This is one of the following seven articles on Multiple Linear Regression in Excel

Basics of Multiple Regression in Excel 2010 and Excel 2013

Complete Multiple Linear Regression Example in 6 Steps in Excel 2010 and Excel 2013

Multiple Linear Regression’s Required Residual Assumptions

Normality Testing of Residuals in Excel 2010 and Excel 2013

Evaluating the Excel Output of Multiple Regression

Estimating the Prediction Interval of Multiple Regression in Excel

Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel

# Step 7 – Evaluate the

Excel Regression Output

The Excel regression output that will now be evaluated is as follows:

*(Click Image To See a Larger Version)*

Interpretation of the most important individual parts of the Excel regression output are as follows:

## Regression Equation

*(Click Image To See a Larger Version)*

The regression equation is shown to be the following:

Y_{i} = b_{0} + b_{1} * X_{1i }+ b_{2} * X_{2i}

Power Consumption (kW) = 37,123,164 + 10.234 (Number of Production Machines X 1,000) + 3.573 (New Employees Added in Last 5 Years X 1,000)

Note that the scaling of the independent variables in Step 2 ensures that the calculated coefficients in the regression equation were of reasonable size (between 1 and 10)

For example, if a company had 10,000 production machines and added 500 new employees in the last 5 years, the company’s annual power consumption would be predicted as follows:

Annual Power Consumption (kW) = 37,123,164 + 10.234 (Number of Production Machines X 1,000) + 3.573 (New Employees Added in Last 5 Years X 1,000)

Annual Power Consumption (kW) = 37,123,164 + 10.234 (10,000 X 1,000) + 3.573 (500 X 1,000)

Annual Power Consumption = 49,143,690 kW

It is very important to note that a regression equation should never be extrapolated outside the range of the original data set used to create the regression equation. The inputs for a regression prediction should not be outside of the following ranges of the original data set:

Number of machine: 442 to 28,345

New employees added in last 5 years: -1,460 to 7,030

A simple example to illustrate why a regression line should never be extrapolated is as follows: Imagine that the height of a child was recorded every six months from ages one to seventeen. Most people stop growing in height at approximately age seventeen. If a regression line was created from that data and then extrapolated to predict that person’s height at age 50, the regression equation might predict that the person would be fifteen feet tall. Conditions are often very different outside the range of the original data set.

Extrapolation of a regression equation beyond the range of the original input data is one of the most common statistical mistakes made.

## R Square –The Equation’s

Overall Predictive Power

*(Click Image To See a Larger Version)*

R Square tells how closely the Regression Equation approximates the data. R Square tells what percentage of the variance of the output variables is explained by the input variables. We would like to see at least .6 or .7 for R Square. The remainder of the variance of the output is unexplained. R Square here is a relatively high value of 0.963. This indicates the 96.3 percent of the total variance in the output variable (annual power consumption) is explained by the variance of the input variables (number of production machines and number of new employees added in the last five years).

## Adjusted R Square

Adjusted R Square is quoted more often than R Square because it is more conservative. Adjusted R Square only increases when new independent variables are added to the regression analysis if those new variables increase an equation’s predictive ability. When you are adding independent variables to the regression equation, add them one at a time and check whether Adjusted R Square has gone up with the addition of the new variable. The value of Adjusted R Square here is 0.959.

## Significance of F -

Overall p Value and

Validity Measure

*(Click Image To See a Larger Version)*

The Significance of F is a p Value. A very small Significance of F confirms the validity of the Regression Equation. The Regression handout has more information about the Significance of F that appears in the Excel Regression output. The significance of F is actually a p Value. If the p value (Significance of F) is nearly zero, then there is almost no chance that the Regression Equation is random. This is very strong evidence of the validity of the overall Regression Equation.

To be more specific, this p value (Significance of F) indicates whether to reject the overall Null Hypothesis of this regression analysis. The overall Null Hypothesis for this regression equation states that all coefficients of the independent variables equal zero. In other words, that for this multiple regression equation:

Y = b** _{0}** + b

**X**

_{1}**+ b**

_{1}**X**

_{2}**+ … + b**

_{2}**X**

_{k}

_{k}The Null Hypothesis for multiple regression states that the coefficients b** _{1}**, b

**, … , b**

_{2}**all equal zero. The Y intercept, b**

_{k}**, is not included in this Null Hypothesis.**

_{0}For this simple regression equation:

Y = b** _{0}** + b

**X**

_{1}The Null Hypothesis for simple regression states that the Coefficient b** _{1}** equals zero. The Y intercept, b

**, is not included in this Null Hypothesis. Coefficient b**

_{0}**is the slope of the regression line in simple regression.**

_{1}In this case, the p Value (Significance of F) is extremely low (6.726657E-13) so we have very strong evidence that this is a valid regression equation. There is almost no probability that the relationship shown to exist between the dependent and independent variables (the nonzero values of coefficient b** _{1}**, b

**, … , b**

_{2}**) was obtained merely by chance.**

_{k}This low p Value (or corresponding high F Value) indicates that there is enough evidence to reject the Null Hypothesis of this regression analysis.

The 95 percent Level of Confidence is usually required to reject the Null Hypothesis. This translates to a 5 percent Level of Significance. The Null Hypothesis is rejected is the p Value (Significance of F) is less than 0.05. If the Null Hypothesis is rejected, the regression output stating that the regression coefficients b** _{1}**, b

**, … , b**

_{2}**do not equal zero is deemed to be statistically significant.**

_{k}

## p Value of Intercept and

Coefficients – Measure of

Their Validity

*(Click Image To See a Larger Version)*

The lower the p-Value for each, more likely that Y-Intercept or coefficient is valid. The Intercept’s low p Value of 0.0003 indicates that there is only a 0.03 chance that this calculated value of the Intercept is a random occurrence.

The extremely low p value for the coefficient for the Number_of_Production_Machines indicates that there is almost no chance that this calculated value of this coefficient is a random occurrence.

The p Value for the Number_of_New_Employees_Added is relatively large. This coefficient cannot be considered statistically significant (reliable) at a 95 percent certainty level. A 95 percent certainty level would be the equivalent of a Level of Significance (Alpha) equal to 0.05. The coefficient for the Number_of_Employees_Added would be considered statistically significant at a 0.05 Level of Significance if its p Value were less than 0.05. This is not the case because this p Value is shown to be 0.2432.

The coefficient for the Number_of_Machines can be considered reliable but not the coefficient for New_Employees_Added.

**The following blog article show how to estimate the Prediction Interval for Multiple Linear Regression.**

**Excel Master Series Blog Directory**

**Excel Master Series Blog Directory**

Statistical Topics and Articles In Each Topic

- Histograms in Excel
- Bar Chart in Excel
- Combinations & Permutations in Excel
- Normal Distribution in Excel
- Overview of the Normal Distribution
- Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
- Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
- Solving Normal Distribution Problems in Excel 2010 and Excel 2013
- Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
- An Important Difference Between the t and Normal Distribution Graphs
- The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
- Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

- t-Distribution in Excel
- Binomial Distribution in Excel
- z-Tests in Excel
- Overview of Hypothesis Tests Using the Normal Distribution in Excel 2010 and Excel 2013
- One-Sample z-Test in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013
- Overview of the Paired (Two-Dependent-Sample) z-Test in 4 Steps in Excel 2010 and Excel 2013

- t-Tests in Excel
- Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
- 1-Sample t-Tests in Excel
- 1-Sample t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
- 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
- 1-Sample t-Test Power With G*Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
- Sign Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013

- 2-Independent-Sample Pooled t-Tests in Excel
- 2-Independent-Sample Pooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
- Two-Independent-Sample Pooled t-Test - All Excel Calculations
- 2- Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test Power With G*Power Utility
- Mann-Whitney U Test in 12 Steps in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013
- 2- Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups

- 2-Independent-Sample Unpooled t-Tests in Excel
- 2-Independent-Sample Unpooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test
- 2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools
- Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility

- Paired (2-Sample Dependent) t-Tests in Excel
- Paired t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing of Paired t-Test Data
- Paired t-Test Excel Calculations, Formulas, and Tools
- Paired t-Test – Effect Size in Excel 2010, and Excel 2013
- Paired t-Test – Test Power With G-Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a Paired t-Test Alternative
- Sign Test in Excel As A Paired t-Test Alternative

- Hypothesis Tests of Proportion in Excel
- Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)
- 1-Sample Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Pooled Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer

- Chi-Square Independence Tests in Excel
- Chi-Square Goodness-Of-Fit Tests in Excel
- F Tests in Excel
- Correlation in Excel
- Pearson Correlation in Excel
- Spearman Correlation in Excel
- Confidence Intervals in Excel
- z-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
- Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013
- Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

- Simple Linear Regression in Excel
- Overview of Simple Linear Regression in Excel 2010 and Excel 2013
- Complete Simple Linear Regression Example in 7 Steps in Excel 2010 and Excel 2013
- Residual Evaluation For Simple Regression in 8 Steps in Excel 2010 and Excel 2013
- Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
- Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
- All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
- Prediction Interval of Simple Regression in Excel 2010 and Excel 2013

- Multiple Linear Regression in Excel
- Basics of Multiple Regression in Excel 2010 and Excel 2013
- Complete Multiple Linear Regression Example in 6 Steps in Excel 2010 and Excel 2013
- Multiple Linear Regression’s Required Residual Assumptions
- Normality Testing of Residuals in Excel 2010 and Excel 2013
- Evaluating the Excel Output of Multiple Regression
- Estimating the Prediction Interval of Multiple Regression in Excel
- Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel

- Logistic Regression in Excel
- Logistic Regression Overview
- Logistic Regression in 6 Steps in Excel 2010 and Excel 2013
- R Square For Logistic Regression Overview
- Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013
- Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013
- Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013
- Hosmer- Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013

- Single-Factor ANOVA in Excel
- Overview of Single-Factor ANOVA
- Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
- Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013
- Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
- Single-Factor ANOVA - All Excel Calculations
- Overview of Post-Hoc Testing For Single-Factor ANOVA
- Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
- Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
- Overview of Effect Size For Single-Factor ANOVA
- ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013
- Power of Single-Factor ANOVA Test Using Free Utility G*Power
- Welch’s ANOVA Test in 8 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
- Brown-Forsythe F-Test in 4 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

- Two-Factor ANOVA With Replication in Excel
- Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013
- Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication – Test Power With G-Power Utility
- Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication

- Two-Factor ANOVA Without Replication in Excel
- Randomized Block Design ANOVA in Excel
- Repeated-Measures ANOVA in Excel
- Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel 2010 and Excel 2013
- Sphericity Testing in 9 Steps For Repeated Measures ANOVA in Excel 2010 and Excel 2013
- Effect Size For Repeated-Measures ANOVA in Excel 2010 and Excel 2013
- Friedman Test in 3 Steps For Repeated-Measures ANOVA in Excel 2010 and Excel 2013

- ANCOVA in Excel
- Normality Testing in Excel
- Creating a Box Plot in 8 Steps in Excel
- Creating a Normal Probability Plot With Adjustable Confidence Interval Bands in 9 Steps in Excel With Formulas and a Bar Chart
- Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel
- Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel

- Nonparametric Testing in Excel
- Mann-Whitney U Test in 12 Steps in Excel
- Wilcoxon Signed-Rank Test in 8 Steps in Excel
- Sign Test in Excel
- Friedman Test in 3 Steps in Excel
- Scheirer-Ray-Hope Test in Excel
- Welch's ANOVA Test in 8 Steps Test in Excel
- Brown-Forsythe F Test in 4 Steps Test in Excel
- Levene's Test and Brown-Forsythe Variance Tests in Excel
- Chi-Square Independence Test in 7 Steps in Excel
- Chi-Square Goodness-of-Fit Tests in Excel
- Chi-Square Population Variance Test in Excel

- Post Hoc Testing in Excel
- Creating Interactive Graphs of Statistical Distributions in Excel
- Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013
- Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013
- Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013

- Solving Problems With Other Distributions in Excel
- Solving Uniform Distribution Problems in Excel 2010 and Excel 2013
- Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013
- Solving Exponential Distribution Problems in Excel 2010 and Excel 2013
- Solving Beta Distribution Problems in Excel 2010 and Excel 2013
- Solving Gamma Distribution Problems in Excel 2010 and Excel 2013
- Solving Poisson Distribution Problems in Excel 2010 and Excel 2013

- Optimization With Excel Solver
- Maximizing Lead Generation With Excel Solver
- Minimizing Cutting Stock Waste With Excel Solver
- Optimal Investment Selection With Excel Solver
- Minimizing the Total Cost of Shipping From Multiple Points To Multiple Points With Excel Solver
- Knapsack Loading Problem in Excel Solver – Optimizing the Loading of a Limited Compartment
- Optimizing a Bond Portfolio With Excel Solver
- Travelling Salesman Problem in Excel Solver – Finding the Shortest Path To Reach All Customers

- Chi-Square Population Variance Test in Excel
- Analyzing Data With Pivot Tables
- SEO Functions in Excel
- Time Series Analysis in Excel
- VLOOKUP

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