverse relationship.
• \(x\) is the independent variable.
• \(\epsilon\) is the error of the estimate. \[ \begin{aligned} y = \beta_0 + \beta_1 x + \epsilon \end{aligned} \]

• With the above equation, we have drawn the Simple Line Regression for the data set ‘cars’

• Code in ggplot showing how Single Line Regression is created

ggplot(data = cars, aes(x = speed, y = dist)) +
  geom_point() +
  geom_smooth(method=lm, se=FALSE)

• Next we’ll be going over the data set ‘penguins’ to see if there’s a relationship between bill length and bill depth.
• Data on adult penguins covering three species found on three islands in the Palmer Archipelago, Antarctica, including their size (flipper length, body mass, bill dimensions), and sex.

## # A tibble: 6 × 7
##   species island    bill_length_mm bill_depth_mm flipper_length_mm body_mass_g
##   <fct>   <fct>              <dbl>         <dbl>             <int>       <int>
## 1 Adelie  Torgersen           39.1          18.7               181        3750
## 2 Adelie  Torgersen           39.5          17.4               186        3800
## 3 Adelie  Torgersen           40.3          18                 195        3250
## 4 Adelie  Torgersen           NA            NA                  NA          NA
## 5 Adelie  Torgersen           36.7          19.3               193        3450
## 6 Adelie  Torgersen           39.3          20.6               190        3650
## # ℹ 1 more variable: sex <fct>

1. Linearity
2. Independence of Errors
3. Normally Distributed
4. Equal Variances

• Using the data set ‘penguins’ we’ll demonstrate what happens when assumptions are incorrect for a data set.
• We’ll begin with a 3D scatter plot to see if there are any relationships between bill dimensions, flipper length, and species.
• From a glance, it looks like there might be something possible.
• When trying to predict bill depth taken from bill length, the values do are not in the same range as known values as seen in the next slides.