Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-1 Chapter 12 Simple Linear Regression Statistics for Managers Using Microsoft ® Excel 4 th Edition

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-2 Chapter Goals After completing this chapter, you should be able to:  Explain the simple linear regression model  Obtain and interpret the simple linear regression equation for a set of data  Evaluate regression residuals for aptness of the fitted model  Understand the assumptions behind regression analysis  Explain measures of variation and determine whether the independent variable is significant

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-3 Chapter Goals After completing this chapter, you should be able to:  Calculate and interpret confidence intervals for the regression coefficients  Use the Durbin-Watson statistic to check for autocorrelation  Form confidence and prediction intervals around an estimated Y value for a given X  Recognize some potential problems if regression analysis is used incorrectly (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-4 Correlation vs. Regression  A scatter plot (or scatter diagram) can be used to show the relationship between two variables  Correlation analysis is used to measure strength of the association (linear relationship) between two variables  Correlation is only concerned with strength of the relationship  No causal effect is implied with correlation  Correlation was first presented in Chapter 3

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-5 Introduction to Regression Analysis  Regression analysis is used to:  Predict the value of a dependent variable based on the value of at least one independent variable  Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to explain Independent variable: the variable used to explain the dependent variable

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-6 Simple Linear Regression Model  Only one independent variable, X  Relationship between X and Y is described by a linear function  Changes in Y are assumed to be caused by changes in X

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-7 Types of Relationships Y X Y X Y Y X X Linear relationshipsCurvilinear relationships

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-8 Types of Relationships Y X Y X Y Y X X Strong relationshipsWeak relationships (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap 12-9 Types of Relationships Y X Y X No relationship (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Linear component Simple Linear Regression Model The population regression model: Population Y intercept Population Slope Coefficient Random Error term Dependent Variable Independent Variable Random Error component

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap (continued) Random Error for this X i value Y X Observed Value of Y for X i Predicted Value of Y for X i XiXi Slope = β 1 Intercept = β 0 εiεi Simple Linear Regression Model

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap The simple linear regression equation provides an estimate of the population regression line Simple Linear Regression Equation Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) Y value for observation i Value of X for observation i The individual random error terms e i have a mean of zero

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Least Squares Method  b 0 and b 1 are obtained by finding the values of b 0 and b 1 that minimize the sum of the squared differences between Y and :

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Finding the Least Squares Equation  The coefficients b 0 and b 1, and other regression results in this chapter, will be found using Excel Formulas are shown in the text at the end of the chapter for those who are interested

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap  b 0 is the estimated average value of Y when the value of X is zero  b 1 is the estimated change in the average value of Y as a result of a one-unit change in X Interpretation of the Slope and the Intercept

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Simple Linear Regression Example  A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)  A random sample of 10 houses is selected  Dependent variable (Y) = house price in $1000s  Independent variable (X) = square feet

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Sample Data for House Price Model House Price in $1000s (Y) Square Feet (X)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Graphical Presentation  House price model: scatter plot

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Regression Using Excel  Tools / Data Analysis / Regression

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations10 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Square Feet The regression equation is:

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Graphical Presentation  House price model: scatter plot and regression line Slope = Intercept =

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Interpretation of the Intercept, b 0  b 0 is the estimated average value of Y when the value of X is zero (if X = 0 is in the range of observed X values)  Here, no houses had 0 square feet, so b 0 = just indicates that, for houses within the range of sizes observed, $98, is the portion of the house price not explained by square feet

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Interpretation of the Slope Coefficient, b 1  b 1 measures the estimated change in the average value of Y as a result of a one- unit change in X  Here, b 1 = tells us that the average value of a house increases by.10977($1000) = $109.77, on average, for each additional one square foot of size

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Predict the price for a house with 2000 square feet: The predicted price for a house with 2000 square feet is ($1,000s) = $317,850 Predictions using Regression Analysis

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Interpolation vs. Extrapolation  When using a regression model for prediction, only predict within the relevant range of data Relevant range for interpolation Do not try to extrapolate beyond the range of observed X’s

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Measures of Variation  Total variation is made up of two parts: Total Sum of Squares Regression Sum of Squares Error Sum of Squares where: = Average value of the dependent variable Y i = Observed values of the dependent variable i = Predicted value of Y for the given X i value

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap  SST = total sum of squares  Measures the variation of the Y i values around their mean Y  SSR = regression sum of squares  Explained variation attributable to the relationship between X and Y  SSE = error sum of squares  Variation attributable to factors other than the relationship between X and Y (continued) Measures of Variation

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap (continued) XiXi Y X YiYi SST =  (Y i - Y) 2 SSE =  (Y i - Y i ) 2  SSR =  (Y i - Y) 2  _ _ _ Y  Y Y _ Y  Measures of Variation

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap  The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable  The coefficient of determination is also called r-squared and is denoted as r 2 Coefficient of Determination, r 2 note:

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap r 2 = 1 Examples of Approximate r 2 Values Y X Y X r 2 = 1 Perfect linear relationship between X and Y: 100% of the variation in Y is explained by variation in X

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Examples of Approximate r 2 Values Y X Y X 0 < r 2 < 1 Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Examples of Approximate r 2 Values r 2 = 0 No linear relationship between X and Y: The value of Y does not depend on X. (None of the variation in Y is explained by variation in X) Y X r 2 = 0

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations10 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Square Feet % of the variation in house prices is explained by variation in square feet

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Standard Error of Estimate  The standard deviation of the variation of observations around the regression line is estimated by Where SSE = error sum of squares n = sample size

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations10 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Square Feet

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Comparing Standard Errors YY X X S YX is a measure of the variation of observed Y values from the regression line The magnitude of S YX should always be judged relative to the size of the Y values in the sample data i.e., S YX = $41.33K is moderately small relative to house prices in the $200 - $300K range

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Assumptions of Regression  Normality of Error  Error values (ε) are normally distributed for any given value of X  Homoscedasticity  The probability distribution of the errors has constant variance  Independence of Errors  Error values are statistically independent

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Residual Analysis  The residual for observation i, e i, is the difference between its observed and predicted value  Check the assumptions of regression by examining the residuals  Examine for linearity assumption  Examine for constant variance for all levels of X (homoscedasticity)  Evaluate normal distribution assumption  Evaluate independence assumption  Graphical Analysis of Residuals  Can plot residuals vs. X

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Residual Analysis for Linearity Not Linear Linear x residuals x Y x Y x

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Residual Analysis for Homoscedasticity Non-constant variance Constant variance xx Y x x Y residuals

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Residual Analysis for Independence Not Independent Independent X X residuals X

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Excel Residual Output RESIDUAL OUTPUT Predicted House PriceResiduals Does not appear to violate any regression assumptions

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap  Used when data are collected over time to detect if autocorrelation is present  Autocorrelation exists if residuals in one time period are related to residuals in another period Measuring Autocorrelation: The Durbin-Watson Statistic

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Autocorrelation  Autocorrelation is correlation of the errors (residuals) over time  Violates the regression assumption that residuals are random and independent  Here, residuals show a cyclic pattern, not random

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap The Durbin-Watson Statistic  The possible range is 0 ≤ D ≤ 4  D should be close to 2 if H 0 is true  D less than 2 may signal positive autocorrelation, D greater than 2 may signal negative autocorrelation  The Durbin-Watson statistic is used to test for autocorrelation H 0 : residuals are not correlated H 1 : autocorrelation is present

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Testing for Positive Autocorrelation  Calculate the Durbin-Watson test statistic = D (The Durbin-Watson Statistic can be found using PHStat in Excel) Decision rule: reject H 0 if D < d L H 0 : positive autocorrelation does not exist H 1 : positive autocorrelation is present 0dUdU 2dLdL Reject H 0 Do not reject H 0  Find the values d L and d U from the Durbin-Watson table (for sample size n and number of independent variables k) Inconclusive

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap  Example with n = 25: Durbin-Watson Calculations Sum of Squared Difference of Residuals Sum of Squared Residuals Durbin-Watson Statistic Testing for Positive Autocorrelation (continued) Excel/PHStat output:

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap  Here, n = 25 and there is k = 1 one independent variable  Using the Durbin-Watson table, d L = 1.29 and d U = 1.45  D = < d L = 1.29, so reject H 0 and conclude that significant positive autocorrelation exists  Therefore the linear model is not the appropriate model to forecast sales Testing for Positive Autocorrelation (continued) Decision: reject H 0 since D = < d L 0 d U = d L =1.29 Reject H 0 Do not reject H 0 Inconclusive

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Inferences About the Slope  The standard error of the regression slope coefficient (b 1 ) is estimated by where: = Estimate of the standard error of the least squares slope = Standard error of the estimate

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations10 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Square Feet

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Comparing Standard Errors of the Slope Y X Y X is a measure of the variation in the slope of regression lines from different possible samples

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Inference about the Slope: t Test  t test for a population slope  Is there a linear relationship between X and Y?  Null and alternative hypotheses H 0 : β 1 = 0(no linear relationship) H 1 : β 1  0(linear relationship does exist)  Test statistic where: b 1 = regression slope coefficient β 1 = hypothesized slope S b1 = standard error of the slope

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap House Price in $1000s (y) Square Feet (x) Estimated Regression Equation: The slope of this model is Does square footage of the house affect its sales price? Inference about the Slope: t Test (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 From Excel output: CoefficientsStandard Errort StatP-value Intercept Square Feet t b1b1

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 Test Statistic: t = There is sufficient evidence that square footage affects house price From Excel output: Reject H 0 CoefficientsStandard Errort StatP-value Intercept Square Feet tb1b1 Decision: Conclusion: Reject H 0  /2=.025 -t α/2 Do not reject H 0 0 t α/2  /2= d.f. = 10-2 = 8 (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1  0 P-value = There is sufficient evidence that square footage affects house price From Excel output: Reject H 0 CoefficientsStandard Errort StatP-value Intercept Square Feet P-value Decision: P-value < α so Conclusion: (continued) This is a two-tail test, so the p-value is P(t > 3.329)+P(t < ) = (for 8 d.f.)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap F-Test for Significance  F Test statistic: where where F follows an F distribution with k numerator and (n – k - 1) denominator degrees of freedom (k = the number of independent variables in the regression model)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations10 ANOVA dfSSMSFSignificance F Regression Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Square Feet With 1 and 8 degrees of freedom P-value for the F-Test

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap H 0 : β 1 = 0 H 1 : β 1 ≠ 0  =.05 df 1 = 1 df 2 = 8 Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is sufficient evidence that house size affects selling price 0  =.05 F.05 = 5.32 Reject H 0 Do not reject H 0 Critical Value: F  = 5.32 F-Test for Significance (continued) F

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: Excel Printout for House Prices: At 95% level of confidence, the confidence interval for the slope is (0.0337, ) CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Square Feet d.f. = n - 2

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $ per square foot of house size CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept Square Feet This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the.05 level of significance Confidence Interval Estimate for the Slope (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap t Test for a Correlation Coefficient  Hypotheses H 0 : ρ = 0 (no correlation between X and Y) H A : ρ ≠ 0 (correlation exists)  Test statistic  (with n – 2 degrees of freedom)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Example: House Prices Is there evidence of a linear relationship between square feet and house price at the.05 level of significance? H 0 : ρ = 0 (No correlation) H 1 : ρ ≠ 0 (correlation exists)  =.05, df = = 8

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Example: Test Solution Conclusion: There is evidence of a linear association at the 5% level of significance Decision: Reject H 0 Reject H 0  /2=.025 -t α/2 Do not reject H 0 0 t α/2  /2= d.f. = 10-2 = 8

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Estimating Mean Values and Predicting Individual Values Y X X i Y = b 0 +b 1 X i  Confidence Interval for the mean of Y, given X i Prediction Interval for an individual Y, given X i Goal: Form intervals around Y to express uncertainty about the value of Y for a given X i Y 

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Confidence Interval for the Average Y, Given X Confidence interval estimate for the mean value of Y given a particular X i Size of interval varies according to distance away from mean, X

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Prediction Interval for an Individual Y, Given X Confidence interval estimate for an Individual value of Y given a particular X i This extra term adds to the interval width to reflect the added uncertainty for an individual case

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Estimation of Mean Values: Example Find the 95% confidence interval for the mean price of 2,000 square-foot houses Predicted Price Y i = ($1,000s)  Confidence Interval Estimate for μ Y|X=X The confidence interval endpoints are and , or from $280,660 to $354,900 i

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Estimation of Individual Values: Example Find the 95% prediction interval for an individual house with 2,000 square feet Predicted Price Y i = ($1,000s)  Prediction Interval Estimate for Y X=X The prediction interval endpoints are and , or from $215,500 to $420,070 i

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Finding Confidence and Prediction Intervals in Excel  In Excel, use PHStat | regression | simple linear regression …  Check the “confidence and prediction interval for X=” box and enter the X-value and confidence level desired

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Input values Finding Confidence and Prediction Intervals in Excel (continued) Confidence Interval Estimate for μ Y|X=Xi Prediction Interval Estimate for Y X=Xi Y 

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Pitfalls of Regression Analysis  Lacking an awareness of the assumptions underlying least-squares regression  Not knowing how to evaluate the assumptions  Not knowing the alternatives to least-squares regression if a particular assumption is violated  Using a regression model without knowledge of the subject matter  Extrapolating outside the relevant range

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Strategies for Avoiding the Pitfalls of Regression  Start with a scatter plot of X on Y to observe possible relationship  Perform residual analysis to check the assumptions  Plot the residuals vs. X to check for violations of assumptions such as homoscedasticity  Use a histogram, stem-and-leaf display, box-and- whisker plot, or normal probability plot of the residuals to uncover possible non-normality

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Strategies for Avoiding the Pitfalls of Regression  If there is violation of any assumption, use alternative methods or models  If there is no evidence of assumption violation, then test for the significance of the regression coefficients and construct confidence intervals and prediction intervals  Avoid making predictions or forecasts outside the relevant range (continued)

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Chapter Summary  Introduced types of regression models  Reviewed assumptions of regression and correlation  Discussed determining the simple linear regression equation  Described measures of variation  Discussed residual analysis  Addressed measuring autocorrelation

Statistics for Managers Using Microsoft Excel, 4e © 2004 Prentice-Hall, Inc. Chap Chapter Summary  Described inference about the slope  Discussed correlation -- measuring the strength of the association  Addressed estimation of mean values and prediction of individual values  Discussed possible pitfalls in regression and recommended strategies to avoid them (continued)