• Simple Linear Regression is used to model the expected relationship between two variables
• The relationship is modeled by fitting a linear line to to the Y vs X data
• Simple Linear Regression is often used to predict data and understand relationships between two variables
• Because of its simplicity compared to other relationship modeling techniques it is a fundamental and often used technique

• The goal is to find a linear equation, where \(\hat{y}\) is the the estimated value of \(y\), \(x\) is the independent variable, \(m\) is the slope, and \(b\) is the y-intercept, where the sum of the squares of \(\hat{y} - y\) for all observed \(x\) values is minimized (Least Squares Model)

\(\hat{y} = mx + b\)

\(b = \bar{y} - m\bar{x}\)

\(m = {\sum(x_i-\bar{x})(y_i-\bar{y}) \over \sum(x_i-\bar{x})^2}\)

• Lets do a walk through on the following data set Salary vs Age

##   Salary Age
## 1  50000  23
## 2  80000  28
## 3 110000  36

• First lets find \(m\), the slope of the linear regression model where \(\bar{x} = 29\) and \(\bar{y} = 80000\)

\(m = {\sum(x_i-\bar{x})(y_i-\bar{y}) \over \sum(x_i-\bar{x})^2}\)

\(= {(23 - 29)(50000-80000) + (28-29)(80000-80000)+(36-29)(110000 - 80000) \over (23 - 29)^2+(28-29)^2+(36-29)^2}\)

\(= {-30000+0+210000 \over 36+1+49} = {180000 \over 86} = 2093\)

• cont. next page

• Next lets find \(b\), the y-intercept of the linear regression model, where \(\bar{x} = 29\) and \(\bar{y} = 80000\) and \(m= 2093\)

\(b = \bar{y} - m\bar{x} = 80000 - 2093(29) = 19303\)

• Now that we have found \(b\) and \(m\) we can format our equation for simple linear regression

\(\hat{y} = 2093x + 19303\)

• In this slide and the Next we Will Use ggplot2 library to Model Linear Regression
• We will use a built in data set, mpg, and model Highway MPG vs Engine Displacement

library(ggplot2)
g <- ggplot(data = mpg, aes(x = displ, y = hwy)) + 
  geom_point() +
  geom_smooth(method ="lm", se = FALSE) +
  xlab("Engine Displacement (Liters)") +
  ylab("Highway MPG (Miles/Gallon)")
print(g)

• First we import the ggplot2 library for use
• Next we create a ggplot object and set the data, x axis, and y axis
• Then we use geom_point() to add the data points on the graph, creating a scatter plot
• After that, geom_smooth() adds a line using the lm method (linear regression)
• Finally we add axis labels and display the plot

• In this Slide we will spice up a simple linear regression plot by adding reference to other variables using color and point shape which can give more insight into trends in the data

library(ggplot2)
g <- ggplot(data = mpg, aes(x = displ, y = hwy)) + 
  geom_point(aes(color = cty, shape = drv)) +
  geom_smooth(method ="lm", se = FALSE) +
  xlab("Engine Displacement (Liters)") +
  ylab("Highway MPG (Miles/Gallon)") +
  scale_color_gradient(low = "red", high = "blue")
print(g)

• The largest difference in the code behind this plot is in the geom_point() function
• This time we specify color as a function of cty values and shape as a function of drv values
• We also specify the color scale using the scale_color_gradient() function
• This graph provides insight into the City MPG and Drive Train data

• So far we have only looked at Simple linear regression with ggplot, there are other ways to do it in R, another common one being using the plotly library
• In the next slide we will show an example of multiple linear regression implemented using the plotly library

library(plotly)
library(dplyr)
data(Orange)
mod <- lm(Orange$circumference ~ Orange$age)

xax <- list(title = "Age of Tree", titlefont = list(family="Modern COmputer Roman"))
yax <- list(title = "Circumference of Tree", titlefont = list(family="Modern COmputer Roman"))

fig <- plot_ly(x = Orange$age, y = Orange$circumference, type = 'scatter', mode = 'markers', name = 'Circumference VS. Age') %>%
  add_lines(x = Orange$age, y = fitted(mod), name = 'Linear Regression Line') %>%
  layout(xaxis = xax, yaxis = yax, title = 'Tree Circumference VS. Age')
print(fig)

• First we start by defining the linear regression formula using lm()
• Next we setup the formatting for the X and Y axis
• Then we plot the scatter plot by calling plot_ly and declaring the type to scatter as well as the point values
• Later we call the add_lines() function to add the linear regression line on top of the scatter plot
• Finally we add the xaxis and yaxis that we prevously defined and print the plot