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Regression Analysis Formula

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Explanation

While running a regression, the main purpose of the researcher is to find out the relationship between the dependent and independent variables. Then, one or multiple independent variables chose to help predict the dependent variable. Regression analysis helps in the process of validating whether the predictor variables are good enough to help in predicting the dependent variable.

Examples

Example #1

Let us try and understand the concept of regression analysis with the help of an example. First, let us try to find out the relation between the distance covered by the truck driver and the age of the truck driver. Then, someone does a regression equation to validate whether what he thinks of the relationship between two variables is also validated by the regression equation.

Below is given data for calculation

For the calculation of regression analysis, go to the "Data" tab in Excel and then select the "Data Analysis" option. For further calculation procedure, refer to the given article here - Analysis ToolPak in Excel

The regression analysis formula for the above example will be

y = MX + b
y= 575.754*-3.121+0
y= -1797

In this particular example, we will see which variable is the dependent variable and which variable is the independent variable. The dependent variable in this regression equation is the distance covered by the truck driver, and the independent variable is the age of the truck driver. The regression for this set of dependent and independent variables proves that the independent variable is a good predictor of the dependent variable with a reasonably high coefficient of determination. In addition, the analysis helps validate that the factors in the form of the independent variable are selected correctly. The snapshot below depicts the regression output for the variables. The data set and the variables are present in the Excel sheet attached.

Example #2

Let us try and understand regression analysis with the help of another example. Let us try to find out the relation between the height of the students of a class and the GPA grade of those students. Then, someone does a regression equation to validate whether what he thinks of the relationship between two variables is also validated by the regression equation.

In this example, Below is given data for calculation in excel

For regression analysis calculation, go to the "Data" tab in Excel and select the "Data Analysis" option.

Regression Analysis Formula Example 2.1jpg

The regression for the above example will be

y = MX + b
y= 2.65*.0034+0
y= 0.009198

In this particular example, we will see which variable is the dependent variable and which variable is the independent variable. The dependent variable in this regression equation is the student's GPA, and the independent variable is the student's height. The regression analysis for this set of dependent and independent variables proves that the independent variable is not a good predictor of the dependent variable as the value for the coefficient of determination is negligible. In this case, we need to find another predictor variable to predict the dependent variable for the regression analysis. The snapshot below depicts the regression output for the variables. The data set and the variables are present in the Excel sheet attached.

Relevance and Uses

Regression is a very useful statistical method. One can validate any business decision to validate a hypothesis that a particular action will increase a division's profitability based on the regression between the dependent and independent variables. Therefore, the regression analysis equation plays a very important role in finance. In addition, a lot of forecasting is performed using regression. For example, one can predict the sales of a particular segment in advance with the help of macroeconomic indicators that have a very good correlation with that segment. Both linear and multiple regressions are useful for practitioners to make predictions of the dependent variables and validate the independent variables as a predictor of the dependent variables.

Frequently Asked Questions (FAQs)

1. What are the assumptions of the regression analysis formula?

Regression analysis relies on several assumptions. First, it assumes a linear relationship between the independent and dependent variables. It also assumes that the observations in the dataset are independent of each other, meaning that one observation does not influence another. The assumption of homoscedasticity states that the variance of the errors or residuals is constant across all levels of the independent variables.

2. What are the limitations of the regression analysis formula?

The regression analysis has limitations to consider. It is only suitable for analyzing variables that exhibit a linear relationship, potentially missing complex nonlinear relationships. While it can identify associations, it cannot establish causation, requiring additional evidence and consideration of other factors. Violating the assumptions can affect the accuracy and reliability of the results. Outliers or influential observations can also disproportionately impact the outcomes, leading to biased estimates.

3. Who discovered regression analysis?

Sir Francis Galton initially developed regression analysis in the late 19th century. Still, it was further refined and formalized by other statisticians, such as Karl Pearson and Ronald Fisher, who made significant contributions to the field.