Multiple Linear Regression (MLP) – Uses, Formula and Examples

Various statistical models help in establishing a relationship between different variables. Multiple linear regression is a statistical model that describes the functioning relationship between one dependent variable and two or more independent variables. This kind of statistical interpretation helps make predictions and acts as a guide to key decisions.

Read on to learn more about this concept and its different working aspects. 

What is the Formula of Multiple Linear Regression?

The multiple regression formula is as follows:

y_i = β₀ + β₁x_i1 + β₂x_i2 + ……… + β_px_ip + ϵ  

For i= n observations  

Whereby y_i represents the dependent variable; 

x_i represents the explanatory variables;

β₀ represents the constant term or y-intercept;

β_p represents slope coefficient for every explanatory variable; 

ϵ represents any model error or residual term.

Using all the data as mentioned in the formula, you will get to know the value of a dependent variable with respect to the independent variable. On plotting these values on a graph, you will see a linear relationship between them. 

How to Calculate Multiple Linear Regression?  

Here are some steps that you can follow to get multiple regression model values: 

Suppose there is one dependent variable named y, and there are two independent variables, namely x₁ and x₂. There are data sets available for all three variables.

Step 1: Compute x₁², x₂², x₁*x_2, x₁*y, x₂*y. 

Step 2: Calculate the summation of all these variables. 

Step 3: The next step entails calculating β₀, β_1, and β₂, respectively, by using the summation calculated in the previous step. 

Step 4: Place all the variables in multiple regression line formula to establish the relationship between y and x₁ and x₂.   

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Example of Multiple Linear Regression 

Let’s understand the working of this concept more clearly with the help of an example: 

Let’s say the values of X1 are 2, 4 and 5; the values of X2 are 1, 3 and 6. 

Meanwhile, values of Y are 10, 12, and 14. 

So, the mean value of X1 = 3.66; mean value of X2 = 3.33, and Y = 12

Sum of squares of X1 = 15, X2 = 15.33. 

The sum of products X1Y = 24; and X2Y = 12 and X1X2 = 32.66

The regression equation is y = β₀ + β₁X1 + β₂X2. Using the formula, 

we get β₀ = 5.6254, β₁ = -.02365 and β₂ = 7.259. 

By putting all these values in the formula, multiple regression comes out as: 

Y = 5.6254 – 0.2365X1 + 7.259X2. 

Therefore, as seen from this example, we can easily establish a relationship that exists between the dependent and independent variables by looking at the graphical representation of the values mentioned above. 

What Are the Assumptions of Multiple Linear Regression Model? 

Every statistical model works on certain assumptions; the multiple linear regression model also works on some assumptions, which are as follows: 

1. Constant Variance 

There is a constant variance of ϵ for all the datasets of Xs. Therefore, we can analyse the residual plots of X’s to get an idea of this variance. Moreover, a rectangular residual plot indicates constant variance. However, if the residual plots have a wedge shape, it means non-constant variance. 

2. Linearity

Another assumption about this model that one should know is linearity. A straight line relationship exists between the variables X and Y. Any curvilinear relation is not considered. You can visually see a linear pattern through the scatter plots. However, residual plots may show some non-linear patterns as well. 

3. Multicollinearity

There exists a linear relation between various sets of independent variables. Therefore, it indicates multicollinearity among these variables. However, as multicollinearity causes several difficulties with regard to regression analysis, we assume that respective datasets of independent variables are not multi co-linear. 

4. Normality

The confidence limits, as well as hypothesis tests, show a normal distribution of residual value, i.e., ϵ. 

How to Use Multiple Regression Analysis Model?

This analysis is quite useful when more than one factor affects a particular variable. For example, by looking at this model, we can get an idea about what factors are affecting sales figures or price changes of a newly launched product. 

This also helps businesses to better plan their projects or product launches by understanding various factors involved and the relationship between them. It may be very helpful for long-term outlook or projections made by a business or firm. 

Individual professionals also use this during research analysis. Therefore, we see that the regression model has several uses for businesses and individuals. One can get a better idea about the behaviour of datasets by looking at the regression line. 

Final Word

Multiple linear regression is a quite popular statistical tool that helps in establishing relationships between dependent and independent variables. It is quite useful in the analysis of research and future planning. Although multiple regression analysis can get quite lengthy and complicated, it offers a better result when compared to other statistical tools. 

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