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- 1. ANOVA andLinear Regression<br />
- 2. Analysis of Variance(ANOVA)<br />
- 3. BUS B272 Unit 1<br />Analysis of Variance<br />The Analysis of Variance (ANOVA) is a procedure that tests to determine whether differences exist between two or more populations.<br />The techniques analyzes the variance of the data to determine whether we can infer that the populations differ.<br />
- 4. One way (Single-factor) analysis of variance<br />ANOVA assumptions<br />F test for difference among k means<br />BUS B272 Unit 1<br />Topics<br />
- 5. BUS B272 Unit 1<br />General Experimental Setting<br />Investigator controls one or more independent variables<br />Called treatments or factors<br />Each treatment contains two or more levels (or categories/classifications)<br />Observe effects on dependent variable<br />Response to different levels of independent variable<br />Experimental design: the plan used to test hypothesis<br />
- 6. BUS B272 Unit 1<br />Completely Randomized Design<br />Experimental units (subjects) are assigned randomly to treatments<br />Subjects are assumed homogeneous<br />Only one factor or independent variable<br />With two or more treatment levels<br />Analyzed by<br />One-way analysis of variance (one-way ANOVA)<br />
- 7. BUS B272 Unit 1<br />Randomized Design Example<br /><br /><br /><br />
- 8. BUS B272 Unit 1<br />One-way Analysis of Variance F Test<br />Evaluate the difference among the mean responses of 2 or more (k) populations<br />e.g. : Several types of tires, oven temperature settings, different types of marketing strategies<br />
- 9. BUS B272 Unit 1<br /><ul><li>Samples are randomly and independently drawn
- 10. This condition must be met
- 11. Populations are normally distributed
- 12. F test is robust to moderate departure from normality
- 13. Populations have equal variances</li></ul>Assumptions of ANOVA<br />
- 14. BUS B272 Unit 1<br />Hypotheses of One-Way ANOVA<br />All population means are equal <br />No treatment effect (no variation in means among groups)<br />At least one population mean is different (others may be the same!) <br />There is treatment effect <br />Does not mean that all population means are different<br />
- 15. BUS B272 Unit 1<br />One-way ANOVA (No Treatment Effect)<br />The Null Hypothesis is True<br />
- 16. BUS B272 Unit 1<br />One-way ANOVA (Treatment Effect Present)<br />The Null Hypothesis is NOT True<br />
- 17. BUS B272 Unit 1<br />One-way ANOVA(Partition of Total Variation)<br />Total Variation SS(Total)<br />Variation Due to Treatment SST<br />Variation Due to Random Sampling SSE<br />+<br />=<br />
- 18. BUS B272 Unit 1<br />ANOVA set-up<br />
- 19. BUS B272 Unit 1<br />Total Variation<br /> : the i-th observation in group j<br /> : the number of observations in group j<br />n : the total number of observations in all groups<br />k : the number of groups<br />the overall or grand mean<br />
- 20. BUS B272 Unit 1<br />Total Variation<br />(continued)<br />
- 21. BUS B272 Unit 1<br />Among-Treatments Variation<br />Variation Due to Differences Among Groups<br />
- 22. BUS B272 Unit 1<br />Among-Treatments Variation<br />(continued)<br />
- 23. BUS B272 Unit 1<br />Summing the variation within each treatment and then adding over all treatments.<br />Within-Treatment Variation<br />
- 24. BUS B272 Unit 1<br />Within-Treatment Variation<br />(continued)<br />
- 25. BUS B272 Unit 1<br />Within-Treatment Variation<br />(continued)<br /><ul><li>If more than 2 groups, use F test.
- 26. For 2 groups, use t-test. F test is more limited.</li></ul>For k = 2, this is the pooled-variance in the t-test.<br />
- 27. BUS B272 Unit 1<br />One-way ANOVAF Test Statistic<br />Test statistic:<br />MST is mean squares among or between variances<br />MSE is mean squares within or error variances<br />Degrees of freedom:<br />
- 28. BUS B272 Unit 1<br />One-way ANOVA Summary Table<br />
- 29. BUS B272 Unit 1<br />Features of One-way ANOVA F Statistic<br />The F statistic is the ratio of the among estimate of variance and the within estimate of variance.<br />The ratio must always be positive<br /> df1 = k -1 will typically be small<br />df2 = n - k will typically be large<br />The ratio should be closed to 1 if the null is true.<br />
- 30. BUS B272 Unit 1<br />One-way ANOVA F Test Example<br />As production manager, you want to see if three filling machines have different mean filling times. You assign 15 similarly trained and experienced workers, five per machine, to the machines. At the 0.05 significance level, is there a difference in mean filling times?<br />Machine1Machine2Machine3 25.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40<br />
- 31. BUS B272 Unit 1<br />One-way ANOVA Example: Scatter Diagram<br />Machine1Machine2Machine3 25.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40<br />Time in Seconds<br />27<br />26<br />25<br />24<br />23<br />22<br />21<br />20<br />19<br />•<br />•<br />•<br />•<br />•<br />•<br />•<br />•<br />•<br />•<br />•<br />•<br />•<br />•<br />•<br />
- 32. BUS B272 Unit 1<br />Machine 1Machine 2Machine 3 25.40 23.40 20.00 26.31 21.80 22.20 24.10 23.50 19.75 23.74 22.75 20.60 25.10 21.60 20.40<br />One-way ANOVA Example Computations<br />
- 33. BUS B272 Unit 1<br />
- 34. BUS B272 Unit 1<br />Summary Table<br />MST/MSE<br />=25.602<br />3-1=2<br />47.1640<br />23.5820<br />15-3=12<br />11.0532<br />0.9211<br />15-1=14<br />58.2172<br />
- 35. BUS B272 Unit 1<br /> = 0.05<br />F<br />0<br />One-way ANOVA Example Solution<br />Critical Value(s):<br />H0: 1 = 2 = 3<br />H1: Not all the means are equal<br />Test Statistic: <br />3.89<br />df1= 2 df2 = 12<br />Reject H0 at = 0.05<br />There is evidence to believe that at least one i differs from the rest.<br />
- 36. BUS B272 Unit 1<br />Computer Application<br />To obtain the Microsoft Excel computer output in the previous page, first enter the data into c columns in an Excel file, then follow the commands:<br /> Tools/ Data Analysis/ Anova: Single Factor<br />
- 37. BUS B272 Unit 1<br />Computer Output using Data Analysis of Excel<br />
- 38. Exercise 1<br />The manager of a large department store wants to test if the average size of customer transactions differs with four types of payment: Visa card, company card, cash or cheque. If there are differences in the average customer transaction size among the four types of payment, the manager will further investigate which types of payment will give rise to higher transaction volumes and hence he will design an appropriate promotional programme. A random sample of 54 customer transactions using various types of payment was drawn during the past two months. With reference to sampled data, the sample statistics are obtained as follows:<br />BUS B272 Unit 1<br />Test if differences of average customer transaction size exist among the four types of payment at a 0.05 level of significance.<br />
- 39. Exercise 1<br />BUS B272 Unit 1<br />One factor is involved, i.e. the type of payment. Under this factor, there are k = 4 treatments (or factor levels) which represent the four types of payment: Visa card, company card, cash and cheque. The experimental units are customer transactions.<br />
- 40. Exercise 1<br />Since the test statistic of 39.16 is greater than the critical value of 2.80, reject H0. At 0.05 level of significance, there is evidence to reveal that the average customer transaction sizes are significantly different among the four types of payment. <br />BUS B272 Unit 1<br />
- 41. Can ANOVA be replaced by t-Test?<br />t-Test : any difference between two population means μ1 and μ2<br />Multiple t-tests are required for more than two population means<br />Conducting multiple tests increases the probability of making Type I errors. <br /> E.g. compare 6 population means, if use ANOVA with significant level 5%, there will be a 5% chance we reject the null hypothesis when it is true. <br /> If we use t-test, we need to perform 15 tests and if same 5% significant level is set, the chance of a Type I error will be<br /> 1 – (1 - 0.05)15 = 0.54<br />BUS B272 Unit 1<br />
- 42. Linear Regression<br />
- 43. BUS B272 Unit 1<br />Linear Regression<br />Origin of regression<br />Determining the simple linear regression equation<br />Assessing the fitness of the model <br />Correlation analysis<br />Estimation and prediction <br />Assumptions of regression and correlation<br />
- 44. BUS B272 Unit 1<br />Origin of Regression<br />“Regression," from a Latin root meaning "going back," is a series of statistical methods used in studying the relationship between two variables and were first employed by Francis Galton in 1877. <br />Galton was interested in studying the relationship between a father’s height and the son’ s height. Making use of the “regression” method, he found that son’s height regress to the overall mean and the method is then called “regression”.<br />
- 45. BUS B272 Unit 1<br />Linear Regression Analysis<br />Linear Regression analysis is used primarily to model and describe linear relationship and provide prediction among variables <br />Predicts the value of a dependent (response) variable based on the value of at least one independent (explanatory) variable<br />Express statistically the effect of the independent variables on the dependent variable<br />
- 46. BUS B272 Unit 1<br />Types of Regression Models<br />Positive Linear Relationship<br />Relationship NOT Linear<br />Negative Linear Relationship<br />No Relationship<br />
- 47. BUS B272 Unit 1<br />Simple Linear Regression Model<br />The relationship between two variables, sayX and Y, is described by a linear function.<br />The change of the variable Y, (called dependent or response variable) is associated with the change in the other variable X(called independent or explanatory variable). <br />Explore the dependency of Y on X.<br />
- 48. (4, 5)<br />(2, 2.5)<br />(3, 2.5)<br />(1, 2)<br />Why Regression?<br />The larger the sum of squares, the poor the estimate.<br />X<br />1<br />2<br />3<br />4<br />Y<br />2<br />2.5<br />2.5<br />5<br />BUS B272 Unit 1<br />
- 49. BUS B272 Unit 1<br />Linear Relationship<br />We wish to study whether there is any association between two quantitative variables, sayX and Y<br />If ‘Y tends to increase as X increases’ <br />If ‘Y tends to decrease as X increases’<br /> If the corresponding magnitude of increase or decrease follows a specific proportion, the relationship identified is said to be a linear one.<br />– apositive relationship<br />– anegative relationship<br />
- 50. BUS B272 Unit 1<br />Scatter Diagram<br />A scatter diagram is a graph plotted for all X-Y pairs of the sample data.<br />By viewing a scatter diagram, one can determine whether a relationship exists between the two variables. It can also suggest the likely mathematical form of that relationship that allow one to judge initially and intuitively whether or not there exists a linear relationship between the two variables involved.<br />
- 51. BUS B272 Unit 1<br />Example<br />The level of air pollution at Kwun Tong and the total number of consultations relating to respiratory diseases in a public clinic in the area were recorded during a specific time period on 14 randomly selected days.<br />
- 52. BUS B272 Unit 1<br />Population Linear Regression<br />Population regression line is a straight line that describes the dependence of the average value (conditional mean) of one variable on the other <br />Random Error<br />Population SlopeCoefficient <br />Population Y intercept <br />Dependent (Response) Variable<br />PopulationRegression<br />Line <br />(conditional mean)<br />Independent (Explanatory) Variable<br />
- 53. BUS B272 Unit 1<br />Population Linear Regression<br />(continued)<br />Random Error (vertical discrepancies or residual for point i )<br />Y<br />(Observed Value of Y) =<br />(Conditional Mean)<br />X<br />Observed Value of Y<br />
- 54. BUS B272 Unit 1<br />Least Squares Method<br />The line fitted by least squares is the one that makes the sum of squares of all those vertical discrepancies (residuals) as small as possible, i.e. minimum of <br />which is the sum of squared residuals.<br />
- 55. BUS B272 Unit 1<br />Sample Y intercept<br />Residual<br />Sample regression line is formed by the point estimates of and , i.e., and . It provides an estimate of the population regression line as well as a predicted value of Y<br />Sample Linear Regression<br />Samplecoefficient of slope<br />Sample regression line <br />(Fitted regression line or predicted value)<br />
- 56. BUS B272 Unit 1<br />Sample Linear Regression<br />(continued)<br />and are obtained by finding the specific values of and that minimizes the sum of the squared residuals<br />
- 57. BUS B272 Unit 1<br />Coefficients of Sample Linear Regression<br />For <br />
- 58. BUS B272 Unit 1<br />Interpretation of the Slope and the Intercept<br />is the average value of Y when the value of X is zero.<br /> measures the change in the average value of Y as a result of a one-unit change in X.<br />
- 59. BUS B272 Unit 1<br />(continued)<br />is the estimated average value of Y when the value of X is zero.<br /> is the estimated change in the average value of Y as a result of one-unit change in X.<br />Interpretation of the Slope and the Intercept<br />
- 60. BUS B272 Unit 1<br />Example 1 : Simple Linear Regression<br />Suppose that you want to examine the linear dependency of the annual sales among seven stores on their size in square footage. Sample data for seven stores were obtained. Find the equation of the straight line that fits the data best.<br />Annual Store Square Sales Feet ($1000)<br /> 1 1,726 3,681<br /> 2 1,542 3,395<br /> 3 2,816 6,653<br /> 4 5,555 9,543<br /> 5 1,292 3,318<br /> 6 2,208 5,563<br /> 7 1,313 3,760 <br />
- 61. BUS B272 Unit 1<br />Example 1 : Scatter Diagram<br />Excel Output<br />
- 62. BUS B272 Unit 1<br />Computation of Regression Coefficient<br /> Annual <br /> Square Sales<br />Store Feet ($1000) <br />XY<br /> 1 1,726 3,681 <br /> 2 1,542 3,395 <br /> 3 2,816 6,653 <br /> 4 5,555 9,543 <br /> 5 1,292 3,318 <br /> 6 2,208 5,563 <br /> 7 1,313 3,760 <br /> 2,979,076<br /> 2,377,764<br /> 7,929,856<br />30,858,025<br /> 1,669,264<br /> 4,875,264<br /> 1,723,969<br />13,549,761<br />11,526,025<br />44,262,409<br />91,068,849<br />11,009,124<br />30,946,969<br />14,137,600<br /> 6,353,406 <br /> 5,235,090<br />18,734,848<br />53,011,365<br /> 4,286,856<br />12,283,104<br /> 4,936,880<br />16,452<br />35,913<br />104,841,549<br />52,413,218<br />216,500,737<br />
- 63. BUS B272 Unit 1<br />Computation of Regression Coefficient<br />
- 64. BUS B272 Unit 1<br />Example 1 : Equation for the Sample Regression Line<br />Yi = 1636.415 +1.487Xi<br /><br />
- 65. BUS B272 Unit 1<br />Example 1 : Interpretation of Results <br />The slope of 1.487 means that for each increase of one unit in X, we predict the average of Y to increase by an estimated 1.487 units.<br />The model estimates that for each increase of one square foot in the size of the store, the expected annual sales are predicted to increase by $1487.<br />
- 66. BUS B272 Unit 1<br />Predicting Annual Sales Based on Square Footage <br />Suppose that we would like to use the fitted model to predict the average annual sales for a store with 4,000 square feet.<br />
- 67. BUS B272 Unit 1<br />Interpolation versus Extrapolation<br />For using regression line for prediction purpose, it is not appropriate to make predictions beyond the relevant range (in the previous example: (1,292, 5,555)) of the independent variable.<br />That is, we may interpolate within the relevant range of X values, but we SHOULD NOT extrapolate beyond the range of X values. For example, it is not appropriate to predict the average annual sales for a store with 7,000 square feet since it is beyond the range of X values, i.e., (1,292, 5,555).<br />
- 68. BUS B272 Unit 1<br />Causal Relationship?<br />In general, when there is a relationship identified between X and Y using regression analysis, we usually would say that ‘X is associated with Y’ instead of saying ‘X causes Y’.<br />We cannot claim that two variables are related by cause and effect just because there is a statistical relationship between the two. In fact, you cannot infer a causal relationship from statistics alone. <br />
- 69. BUS B272 Unit 1<br />For example, the price of dog food and houses, may well be positively correlated over time. <br />When you collect data concerning the price of dog food and the price of houses over time, you might end up with an inference that they have a positive relationship, but can you conclude that an increase in the price of dog food would directly cause the price of houses to increase too? <br />It might be that an inflationary force is influencing both and hence they can be seen to move in the same general direction over time. <br />
- 70. BUS B272 Unit 1<br />Computer Application<br />Import the data into two adjacent columns in an Excel file and then click Tools/Data Analysis/ Regression(See page 624-5 for detail description).<br />
- 71. BUS B272 Unit 1<br />Example 1: Computer Output<br />
- 72. BUS B272 Unit 1<br />Exercise 2<br />Consider the example about the level of air pollution at Kwun Tong and the total number of consultations that relate to respiratory diseases in a public clinic in the area. The corresponding data were given as follows:<br />
- 73. BUS B272 Unit 1<br />Exercise 1<br />(a) Determine the sample regression line to predict the number of consultations by the level of pollution.<br />(b) Interpret the coefficients.<br />Solution:<br />
- 74. BUS B272 Unit 1<br />Exercise 1<br />For , each additional increase in pollution level, the number of consultations increases, on average by 0.456701074. <br />No meaningful interpretation for can be made, as the range of x does not include zero.<br />
- 75. BUS B272 Unit 1<br />Assessing the simple linear regression model<br />From time to time, after we have set up a linear regression model, we wish to assess the fitness of the model. That is, we wish to find out how well the model fit to the given data. For a good fit, the data as a whole should be quite close to the regression line and the independent variable can thus be used to predict the value of the dependent variable with high accuracy. <br />To examine how well the independent variable predicts the dependent variable, we need to develop several measures of variation.<br />
- 76. BUS B272 Unit 1<br />Total Sample Variability<br />Unexplained Variability<br />=<br />Explained Variability<br />+<br />Measure of Variation: The Sum of Squares<br />SS(Total) =SSR + SSE<br />
- 77. BUS B272 Unit 1<br />Measure of Variation: The Sum of Squares<br />SS(Total) = total sum of squares <br />Measures the variation of the Yi values around their mean Y<br />SSR = regression sum of squares <br />Explained variation attributable to the relationship between X and Y<br />SSE = error sum of squares <br />Variation attributable to factors other than the relationship between X and Y (Unexplained variation)<br />(continued)<br />
- 78. BUS B272 Unit 1<br />Measure of Variation: The Sum of Squares<br />_<br />SS(Total) = (Yi – Y )2<br />(continued)<br />Y<br />Yi<br /><br />SSE=(Yi - Yi)2<br />_<br /><br />_<br />SSR = (Yi - Y)2<br />_<br />Y<br />X<br />Xi<br />
- 79. BUS B272 Unit 1<br />
- 80. BUS B272 Unit 1<br />Standard Error of Estimate<br />The standard deviation of the variation of observations around the regression line. <br />
- 81. The smallest value that can assume is 0, which occurs when SSE = 0, that is, when all the points fall on the regression line. Thus, when is small, the fit is excellent, and the linear regression model is likely to be an effective analytical and forecasting tool.<br />When is large, the regression model is a poor one, it is of little value to be used.<br />BUS B272 Unit 1<br />Standard Error of Estimate<br />
- 82. BUS B272 Unit 1<br />The Coefficient of Determination (r 2 or R 2 )<br />By themselves, SSR, SSE and SS(Total) provide little that can be directly interpreted. A simple ratio of SSR and SS(Total) provides a measure of the usefulness of the regression equation.<br />Measures the proportion of variation in Y that is explained by the independent variable X in the regression model <br />
- 83. BUS B272 Unit 1<br />Coefficients of Determination (r 2)<br />r2 = 1<br />Y<br />Y<br />r2 = 1<br />^<br />Y<br /> = <br />b<br /> + <br />b<br />X<br />i<br />0<br />1<br />i<br />^<br />Y<br /> = <br />b<br /> + <br />b<br />X<br />i<br />0<br />1<br />i<br />X<br />X<br />r2 = 0<br />r2 = 0.8<br />Y<br />Y<br />^<br />^<br />Y<br /> = <br />b<br /> + <br />b<br />X<br />Y<br /> = <br />b<br /> + <br />b<br />X<br />i<br />0<br />1<br />i<br />i<br />0<br />1<br />i<br />X<br />X<br />
- 84. BUS B272 Unit 1<br />Coefficient of Correlation<br />Coefficient of correlation is used to measure strength of association (linear relationship) between two numerical variables)<br />Only concerned with strength of the relationship<br />No causal effect is implied<br />
- 85. BUS B272 Unit 1<br />(continued)<br />Population correlation coefficient is denoted by (Rho).<br />Sample correlation coefficient is denoted by r . It is an estimate of and is used to measure the strength of the linear relationship in the sample observations.<br />Coefficient of Correlation<br />
- 86. BUS B272 Unit 1<br />Coefficient of Correlation<br />
- 87. BUS B272 Unit 1<br />Sample of Observations from Various r Values<br />Y<br />Y<br />Y<br />X<br />X<br />X<br />r = –1<br />r = –0.6<br />r = 0<br />Y<br />Y<br />X<br />X<br />r = 0.6<br />r = 1<br />
- 88. BUS B272 Unit 1<br />Features of r and r<br />Unit free<br />Range between –1 and 1<br />The closer to –1, the stronger the negative linear relationship<br />The closer to 1, the stronger the positive linear relationship<br />The closer to 0, the weaker the linear relationship<br />
- 89. BUS B272 Unit 1<br />There is also a more systematic way to assess model fitness, i.e., to perform a hypothesis testing on the slope of the regression line.<br />Inference about the Slope<br />If the two variables involved are not at all linearly related, one could observe from the scatter diagram shown on the right that the slope of the regression line will be zero.<br />
- 90. BUS B272 Unit 1<br />Hence, we can determine whether a significant relationship between the variables X and Y exists by testing whether (the true slope) is equal to zero.<br />Inference about the Slope<br />(There is no linear relationship)<br />(There is a linear relationship)<br />If is rejected, there is evidence to believe that a linear relationship exists between X and Y.<br />
- 91. BUS B272 Unit 1<br />The standard error of the slope<br />The estimated standard error of . <br />
- 92. BUS B272 Unit 1<br />Inference about the Slope: t Test<br />t test for a population slope<br />Is there a linear dependency of Y on X ?<br />Null and alternative hypotheses<br />H0: 1 = 0 (no linear dependency)<br />H1: 1 0 (linear dependency)<br />Test statistic:<br />
- 93. BUS B272 Unit 1<br />Example: Store Sales<br />Data for Seven Stores:<br />Estimated Regression Equation:<br />Annual Store Square Sales Feet ($000)<br /> 1 1,726 3,681<br /> 2 1,542 3,395<br /> 3 2,816 6,653<br /> 4 5,555 9,543<br /> 5 1,292 3,318<br /> 6 2,208 5,563<br /> 7 1,313 3,760 <br /><br />Yi = 1636.415 +1.487Xi<br />The slope of this model is 1.487. <br />Is square footage of the store affecting its annual sales?<br />
- 94. H0: 1 = 0 0.05<br />H1: 1 0 df7 - 2 = 5<br />Test Statistic: <br />BUS B272 Unit 1<br />
- 95. BUS B272 Unit 1<br />Inferences about the Slope: t Test Example<br />Reject<br />Reject<br />0.025<br />0.025<br />0<br />2.5706<br />-2.5706<br />Decision:<br />Conclusion:<br />Critical Value(s):<br />Reject H0<br />At 5% level of significance, there is evidence to reveal that square footage is associated with annual sales.<br />
- 96. BUS B272 Unit 1<br />(No linear relationship)<br />(A linear relationship)<br />(No positive linear relationship)<br />(A positive linear relationship)<br />(No negative linear relationship)<br />(A negative linear relationship)<br />Inferences about the Slope<br />
- 97. BUS B272 Unit 1<br />Exercise 3<br /> Consider the data of Exercise 2 about the level of air pollution at Kwun Tong and the total number of consultations that relate to respiratory diseases in a public clinic in the area. <br />Test at the 5% level of significance to determine whether level of air pollution and the total number of consultations are positively linearly related.<br />
- 98. BUS B272 Unit 1<br />Solution:<br />0.05; df14 - 2 = 12<br />
- 99. BUS B272 Unit 1<br />Exercise 3<br />
- 100. BUS B272 Unit 1<br />Computer Output<br />For two-tailed test<br />
- 101. BUS B272 Unit 1<br />Exercise 3<br />Decision:<br />Conclusion:<br />Reject H0<br />Critical Value(s):<br />Reject H0<br />At 5% level of significance, there is evidence to believe that level of air pollution and total number of consultations are positively linearly related.<br />0.05<br />0<br />1.7823<br />
- 102. BUS B272 Unit 1<br />You have seen how can we assess the model fitness. If the model fits satisfactorily, we can use it to forecast and estimate values of the dependent variable. <br />We can obtain a point prediction of Y with a given value of X using the linear regression line.<br />Confidence interval about the particular value of Y or the average of Y for a given value of X can also be computed if desired.<br />Estimation of Mean Values<br />
- 103. BUS B272 Unit 1<br />Estimation of Mean Values<br />Confidence interval estimate for :<br />The mean of Y given a particular <br />Size of interval varies according to distance away from mean, <br />Standard error of the estimate<br />t value from table with df = n - 2<br />
- 104. BUS B272 Unit 1<br />Prediction of Individual Values<br />Prediction interval for individual response Yi at a particular <br />Addition of one increases width of interval from that for the mean of Y<br />
- 105. BUS B272 Unit 1<br />Interval Estimates for Different Values of X<br />Confidence Interval for the mean of Y<br />Prediction Interval for a individual Yi<br />Y<br /><br />Yi = b0 + b1Xi<br />X<br />Y given X<br />
- 106. BUS B272 Unit 1<br />Example: Stores Sales<br />Data for seven stores:<br />Predict the annual sales for a store with 2000 square feet.<br />Annual Store Square Sales Feet ($000)<br /> 1 1,726 3,681<br /> 2 1,542 3,395<br /> 3 2,816 6,653<br /> 4 5,555 9,543<br /> 5 1,292 3,318<br /> 6 2,208 5,563<br /> 7 1,313 3,760 <br />Regression Model Obtained:<br /><br />Yi = 1636.415 +1.487Xi<br />
- 107. Estimation of Mean Values: Example<br />Confidence Interval Estimate for<br />Find the 95% confidence interval for the average annual sales for a 2,000 square-foot store.<br /><br />Predicted Sales Yi = 1636.415 +1.487Xi = 4609.68 ($000)<br />tn-2 = t5 = 2.571<br />X = 2350.29<br />BUS B272 Unit 1<br />
- 108. Prediction Interval for Y : Example<br />Prediction Interval for Individual Y<br />Find the 95% prediction interval for the annual sales of a 2,000 square-foot store<br /><br />Predicted Sales Yi = 1636.415 +1.487Xi = 4609.68 ($000)<br />tn-2 = t5 = 2.571<br />X = 2350.29<br />BUS B272 Unit 1<br />
- 109. BUS B272 Unit 1<br />Computer Application<br />Commands:Tools/ Data Analysis Plus/ Prediction Interval.<br />
- 110. BUS B272 Unit 1<br />Computer Output<br />
- 111. BUS B272 Unit 1<br />Linear Regression Assumptions<br />1. Normality<br />Y values are normally distributed for each X<br />Probability distribution of error is normal<br />2. Homoscedasticity (Constant Variance)<br />3. Independence of Errors<br />
- 112. BUS B272 Unit 1<br /><ul><li>Y values are normally distributed around the regression line.
- 113. For each X value, the “spread” or variance around the regression line is the same.</li></ul>Variation of Errors around the Regression Line<br />f(e)<br />Y<br />X2<br />X1<br />X<br />Sample Regression Line<br />.<br />
- 114. Multiple Regression<br />
- 115. BUS B272 Unit 1<br />Introduction<br />Extension of the simple linear regression model to allow for any fixed number of independent variables. That is, the number of independent variables could be more than one.<br />
- 116. BUS B272 Unit 1<br />Multiple Linear Regression<br />To make use of computer printout to <br />Assess the model<br />How well it fits the data<br />Is it useful<br />Are any required conditions violated?<br />Employ the model<br />Interpreting the coefficients<br />Predictions using the prediction equation<br />Estimating the expected value of the dependent variable<br />
- 117. BUS B272 Unit 1<br />Allow for k independent variables to potentially be related to the dependent variable<br />y = b0 + b1x1+ b2x2 + …+ bkxk + e<br />Regression<br />Coefficients<br />Random error <br />variable<br />Dependent variable<br />Independent variables<br />Model and Required Conditions<br />
- 118. Multiple Regression for k = 2, Graphical Demonstration<br />X<br />1<br />The simple linear regression model<br />allows for one independent variable, “x”<br />for y = b0 + b1x + e<br />y<br />y = b0 + b1x1 + b2x2<br />y = b0 + b1x1 + b2x2<br />y = b0 + b1x1 + b2x2<br />y = b0 + b1x1 + b2x2<br />y = b0 + b1x1 + b2x2<br />y = b0 + b1x1 + b2x2<br />y = b0 + b1x1 + b2x2<br />The multiple linear regression model<br />allows for more than one independent variable.<br />Y = b0 + b1x1 + b2x2 + e<br />X2<br />BUS B272 Unit 1<br />
- 119. BUS B272 Unit 1<br />The errore is normally distributed.<br />The mean is equal to zero and the standard deviation is constant (se)for all values of y. <br />The errors are independent.<br />Required conditions for the error variable<br />
- 120. BUS B272 Unit 1<br />Estimating the Coefficients andAssessing the Model<br />The procedure used to perform multiple regression analysis:<br /><ul><li>Obtain the model coefficients and statistics using a statistical software.
- 121. Assess the model fitness using statistics obtained from the sample.
- 122. If the model assessment indicates good fit to the data, use it to interpret the coefficients and generate predictions.</li></li></ul><li>BUS B272 Unit 1<br />Estimating the Coefficients and Assessing the Model, Example<br />Example 18.1 Keller: Where to locate a new motor inn?<br />La Quinta Motor Inns is planning to build new inns.<br />Management wishes to predict which sites are likely to be profitable.<br />Several areas where predictors of profitability (operating margin) can be identified are:<br />Competition<br />Market awareness<br />Demand generators<br />Demographics<br />Physical quality<br />La Quinta defines profitable inns as those with an operating margin in excess of 50% and unprofitable ones with margins of less than 30%.<br />
- 123. Estimating the Coefficients and Assessing the Model, Example<br />Physical<br />Profitability<br />Margin (%)<br />Market <br />awareness<br />Competition<br />Customers<br />Community<br />Number<br />Office<br />space<br />Income<br />Distance<br />Nearest<br />Enrollment<br />Median<br />household<br />income <br />of nearby<br />area (in $thousands)<br />Number of <br />hotels/motels<br />rooms within <br />3 miles from <br />the site<br />Enrollemnt in nearby university or college (in thousands)<br />Distance to<br /> the downtown<br />core (in miles)<br />Number of miles to closest competition<br />Office space in nearby community<br />BUS B272 Unit 1<br />
- 124. BUS B272 Unit 1<br />Estimating the Coefficients and Assessing the Model, Example<br />Data were collected from randomly selected 100 inns that belong to La Quinta, and ran for the following suggested model:<br />Margin = b0 + b1Rooms + b2Nearest + b3Office + b4College + b5Income + b6Disttwn<br />Xm18-01<br />
- 125. BUS B272 Unit 1<br />Regression Analysis, Excel Output<br />Margin = 38.14 - 0.0076Number +1.65Nearest<br />+ 0.020Office Space +0.21Enrollment<br />+ 0.41Income - 0.23Distance<br />This is the sample regression equation <br />(sometimes called the prediction equation)<br />
- 126. BUS B272 Unit 1<br />Model Assessment<br />The model is assessed using two tools:<br />The coefficient of determination<br />The F -test of the analysis of variance<br />The standard error of estimates participates in building the above tools.<br />
- 127. BUS B272 Unit 1<br />Standard Error of Estimate<br />The standard deviation of the error is estimated by the Standard Error of Estimate:<br />The magnitude of seis judged by comparing it to <br />
- 128. BUS B272 Unit 1<br />From the printout, se = 5.51 <br />Calculating the mean value of y, we have<br />It seems se is not particularly small. <br />Question:Can we conclude the model does not fit the data well? <br />Standard Error of Estimate<br />
- 129. BUS B272 Unit 1<br />Coefficient of Determination<br />The definition is:<br />From the printout, r 2 = 0.5251<br />52.51% of the variation in operating margin is explained by the six independent variables. 47.49% remains unexplained.<br />
- 130. BUS B272 Unit 1<br />Testing the Validity of the Model<br />For testing the validity of the model, the following question is asked:<br /> Is there at least one independent variable linearly related to the dependent variable? <br />To answer the question we test the hypothesis<br />H0: b1 = b2 = … = bk = 0<br />H1: At least one bi is not equal to zero.<br />If at least one bi is not equal to zero, the model has some validity or usefulness. <br />
- 131. BUS B272 Unit 1<br />Testing the Validity of the La Quinta Inns Regression Model<br />The hypotheses are tested by an ANOVA procedure ( the Excel output)<br />MSR / MSE<br />k =<br />n–k–1 =<br /> n-1 = <br />SSR<br />MSR=SSR / k<br />SSE<br />MSE=SSE / (n-k-1)<br />
- 132. BUS B272 Unit 1<br />Testing the Validity of the La Quinta Inns Regression Model<br /> [Total variation in y] SS(Total) = SSR + SSE. <br /> Large F results from a large SSR. That implies much of the variation in y can be explained by the regression model; the model is useful, and thus, the null hypothesis should be rejected. Therefore, the rejection region is:<br />F > Fa, k, n – k – 1<br />while the test statistic is:<br />
- 133. BUS B272 Unit 1<br />Testing the Validity of the La Quinta Inns Regression Model<br />Fa, k, n-k-1 = F0.05,6,100-6 -1 = 2.17<br />F = 17.14 > 2.17<br />Conclusion: There is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. At least one of the bi is not equal to zero. Thus, at least one independent variable is linearly related to y. This linear regression model is valid.<br />Also, the p-value (Significance F) = 0.0000; Reject the null hypothesis. <br />
- 134. BUS B272 Unit 1<br />Interpreting the Coefficients<br />b0 = 38.14. This is the intercept, the value of y when all the variables take the value zero. Since the data range of all the independent variables do not cover the value zero, do not interpret the intercept. <br />b1 = – 0.0076. In this model, for each additional room within 3 mile of the La Quinta inn, the operating margin decreases on average by 0.0076% (assuming the other variables are held constant).<br />
- 135. BUS B272 Unit 1<br />Interpreting the Coefficients<br />b2 = 1.65. In this model, for each additional mile that the nearest competitor is to a La Quinta inn, the operating margin increases on average by 1.65% when the other variables are held constant. <br />b3 = 0.020.For each additional 1000 sq-ft of office space, the operating margin will increase on average by 0.02% when the other variables are held constant. <br />b4 = 0.21. For each additional thousand students the operating margin increases on average by 0.21% when the other variables are held constant. <br />
- 136. BUS B272 Unit 1<br />Interpreting the Coefficients<br />b5 = 0.41. For additional $1000 increase in median household income, the operating margin increases on average by 0.41%, when the other variables remain constant.<br />b6 = -0.23. For each additional mile to the downtown center, the operating margin decreases on average by 0.23% when the other variables are held constant.<br />
- 137. BUS B272 Unit 1<br />Testing the Coefficients<br />The hypothesis for each bi is<br />Excel printout<br />Test statistic:<br />H0: bi= 0<br />H1: bi¹ 0<br />d.f. = n - k -1<br />
- 138. BUS B272 Unit 1<br />Using the Linear Regression Equation<br />The model can be used for making predictions by<br />Producing prediction interval estimate for the particular value of y, for a given set of values of xi.<br />Producing a confidence interval estimate for the expected value of y, for a given set of values of xi.<br />The model can be used to learn about relationships between the independent variables xi, and the dependent variable y, by interpreting the coefficients bi<br />
- 139. BUS B272 Unit 1<br />La Quinta Inns, Predictions<br />Xm18-01<br />Predict the average operating margin of an inn at a site with the following characteristics:<br />3815 rooms within 3 miles,<br />Closet competitor 0.9 miles away,<br />476,000 sq-ft of office space,<br />24,500 college students,<br />$35,000 median household income,<br />11.2 miles away from downtown center.<br />MARGIN = 38.14 - 0.0076(3815)+1.65(0.9) + 0.020(476)<br /> +0.21(24.5) + 0.41(35) - 0.23(11.2) = 37.1%<br />
- 140. BUS B272 Unit 1<br />La Quinta Inns, Predictions<br />Interval estimates by Excel (Data Analysis Plus)<br />It is predicted, with 95% confidence that the operating margin will lie between 25.4% and 48.8%.<br />It is estimated the average operating margin of all sites that fit this category falls within 33% and 41.2%.<br />Both of them suggested that the given site would not be profitable (less than 50%).<br />

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