© 2003 Prentice-Hall, Inc.Chap 14-1 Basic Business Statistics (9 th Edition) Chapter 14 Introduction to Multiple Regression

© 2003 Prentice-Hall, Inc. Chap 14-2 Chapter Topics The Multiple Regression Model Residual Analysis Testing for the Significance of the Regression Model Inferences on the Population Regression Coefficients Testing Portions of the Multiple Regression Model Dummy-Variables and Interaction Terms

© 2003 Prentice-Hall, Inc. Chap 14-3 Population Y-intercept Population slopesRandom error The Multiple Regression Model Relationship between 1 dependent & 2 or more independent variables is a linear function Dependent (Response) variable Independent (Explanatory) variables

© 2003 Prentice-Hall, Inc. Chap 14-4 Multiple Regression Model Bivariate model

© 2003 Prentice-Hall, Inc. Chap 14-5 Multiple Regression Equation Bivariate model Multiple Regression Equation

© 2003 Prentice-Hall, Inc. Chap 14-6 Multiple Regression Equation Too complicated by hand! Ouch!

© 2003 Prentice-Hall, Inc. Chap 14-7 Interpretation of Estimated Coefficients Slope ( b j ) Estimated that the average value of Y changes by b j for each 1 unit increase in X j, holding all other variables constant (ceterus paribus) Example: If b 1 = -2, then fuel oil usage ( Y ) is expected to decrease by an estimated 2 gallons for each 1 degree increase in temperature ( X 1 ), given the inches of insulation ( X 2 ) Y-Intercept ( b 0 ) The estimated average value of Y when all X j = 0

© 2003 Prentice-Hall, Inc. Chap 14-8 Multiple Regression Model: Example ( 0 F) Develop a model for estimating heating oil used for a single family home in the month of January, based on average temperature and amount of insulation in inches.

© 2003 Prentice-Hall, Inc. Chap 14-9 Multiple Regression Equation: Example Excel Output For each degree increase in temperature, the estimated average amount of heating oil used is decreased by gallons, holding insulation constant. For each increase in one inch of insulation, the estimated average use of heating oil is decreased by gallons, holding temperature constant.

© 2003 Prentice-Hall, Inc. Chap Multiple Regression in PHStat PHStat | Regression | Multiple Regression … Excel spreadsheet for the heating oil example

© 2003 Prentice-Hall, Inc. Chap Venn Diagrams and Explanatory Power of Regression Oil Temp Variations in Oil explained by Temp or variations in Temp used in explaining variation in Oil Variations in Oil explained by the error term Variations in Temp not used in explaining variation in Oil

© 2003 Prentice-Hall, Inc. Chap Venn Diagrams and Explanatory Power of Regression Oil Temp (continued)

© 2003 Prentice-Hall, Inc. Chap Venn Diagrams and Explanatory Power of Regression Oil Temp Insulation Overlapping variation NOT estimation Overlapping variation in both Temp and Insulation are used in explaining the variation in Oil but NOT in the estimation of nor NOT Variation NOT explained by Temp nor Insulation

© 2003 Prentice-Hall, Inc. Chap Coefficient of Multiple Determination Proportion of Total Variation in Y Explained by All X Variables Taken Together Never Decreases When a New X Variable is Added to Model Disadvantage when comparing among models

© 2003 Prentice-Hall, Inc. Chap Venn Diagrams and Explanatory Power of Regression Oil Temp Insulation

© 2003 Prentice-Hall, Inc. Chap Adjusted Coefficient of Multiple Determination Proportion of Variation in Y Explained by All the X Variables Adjusted for the Sample Size and the Number of X Variables Used Penalizes excessive use of independent variables Smaller than Useful in comparing among models Can decrease if an insignificant new X variable is added to the model

© 2003 Prentice-Hall, Inc. Chap Coefficient of Multiple Determination Excel Output Adjusted r 2  reflects the number of explanatory variables and sample size  is smaller than r 2

© 2003 Prentice-Hall, Inc. Chap Interpretation of Coefficient of Multiple Determination 96.56% of the total variation in heating oil can be explained by temperature and amount of insulation 95.99% of the total fluctuation in heating oil can be explained by temperature and amount of insulation after adjusting for the number of explanatory variables and sample size

© 2003 Prentice-Hall, Inc. Chap Simple and Multiple Regression Compared simple The slope coefficient in a simple regression picks up the impact of the independent variable plus the impacts of other variables that are excluded from the model, but are correlated with the included independent variable and the dependent variable multiple Coefficients in a multiple regression net out the impacts of other variables in the equation Hence, they are called the net regression coefficients They still pick up the effects of other variables that are excluded from the model, but are correlated with the included independent variables and the dependent variable

© 2003 Prentice-Hall, Inc. Chap Simple and Multiple Regression Compared: Example Two Simple Regressions: Multiple Regression:

© 2003 Prentice-Hall, Inc. Chap Simple and Multiple Regression Compared: Slope Coefficients

© 2003 Prentice-Hall, Inc. Chap Simple and Multiple Regression Compared: r 2 

© 2003 Prentice-Hall, Inc. Chap Example: Adjusted r 2 Can Decrease Adjusted r 2 decreases when k increases from 2 to 3 Color is not useful in explaining the variation in oil consumption.

© 2003 Prentice-Hall, Inc. Chap Using the Regression Equation to Make Predictions Predict the amount of heating oil used for a home if the average temperature is 30 0 and the insulation is 6 inches. The predicted heating oil used is gallons.

© 2003 Prentice-Hall, Inc. Chap Predictions in PHStat PHStat | Regression | Multiple Regression … Check the “Confidence and Prediction Interval Estimate” box Excel spreadsheet for the heating oil example

© 2003 Prentice-Hall, Inc. Chap Residual Plots Residuals Vs May need to transform Y variable Residuals Vs May need to transform variable Residuals Vs May need to transform variable Residuals Vs Time May have autocorrelation

© 2003 Prentice-Hall, Inc. Chap Residual Plots: Example No Discernable Pattern Maybe some non- linear relationship

© 2003 Prentice-Hall, Inc. Chap Testing for Overall Significance Shows if Y Depends Linearly on All of the X Variables Together as a Group Use F Test Statistic Hypotheses: H 0 :      …  k = 0 (No linear relationship) H 1 : At least one  i  ( At least one independent variable affects Y ) The Null Hypothesis is a Very Strong Statement The Null Hypothesis is Almost Always Rejected

© 2003 Prentice-Hall, Inc. Chap Testing for Overall Significance Test Statistic: Where F has k numerator and ( n-k-1 ) denominator degrees of freedom (continued)

© 2003 Prentice-Hall, Inc. Chap Test for Overall Significance Excel Output: Example k = 2, the number of explanatory variables n - 1 p -value

© 2003 Prentice-Hall, Inc. Chap Test for Overall Significance: Example Solution F H 0 :  1 =  2 = … =  k = 0 H 1 : At least one  j  0  =.05 df = 2 and 12 Critical Value : Test Statistic: Decision: Conclusion: Reject at  = There is evidence that at least one independent variable affects Y.  = 0.05 F  (Excel Output)

© 2003 Prentice-Hall, Inc. Chap Test for Significance: Individual Variables Show If Y Depends Linearly on a Single X j Individually While Holding the Effects of Other X’ s Fixed Use t Test Statistic Hypotheses: H 0 :  j  0 (No linear relationship) H 1 :  j  0 (Linear relationship between X j and Y )

© 2003 Prentice-Hall, Inc. Chap t Test Statistic Excel Output: Example t Test Statistic for X 1 (Temperature) t Test Statistic for X 2 (Insulation)

© 2003 Prentice-Hall, Inc. Chap t Test : Example Solution H 0 :  1 = 0 H 1 :  1  0 df = 12 Critical Values: Test Statistic: Decision: Conclusion: Reject H 0 at  = There is evidence of a significant effect of temperature on oil consumption holding constant the effect of insulation. t Reject H Does temperature have a significant effect on monthly consumption of heating oil? Test at  = t Test Statistic =

© 2003 Prentice-Hall, Inc. Chap Venn Diagrams and Estimation of Regression Model Oil Temp Insulation Only this information is used in the estimation of This information is NOT used in the estimation of nor

© 2003 Prentice-Hall, Inc. Chap Confidence Interval Estimate for the Slope Provide the 95% confidence interval for the population slope  1 (the effect of temperature on oil consumption)   1  We are 95% confident that the estimated average consumption of oil is reduced by between 4.7 gallons to 6.17 gallons per each increase of 1 0 F holding insulation constant. We can also perform the test for the significance of individual variables, H 0 :  1 = 0 vs. H 1 :  1  0, using this confidence interval.

© 2003 Prentice-Hall, Inc. Chap Contribution of a Single Independent Variable Let X j Be the Independent Variable of Interest Measures the additional contribution of X j in explaining the total variation in Y with the inclusion of all the remaining independent variables

© 2003 Prentice-Hall, Inc. Chap Contribution of a Single Independent Variable Measures the additional contribution of X 1 in explaining Y with the inclusion of X 2 and X 3. From ANOVA section of regression for

© 2003 Prentice-Hall, Inc. Chap Coefficient of Partial Determination of Measures the proportion of variation in the dependent variable that is explained by X j while controlling for (holding constant) the other independent variables

© 2003 Prentice-Hall, Inc. Chap Coefficient of Partial Determination for (continued) Example: Model with two independent variables

© 2003 Prentice-Hall, Inc. Chap Venn Diagrams and Coefficient of Partial Determination for Oil Temp Insulation =

© 2003 Prentice-Hall, Inc. Chap Coefficient of Partial Determination in PHStat PHStat | Regression | Multiple Regression … Check the “Coefficient of Partial Determination” box Excel spreadsheet for the heating oil example

© 2003 Prentice-Hall, Inc. Chap Contribution of a Subset of Independent Variables Let X s Be the Subset of Independent Variables of Interest Measures the contribution of the subset X s in explaining SST with the inclusion of the remaining independent variables

© 2003 Prentice-Hall, Inc. Chap Contribution of a Subset of Independent Variables: Example Let X s be X 1 and X 3 From ANOVA section of regression for

© 2003 Prentice-Hall, Inc. Chap Testing Portions of Model Examines the Contribution of a Subset X s of Explanatory Variables to the Relationship with Y Null Hypothesis: Variables in the subset do not improve the model significantly when all other variables are included Alternative Hypothesis: At least one variable in the subset is significant when all other variables are included

© 2003 Prentice-Hall, Inc. Chap Testing Portions of Model One-Tailed Rejection Region Requires Comparison of Two Regressions One regression includes everything Another regression includes everything except the portion to be tested (continued)

© 2003 Prentice-Hall, Inc. Chap Partial F Test for the Contribution of a Subset of X Variables Hypotheses: H 0 : Variables X s do not significantly improve the model given all other variables included H 1 : Variables X s significantly improve the model given all others included Test Statistic: with df = m and ( n-k-1 ) m = # of variables in the subset X s

© 2003 Prentice-Hall, Inc. Chap Partial F Test for the Contribution of a Single Hypotheses: H 0 : Variable X j does not significantly improve the model given all others included H 1 : Variable X j significantly improves the model given all others included Test Statistic: with df = 1 and ( n-k-1 ) m = 1 here

© 2003 Prentice-Hall, Inc. Chap Testing Portions of Model: Example Test at the  =.05 level to determine if the variable of average temperature significantly improves the model, given that insulation is included.

© 2003 Prentice-Hall, Inc. Chap Testing Portions of Model: Example H 0 : X 1 (temperature) does not improve model with X 2 (insulation) included H 1 : X 1 does improve model  =.05, df = 1 and 12 Critical Value = 4.75 (For X 1 and X 2 )(For X 2 ) Conclusion: Reject H 0 ; X 1 does improve model.

© 2003 Prentice-Hall, Inc. Chap Testing Portions of Model in PHStat PHStat | Regression | Multiple Regression … Check the “Coefficient of Partial Determination” box Excel spreadsheet for the heating oil example

© 2003 Prentice-Hall, Inc. Chap Do We Need to Do This for One Variable? The F Test for the Contribution of a Single Variable After All Other Variables are Included in the Model is IDENTICAL to the t Test of the Slope for that Variable The Only Reason to Perform an F Test is to Test Several Variables Together

© 2003 Prentice-Hall, Inc. Chap Dummy-Variable Models Categorical Explanatory Variable with 2 or More Levels Yes or No, On or Off, Male or Female, Use Dummy-Variables (Coded as 0 or 1) Only Intercepts are Different Assumes Equal Slopes Across Categories The Number of Dummy-Variables Needed is (# of Levels - 1) Regression Model Has Same Form:

© 2003 Prentice-Hall, Inc. Chap Dummy-Variable Models (with 2 Levels) Given: Y = Assessed Value of House X 1 = Square Footage of House X 2 = Desirability of Neighborhood = Desirable ( X 2 = 1) Undesirable ( X 2 = 0) 0 if undesirable 1 if desirable Same slopes

© 2003 Prentice-Hall, Inc. Chap Undesirable Desirable Location Dummy-Variable Models (with 2 Levels) (continued) X 1 (Square footage) Y (Assessed Value) b 0 + b 2 b0b0 Same slopes Intercepts different

© 2003 Prentice-Hall, Inc. Chap Interpretation of the Dummy- Variable Coefficient (with 2 Levels) Example: : GPA 0 non-business degree 1 business degree : Annual salary of college graduate in thousand $ With the same GPA, college graduates with a business degree are making an estimated 6 thousand dollars more than graduates with a non-business degree, on average. :

© 2003 Prentice-Hall, Inc. Chap Dummy-Variable Models (with 3 Levels)

© 2003 Prentice-Hall, Inc. Chap Interpretation of the Dummy- Variable Coefficients (with 3 Levels) With the same footage, a Split- level will have an estimated average assessed value of thousand dollars more than a Condo. With the same footage, a Ranch will have an estimated average assessed value of thousand dollars more than a Condo.

© 2003 Prentice-Hall, Inc. Chap Regression Model Containing an Interaction Term Hypothesizes Interaction between a Pair of X Variables Response to one X variable varies at different levels of another X variable Contains a Cross-Product Term Can Be Combined with Other Models E.g., Dummy-Variable Model

© 2003 Prentice-Hall, Inc. Chap Effect of Interaction Given: Without Interaction Term, Effect of X 1 on Y is Measured by  1 With Interaction Term, Effect of X 1 on Y is Measured by  1 +  3 X 2 Effect Changes as X 2 Changes

© 2003 Prentice-Hall, Inc. Chap Y = 1 + 2X 1 + 3(1) + 4X 1 (1) = 4 + 6X 1 Y = 1 + 2X 1 + 3(0) + 4X 1 (0) = 1 + 2X 1 Interaction Example Effect (slope) of X 1 on Y depends on X 2 value X1X Y Y = 1 + 2X 1 + 3X 2 + 4X 1 X 2

© 2003 Prentice-Hall, Inc. Chap Interaction Regression Model Worksheet Multiply X 1 by X 2 to get X 1 X 2 Run regression with Y, X 1, X 2, X 1 X 2 Case, iYiYi X 1i X 2i X 1i X 2i :::::

© 2003 Prentice-Hall, Inc. Chap Interpretation When There Are 3+ Levels MALE = 0 if female and 1 if male MARRIED = 1 if married; 0 if not DIVORCED = 1 if divorced; 0 if not MALEMARRIED = 1 if male married; 0 otherwise = (MALE times MARRIED) MALEDIVORCED = 1 if male divorced; 0 otherwise = (MALE times DIVORCED)

© 2003 Prentice-Hall, Inc. Chap Interpretation When There Are 3+ Levels (continued)

© 2003 Prentice-Hall, Inc. Chap Interpreting Results FEMALE Single: Married: Divorced: MALE Single: Married: Divorced: Main Effects : MALE, MARRIED and DIVORCED Interaction Effects : MALEMARRIED and MALEDIVORCED Difference

© 2003 Prentice-Hall, Inc. Chap Suppose X 1 and X 2 are Numerical Variables and X 3 is a Dummy-Variable To Test if the Slope of Y with X 1 and/or X 2 are the Same for the Two Levels of X 3 Model: Hypotheses: H 0 :   =   = 0 (No Interaction between X 1 and X 3 or X 2 and X 3 ) H 1 :  4 and/or  5  0 ( X 1 and/or X 2 Interacts with X 3 ) Perform a Partial F Test Evaluating the Presence of Interaction with Dummy-Variable

© 2003 Prentice-Hall, Inc. Chap Evaluating the Presence of Interaction with Numerical Variables Suppose X 1, X 2 and X 3 are Numerical Variables To Test If the Independent Variables Interact with Each Other Model: Hypotheses: H 0 :   =   =   = 0 (no interaction among X 1, X 2 and X 3 ) H 1 : at least one of  4,  5,  6  0 (at least one pair of X 1, X 2, X 3 interact with each other) Perform a Partial F Test

© 2003 Prentice-Hall, Inc. Chap Chapter Summary Developed the Multiple Regression Model Discussed Residual Plots Addressed Testing the Significance of the Multiple Regression Model Discussed Inferences on Population Regression Coefficients Addressed Testing Portions of the Multiple Regression Model Discussed Dummy-Variables and Interaction Terms