Multiple Regression

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What is Multiple Regression?

Multiple regression is an extension of the simple linear equation. Simple linear analysis is to study two variables, where one variable is the independent variable (X), and the other is the dependent variable. It serves to predict the change in the dependent variable based on the difference in the independent variable; this could also be called a Multiple regression line. In this blog, you’ll also learn how to create multiple linear regression with r on excel as well as 3 multiple regression equation calculator webpages that will help you create them quickly.

What are the applications?

Multiple Regression helps in a wide range of fields. Human Resource professionals might collect data about an employee’s salary based on factors such as work experience and competence. Your data can be used to build a model and determine employee wages. Is it more for certain groups of employees than the norm? Are there any employees or groups that are paid less than the norm?
Similar to the above, different researchers may use regression to determine which variables can best predict a specific outcome. To best match the results, deciding on what independent variables are required is necessary. Which factors determine how schools score on their tests? What factors affect the productivity of a supply chain?

When to use the multiple regression line model

Multiple regression refers to the study of more than one variable. The main difference between numerous and simple regression is the explanatory variables. Multiple regression is different from simple linear because there are countless independent variables (X), but these independent variables are used to predict one dependent variable (Y). The changes in independent variables indicate the evolution of the dependent variable (Y).

How should Six Sigma practitioners deal with situations where more than one X influences a Y? They use multiple-linear regression because it is more common than just one affecting variable. You can create equations that include more than one variable, such as Y = (X1, X2) , Xn).

The multiple linear regression model can be described as an extension of the simple linear model. For example, if X1 andX2 contribute to the same Y value, then multiple linear models can be used.

Yi = b0 + B1X1 +B11X12 +B2X2 +B22X22 +B12X1X2+e

The equation includes five types of terms:

  • b : This is the overall effect. It determines the starting point for all other effects regardless of the X variables.
  • biXi: These are the main effects in terms of the equation. They are the b1X1 or b2X2 pieces. These terms, just like the simple linear regression model’s main effects terms, capture the linear effect that each Xi has upon the output Y.
  • biiXi2 – b11X12, b22X22 and b22X22 respectively are the squared or second-order effects for each of these Xs. The effect is quadratic, rather than linear, because the variable is increased to the second power. These second-order effects can be identified by the associated b and ii coefficients.
  • This effect is known as the interaction effect. This term allows input variables to have an inter- or combined effect on the outcome of Y. The b12 coefficient captures the magnitude and direction.
  • e: This term describes all random variation that other terms cannot explain. e is a normal distribution that has its center at zero.
multiple regression line
multiple regression line

Multiple regression line Equations

The results from a multiple linear regression equation calculator can be more than just a straight line. They can also accommodate three-dimensional surfaces and abstract relationships in n-dimensional spaces. Using a multiple regression equation calculator can be intimidating. Multiple regression line can handle almost anything, they’ve performed in the same way as simple linear regression:

  1. Collect the Xs and Ys data.
  2. Calculate multiple linear regression coefficients. Having more than one X variable can make the equations to calculate the bs very complicated and tedious. Statistical analysis software tools can automatically calculate these equations. If you don’t want to buy number 2 pencils, grab a few and get ready to roll! The bs will just come out. To confirm that the residual values meet the initial assumptions of the multiple-linear regression model, check them.
    It is crucial to ensure that the residuals are not abnormal. The multiple linear regression model’s starting assumptions are invalid if the variance of the residuals doesn’t center on zero or if it is not random and normal.
  3. To determine which terms of multiple linear regression equation terms should be retained in the model and which ones are not, perform statistical tests. Some terms of the multiple regression equation don’t have significant significance. To find which words are not substantial, you can perform an F check for each word of the equation. If the term’s variation contribution is low relative to its residual variation, it won’t pass F and can be removed from the equation.
  4. The goal of the regression equation is to be as simple as possible and maximize the R HTML2 metric. Simpler is better. If you have two equations with the same HTML2 value, then you should settle for the one with the simplest terms.
    The higher-order terms usually go first. It is less likely that a squared or interaction term will be statistically significant.
  5. Calculate the final coefficient for determination R HTML2_1_1_2 in the multiple linear regression model.
    To quantify the amount of observed variation in your equation, use the R HTML2 metric.

What is the adjusted ratio R squared?

The adjusted R squared measures the explanatory power for regression models with different numbers of predictors.

Let’s say you want to compare a five-predictor prediction model with a higher R squared to a one predictor model. Is the R-squared higher for the five-predictor models because they are better? Is the R squared higher simply because there are more predictors? To find out, simply compare the adjusted R squared values!

The adjusted R squared is a modified R squared, which has been adjusted to account for the number of predictors in a model. The adjusted R squared will increase if the new term makes the model better than expected. It drops if a predictor makes the model more accurate than it would be by chance. Although the adjusted R-squared may be harmful, it is usually not. It is always lower than the R squared.

Below is a simplified output of Best Subsets Regression. You can see the peak and declines in the adjusted R squared. Nevertheless, R squared is increasing.

linear regression on excel
linear regression with r

This model might only include three predictors. We saw in my previous blog how a poorly-specified model can lead to biased estimates. An oversimplified model (one with too many variables) will result in lower precision estimates and predictions. Therefore, don’t include terms that are not necessary in your model.

What is the Predicted R-Squared?

The predicted R squared is a measure of how accurately a regression model predicts new observations. This statistic can be used to determine if the model is able to predict new observations, but not the original data. 

Mini-tab calculates predicted R squared by subtracting each observation from the data, then estimating the regression equation and determining how well it predicts the missing observation. Predicted R-squared is similar to adjusted R-squared and can be negative. It is always lower than R squared.

Even if the model isn’t used for predictions, the R squared predicted still offers crucial information. It has the key advantage of preventing you from overfitting models. Overfitting a model with too many predictors can lead to it modeling random noise.

It’s impossible to predict random noise so the R-squared for an overfit model must decrease. A predicted R squared lower than the regular R squared is almost always a sign that there are too many terms in your model.

Multiple linear regression on excel

This tutorial from Statology explains how to perform multiple linear regression on excel.

Note: If you only have one explanatory variable, you should instead perform simple linear regression on excel.

Step 1: Enter the data.

multiple regression on excel
linear regression with r

Step 2: Perform multiple linear regression.

Along with the top ribbon in Excel, go to the Data tab and click on Data Analysis. If you don’t see this option, then you need to first install the free Analysis ToolPak.

multiple regression on excel
linear regression on excel

Once you click on Data Analysis, a new window will pop up. Select Regression and click OK.

linear regression on excel
linear regression with r

For Input Y Range, fill in the array of values for the response variable. For Input X Range, fill in the array of values for the two explanatory variables. Check the box next to Labels so Excel knows that we included the variable names in the input ranges. For Output Range, select a cell where you would like the output of the regression to appear. Then click OK.

multiple regression on excel
multiple regression on excel

The following output will automatically appear:

linear regression on excel
linear regression on excel

Step 3: Interpret the output.

Online Multiple Regression Calculators

multiple regression calculator
multiple regression equation calculator

Social Science Statistics:

This simple multiple regression line calculator uses the least-squares method to find the line of best fit for data comprising two independent X values and one dependent Y value, allowing you to estimate the value of a dependent variable (Y) from two given independent (or explanatory) variables (X1 and X2).

Statistics Kingdom:

This multiple regression calculator makes use of variable transformations and calculates R, the Linear equation, and the p-value. It also calculates outliers and the adjusted Fisher-Pearson coefficient for skewness. The program interprets the results after confirming the residuals’ normality and multicollinearity. It then draws a histogram and residuals QQ-plots, a correlation matrix, and a distribution graph. You can transform the variables, exclude any predictor, or run backward stepwise and select automatically based upon the predictor’s predicted p-value.

Stats Solver:

Stats Solver is a multiple regression equation calculator, their goal is to solve any statistical problem quickly and easily. Use our intuitive interface to enter your problem and receive a step-by, step solution. Even if you don’t have an issue to solve, there are still ways you can learn from other examples. This multiple regression calculator will allow you to see if the solution changes, and you can change values quickly. To find out more about the topic, click on definitions, formulas, and explanations at the bottom of each page.

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