M24- Std Error & r-square 1  Department of ISM, University of Alabama, 1992-2003 Lesson Objectives  Understand how to calculate and interpret the “r-square”

M24- Std Error & r-square 1  Department of ISM, University of Alabama, 1992-2003 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, 1992-2003 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, 1992-2003 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, 1992-2003 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, 1992-2003 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, 1992-2003 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, 1992-2003 Weight vs. Height example: SSE = 868.06 SST = 4858.00 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, 1992-2003 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, 1992-2003 Mean Square Error (MSE) SSE n - 2 MSE = Standard Error of Estimation: MSE = 289.3 = 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, 1992-2003 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, 1992-2003 0  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, 1992-2003 Calculating r 2, for Wt vs. Ht or, have the computer do it for you! SSR SST r 2 = 3989.94 4858 = =.8213 Example 1, continued

M24- Std Error & r-square 13  Department of ISM, University of Alabama, 1992-2003 Equivalently, r 2 = 1.0 - SSE SST total variation “UNaccounted for”

M24- Std Error & r-square 14  Department of ISM, University of Alabama, 1992-2003 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, 1992-2003 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, 1992-2003 “ % 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, 1992-2003 Std. Error of Estimation: MSE = 289.4 = 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, 1992-2003 “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, 1992-2003 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 134134 3989.94 868.06 4858.00 3989.94 289.35 1214.50 SourceDFSS MS = SS df F = MSR MSE 13.79 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, 1992-2003 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, 1992-2003 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, 1992-2003 Y X Y Lower r 2, Higher Std. Err.

Y X Y r 2 =, Std. Err. = Why? ee i 2 =  SSE = SST =

M24- Std Error & r-square 25  Department of ISM, University of Alabama, 1992-2003 Regression Analysis in Minitab More

M24- Std Error & r-square 26  Department of ISM, University of Alabama, 1992-2003 Example 4 Can the “depth” of lakes be estimated using “surface area”? Lakes in Vilas and Oneida counties in northern Wisconsin from the years 1959-1963.

M24- Std Error & r-square 27  Department of ISM, University of Alabama, 1992-2003 Regression Analysis The regression equation is Depth = 28.2 + 0.00726 Area Predictor Coef StDev T P Constant 28.187 2.443 11.54 0.000 Area 0.007262 0.004277 1.70 0.094 S = 17.81 R-Sq = 4.0% R-Sq(adj) = 2.6% Analysis of Variance Source DF SS MS F P Regression 1 914.9 914.9 2.88 0.094 Error 69 21891.0 317.3 Total 70 22805.9 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, 1992-2003 Regression Analysis The regression equation is Depth = 28.2 + 0.00726 Area Predictor Coef StDev T P Constant 28.187 2.443 11.54 0.000 Area 0.007262 0.004277 1.70 0.094 S = 17.81 R-Sq = 4.0% R-Sq(adj) = 2.6% Analysis of Variance Source DF SS MS F P Regression 1 914.9 914.9 2.88 0.094 Error 69 21891.0 317.3 Total 70 22805.9 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, 1992-2003 0  2s Example 4 Depth of Lakes (feet) vs. Surface Area (acres)

M24- Std Error & r-square 30  Department of ISM, University of Alabama, 1992-2003 Example 4 Estimate depth of lakes …

M24- Std Error & r-square 31  Department of ISM, University of Alabama, 1992-2003 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, 1992-2003 Regression Analysis The regression equation is Depth = 28.2 + 0.00726 Area Predictor Coef StDev T P Constant 28.187 2.443 11.54 0.000 Area 0.007262 0.004277 1.70 0.094 S = 17.81 R-Sq = 4.0% R-Sq(adj) = 2.6% Analysis of Variance Source DF SS MS F P Regression 1 914.9 914.9 2.88 0.094 Error 69 21891.0 317.3 Total 70 22805.9 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, 1992-2003 Analysis Diary Step Y X s r-sqr Comments 1 Depth Area 17.81 4.00% 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, 1992-2003 The regression equation is Depth = 25.3 + 0.0226 Area Predictor Coef SE Coef T P Constant 25.325 3.380 7.49 0.000 Area 0.02265 0.01454 1.56 0.124 S = 18.00 R-Sq = 3.7% Analysis of Variance Source DF SS MS F P Regression 1 785.8 785.8 2.43 0.124 Residual Error 64 20726.1 323.8 Total 65 21511.9 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, 1992-2003 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, 1992-2003 Analysis Diary Step Y X s r-sqr Comments 1 Depth Area 17.81 4.00% 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 18.00 3.70% n = 71 lakes Lakes larger than 900 acres in surface area are removed and the population is redefined. The p-value for “area” is 0.124. “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, 1992-2003 How helpful is “engine size” for estimating “mpg”? Example 5

M24- Std Error & r-square 39  Department of ISM, University of Alabama, 1992-2003 How helpful is engine size for estimating mpg? Regression Analysis The regression equation is mpg_city = 29.3 - 0.0480 displace 113 cases used 4 cases contain missing values Predictor Coef StDev T P Constant 29.2651 0.7076 41.36 0.000 displace 0.047967 0.004154 -11.55 0.000 S = 2.880 R-Sq = 54.6% R-Sq(adj) = 54.2% Analysis of Variance Source DF SS MS F P Regression 1 1106.1 1106.1 133.33 0.000 Error 111 920.8 8.3 Total 112 2026.9 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, 1992-2003 How helpful is engine size for estimating mpg? Regression Analysis The regression equation is mpg_city = 29.3 - 0.0480 displace 113 cases used 4 cases contain missing values Predictor Coef StDev T P Constant 29.2651 0.7076 41.36 0.000 displace 0.047967 0.004154 -11.55 0.000 S = 2.880 R-Sq = 54.6% R-Sq(adj) = 54.2% Analysis of Variance Source DF SS MS F P Regression 1 1106.1 1106.1 133.33 0.000 Error 111 920.8 8.3 Total 112 2026.9 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, 1992-2003 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, 1992-2003 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, 1992-2003 Analysis Diary Step Y X s r-sqr Comments 1 mpg displac 2.880 54.6% 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, 1992-2003 Which variable is a better predictor of the rating of professional football quarterbacks, percent of touchdown passes or percent of interceptions? Page 626, Problem 15.23 Example 6

M24- Std Error & r-square 45  Department of ISM, University of Alabama, 1992-2003 Rating TD% Inter% 96.8 5.6 2.6 92.3 5.1 2.6 87.1 5.4 3.2 86.4 5.0 3.0 85.4 4.0 2.4 84.4 5.0 3.7 83.4 5.2 3.7 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, 1992-2003 Regression Analysis: Rating versus TD% The regression equation is Rating = 65.2 + 4.52 TD% Predictor Coeff SE Coef T P Constant 65.18 18.90 3.45 0.018 TD% 4.520 3.731 1.21 0.280 S = 4.655 R-Sq = 22.7% R-Sq(adj) = 7.2% Analysis of Variance Source DF SS MS F P Regression 1 31.82 31.82 1.47 0.280 Residual Error 5 108.36 21.67 Total 6 140.17 Problem 15.23, Page 626Example 6

M24- Std Error & r-square 47  Department of ISM, University of Alabama, 1992-2003 Regression Analysis: Rating vs. Interception% The regression equation is Rating = 105 - 5.66 Inter% Predictor Coef SE Coef T P Constant 105.121 9.767 10.76 0.000 Inter% -5.663 3.183 -1.78 0.135 S = 4.144 R-Sq = 38.8% R-Sq(adj) = 26.5% Analysis of Variance Source DF SS MS F P Regression 1 54.33 54.33 3.16 0.135 Residual Error 5 85.84 17.17 Total 6 140.17 Problem 15.23, Page 626Example 6

M24- Std Error & r-square 48  Department of ISM, University of Alabama, 1992-2003 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, 1992-2003 TD% Inter% Problem 15.23, Page 626Example 6

M24- Std Error & r-square 50  Department of ISM, University of Alabama, 1992-2003 Regression Analysis: Rating versus TD%, Inter% The regression equation is Rating = 75.5 + 7.23 TD% - 7.93 Inter% Predictor Coef SE Coef T P Constant 75.545 7.632 9.90 0.001 TD% 7.226 1.543 4.68 0.009 Inter% -7.929 1.479 -5.36 0.006 S = 1.819 R-Sq = 90.6% R-Sq(adj) = 85.8% Analysis of Variance Source DF SS MS F P Regression 2 126.940 63.470 19.18 0.009 Residual Error 4 13.235 3.309 Total 6 140.174 This is a “multiple regression” Problem 15.23, Page 626Example 7

M24- Std Error & r-square 51  Department of ISM, University of Alabama, 1992-2003 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, 1992-2003 Rating = 75.5 + 7.23 TD% - 7.93 Inter% Std Error = 1.8190, 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, 1992-2003 NFL Quarterback Ratings for 2002 season. NFL Quarterback Ratings.MTW D:\Edd\Edd\Classes\ST260\data sets http://espn.go.com/nfl/statistics/glossary.html Source: 12 NFL QB Ratings, 2002 SeasonExample 8

M24- Std Error & r-square 54  Department of ISM, University of Alabama, 1992-2003 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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 NFL QB Ratings, 2002 SeasonExample 8 n = 32 Cases

M24- Std Error & r-square 55  Department of ISM, University of Alabama, 1992-2003 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, 1992-2003 Analysis of Variance Source DF SS MS F P Regression 12 2895.52 241.29 12347.02 0.000 Residual Error 19 0.37 0.02 Total 31 2895.89 The regression equation is QB Rating = 0.30 - 0.0170 COM + 0.00266 ATT + 0.920 COM% + 0.00123 YDS + 3.59 YPA + 0.00342 LNG - 0.0236 TD + 3.41 TD% - 0.0233 INT - 4.09 INT% - 0.00699 SK + 0.00181 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, 1992-2003 Example 8 All k = 12 X-vars. included Is there a non-random pattern? ________

Predictor Coef SE Coef T P Constant 0.302 1.641 0.18 0.856 COM -0.016982 0.007611 -2.23 0.038 ATT 0.002657 0.003952 0.67 0.509 COM% 0.91976 0.03536 26.01 0.000 YDS 0.0012311 0.0005133 2.40 0.027 YPA 3.5921 0.2369 15.16 0.000 LNG 0.003418 0.002331 1.47 0.159 TD -0.02358 0.04552 -0.52 0.610 TD% 3.4065 0.2086 16.33 0.000 INT -0.02333 0.04706 -0.50 0.626 INT% -4.0876 0.2165 -18.88 0.000 SK -0.006990 0.007671 -0.91 0.374 SYD 0.001806 0.001335 1.35 0.192 S = 0.1398 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, 1992-2003 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, 1992-2003 NFL QB Ratings, 2002 SeasonExample 8 Backward Elimination Process Step Var. Out? t p s R 2 Action 1INT-.023.626.14099.99 2ATT.48.639.13799.99 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, 1992-2003 NFL QB Ratings, 2002 SeasonExample 8 Backward Elimination Process Step Var. Out? t p s R 2 Action 1INT-.023.626.14099.99 2ATT.48.639.13799.99 6LNG2.04.053.13699.98 3TD-.43.674.135 99.99 4SK-.87.394.132 99.99 5SYD1.61.121.131 99.99 10INT%-8.09.0002.03096.02 9YPA66.01.000.16299.98 8COM-1.36.186.16099.98 7YDS2.65.014.14499.98 13 Constant 9.670.000 12COM%8.63.0005.26073.00 11TD%5.80.0003.64086.71 Delete

M24- Std Error & r-square 62  Department of ISM, University of Alabama, 1992-2003 The regression equation is QB Rating = 2.13 + 0.833 COM% + 4.20 YPA + 3.29 TD% - 4.19 INT% Predictor Coef SE Coef T P Constant 2.1295 0.4179 5.10 0.000 COM%.832514 0.008313 100.14 0.000 YPA 4.19967 0.06362 66.01 0.000 TD% 3.29368 0.04121 79.92 0.000 INT% -4.18577 0.03856 -108.56 0.000 S = 0.1622 R-Sq = 100.0% Analysis of Variance Source DF SS MS F P Regression 4 2895.18 23.79 27500.39 0.000 Residual Error 27 0.71 0.03 Total 31 2895.89 Example 8Result after dropping 8 X-variables:

M24- Std Error & r-square 63  Department of ISM, University of Alabama, 1992-2003 k = 4 X-vars. included. Is there a non-random pattern? No

M24- Std Error & r-square 64  Department of ISM, University of Alabama, 1992-2003 2.1295 0.8325 4.1997 3.2937 -4.1858 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, 1992-2003 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 2.375 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, 1992-2003 Regression Estimates 2.1295 0.8325 4.1997 3.2937 -4.1858 2.0833 0.8333 4.1667 3.3333 -4.1667 Actual Values* * Ignoring limits for each part. NFL QB Ratings, 2002 SeasonExample 8 Constant COMP% YPA TD% INT% Comparison of true to estimates

M24- Std Error & r-square 67  Department of ISM, University of Alabama, 1992-2003

M24- Std Error & r-square 151  Department of ISM, University of Alabama, 1992-2003 Extrapolation: Predicting outside the range your of X values. Warning 1:

M24- Std Error & r-square 152  Department of ISM, University of Alabama, 1992-2003 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, 1992-2003 Warning 3: Be sure your model looks reasonable! Remember to DTDP.

M24- Std Error & r-square 154  Department of ISM, University of Alabama, 1992-2003 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:

M24- Std Error & r-square 1  Department of ISM, University of Alabama, 1992-2003 Lesson Objectives  Understand how to calculate and interpret the “r-square”

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M24- Std Error & r-square 1  Department of ISM, University of Alabama, 1992-2003 Lesson Objectives  Understand how to calculate and interpret the “r-square”

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