Residuals. Residuals = ª Department of ISM, University of Alabama, ST 260, M23 Residuals & Minitab. ^ e i = y i  y i


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1 A continuation of regression analysis Lesson Objectives Continue to build on regression analysis. Learn how residual plots help identify problems with the analysis. M231 M232 Example 1: continued Case Y Sample of n = 5 students, Y = Weight in pounds, = Height in inches. Prediction equation: Wt ^ = Ht rsquare =? Std. error =? To be found later. M233 Example 1, continued WEIGHT ^ Y = = distance from point to line, measured parallel to Y axis HEIGHT M234 Calculation: For each case, residual = observed value For the i th case, ^ e i = y i  y i estimated mean M235 Example 1, continued Compute the fitted value and residual for the 4 th person in the sample; i.e., = 72 inches, Y = 207 lbs. fitted value = ^y 4 = ( ) residual = e 4 = = = ^ y 4  y 4 = M236 1
2 Residual Plots Scatterplot of residuals vs. the predicted means of Y, Y; ^ or an variable. M237 Example 1, continued WEIGHT ^ Y = e 4 = = distance from point to line, measured parallel to Y axis HEIGHT M238 Example 1, continued Residual Plot Regression line from previous plot is rotated to horizontal. e 4 is the residual for the 4 th case, = HEIGHT M239 Residual Plot Scatterplot of residuals versus the predicted means of Y, Y; ^ or an variable, or Time. Expect random dispersion around a horizontal line at zero. Problems occur if: Unusual patterns Unusual cases M versus Good random pattern, or time M versus Outliers? Next step: to determine if a recording error has occurred., or time M
3 0 versus Nonlinear relationship Next step: Add a quadratic term, or use., or time M versus Next step: Stabilize variance by using. Variance is increasing, or time M2314 Residual Plots help identify Unusual patterns: Possible curvature in the data. Variances that are not constant as changes. Unusual cases: Outliers High leverage cases Influential cases Three properties of illustrated with some computations. M2315 M2316 Y = Weight = Height Property 1. ^ Y = Y Y^ e = Y Y^ Find the sum of the residuals. roundoff error Properties of Least Squares Line 1. always sum to zero. Se i = 0. M
4 Y = Weight = Height Y Y^ ^ Y = Property 2. e = Y Y^ e Properties of Least Squares Line 1. always sum to zero. 2. This least squares line produces a smaller Sum of squared residuals than any other straight line can. Se 2 SSE for Find the sum of squares i = SSE = < any other line. of the residuals. M2320 WEIGHT Y = 68.4, Y = 159 Property HEIGHT M2321 Properties of Least Squares Line 1. always sum to zero. 2. This least squares line produces a smaller Sum of squared residuals than any other straight line can. 3. Line always passes through the point ( x, y ). M2322 Illustration of unusual cases: Outliers Leverage Influential Y outlier Unusual point does not follow pattern. It s near the mean; the entire line pulled toward it. M2323 M
5 Y outlier Unusual point does not follow pattern. The line is pulled down and twisted slightly. M2325 Y Unusual point is far from the mean, but still follows the pattern. High leverage M2326 Y Unusual point is far from the mean, but does not follow the pattern. Line really twists! leverage & outlier, influential M2327 Definitions: Outlier: An unusual yvalue relative to the pattern of the other cases. Usually has a large residual. High Leverage Case: An extreme value value relative to the other values. M2328 Definitions: continued Influential Case has an unusually largeeffect on the slope of the least squares line. M2329 Definitions: continued Conclusion: High leverage potentially influential. High leverage & Outlier influential!! M
6 Why do we care about identifying unusual cases? The least squares regression line is not resistant to unusual cases. Regression Analysis in Minitab M2331 M2332 Lesson Objectives Learn two ways to use Minitabto run a regression analysis. Learn how to read output from Minitab. Can height be predicted using shoe size? Step 1? DTDP M2333 M
7 Graph Plot Height Scatterplot The scatter for each subpopulation is about the same; i.e., there is constant variance Shoe Size 13 Jitter added in direction. 14 Female Male 15 M2335 Stat Method 1 Regression Regression Y = a + b M
8 Copied from Session Window. Regression Analysis: Height versus Shoe Size The regression equation is Height = Shoe Size Predictor Coef SE Coef T P Constant Shoe Siz S = RSq = 79.1% RSq(adj) = 79.0% Analysis of Variance Source DF SS MS F P Regression Error Total M2337 Regression Analysis: Height versus Shoe Size The regression equation is Height = Shoe Size Predictor Coef SE Coef T P Constant Shoe Siz S = RSq = 79.1% RSq(adj) = 79.0% Analysis of Variance Source DF SS MS F P Regression Error Total Least squares estimated coefficients. Total Degrees of Freedom = Number of cases  1 M
9 Regression Analysis: Height versus Shoe Size The regression equation is Height = Shoe Size RSq = SSR Predictor Coef SE Coef T P Constant TSS0.000 Shoe Siz = S = RSq = 79.1% RSq(adj) = 79.0% Analysis of Variance Source DF SS MS F P Regression Error Total M2339 Regression Analysis: Height versus Shoe Size The regression equation is Height Standard = 50.5 Error + of 1.87 Regression. Shoe Size Measure of variation around Predictor Coef SE Coef T P Constant the regression line Shoe Siz S = RSq = 79.1% RSq(adj) = 79.0% Sum of squared residuals Analysis of Variance Source DF SS MS F P Regression Error Error Total MSE S = MSE = 3.8 Mean Squared M
10 5 Versus Shoe Siz (response is Height) Are there any problems visible in this plot? Residual Shoe Siz 15 No Jitter added. M2341 Least squares regression equation: Height = Shoe rsquare = 79.1%, Std. error = inches The two summary measures that should always be given with the equation. M
11 Stat Method 2 Regression Fitted Line Plot Y = a + b This program gives a scatterplot with the regression superimposed on it. M2343 Regression Plot Height = Shoe Size S = RSq = 79.1 % RSq(adj) = 79.0 % 80 The fit looks Height Shoe Size M
12 Regression Analysis: Height versus Shoe Size The regression equation is Height = Shoe Size What information do these values provide? Predictor Coef SE Coef T P Constant Shoe Siz S = RSq = 79.1% RSq(adj) = 79.0% Analysis of Variance Source DF SS MS F P Regression Error Total M2345 How do you determine if the variable is a useful predictor? Use the tstatistic or the Fstat. t measures how many standard errors the estimated coefficient is from zero. F = t 2 for simple regression. 1 M
13 How do you determine if the variable is a useful predictor? 2 A Pvalue is associated with t and F. The further t and F are from zero, in either direction, the smaller the corresponding Pvalue will be. Pvalue: a measure of the likelihood that the true coefficient IS ZERO. M2347 If the Pvalue IS SMALL (typically < 0.10 ), then conclude: 3 1. It is unlikely that the true coefficient is really zero, and therefore, 2. The variable IS a useful predictor for the Y variable. Keep the variable! If the Pvalue is NOT SMALL (i.e., > 0.10 ), then conclude: 1. For all practical purposes the true coefficient MAY BE ZERO; therefore 2. The variable IS NOT a useful predictor of the Y variable. Don t use it. M
14 Could shoe size have a true coefficient that is actually zero? Regression Analysis: Height versus Shoe Size t measures how many standard The regression equation errors is the estimated coefficient Height = Shoe is from Size zero. Predictor Coef SE Coef T P Constant Shoe Siz S = RSq = Pvalue: 79.1% a measure RSq(adj) of the = likelihood 79.0% that the true coefficient is zero. Analysis of Variance The Pvalue for Shoe Size IS SMALL (< 0.10). Conclusion: Source DF SS MS F P Regression Error Total Shoe 256 size IS a useful predictor The shoe size coefficient is NOT zero! of the mean of height. M2349 The logic just explained is statistical inference. This will be covered in more detail during the last three weeks of the course. M
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