Statistical Inference via Data Science A ModernDive into R and the Tidyverse (Second Edition)

We have introduced data visualization in Chapter 2, data wrangling in Chapter 3, and data importing and “tidy” data in Chapter 4. In this chapter, we work with regression, a method that helps us study the relationship between an outcome variable or response and one or more explanatory variables or regressors. The method starts by proposing a statistical model. Data is then collected and used to estimate the coefficients or parameters for the model, and these results are typically used for two purposes:

As an illustration, assume that we want to study the relationship between blood pressure and potential risk factors such as daily salt intake, age, and physical activity levels. The response is blood pressure, and the regressors are the risk factors. If we use linear regression for explanation, we may want to determine whether reducing daily salt intake has a real effect on lowering blood pressure, or by how much blood pressure decreases if an individual reduces their salt intake by half. This information may help target individuals of a specific age group with advice on dietary changes to manage blood pressure. On the other hand, if we use linear regression for prediction, we would like to determine, as accurately as possible, the blood pressure of a given individual based on the data collected about their salt intake, age, and physical activity levels. In this chapter, we will use linear regression for explanation.

The most basic and commonly-used type of regression is linear regression. Linear regression involves a numerical response and one or more regressors that can be numerical or categorical. It is called linear regression because the statistical model that describes the relationship between the expected response and the regressors is assumed to be linear. In particular, when the model has a single regressor, the linear regression is the equation of a line. Linear regression is the foundation for almost any other type of regression or related method.

In Chapter 5, we introduce linear regression with only one regressor. In Section 5.1, the explanatory variable is numerical. This scenario is known as simple linear regression. In Section 5.2, the explanatory variable is categorical.

In Chapter 6 on multiple regression, we extend these ideas and work with models with two explanatory variables. In that chapter, we work with two numerical explanatory variables as well as one numerical and one categorical explanatory variable and study the model with and without interactions.

In Chapter 10 on inference for regression, we revisit the regression models and analyze the results using statistical inference, a method discussed in later chapters on sampling, bootstrapping and confidence intervals, and hypothesis testing and \(p\)-values, respectively. The focus there is also on using linear regression for prediction instead of explanation.

We begin with regression with a single explanatory variable. We also introduce the correlation coefficient, discuss “correlation versus causation,” and determine whether the model fits the data observed.

We now load all the packages needed for this chapter (this assumes you’ve already installed them). In this chapter, we introduce some new packages:

If needed, read the earlier section on packages for information on how to install and load R packages.

library(tidyverse)
library(moderndive)