An introduction to linear regression

Estimated time: 20 minutes

Overview

Questions

• What type of variables are required for simple linear regression?
• What do each of the components in the equation of a simple linear regression model represent?

Objectives

• Identify questions that can be addressed with a simple linear regression model.
• Describe the components that are involved in simple linear regression.

Questions that can be addressed with simple linear regression

---

Simple linear regression is commonly used, but when is it appropriate to apply this method? Broadly speaking, simple linear regression may be suitable when the following conditions hold:

• You seek a model of the relationship between one continuous dependent variable and one continuous or categorical explanatory variable.
• Your data and simple linear regression model do not violate the assumptions of the simple linear regression model. We will cover these assumptions in the final episode of this lesson.

Exercise

A colleague has started working with the NHANES data set. They approach you for advice on the use of simple linear regression on these data. Assuming that the assumptions of the simple linear regression model hold, which of the following questions could potentially be tackled with a simple linear regression model? Think closely about the outcome and explanatory variables, between which a relationship will be modelled to answer the research question.

1. Does home ownership (whether a participant’s home is owned or rented) vary across income brackets in the general US population?
   
2. Is there an association between BMI and pulse rate in the general US population?
   
3. Do teenagers on average have a higher pulse rate than adults in the general US population?

Show me the solution

1. The outcome variable is home ownership and the explanatory variable is income bracket. Since home ownership is a categorical outcome variable, simple linear regression is not a suitable way to answer this question.
   
2. Since both variables are continous, simple linear regression may be a suitable way to answer this question.
   
3. The outcome variable is pulse rate and the explanatory variable is age group (teenager vs adult). Since the outcome variable is continuous and the explanatory variable is categorical, simple linear regression may be a suitable way to answer this question.

The simple linear regression equation

---

The simple linear regression model can be described by the following equation:

\[ E(y) = \beta_0 + \beta_1 \times x_1. \]

The outcome variable is denoted by \(y\) and the explanatory variable is denoted by \(x_1\). Simple linear regression models the expectation of \(y\), i.e. \(E(y)\). This is another way of referring to the mean of \(y\). The expectation of \(y\) is a function of \(\beta_0\) and \(\beta_1 \times x_1\). The intercept is denoted by \(\beta_0\) - this is the value of \(E(y)\) when the explanatory variable, \(x_1\), equals 0. The effect of our explanatory variable is denoted by \(\beta_1\) - for every one-unit increase in \(x_1\), \(E(y)\) changes by \(\beta_1\).

Before fitting the model, we have access to \(y\) and \(x_1\) values for each observation in our data. For example, we may want to model the relationship between weight and height. \(y\) would represent weight and \(x_1\) would represent height. After we fit the model, R will return to us values of \(\beta_0\) and \(\beta_1\) - these are estimated using our data.

Exercise

We are asked to study the effect of participant’s age on their BMI. We are given the following equation of a simple linear regression to use:

\[E(y) = \\beta\_0 + \\beta\_1 \\times x\_1\].

Match the following components of this simple linear regression model to their descriptions:

1. \(E(y)\)
2. \({\beta}_0\)
3. \(x_1\)
4. \({\beta}_1\)

1. Mean BMI at a particular value of age.
   
2. The expected BMI when the age equals 0.
   
3. The expected change in BMI with a one-unit increase in age.
   
4. A specific value of age.

Show me the solution

1. 1
   
2. 2
   
3. 4
   
4. 3

Key Points

• Simple linear regression requires one continuous dependent variable and one continuous or categorical explanatory variable. In addition, the assumptions of the model must hold.
• The components of the model describe the mean of the dependent variable as a function of the explanatory variables, the mean of the dependent variable at the 0-point of the explanatory variable and the effect of the explanatory variable on the mean of dependent variable.