Simple linear regression is a statistical method used to estimate the relationship between two quantitative variables.
By using simple linear regression, we can create an equation that predicts the value of the independent variable based on the dependent variable we input.
\(y=\alpha+ \beta x\), where
\(y\) = y-coordinate
\(\alpha\) = y-intercept
\(\beta\) = slope
\(x\) = x-coordinate
In this lesson, we will be using a built in R data set, “faithful”, to easily demonstrate simple linear regression.
Here, we can see that the data set provides eruption durations and waiting times for Old Faithful.
How can we use simple linear regression to predict the waiting times based on the eruption duration?
## eruptions waiting ## 1 3.600 79 ## 2 1.800 54 ## 3 3.333 74 ## 4 2.283 62 ## 5 4.533 85 ## 6 2.883 55
There is a positive linear relationship, which seems relatively constant. This would make a good instance to use simple linear regression.
faithfulGraph = ggplot(faithful, aes(x=eruptions, y = waiting)) +
geom_smooth(method="lm", se = FALSE) +
geom_point() +
labs(title = "Eruption Duration and Waiting Times",
x = "Eruption Duration", y = "Waiting Times")
## `geom_smooth()` using formula = 'y ~ x'
How can get the actual equation?
equation = lm(waiting ~ eruptions, faithful) equation
## ## Call: ## lm(formula = waiting ~ eruptions, data = faithful) ## ## Coefficients: ## (Intercept) eruptions ## 33.47 10.73
As previously mentioned, \(y=\alpha+ \beta x\)
Therefore, for the old faithful data,
\(y=33.47+ 10.73 x\)
\(y=33.47+ 10.73 x\)