What Is Multiple Regression? (With an Example Formula)

Updated 30 September 2022

Multiple regression, also known as multiple linear regression (MLR), refers to explanatory variables and their effect on response variables. Individuals and organisations can utilise the MLR calculation across any industry because variables can technically refer to anything. Understanding how to calculate the MLR can help you apply this technique in your career. In this article, we discuss what multiple regression is, explain the variables involved, provide reasons for calculating the MLR, detail the formula and provide examples of professionals using the MLR equation.

What is multiple regression?

Multiple regression is a calculation that determines the effects of explanatory variables on a single response variable. Identifying the relationship between the explanatory and response variables is usually the primary purpose of conducting the MLR. For example, a production manager may calculate the effects of additional employees on the rate of production. The number of employees is the explanatory variable and the production rate is the response variable. From this calculation, the production manager can identify the relationship between employees and production rates.

The formula for calculating a multiple linear regression can vary depending on the types and quantities of variables. The more variables in an equation, the more complex the calculation becomes. Most professionals that use MLR often have experience in statistical and data analysis.

Types of variables

To understand how the multiple linear regression method works, it can be a good idea to review the variables involved in the equation. Below, you can find more details on the variables included in the MLR technique:

Explanatory variable

The explanatory variable is a variable that changes the response variable. For example, if you want to determine how age affects physical health, age is the explanatory variable because it affects physical health, which makes physical health the response variable. There's typically more than one explanatory variable in a multiple linear regression model.

Some people may refer to explanatory variables as independent variables, but explanatory variables aren't always independent. Consider you want to determine how diet and sleep can affect your physical health. Diet and sleep aren't independent variables, because they indirectly affect each other, but they're explanatory variables because they affect your physical health. Multiple linear regression typically involves multiple factors that can indirectly or directly affect each other, which is why professionals usually prefer the term explanatory, rather than independent.

Response variable

A response variable is a variable that changes based on the explanatory variables. Consider you want to identify how sleep and study can affect your test grades. Your test grades are the response variable because you're measuring how sleep and study can affect it. In multiple linear regression models, there's only one response variable.

A response variable is technically a dependent variable, but professionals typically prefer the term response variable when calculating the MLR. This is because multiple linear regression utilises explanatory variables, which differ slightly from independent variables. Professionals may use response variable because it relates more appropriately to explanatory variables.

Why calculate multiple linear regression?

The main reason for calculating multiple linear regression is to identify the relationship between explanatory and response variables. The purpose of these relationships can vary depending on the variables. Anything can be a response variable, so the reason you might use a multiple linear regression equation entirely depends on your response variable.

If you're a marketer, you can use an MLR equation to determine how event marketing and social media strategies affect lead generation. If you're a business analyst, you can determine how market movements and interest rates affect business sales. The MLR formula can generally help you calculate the relationship between anything that you can logically assign a value to.

Multiple linear regression formula

It can be important to understand that most professionals who calculate a multiple linear regression utilise specialised statistic software because the calculations are often extremely complex. The results of the MLR formula can also require statistical knowledge to interpret. Depending on the variables you use, there may be variations in the formula. Below, you can find a basic formula for calculating the multiple linear regression of variables:

Y = B0 + B1X1 + B2X2 + BnXn + E

The following list explains what each value of the formula represents:

Y: the value of the response variable
B0: the value of the response variable with no explanatory variables
B1: the regression coefficient of the first explanatory variable
X1: the first explanatory variable
B2: the regression coefficient of the second explanatory variable
X2: the second explanatory variable
Bn: the final regression coefficient of the final explanatory variable
Xn: the final explanatory variable
E: the variation between the prediction and actual value of the response variable

Examples of multiple linear regression

Below, you can find some examples of when professionals in different industries might utilise multiple linear regression:

Real estate

Below is an example of a real estate manager using multiple linear regression:

Example: Lynn is a real estate manager who wants to determine an appropriate time for selling property. To determine this, Lynn wants to identify when the property may be at its maximum price. Lynn identifies the factors affecting housing prices and begins planning a multiple linear regression model. The maximum housing price is the response variable and the factors affecting housing prices are the explanatory variables.

Lynn assigns interest rates, economic growth, speculative demand and consumer confidence as the explanatory variables. After completing the formula, Lynn discovers that housing prices peak when the explanatory variables reach a specific value. By identifying the values of the explanatory variables, Lynn can then predict an appropriate time to sell the property.

Medical

Here you can find an example of a research scientist using multiple linear regression:

Example: Toby is a research scientist who wants to develop an effective recovery plan for patients suffering from a specific virus. Toby implements a multiple linear regression model to create an effective recovery plan. This model helps Toby identify the typical recovery rate and the factors most likely to affect the rate.

Toby assigns diet, rest, environment and attitude as the main explanatory variables affecting patient recovery rates, which is the response variable. From the calculations, Toby discovers the extent that each explanatory variable affects patient recovery. Toby identifies rest as the largest contributor to a quick recovery. Toby then utilises the data from the multiple linear regression model to create an effective recovery plan that targets the most important factors affecting recovery.

Sports

Below, you can find an example of a sports manager utilising multiple linear regression:

Example: Vince is a manager for an international sports club who wants to implement new training programs for the athletes. To create an effective training program, Vince wants to discover the factors that affect their athletes' performances. Vince utilises a multiple linear regression model to determine these factors and the level of influence they have on an athlete's performance.

Vince determines the explanatory variables are strength, height, flexibility and diet. The response variable is the athletic performance of club members. After calculating the multiple linear regression, Vince discovers that strength and diet improve the performance drastically. From these results, Vince implements a training program that focuses on strength training and dietary needs.

Finance

Here, you can explore an example of a financial analyst utilising the multiple linear regression method:

Example: Bridget is an investment analyst who wants to identify businesses with high potential for investment returns. To determine this, Bridget identifies the factors that can affect a business's securities value. Bridget implements a multiple linear regression equation that identifies the impact of market aspects on a business's stock value.

Bridget includes perceived risk, business growth, earnings and consumer confidence as the explanatory variables of the equation. The response variable is the value of a business's stocks. Bridget completes the multiple linear regression formula and discovers business growth to be the largest contributor to a business's stock value. From this information, Bridget analyses businesses that have grown exponentially in the past month and determines their stocks as potential investment options.

Business administration

Below, you can find an example of an office manager utilising a multiple linear regression model:

Example: Stacy is an office manager who wants to improve productivity in the workplace. Stacy is developing workplace procedures that address common aspects that they believe reduce productivity. Stacy conducts a multiple linear regression calculation to identify the most disruptive aspects of the workplace.

Stacy assigns hydration, air quality, office layout and temperature as the explanatory variables of the equation. The response variable in this experiment is the productivity of team members. After the calculations, Stacy discovers the office layout and temperature of the office impact the team's productivity the most. From these results, Stacy organises a technician to install new air conditioning units. Stacy also rearranges the office layout to provide team members with more privacy when completing their responsibilities.

What Is Multiple Regression? (With an Example Formula)