The formula for simple linear regression is:

\(y = \beta_0 + \beta_1 x\)

where \(y\) is the dependent variable, \(\beta_0\) is the constant or intercept, \(\beta_1\) is \(x\)’s slope, and \(x\) is the independent variable. Linear regression tells us the value of unknown data by using known data values.

Here we show the relationship between the girth and volume of trees from the trees data set.

## `geom_smooth()` using formula = 'y ~ x'

library(ggplot2)
g <- ggplot(data = trees, aes(x = Girth, y = Volume)) + geom_point()
g + geom_smooth(method = "lm")

## `geom_smooth()` using formula = 'y ~ x'

model <- lm(trees$Volume ~ trees$Girth)
coef(model)

## (Intercept) trees$Girth 
##  -36.943459    5.065856

This tells us that our formula is volume \(= -32.94 + 5.07 \times\) girth. So this can be written as: \(y = -32.94 + 5.07 \times x\)

## `geom_smooth()` using formula = 'y ~ x'

model2 <- lm(mtcars$mpg ~ mtcars$wt)
coef(model2)

## (Intercept)   mtcars$wt 
##   37.285126   -5.344472

This tells us that our formula is MPG = \(37.29 - 5.34 \times\) weight. So this can be written as: \(y = 37.29 - 5.34 \times x\)

We can find the correlation coefficient between 2 variables by using cor().

cor(trees$Volume, trees$Girth)

## [1] 0.9671194

This is near perfect positive correlation since this number is very close to +1.

cor(mtcars$mpg, mtcars$wt)

## [1] -0.8676594

This is very close to -1, which is perfect negative correlation. Both of these values are not close to 0, which suggests a strong relationship between variables.