M24- Std Error & r-square 1 Department of ISM, University of Alabama, Lesson Objectives Understand how to calculate and interpret the “r-square” value. Understand how to calculate and interpret the “standard error of regression”. Learn more about doing regression in Minitab.
M24- Std Error & r-square 2 Department of ISM, University of Alabama, Two measures of “How Well Does the Line Fit the Data?” 1. Standard Error of Estimation, = SQRT of ( Mean Square Error ) 2. r- Square
M24- Std Error & r-square 3 Department of ISM, University of Alabama, Variation in the Y values SST = SSR + SSE total = variation + variation variation accounted unaccounted in Y for by the for by the regression regression can be split into identifiable parts:
M24- Std Error & r-square 4 Department of ISM, University of Alabama, Y X-axis Without X variable information: SST is the sum of squared deviations from the mean of Y. Y Note: This is the concept. You will NOT calculate this way.
M24- Std Error & r-square 5 Department of ISM, University of Alabama, Y ^ Using X variable information: Y X-axis SSE is the sum of squared deviations from the regression line. Note: This is the concept. You will NOT calculate this way. Each deviation is a “residual”.
M24- Std Error & r-square 6 Department of ISM, University of Alabama, Calculations SST = (n–1)s y 2 SSE SSR = Total Variation: Unaccounted for by regression: Accounted for by regression: e 2i2i = SST - SSE 3 =
M24- Std Error & r-square 7 Department of ISM, University of Alabama, Weight vs. Height example: SSE = SST = SSR = See file M22 & file M23; or use computer output! Example 1, continued
M24- Std Error & r-square 8 Department of ISM, University of Alabama, n - 2 e e 2 i Mean Square Error (MSE) MSE = Example 1, continued
M24- Std Error & r-square 9 Department of ISM, University of Alabama, Mean Square Error (MSE) SSE n - 2 MSE = Standard Error of Estimation: MSE = = 17.0 lb. Estimate of “Std. Dev. around the fitted line.” = = Example 1, continued
M24- Std Error & r-square 10 Department of ISM, University of Alabama, r 2 = the “r-square” value “is the fraction of the total variation of Y accounted for by using regression.” variation of Y “accounted for” total variation of Y r 2 = SSR SST r 2 = or
M24- Std Error & r-square 11 Department of ISM, University of Alabama, r 2 1.0 r 2 = 0.0 no regression effect; X is NOT useful. r 2 = 1.0 perfect fit to the data; X is USEFUL!
M24- Std Error & r-square 12 Department of ISM, University of Alabama, Calculating r 2, for Wt vs. Ht or, have the computer do it for you! SSR SST r 2 = = =.8213 Example 1, continued
M24- Std Error & r-square 13 Department of ISM, University of Alabama, Equivalently, r 2 = SSE SST total variation “UNaccounted for”
M24- Std Error & r-square 14 Department of ISM, University of Alabama, r 2 (correlation) 2 =.8213 = (.9063) 2 Equivalently, “coefficient of determination” r 2 is also called the “coefficient of determination” For the weight-height data: Example 1, continued
M24- Std Error & r-square 15 Department of ISM, University of Alabama, For the weight-height data: “82.1% of the total variation of the body weights is accounted for by using height as a predictor variable.” r 2 =.8213 Interpretation: Example 1, continued L.O.P.
M24- Std Error & r-square 16 Department of ISM, University of Alabama, “ % of the total variation of the Y-variable is accounted for by using the X-variable as a predictor variable.” r 2 interpretation in general: L.O.P.
M24- Std Error & r-square 17 Department of ISM, University of Alabama, Std. Error of Estimation: MSE = = 17.0 lb. “The estimated std. dev. of body weights around the regression line is 17.0 pounds.” Interpretation: Example 1, continued L.O.P.
M24- Std Error & r-square 18 Department of ISM, University of Alabama, “The estimated std. dev. of the Y-variable around the regression line is units.” L.O.P. estimation the regression variation around the regression line interpretation in general: Std. Error of
M24- Std Error & r-square 19 Department of ISM, University of Alabama, Regression Error Total Source of Variation degrees of freedom Sum of Squares Mean Squares F-Ratio 1* n – 2** n - 1 *Number of X-variables used, “k” **n – 1 - k SSR SSE SST MSR MSE S Y 2 SourceDFSS MS = SS df F = MSR MSE F Analysis of Variance Table
Regression Error Total Source of Variation degrees of freedom Sum of Squares Mean Squares F-Ratio SourceDFSS MS = SS df F = MSR MSE Variance of Y without X: Variance of Y with X: Example 1, continued
M24- Std Error & r-square 21 Department of ISM, University of Alabama, Y If we have data for the response variable, but no knowledge of an X-variable, what is the best estimate of the mean of Y?
M24- Std Error & r-square 22 Department of ISM, University of Alabama, Y X Y “High” r 2, Low Std. Err. We now have data for both Y and X. What is the best estimate of the mean of Y?
M24- Std Error & r-square 23 Department of ISM, University of Alabama, Y X Y Lower r 2, Higher Std. Err.
Y X Y r 2 =, Std. Err. = Why? ee i 2 = SSE = SST =
M24- Std Error & r-square 25 Department of ISM, University of Alabama, Regression Analysis in Minitab More
M24- Std Error & r-square 26 Department of ISM, University of Alabama, Example 4 Can the “depth” of lakes be estimated using “surface area”? Lakes in Vilas and Oneida counties in northern Wisconsin from the years
M24- Std Error & r-square 27 Department of ISM, University of Alabama, Regression Analysis The regression equation is Depth = Area Predictor Coef StDev T P Constant Area S = R-Sq = 4.0% R-Sq(adj) = 2.6% Analysis of Variance Source DF SS MS F P Regression Error Total Max. depth in feet surface area acres Data in Mtbwin/data/lake. Example 4 Estimate depth of lakes using surface area?
M24- Std Error & r-square 28 Department of ISM, University of Alabama, Regression Analysis The regression equation is Depth = Area Predictor Coef StDev T P Constant Area S = R-Sq = 4.0% R-Sq(adj) = 2.6% Analysis of Variance Source DF SS MS F P Regression Error Total Max. depth in feet surface area acres Data in Mtbwin/data/lake. “t” measures how many standard errors the estimated coefficient is from “zero.” P-value: a measure of the likelihood that the true coefficient is “zero.” Example 4 Estimate depth of lakes using surface area?
M24- Std Error & r-square 29 Department of ISM, University of Alabama, 2s Example 4 Depth of Lakes (feet) vs. Surface Area (acres)
M24- Std Error & r-square 30 Department of ISM, University of Alabama, Example 4 Estimate depth of lakes …
M24- Std Error & r-square 31 Department of ISM, University of Alabama, How do you determine if the X-variable is a useful predictor? 3 See slides 21 in the previous section.
M24- Std Error & r-square 32 Department of ISM, University of Alabama, Regression Analysis The regression equation is Depth = Area Predictor Coef StDev T P Constant Area S = R-Sq = 4.0% R-Sq(adj) = 2.6% Analysis of Variance Source DF SS MS F P Regression Error Total Max. depth in feet surface area in acres Data in Mtbwin/data/lake. Example 4 Estimate depth of lakes using surface area? The P-value for “surface area” IS SMALL (<.10). Conclusion: The “area” coefficient is NOT zero! “Surface area” IS a useful predictor of the mean of “depth”. Could “area” have a true coefficient that is actually “zero”?
Depth of Lakes (feet) vs. Surface Area (acres) 0 2s Where would the line be if the outlier is removed? ______________. Example 4
M24- Std Error & r-square 34 Department of ISM, University of Alabama, Analysis Diary Step Y X s r-sqr Comments 1 Depth Area % Most lakes have area less than 900 acres. Large lakes dominate the line. Although p-value is small, the line does not fit the points well. Eliminate large lakes; re-run. Example 4Lakes in northern Wisconsin n = 71 lakes
M24- Std Error & r-square 35 Department of ISM, University of Alabama, The regression equation is Depth = Area Predictor Coef SE Coef T P Constant Area S = R-Sq = 3.7% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Max. depth in feet surface area in acres Data in Mtbwin/data/lake. Example 4 Estimate depth of lakes using surface area? n = 66 lakes
M24- Std Error & r-square 36 Department of ISM, University of Alabama, Example 4 Estimate depth of lakes using surface area? n = 66 lakes 2s
M24- Std Error & r-square 37 Department of ISM, University of Alabama, Analysis Diary Step Y X s r-sqr Comments 1 Depth Area % Most lakes have area less than 900 acres. Large lakes dominate. Although p-value is small, the line does not fit the points well. Eliminate large lakes; re-run. 2 Depth Area % n = 71 lakes Lakes larger than 900 acres in surface area are removed and the population is redefined. The p-value for “area” is “Surface area” is NOT a good predictor of lake “depth.” n = 66 lakes Example 4Lakes < 900 acres in northern Wisconsin
M24- Std Error & r-square 38 Department of ISM, University of Alabama, How helpful is “engine size” for estimating “mpg”? Example 5
M24- Std Error & r-square 39 Department of ISM, University of Alabama, How helpful is engine size for estimating mpg? Regression Analysis The regression equation is mpg_city = displace 113 cases used 4 cases contain missing values Predictor Coef StDev T P Constant displace S = R-Sq = 54.6% R-Sq(adj) = 54.2% Analysis of Variance Source DF SS MS F P Regression Error Total displacement in cubic in. mpg_city in ??? Data in Car89 Data Example 5
M24- Std Error & r-square 40 Department of ISM, University of Alabama, How helpful is engine size for estimating mpg? Regression Analysis The regression equation is mpg_city = displace 113 cases used 4 cases contain missing values Predictor Coef StDev T P Constant displace S = R-Sq = 54.6% R-Sq(adj) = 54.2% Analysis of Variance Source DF SS MS F P Regression Error Total displacement in cubic in. mpg_city in ??? Data in Car89 Data Example 5 The P-value for “displacement” IS SMALL (<.10). Conclusion: The “displacement” coefficient is NOT zero! “Displacement” IS a useful predictor of the mean of “mpg_city”. (But, … “t” measures how many standard errors the estimated coefficient is from “zero.” P-value: a measure of the likelihood that the true coefficient is “zero.”
M24- Std Error & r-square 41 Department of ISM, University of Alabama, mpg_city vs. displacement S = 2.88 Is this a good fit? The data pattern appears curved; we can do better! Example 5
M24- Std Error & r-square 42 Department of ISM, University of Alabama, Plot of residuals vs. Y-hats S = 2.88 mpg_city vs. displacement Example 5 Apply a transformation in the next section.
M24- Std Error & r-square 43 Department of ISM, University of Alabama, Analysis Diary Step Y X s r-sqr Comments 1 mpg displac % Slope of “displacement” in not zero; but plot indicates a curved pattern. Transform a variable and re-run. Example 5“mpg_city” versus engine “displacement” 2 to be done in next section.
M24- Std Error & r-square 44 Department of ISM, University of Alabama, Which variable is a better predictor of the rating of professional football quarterbacks, percent of touchdown passes or percent of interceptions? Page 626, Problem Example 6
M24- Std Error & r-square 45 Department of ISM, University of Alabama, Rating TD% Inter% Problem 15.23, Page 626 Quarterback Steve Young Joe Montana Brett Favre Dan Marino Mark Brunnell Jim Kelly Roger Staubach Example 6
M24- Std Error & r-square 46 Department of ISM, University of Alabama, Regression Analysis: Rating versus TD% The regression equation is Rating = TD% Predictor Coeff SE Coef T P Constant TD% S = R-Sq = 22.7% R-Sq(adj) = 7.2% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Problem 15.23, Page 626Example 6
M24- Std Error & r-square 47 Department of ISM, University of Alabama, Regression Analysis: Rating vs. Interception% The regression equation is Rating = Inter% Predictor Coef SE Coef T P Constant Inter% S = R-Sq = 38.8% R-Sq(adj) = 26.5% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Problem 15.23, Page 626Example 6
M24- Std Error & r-square 48 Department of ISM, University of Alabama, Which X variable is better for predicting the mean of “Rating”? What criteria should be used? TD% Inter% Std Error R-Square ______ _______ Problem 15.23, Page 626Example 6 Neither is great; look at plots.
M24- Std Error & r-square 49 Department of ISM, University of Alabama, TD% Inter% Problem 15.23, Page 626Example 6
M24- Std Error & r-square 50 Department of ISM, University of Alabama, Regression Analysis: Rating versus TD%, Inter% The regression equation is Rating = TD% Inter% Predictor Coef SE Coef T P Constant TD% Inter% S = R-Sq = 90.6% R-Sq(adj) = 85.8% Analysis of Variance Source DF SS MS F P Regression Residual Error Total This is a “multiple regression” Problem 15.23, Page 626Example 7
M24- Std Error & r-square 51 Department of ISM, University of Alabama, Which X variable is better for predicting the mean of “Rating”? TD% Inter% Std Error R-Square _______ ________ TD% & Inter% _______ ________ Together, the two variables predict much better than either one individually. Problem 15.23, Page 626Example 7
M24- Std Error & r-square 52 Department of ISM, University of Alabama, Rating = TD% Inter% Std Error = , R-Square = 90.6% Prediction model for QB Ratings Notes: Model is based on only n = 7 quarterbacks who played over a 30 year period. Problem 15.23, Page 626Example 7 Final Model:
M24- Std Error & r-square 53 Department of ISM, University of Alabama, NFL Quarterback Ratings for 2002 season. NFL Quarterback Ratings.MTW D:\Edd\Edd\Classes\ST260\data sets Source: 12 NFL QB Ratings, 2002 SeasonExample 8
M24- Std Error & r-square 54 Department of ISM, University of Alabama, C. Pennington, NYJ R. Gannon, OAK B. Johnson, TB T. Green, KC P. Manning, IND M. Hasselbeck, SEA D. McNabb, PHI D. Bledsoe, BUF Tom Brady, NE M. Brunell, JAC J. Garcia, SF B. Favre, GB B. Griese, DEN K. Collins, NYG J. Fiedler, MIA T. Maddox, PIT S. McNair, TEN M. Vick, ATL A. Brooks, NO Jon Kitna, CIN Jim Miller, CHI R. Peete, CAR Jeff Blake, BAL Drew Brees, SD Tim Couch, CLE D. Culpepper, MIN S. Matthews, WAS P. Ramsey, WAS C. Hutchinson, DAL J. Plummer, ARI David Carr, HOU J. Harrington, DET NFL QB Ratings, 2002 SeasonExample 8 n = 32 Cases
M24- Std Error & r-square 55 Department of ISM, University of Alabama, COMCompletions ATTAttempts COM%Percentage of completed passes YDSTotal Yards YPAYards per attempt LNGLongest pass play TDTouchdown passes TD%Touchdown percentage TD passes / pass attempts INTInterceptions thrown INT%Interception percentage Interceptions / pass attempts SKSacks SYDSacked yards lost RATPasser (QB) Rating Variables Measured NFL QB Ratings, 2002 SeasonExample 8 k = 12 X-variables
M24- Std Error & r-square 56 Department of ISM, University of Alabama, Analysis of Variance Source DF SS MS F P Regression Residual Error Total The regression equation is QB Rating = COM ATT COM% YDS YPA LNG TD TD% INT INT% SK SYD NFL QB Ratings, 2002 SeasonExample 8 Minitab output What is the R-Square? How many X-variables? How many cases?
M24- Std Error & r-square 57 Department of ISM, University of Alabama, Example 8 All k = 12 X-vars. included Is there a non-random pattern? ________
Predictor Coef SE Coef T P Constant COM ATT COM% YDS YPA LNG TD TD% INT INT% SK SYD S = R-Sq = 100.0% R-Sq(adj) = 100.0% NFL QB Ratings, 2002 SeasonExample 8 Minitab output All k = 12 X-vars. included Do we need all 12 variables?Which is least useful?
M24- Std Error & r-square 59 Department of ISM, University of Alabama, NFL QB Ratings, 2002 SeasonExample 8 Comments: constant term 1.Always leave the constant term in the model. 2.Never delete more than ONE 2.Never delete more than ONE X-variable per run; re-run the regression at each step, each time deleing only one variable. backward elimination 3.A “backward elimination” can speed-up the process. re-assess your model 4.At the last step, re-assess your model by checking the residual plots again.
M24- Std Error & r-square 60 Department of ISM, University of Alabama, NFL QB Ratings, 2002 SeasonExample 8 Backward Elimination Process Step Var. Out? t p s R 2 Action 1INT ATT Delete Re-run regression with one less variable; determine the least useful of the remaining variables. (Look for the largest P-value).
M24- Std Error & r-square 61 Department of ISM, University of Alabama, NFL QB Ratings, 2002 SeasonExample 8 Backward Elimination Process Step Var. Out? t p s R 2 Action 1INT ATT LNG TD SK SYD INT% YPA COM YDS Constant COM% TD% Delete
M24- Std Error & r-square 62 Department of ISM, University of Alabama, The regression equation is QB Rating = COM% YPA TD% INT% Predictor Coef SE Coef T P Constant COM% YPA TD% INT% S = R-Sq = 100.0% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Example 8Result after dropping 8 X-variables:
M24- Std Error & r-square 63 Department of ISM, University of Alabama, k = 4 X-vars. included. Is there a non-random pattern? No
M24- Std Error & r-square 64 Department of ISM, University of Alabama, Constant Completions per Attempt Yards per Attempt TD per Attempt Interception per Attempt Regression Estimates Final Prediction Model NFL QB Ratings, 2002 Example 8 Std Error = 0.160, R-Square = 99.98% Variables Result using 4 X-variables:
M24- Std Error & r-square 65 Department of ISM, University of Alabama, Step 1: Complete passes divided by pass attempts. Subtract 0.3, then divide by 0.2 Step 2: Passing yards divided by pass attempts. Subtract 3, then divide by 4. Step 3: Touchdown passes divided by pass attempts, then divide by.05. Step 4: Start with.095, and subtract interceptions divided by attempts. Divide the difference by.04. The sum of each step cannot be greater than or less than zero. Add the sum of the Steps 1 through 4, multiply by 100 and divide by 6. Actual Rating Formula: NFL QB Ratings, 2002 SeasonExample 8
M24- Std Error & r-square 66 Department of ISM, University of Alabama, Regression Estimates Actual Values* * Ignoring limits for each part. NFL QB Ratings, 2002 SeasonExample 8 Constant COMP% YPA TD% INT% Comparison of true to estimates
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M24- Std Error & r-square 151 Department of ISM, University of Alabama, Extrapolation: Predicting outside the range your of X values. Warning 1:
M24- Std Error & r-square 152 Department of ISM, University of Alabama, A strong relationship between Y and X does not imply “cause and effect.” Warning 2:
M24- Std Error & r-square 153 Department of ISM, University of Alabama, Warning 3: Be sure your model looks reasonable! Remember to DTDP.
M24- Std Error & r-square 154 Department of ISM, University of Alabama, Summarizing the relationship between X and Y Estimating the mean level of Y for a given value of X Predicting future values of Y for given values of X Uses of the regression line: