Statr session 23 and 24

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Statr session 23 and 24

  1. 1. Simple Regression Analysis • Bivariate (two variables) linear regression -- the most elementary regression model – dependent variable, the variable to be predicted, usually called Y – independent variable, the predictor or explanatory variable, usually called X – Usually the first step in this analysis is to construct a scatter plot of the data • Nonlinear relationships and regression models with more than one independent variable can be explored by using multiple regression models
  2. 2. Linear Regression Models • Deterministic Regression Model - - produces an exact output: • Probabilistic Regression Model • 0 and 1 are population parameters • 0 and 1 are estimated by sample statistics b0 and b1 0 1 ˆy x   0 1 ˆy x    
  3. 3. Equation of the Simple Regression Line
  4. 4. A typical regression line X Y 𝑏0 ϴ Slope = 𝑏1 = 𝑡𝑎𝑛𝜃 y-intercept = 𝑏0
  5. 5. Hypothesis Tests for the Slope of the Regression Model • A hypothesis test can be conducted on the sample slope of the regression model to determine whether the population slope is significantly different from zero. • Using the non-regression model (the 𝑦 model) as a worst case, the researcher can analyze the regression line to determine whether it adds a more significant amount of predictability of y than does the model.
  6. 6. Hypothesis Tests for the Slope of the Regression Model • As the slope of the regression line diverges from zero, the regression model is adding predictability that the line is not generating. • Testing the slope of the regression line to determine whether the slope is different from zero is important. • If the slope is not different from zero, the regression line is doing nothing more than the average line of y predicting y 𝑦 model
  7. 7. Hypothesis Tests for the Slope of the Regression Model
  8. 8. Solving for 𝑏1 and 𝑏0 of the Regression Line: Airline Cost Data Airlines Cost Data include the costs and associated number of passengers for twelve 500-mile commercial airline flights using Boeing 737s during the same season of the year. Number of Cost Passengers ($1,000) 61 4,280 63 4,080 67 4,420 69 4,170 70 4,480 74 4,300 76 4,820 81 4,700 86 5,110 91 5,130 95 5,640 97 5,560
  9. 9. Hypothesis Test: Airline Cost Example 0 0 10,025. Hrejectnotdo,228.2228.2 Hreject,228.2|| 228.2 05. 102102      tIf tIf ndf t 
  10. 10. Hypothesis Test: Airline Cost Example |t| = 9.44 > 2.228 so reject H0 Note: P-value = 0.000
  11. 11. Hypothesis Test: Airline Cost Example • The t value calculated from the sample slope falls in the rejection region and the p-value is .00000014. • The null hypothesis that the population slope is zero is rejected. • This linear regression model is adding significantly more predictive information to the model (no regression).
  12. 12. Comparison of F and t values • ANOVA can be used to test hypotheses about the difference in two means • Analysis of data from two samples by both a t test and ANOVA show that Observed F = Square of Observed t for dfc = 1 • The t test for two independent samples is a special case one-way ANOVA when there are two treatment levels (dfc = 1)
  13. 13. Testing the Overall Model • It is common in regression analysis to compute an F test to determine the overall significance of the model. • In multiple regression, this test determines whether at least one of the regression coefficients (from multiple predictors) is different from zero. • Simple regression provides only one predictor and only one regression coefficient to test. • Because the regression coefficient is the slope of the regression line, the F test for overall significance is testing the same thing as the t test in simple regression
  14. 14. Testing the Overall Model
  15. 15. Testing the Overall Model F = 89.09 > 4.96 so reject H0 Note: P-value = 0.000
  16. 16. Testing the Overall Model • The difference between the F value (89.09) and the value obtained by squaring the t statistic (88.92) is due to rounding error. • The probability of obtaining an F value this large or larger by chance if there is no regression prediction in this model is .000 according to the ANOVA output (the p-value).
  17. 17. Estimation • One of the main uses of regression analysis is as a prediction tool. • If the regression function is a good model, the researcher can use the regression equation to determine values of the dependent variable from various values of the independent variable. • In simple regression analysis, a point estimate prediction of y can be made by substituting the associated value of x into the regression equation and solving for y.
  18. 18. Point Estimation for the Airline Cost Example
  19. 19. Confidence Interval of Estimate of the Conditional Mean of y • The regression line is determined by a sample set of points. For different samples, the regression equations will be different, yielding different Point Estimates. • Hence a Confidence Interval (CI) of estimation is often useful because for any value of independent variable (x), there can be many values of dependent variable (y). • One type of C.I. is an estimate of the average value of y for a given value of x and is designated as E(yx)
  20. 20. Confidence Interval of Estimate of the Conditional Mean of y • The regression line is determined by a sample set of points. For different samples, the regression equations will be different, yielding different Point Estimates. • Hence a Confidence Interval (CI) of estimation is often useful because for any value of independent variable (x), there can be many values of dependent variable (y). • One type of C.I. is an estimate of the average value of y for a given value of x and is designated as E(yx)
  21. 21. Prediction Interval of Estimate of a Single Value y • The second type of interval in regression estimation to estimate a single value of y for a given value of x • The P.I. is wider than C.I. • The P.I. takes into account all the y values for a given x
  22. 22. Intervals for Estimation: Airline Cost Example
  23. 23. Multiple Regression Models Regression analysis with two or more independent variables or with at least one nonlinear predictor is called multiple regression analysis.
  24. 24. Regression Models Probabilistic Multiple Regression Model Y = 0 + 1X1 + 2X2 + 3X3 + . . . + kXk+  Y = the value of the dependent (response) variable 0 = the regression constant 1 = the partial regression coefficient of independent variable 1 2 = the partial regression coefficient of independent variable 2 k = the partial regression coefficient of independent variable k k = the number of independent variables  = the error of prediction
  25. 25. Regression Models • In multiple regression analysis, the dependent variable y is sometimes referred to as the response variable. • The partial regression coefficient of an independent variable βi represents the increase that will occur in the value of y from a one-unit increase in that independent variable if all other variables are held constant. • The partial regression coefficients occur because more than one predictor is included in a model.
  26. 26. Estimated Regression Models
  27. 27. Multiple Regression Model with 2 Independent Variables (First-Order) • The simplest multiple regression model is one constructed with two independent variables, where the highest power of either variable is 1 (first-order regression model). • In multiple regression analysis, the resulting model produces a response surface.
  28. 28. Multiple Regression Model with 2 Independent Variables (First-Order) 1 20 1 2 0 1 2 : = the regression constant the partial regression coefficient for independent variable 1 the partial regression coefficient for independent variable 2 = the error of pred where Y X X              1 20 1 2 0 1 2 iction ˆ: predicted value of Y estimate of regression constant estimate of regression coefficient 1 estimate of regression coefficient 2 ˆ where Y Y b b bX X b b b        Population Model Estimated Model
  29. 29. Response Plane for First-Order Two-Predictor Multiple Regression Model • In multiple regression analysis, the resulting model produces a response surface. • In the multiple regression model shown on the next slide with two independent first-order variables, the response surface is a response plane. • The response plane for such a model is fit in a three-dimensional space (x1, x2, y).
  30. 30. Response Plane for First-Order Two-Predictor Multiple Regression Model
  31. 31. Determining the Multiple Regression Equation • The simple regression equations for determining the sample slope and intercept given in earlier material are the result of using methods of calculus to minimize the sum of squares of error for the regression model. • The formulas are established to meet an objective of minimizing the sum of squares of error for the model. • The regression analysis shown here is referred to as least squares analysis. Methods of calculus are applied, resulting in k + 1 equations with k + 1 unknowns for multiple regression analyses with k independent variables.
  32. 32. Least Squares Equations for k = 2
  33. 33. Multiple Regression Model • A real estate study was conducted in a small Louisiana city to determine what variables, if any, are related to the market price of a home. • Suppose the researcher wants to develop a regression model to predict the market price of a home by two variables, “total number of square feet in the house” and “the age of the house.”
  34. 34. Real Estate Data Observation Y X1 X2 Observation Y X1 X2 1 63.0 65.1 1,605 35 13 79.7 2,121 14 2 2,489 45 14 84.5 2,485 9 3 69.9 7 1,553 20 15 96.0 2,300 19 4 76.8 2,404 32 16 109.5 2,714 4 5 73.9 1,884 25 17 102.5 2,463 5 6 77.9 1,558 14 18 121.0 3,076 7 7 74.9 1,748 8 19 104.9 3,048 3 8 78.0 3,105 10 20 128.0 3,267 6 9 79.0 1,682 28 21 129.0 3,069 10 10 63.4 2,470 30 22 117.9 4,765 11 11 79.5 1,820 2 23 140.0 4,540 8 12 83.9 2,143 6 Market Price ($1,000) Square Feet Age (Years) Market Price ($1,000) Square Feet Age (Years)
  35. 35. Package Output for the Real Estate Example The regression equation is Price = 57.4 + 0.0177 Sq.Feet - 0.666 Age Predictor Coef StDev T P Constant 57.35 10.01 5.73 0.000 Sq.Feet 0.017718 0.003146 5.63 0.000 Age -0.6663 0.2280 -2.92 0.008 S = 11.96 R-Sq = 74.1% R-Sq(adj) = 71.5% Analysis of Variance Source DF SS MS F P Regression 2 8189.7 4094.9 28.63 0.000 Residual Error 20 2861.0 143.1 Total 22 11050.7
  36. 36. Predicting the Price of Home
  37. 37. Evaluating the Multiple Regression Model H H k a 0 1 2 3 0: :           At least one of the regression coefficients is 0 H H H H H H H H a a a k a k 0 1 1 0 3 3 0 2 2 0 0 0 0 0 0 0 0 0 : : : : : : : :                  Significance Tests for Individual Regression Coefficients Testing the Overall Model
  38. 38. Testing the Overall Model for the Real Estate Example • It is important to test the model to determine whether it fits the data well and the assumptions underlying regression analysis are met. • With simple regression, a t test of the slope of the regression line is used to determine whether the population slope of the regression line is different from zero. • Fail to reject the null hypothesis - the regression model has no significant predictability for the dependent variable.
  39. 39. Testing the Overall Model for the Real Estate Example • A rejection of the null hypothesis indicates that at least one of the independent variables is adding significant predictability for y. • The F value is 28.63; because p = 0.000, the F value is significant at = 0.001. • The null hypothesis is rejected, and there is at least one significant predictor of house price in this analysis.
  40. 40. Testing the Overall Model for the Real Estate Example ANOVA df SS MS F p Regression 2 8189.723 4094.86 28.63 .000 Residual (Error) 20 2861.017 143.1 Total 22 11050.74
  41. 41. Significance Test: Regression Coefficients for the Real Estate Example t.025,20 = 2.086 tCal = 5.63 > 2.086, reject H0. Coefficients Std Dev t Stat p x1 (Sq.Feet) 0.0177 0.003146 5.63 .000 x2 (Age) -0.666 0.2280 -2.92 .008
  42. 42. Residuals • The residual, or error, of the regression model is the difference between the actual 𝑦 value and its predicted value 𝑦 which is 𝑦 - 𝑦 • The residuals for a multiple regression model are solved for in the same manner as they are with simple regression. • First, a predicted value of 𝑦 is determined by entering the value for each independent variable for a given set of observations into the multiple regression equation.
  43. 43. Residuals • Residuals are also helpful in locating outliers. • Outliers are data points that are apart, or far, from the mainstream of the other data. • They are sometimes data points that were mistakenly recorded or measured. • Because every data point influences the regression model, outliers can exert an overly important influence on the model based on their distance from other points.
  44. 44. Sum of Squares Error • In an effort to compute a single statistic that can represent the error in a regression analysis, the zero-sum property can be overcome by squaring the residuals and then summing the squares. • Such an operation produces the sum of squares of error (SSE).
  45. 45. Residuals and Sum of Squares Error for the Real Estate Example SSE Observation Y Observation Y 1 43.0 42.466 0.534 0.285 13 59.7 65.602 -5.902 34.832 2 45.1 51.465 -6.365 40.517 14 64.5 75.383 -10.883 118.438 3 49.9 51.540 -1.640 2.689 15 76.0 65.442 10.558 111.479 4 56.8 58.622 -1.822 3.319 16 89.5 82.772 6.728 45.265 5 53.9 54.073 -0.173 0.030 17 82.5 77.659 4.841 23.440 6 57.9 55.627 2.273 5.168 18 101.0 87.187 13.813 190.799 7 54.9 62.991 -8.091 65.466 19 84.9 89.356 -4.456 19.858 8 58.0 85.702 -27.702 767.388 20 108.0 91.237 16.763 280.982 9 59.0 48.495 10.505 110.360 21 109.0 85.064 23.936 572.936 10 63.4 61.124 2.276 5.181 22 97.9 114.447 -16.547 273.815 11 59.5 68.265 -8.765 76.823 23 120.0 112.460 7.540 56.854 12 63.9 71.322 -7.422 55.092 2861.017 Y Y Y    2 Y Y  Y Y Y    2 Y Y 
  46. 46. General Linear Regression Model Regression models presented thus far are based on the general linear regression model, which has the form Y = 0 + 1X1 + 2X2 + 3X3 + . . . + kXk+  Y = the value of the dependent (response) variable 0 = the regression constant 1 = the partial regression coefficient of independent variable 1 2 = the partial regression coefficient of independent variable 2 k = the partial regression coefficient of independent variable k k = the number of independent variables  = the error of prediction
  47. 47. General Linear Regression Model • In the general linear model, the parameters, βi, are linear. • However, dependent variable, y, is not necessarily linearly related to the predictor variables. • Multiple regression response surfaces are not restricted to linear surfaces and may be curvilinear. • Regression models can be developed for more than two predictors.
  48. 48. Polynomial Regression • Regression models in which the highest power of any predictor variable is 1 and in which there are no interaction terms are referred to as first-order models • If a second independent variable is added, the model is referred to as a first-order model with two independent variables • Polynomial regression models are regression models that are second- or higher-order models - contain squared, cubed, or higher powers of the predictor variable(s)
  49. 49. Non Linear Models: Mathematical Transformation Y X X   0 1 1 2 2    First-order with Two Independent Variables Second-order with One Independent Variable Second-order with an Interaction Term Second-order with Two Independent Variables Y X X   0 1 1 2 1 2     Y X X X X    0 1 1 2 2 3 1 2     Y X X X X X X      0 1 1 2 2 3 1 2 4 2 2 5 1 2      
  50. 50. Sales Data and Scatter Plot for 13 Manufacturing Companies • Consider the table in the next slide. • The table contains sales for 13 manufacturing companies along with the number of manufacturer representatives associated with each firm. • A simple regression analysis to predict sales by the number of manufacturer’s representatives results in the Excel output.
  51. 51. Sales Data and Scatter Plot for 13 Manufacturing Companies 0 50 100 150 200 250 300 350 400 450 500 0 2 4 6 8 10 12 Number of Representatives Sales Manufacturer Sales ($1,000,000) Number of Manufacturing Representatives 1 2.1 2 2 3.6 1 3 6.2 2 4 10.4 3 5 22.8 4 6 35.6 4 7 57.1 5 8 83.5 5 9 109.4 6 10 128.6 7 11 196.8 8 12 280.0 10 13 462.3 11
  52. 52. Excel Simple Linear Regression Output for the Manufacturing Example Regression Statistics Multiple R 0.933 R Square 0.870 Adjusted R Square 0.858 Standard Error 51.10 Observations 13 Coefficients Standard Error t Stat P-value Intercept -107.03 28.737 -3.72 0.003 numbers 41.026 4.779 8.58 0.000 ANOVA df SS MS F Significance F Regression 1 192395 192395 73.69 0.000 Residual 11 28721 2611 Total 12 221117
  53. 53. Sales Data and Scatter Plot for 13 Manufacturing Companies • Researcher creates a second predictor variable, (number of manufacturer’s representatives2) to use in the regression analysis to predict sales along with number of manufacturer’s representatives • This variable can be created to explore second- order parabolic relationships by squaring the data from the independent variable of the linear model and entering it into the analysis • With the new data, a multiple regression model can be developed
  54. 54. Manufacturing Data with Newly Created Variable Manufacturer Sales ($1,000,000) Number of Mgfr Reps X1 (No. Mgfr Reps)2 X2 = (X1)2 1 2.1 2 4 2 3.6 1 1 3 6.2 2 4 4 10.4 3 9 5 22.8 4 16 6 35.6 4 16 7 57.1 5 25 8 83.5 5 25 9 109.4 6 36 10 128.6 7 49 11 196.8 8 64 12 280.0 10 100 13 462.3 11 121
  55. 55. Package output for Quadratic Model to Predict Sales Regression Statistics Multiple R 0.986 R Square 0.973 Adjusted R Square 0.967 Standard Error 24.593 Observations 13 Coefficients Standard Error t Stat P-value Intercept 18.067 24.673 0.73 0.481 MfgrRp -15.723 9.5450 - 1.65 0.131 MfgrRpSq 4.750 0.776 6.12 0.000 ANOVA df SS MS F Significance F Regression 2 215069 107534 177.79 0.000 Residual 10 6048 605 Total 12 221117
  56. 56. Tukey’s Ladder of Transformations • Tukey’s ladder of expressions can be used to straighten out a plot of x and y. • Tukey used a four-quadrant approach to show which expressions on the ladder are more appropriate for a given situation. • If the scatter plot of x and y indicates a shape like that shown in the upper left quadrant, recoding should move “down the ladder” for the x variable toward or “up the ladder” for the y variable toward. • If the scatter plot of x and y indicates a shape like that of the lower right quadrant, the recoding should move “up the ladder” for the x variable toward or “down the ladder” for the y variable toward.
  57. 57. Tukey’s Four Quadrant Approach
  58. 58. Regression Models with Interaction • When two different independent variables are used in a regression analysis, an interaction occurs between the two variables • Interaction can be examined as a separate independent variable • An interaction predictor variable can be designed by multiplying the data values of one variable by the values of another variable, thereby creating a new variable
  59. 59. Example – Three Stocks Suppose the data in the following table represent the closing stock prices for three corporations over a period of 15 months. An investment firm wants to use the prices for stocks 2 and 3 to develop a regression model to predict the price of stock 1.
  60. 60. Prices of Three Stocks over a 15-Month Period Stock 1 Stock 2 Stock 3 41 36 35 39 36 35 38 38 32 45 51 41 41 52 39 43 55 55 47 57 52 49 58 54 41 62 65 35 70 77 36 72 75 39 74 74 33 83 81 28 101 92 31 107 91
  61. 61. Regression Models for the Three Stocks First-order with Two Independent Variables Second-order with an Interaction Term
  62. 62. Regression for Three Stocks: First-order, Two Independent Variables The regression equation is Stock 1 = 50.9 - 0.119 Stock 2 - 0.071 Stock 3 Predictor Coef StDev T P Constant 50.855 3.791 13.41 0.000 Stock 2 -0.1190 0.1931 -0.62 0.549 Stock 3 -0.0708 0.1990 -0.36 0.728 S = 4.570 R-Sq = 47.2% R-Sq(adj) = 38.4% Analysis of Variance Source DF SS MS F P Regression 2 224.29 112.15 5.37 0.022 Error 12 250.64 20.89 Total 14 474.93
  63. 63. Regression for Three Stocks: Second-order With an Interaction Term The regression equation is Stock 1 = 12.0 - 0.879 Stock 2 - 0.220 Stock 3 – 0.00998 Inter Predictor Coef StDev T P Constant 12.046 9.312 1.29 0.222 Stock 2 0.8788 0.2619 3.36 0.006 Stock 3 0.2205 0.1435 1.54 0.153 Inter -0.009985 0.002314 -4.31 0.001 S = 2.909 R-Sq = 80.4% R-Sq(adj) = 25.1% Analysis of Variance Source DF SS MS F P Regression 3 381.85 127.28 15.04 0.000 Error 11 93.09 8.46 Total 14 474.93
  64. 64. Regression for Three Stocks: Comparison of two models • The introduction of the interaction term caused the R-squared to increase from 47.2% to 80.4% • The standard error of the estimate decreased from 4.570 in the first model to 2.909 in the second model • The t ratios of the x term and the interaction term are statistically significant in the second model • Inclusion of the interaction term helped the model account for a substantially greater amount of the dependent variable.
  65. 65. Nonlinear Regression Models: Model Transformation
  66. 66. Data Set for Model Transformation Example Company Y X 1 2580 1.2 2 11942 2.6 3 9845 2.2 4 27800 3.2 5 18926 2.9 6 4800 1.5 7 14550 2.7 Company LOG Y X 1 3.41162 1.2 2 4.077077 2.6 3 3.993216 2.2 4 4.444045 3.2 5 4.277059 2.9 6 3.681241 1.5 7 4.162863 2.7 ORIGINAL DATA TRANSFORMED DATA Y = Sales ($ million/year) X = Advertising ($ million/year)
  67. 67. Regression Output for Model Transformation Example Regression Statistics Multiple R 0.990 R Square 0.980 Adjusted R Square 0.977 Standard Error 0.054 Observations 7 Coefficients Standard Error t Stat P-value Intercept 2.9003 0.0729 39.80 0.000 X 0.4751 0.0300 15.82 0.000 ANOVA df SS MS F Significance F Regression 1 0.7392 0.7392 250.36 0.000 Residual 5 0.0148 0.0030 Total 6 0.7540
  68. 68. Prediction with the Transformed Model
  69. 69. Indicator (Dummy) Variables • Some variables are referred to as Qualitative variables  Qualitative variables do not yield quantifiable outcomes  Qualitative variables yield nominal- or ordinal- level information; used more to categorize items. • Qualitative variables are referred to as indicator or dummy variables • If a dummy variable has c categories, then c – 1 dummy variables must be created
  70. 70. Monthly Salary Example As an example, consider the issue of sex discrimination in the salary earnings of workers in some industries. In examining this issue, suppose a random sample of 15 workers is drawn from a pool of employed laborers in a particular industry and the workers’ average monthly salaries are determined, along with their age and gender. The data are shown in the following table. As sex can be only male or female, this variable is coded as a dummy variable with 0 = female, 1 = male.
  71. 71. Data for the Monthly Salary Example
  72. 72. Regression Output for the Monthly Salary Example The regression equation is Salary = 1.732 + 0.111 Age + 0.459 Gender Predictor Coef StDev T P Constant 1.7321 0.2356 7.35 0.000 Age 0.11122 0.07208 1.54 0.149 Gender 0.45868 0.05346 8.58 0.000 S = 0.09679 R-Sq = 89.0% R-Sq(adj) = 87.2% Analysis of Variance Source DF SS MS F P Regression 2 0.90949 0.45474 48.54 0.000 Error 12 0.11242 0.00937 Total 14 1.02191
  73. 73. Regression Output for the Monthly Salary Example
  74. 74. MODEL-BUILDING Suppose a researcher wants to develop a multiple regression model to predict the world production of crude oil. The researcher decides to use as predictors the following five independent variables. • U.S. energy consumption • Gross U.S. nuclear electricity generation • U.S. coal production • Total U.S. dry gas (natural gas) production • Fuel rate of U.S.-owned automobiles
  75. 75. Data for Multiple Regression to Predict Crude Oil Production Y World Crude Oil Production X1 U.S. Energy Consumption X2 U.S. Nuclear Generation X3 U.S. Coal Production X4 U.S. Dry Gas Production X5 U.S. Fuel Rate for Autos Y X1 X2 X3 X4 X5 55.7 74.3 83.5 598.6 21.7 13.30 55.7 72.5 114.0 610.0 20.7 13.42 52.8 70.5 172.5 654.6 19.2 13.52 57.3 74.4 191.1 684.9 19.1 13.53 59.7 76.3 250.9 697.2 19.2 13.80 60.2 78.1 276.4 670.2 19.1 14.04 62.7 78.9 255.2 781.1 19.7 14.41 59.6 76.0 251.1 829.7 19.4 15.46 56.1 74.0 272.7 823.8 19.2 15.94 53.5 70.8 282.8 838.1 17.8 16.65 53.3 70.5 293.7 782.1 16.1 17.14 54.5 74.1 327.6 895.9 17.5 17.83 54.0 74.0 383.7 883.6 16.5 18.20 56.2 74.3 414.0 890.3 16.1 18.27 56.7 76.9 455.3 918.8 16.6 19.20 58.7 80.2 527.0 950.3 17.1 19.87 59.9 81.3 529.4 980.7 17.3 20.31 60.6 81.3 576.9 1029.1 17.8 21.02 60.2 81.1 612.6 996.0 17.7 21.69 60.2 82.1 618.8 997.5 17.8 21.68 60.6 83.9 610.3 945.4 18.2 21.04 60.9 85.6 640.4 1033.5 18.9 21.48
  76. 76. Regression Analysis for Crude Oil Production
  77. 77. MODEL-BUILDING : Objectives • To develop a regression model that accounts for the most variation of the dependent variable • To make the model simple and economical at the same time
  78. 78. All Possible Regressions with Five Independent Variables Four Predictors X1,X2,X3,X4 X1,X2,X3,X5 X1,X2,X4,X5 X1,X3,X4,X5 X2,X3,X4,X5 Single Predictor X1 X2 X3 X4 X5 Two Predictors X1,X2 X1,X3 X1,X4 X1,X5 X2,X3 X2,X4 X2,X5 X3,X4 X3,X5 X4,X5 Three Predictors X1,X2,X3 X1,X2,X4 X1,X2,X5 X1,X3,X4 X1,X3,X5 X1,X4,X5 X2,X3,X4 X2,X3,X5 X2,X4,X5 X3,X4,X5 Five Predictors X1,X2,X3,X4,X5
  79. 79. MODEL-BUILDING : Search Procedures Search procedures are processes whereby more than one multiple regression model is developed for a given database, and the models are compared and sorted by different criteria, depending on the given procedure: • All Possible Regressions • Stepwise Regression • Forward Selection • Backward Elimination
  80. 80. MODEL-BUILDING : Stepwise Regression • Stepwise regression is a step-by-step process that begins by developing a regression model with a single predictor variable and adds and deletes predictors one step at a time. • Perform k simple regressions; and select the best as the initial model. • Evaluate each variable not in the model  If none meets the criterion, stop  Add the best variable to the model; reevaluate previous variables, and drop any which are not significant • Return to previous step.
  81. 81. Stepwise: Step 1 - Simple Regression Results for Each Independent Variable Dependent Variable Independent Variable t-Ratio R2 Y X1 11.77 85.2% Y X2 4.43 45.0% Y X3 3.91 38.9% Y X4 1.08 4.6% Y X5 3.54 34.2%
  82. 82. Stepwise Regression Step 2: Two Predictors Step 3: Three Predictors
  83. 83. MODEL-BUILDING : Forward Selection • Forward selection is like stepwise regression, but once a variable is entered into the process, it is never dropped out. • Forward selection begins by finding the independent variable that will produce the largest absolute value of t (and largest R2) in predicting y.
  84. 84. MODEL-BUILDING : Backward Elimination • Start with the “full model” (all k predictors). • If all predictors are significant, stop. • Otherwise, eliminate the most non-significant predictor; return to previous step.
  85. 85. MODEL-BUILDING : Backward Elimination Step 1: Step 2:
  86. 86. MODEL-BUILDING : Backward Elimination Step 3: Step 4:

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