Linear regression is a statistical technique used to measure the average relationship between two or more variables in their original units. Regression analysis involves two types of variables: the dependent variable, which is the variable being predicted or changed, and the independent variable, which changes the values or aids in prediction (SC Gupta & V.K. Kapoor, 1970, pg 10.49). When there is a linear correlation between two quantitative variables, a linear model can be created that helps estimate the value of the dependent variable using the independent variable.

Linear regression equations are of particular interest for several reasons. Firstly, they lend themselves to additional mathematical operations. Secondly, they are often used to approximate otherwise difficult progression equations. Thirdly, the regression equations for bivariate normal distributions are linear (Miller & Miller, 2004, p. 438). Additionally, the linear regression equation is only applicable when there is a significant linear connection between the data points. In many cases, a set of paired data indicates that the regression is linear, but the joint distribution of the random variable under consideration is unknown, and the “m” and “b” regression coefficients in the equation y = mx + b need to be calculated (b = y - mx) (1).

Moreover, the linear regression line always passes through the point (x, y) according to equation (1). The method of least squares is the process of finding the line of best fit for a set of data points by reducing the sum of squares of the offsets, or the residual part of points from the curve. The method of least squares in linear regression defines the solution for the minimization of the sum of squares of deviation or the errors in the result of each equation.

Consider the example of fitting a straight line to the data below:

X 2 3 5 7 9

Y 4 5 7 10 5

Before attempting to solve the example, it is important to understand the steps involved in the least squares method. The first step is to calculate x^2 and xy for each (x, y) point. The second step involves adding up the values of x, y, x^2 and xy, which yields Ʃx, Ʃy, Ʃx^2, and Ʃxy (Ʃ refers to summation). The third step entails calculating the slope, m:

$m = \frac{n(Ʃxy) - (Ʃx)(Ʃy)}{n}$ (2)

where n is the number of arguments. The fourth step is to calculate the intercept, b:

$b = \frac{Ʃy - m(Ʃx)}{n}$ (3)

The final step is to assemble the equation of a line, y = mx + b. For the example listed above, Ʃx = 26, Ʃy = 41, Ʃxy = 263, Ʃx^2 = 168, and n = 5. After substituting these values into equation (2), we get m = 1.5183 and, after plugging the values into equation (3), we get b = 0.3049. Substituting this into the equation of the line, we get:

y = 1.5183 x + 0.3049

It is important to note that the coefficients m = 1.5183 and b = 0.3049 are based on sample data and are estimates of the true population regression line coefficient.

Finally, predictions can be made using regression equations when the following conditions are met:

1) The variables must have a linear relationship with each other.

2) Only predictions for the dependent variable are made.

3) The values used for ‘x’ are within the data set’s value domain, as taught by Meghan Corcoran.

To gain a deeper understanding of Linear Regression, let's explore some real-world examples and how we can apply it in our daily lives.

1. Businesses often use Linear Regression to determine their advertising expenses and revenue. For instance, they can utilize a basic linear regression model to predict the amount of revenue they can obtain from their advertising spending. The model generally takes the form of:

revenue = β₀ + β₁(ad spending)

With β₀ representing the total expected revenue when ad spending is zero. When advertising expenditure increases by one unit, the coefficient β₁ represents the average change in total revenue (such as one dollar). A negative β₁ implies that increasing advertising expenditure produces less revenue, and a near-zero β₁ suggests that advertising spending has little effect on revenue. On the other hand, a positive β₁ suggests that increased ad expenditure is linked to higher revenue. A company can adjust its advertising expenditure based on the value of β₁.

2. Medical researchers frequently use Linear Regression to determine the relationship between drug dosage and patient blood pressure. By studying patients who have received varying doses of a particular medication, researchers can identify the link between dosage and blood pressure. The simple linear regression model they may utilize looks like this:

blood pressure = β₀ + β₁(dosage)

The coefficient β₀ represents the expected blood pressure when the dosage is zero, whereas the coefficient β₁ represents the average change in blood pressure when the dosage is increased by one unit. A negative β₁ signifies that increasing dosage results in a drop in blood pressure, while a near-zero β₁ indicates that there is no connection between an increase in dosage and a change in blood pressure. In contrast, a positive β₁ implies that dosage contributes to an increase in blood pressure. Based on the value of β₁, researchers may choose to adjust a patient's dosage.

3. Linear Regression is also employed to evaluate the impact of marketing, pricing, and promotions on product sales. For example, a company can utilize Linear Regression to determine the effect of a marketing campaign on a particular brand's sales. One of the benefits of Linear Regression is that it allows us to isolate the effect of each marketing campaign while controlling for variables that might influence sales. In the real world, various advertising campaigns run simultaneously, but a linear regression model can capture both the isolated and combined impact of different promotions.

4. In the finance and insurance industries, Linear Regression can be used to evaluate risk. For instance, a car insurance company may use Linear Regression to determine the expected claims to insured declared value ratio and create a proposed premium table. They can calculate risk based on the car's characteristics, driver information, or demographics. The findings of such a study could be used to make critical business decisions. In the credit card sector, a financial business can use Linear Regression to assess the top five variables that cause a customer to default, based on which they may implement specific EMI solutions to mitigate the risk of default.

References:

Gupta, S. C., & Kapoor, V. K. (2000). Fundamentals of Mathematical Statistics (10th Edition). Sultan Chand.

Freund, J. E., Miller, M., & Miller, M. (2004). John E. Freund's Mathematical Statistics with Applications (7th Edition). Prentice Hall.

Z. (2020, May 19). 4 Examples of Using Linear Regression in Real Life. Statology. Retrieved April 11, 2022, from https://www.statology.org/linear-regression-real-life-examples/

Dusane, J. (2020, May 28). Popular Applications of Linear Regression for Businesses. Jigsaw Academy. Retrieved April 11, 2022, from https://www.jigsawacademy.com/popular-applications-of-linear-regression-for-businesses/