# SAS LOGISTIC

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SAS HATCO Logistic Regression – HW

2-Group Analysis:

Perform a Logistic Regression Analysis paralleling the HBAT analysis performed in the textbook and in class, but now use the HATCO dataset (see HATCO_Split60 data file).

§ Model: X11 = X1 – X7

§ Split 60-40 Variable: Split60

§ Stepwise Analysis

§ Classification and Prediction Analysis including Holdout Sample

§ Fully Discuss the Analyses and Solutions

Provide a brief 2-3 page management summary

including key selected results along with the SAS program and output as an appendix.

SAS LOGISTIC

*; *; * HBAT – Logistic Regression Analysis; *; *; ods graphics on ; *; options ls =80 ps =50 nodate pageno= 1; *; Title ‘Chapter 6 Logistic Regression Example’ ; *; * Input HBAT ; *; Data HBAT; Infile ‘C: Documents and Settings Thomas F Brantle My Documents Stevens_2006 Stevens_Teaching BIA_652_Multivariate_2014_Spring Class_09 Chapter 5-6HBAT_Split60.txt’ DLM = ’09’X TRUNCOVER ; Input ID Split60 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23; *; Data HBAT; Se t HBAT (Keep = ID Split60 X4 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18); Label ID = ‘ID – Identification Number’ Split60 = ‘Split60’ X4 = ‘X4 – Region’ X6 = ‘X6 – Product Quality’ X7 = ‘X7 – E-Commerce’ X8 = ‘X8 – Techni cal Support’ X9 = ‘X9 – Complaint Resolution’ X10 = ‘X10 – Advertizing’ X11 = ‘X11 – Product Line’ X12 = ‘X12 – Salesforce Image’ X13 = ‘X13 – Competitive Pricing’ X14 = ‘X14 – Warranty & Claims’ X15 = ‘X15 – New Products’ X16 = ‘X16 – Order & Billing’ X17 = ‘X17 – Price Flexibility’ X18 = ‘X18 – Delivery Speed’ ; *; * Create HBAT Split 60 (Original/Initial) and Split 40 (Validation/Holdout) Datasets ; *; Data HBAT60; Set HBAT; If Split60 = 0; *; Data HBAT40; Set HBAT; If Split60 = 1; *; Proc Print Data = HBAT60; *; Proc Print Data = HBAT40; *; *; * Stepwise Logistic Regression Analysis – X4 = X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18; *; * EVENT=’category’ | keyword * specifies the event category for the binary response model. *; * SELECTION = option specifies the method used to select the explanatory variables in the model. * STEPWISE requests stepwise selection; *; * SLENTRY = option specifies the significance level for entry into the model * SLSTAY = option specifies the significance level for staying in the model *; * DETAILS option produces detailed printout at each step of the model -building process *; * LACKFIT requests Hosmer and Lemeshow goodness -of -fit test *; * RSQUARE displays generalized R^2 *; * CTABLE option requests the printing of a classification table for the final model produced by the procedure. *; * PPROB = option specifies possibly multiple cutpoi nts used to classify observations for the CTABLE option. * The values must be between 0 and 1. If the PPROB= option is not specified, the * default is to print the classification for a range of probabilities from the smallest estimated * probability (rounded below to the nearest .02) to the highest estimated probability (rounded above * to the nearest .02) with 0.02 increments. Note that the PPROB= option has no effect unless the * CTABLE option is also specified. *; *; Proc Logistic Data = HBAT60; Model X4( event =’0′ ) = X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 / Selection =Stepwise SLEntry =0.05 SLStay =0.05 Details LackFit RSquare CTable PProb =( 0 to 1 by .10 ); *; * Final Resultant Model and Output Model; *; Proc Logistic Data = HBAT60 OutModel =Logistic60; Model X4( event =’0′ ) = X13 X17 / LackFit RSquare CTable PProb =( 0.40 to 0.60 by .01 ); *; * Original Split60 Logistic Model Fitted to Split40 validation Data; *; Proc Logistic InModel =Logistic60; Score Data = HBAT60 (Keep = X4 X13 X17) Out = HBAT60Score; *; * Proc Freq Crosstabulations Original and Holdout Validation Datasets; *; Proc Print Data = HBAT60Score; Proc Freq Data = HBAT60Score; Table F_X4 * I_X4; *; Proc Logistic InModel =Logistic60; Score Data = HBAT40 (Keep = X4 X13 X17) Out = HBAT40Score; Proc Print Data = HBAT40Score; Proc Freq Data = HBAT40Score; Table F_X4 * I_X4; *; *; * ods graphics off; *; *; Run ; Quit ;

SAS LOGISTIC

SAS Output Chapter 6 Logistic Regression Example Obs ID Split60 X4 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X181 1 0 1 8.5 3.9 2.5 5.9 4.8 4.9 6.0 6.8 4.7 4.3 5.0 5.1 3.7 2 2 0 0 8.2 2.7 5.1 7.2 3.4 7.9 3.1 5.3 5.5 4.0 3.9 4.3 4.9 3 3 0 1 9.2 3.4 5.6 5.6 5.4 7.4 5.8 4.5 6.2 4.6 5.4 4.0 4.5 4 4 0 1 6.4 3.3 7.0 3.7 4.7 4.7 4.5 8.8 7.0 3.6 4.3 4.1 3.0 5 5 0 0 9.0 3.4 5.2 4.6 2.2 6.0 4.5 6.8 6.1 4.5 4.5 3.5 3.5 6 6 0 1 6.5 2.8 3.1 4.1 4.0 4.3 3.7 8.5 5.1 9.5 3.6 4.7 3.3 7 7 0 1 6.9 3.7 5.0 2.6 2.1 2.3 5.4 8.9 4.8 2.5 2.1 4.2 2.0 8 8 0 1 6.2 3.3 3.9 4.8 4.6 3.6 5.1 6.9 5.4 4.8 4.3 6.3 3.7 9 10 0 1 6.4 4.5 5.1 6.1 4.7 5.7 5.7 8.4 5.4 5.3 4.1 5.8 4.4 10 11 0 0 8.7 3.2 4.6 4.8 2.7 6.8 4.6 6.8 5.8 7.5 3.8 3.7 4.0 11 12 0 1 6.1 4.9 6.3 3.9 4.4 3.9 6.4 8.2 5.8 5.9 3.0 4.9 3.2 12 14 0 0 9.2 3.9 5.7 5.5 2.4 8.4 4.8 7.1 6.7 3.0 4.5 2.6 4.2 13 15 0 1 6.3 4.5 4.7 6.9 4.5 6.8 5.9 8.8 6.0 5.4 4.8 6.2 5.2 14 16 0 0 8.7 3.2 4.0 6.8 3.2 7.8 3.8 4.9 6.1 5.0 4.3 3.9 4.5 15 17 0 1 5.7 4.0 6.7 6.0 3.3 5.5 5.1 6.2 6.7 5.4 4.2 6.2 4.5 16 20 0 1 9.1 4.5 3.6 6.4 5.3 5.3 7.1 8.4 5.8 6.7 4.5 6.1 4.4 17 24 0 1 9.3 2.4 2.6 7.2 2.2 7.2 4.5 6.2 6.4 4.2 6.7 4.4 4.5 18 27 0 0 8.5 3.0 7.2 5.8 4.1 7.6 3.7 4.8 6.9 6.7 5.3 3.8 4.4 19 29 0 0 8.5 3.0 5.7 6.0 2.3 7.6 3.7 4.8 5.8 6.0 5.7 3.8 4.4 20 30 0 1 7.6 3.6 3.0 4.0 5.1 4.2 4.6 7.7 4.9 7.2 4.7 5.5 3.5 21 31 0 0 6.9 3.4 8.5 4.3 4.5 6.4 4.7 5.2 7.7 3.3 3.7 2.7 3.3 22 32 0 1 8.1 2.5 7.2 4.5 2.3 5.1 3.8 6.6 6.8 6.1 3.0 3.5 3.0 23 33 0 1 6.7 3.7 6.5 5.3 5.3 5.1 4.9 9.2 5.7 4.2 3.5 4.5 3.4 24 35 0 1 6.7 4.0 5.2 3.9 3.0 5.4 6.8 8.4 6.2 6.0 2.5 4.3 3.5 25 36 0 0 8.7 3.2 6.1 4.3 3.5 6.1 2.9 5.6 6.1 6.5 3.1 2.9 2.5 26 37 0 0 9.0 3.4 5.9 4.6 3.9 6.0 4.5 6.8 6.4 4.3 3.9 3.5 3.5 27 38 0 1 9.6 4.1 6.2 7.3 2.9 7.7 5.5 7.7 6.1 4.4 5.2 4.6 4.9 28 43 0 0 9.3 5.1 4.6 6.8 5.8 6.6 6.3 7.4 5.1 4.1 4.6 4.6 4.3 29 44 0 1 5.1 5.1 6.6 6.9 4.4 5.4 7.8 5.9 7.2 5.2 4.9 6.3 4.5 30 45 0 0 8.0 2.5 4.7 7.1 3.6 7.7 3.0 5.2 5.1 3.9 4.3 4.2 4.7 31 46 0 1 5.9 4.1 5.7 5.9 5.8 6.4 5.5 8.4 6.4 5.1 5.2 5.8 4.8 32 47 0 0 10.0 4.3 7.1 6.3 2.9 5.4 4.5 3.8 6.7 3.7 5.0 4.0 3.5 33 48 0 1 5.7 3.8 6.8 7.5 5.7 5.7 6.0 8.2 6.6 4.8 6.5 7.3 5.2 34 49 0 1 9.9 3.7 3.7 6.1 4.2 7.0 6.7 6.8 5.9 7.2 4.5 3.4 3.9 35 50 0 0 7.9 3.9 4.3 5.8 4.4 6.9 5.8 4.7 5.2 3.6 4.1 4.2 4.3 36 52 0 0 8.2 2.7 3.7 7.4 2.7 7.9 3.1 5.3 5.3 5.0 4.5 4.3 4.9 37 53 0 1 9.4 2.5 4.8 6.1 3.2 7.3 4.6 6.3 6.3 9.2 4.7 4.6 4.6 38 54 0 0 6.9 3.4 5.7 4.4 3.3 6.4 4.7 5.2 6.4 4.4 3.2 2.7 3.3 39 56 0 0 9.3 3.8 7.3 5.7 3.7 6.4 5.5 7.4 6.6 5.9 4.1 3.2 3.4 40 58 0 0 7.6 3.6 5.2 5.8 5.6 6.6 5.4 4.4 6.7 6.4 4.6 3.9 4.0 41 60 0 1 9.9 2.8 7.2 6.9 2.6 5.8 3.5 5.4 6.2 7.0 5.6 4.9 4.0 SAS Output 4261 0 0 8.7 3.2 8.4 6.1 2.8 7.8 3.8 4.9 7.2 4.5 5.4 3.9 4.5 4363 0 0 8.8 3.9 3.8 5.1 4.3 4.7 4.8 5.8 5.0 7.2 4.4 3.7 2.9 44 64 0 1 7.7 2.2 6.3 4.5 2.4 4.7 3.4 6.2 6.0 4.7 3.3 3.1 2.6 45 65 0 1 6.6 3.6 5.8 4.1 4.9 4.7 4.8 7.2 6.5 3.9 3.5 3.6 2.8 46 67 0 1 5.7 4.0 7.9 6.4 2.7 5.5 5.1 6.2 7.5 6.4 5.0 6.2 4.5 47 68 0 1 5.5 3.7 4.7 5.4 4.3 5.3 4.9 6.0 5.6 2.5 4.5 5.9 4.3 48 69 0 1 7.5 3.5 3.8 3.5 2.9 4.1 4.5 7.6 5.1 5.2 4.0 5.4 3.4 49 72 0 0 6.7 3.2 3.0 3.7 4.8 6.3 4.5 5.0 5.2 2.5 2.9 2.6 3.1 50 79 0 0 9.3 3.5 6.3 7.6 5.5 7.5 5.9 4.6 6.6 3.1 5.2 4.1 4.6 51 80 0 1 7.1 3.4 4.9 4.1 4.0 5.0 5.9 7.8 6.1 3.5 2.6 3.1 2.7 52 81 0 0 9.9 3.0 7.4 4.8 4.0 5.9 4.8 4.9 5.9 6.9 3.2 4.3 3.8 53 86 0 1 7.5 3.5 4.1 4.5 3.5 4.1 4.5 7.6 4.9 2.8 3.4 5.4 3.4 54 87 0 1 5.0 3.6 1.3 3.0 3.5 4.2 4.9 8.2 4.3 7.6 2.4 4.8 3.1 55 88 0 0 7.7 2.6 8.0 6.7 3.5 7.2 4.3 5.9 6.9 7.7 5.1 3.9 4.3 56 92 0 1 7.1 4.2 4.1 2.6 2.1 3.3 4.5 9.9 5.5 3.5 2.0 4.0 2.4 57 94 0 1 9.3 3.5 5.4 7.8 4.6 7.5 5.9 4.6 6.4 4.9 4.8 4.1 4.6 58 95 0 0 9.3 3.8 4.0 4.6 4.7 6.4 5.5 7.4 5.3 4.8 3.6 3.2 3.4 59 98 0 0 8.7 3.2 3.3 3.2 3.1 6.1 2.9 5.6 5.0 4.3 3.1 2.9 2.5 60 100 0 1 7.9 3.0 4.4 5.1 5.9 4.2 4.8 9.7 5.7 5.8 3.4 5.4 3.5 Chapter 6 Logistic Regression Example Obs ID Split60 X4 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 1 9 1 1 5.8 3.6 5.1 6.7 3.7 5.9 5.8 9.3 5.9 4.4 4.4 6.1 4.6 2 13 1 0 9.5 5.6 4.6 6.9 5.0 6.9 6.6 7.6 6.5 5.3 5.1 4.5 4.4 3 18 1 1 5.9 4.1 5.5 7.2 3.5 6.4 5.5 8.4 6.2 6.3 5.7 5.8 4.8 4 19 1 1 5.6 3.4 5.1 6.4 3.7 5.7 5.6 9.1 5.4 6.1 5.0 6.0 4.5 5 21 1 1 5.2 3.8 7.1 5.2 3.9 4.3 5.0 8.4 7.1 4.6 3.3 4.9 3.3 6 22 1 1 9.6 5.7 6.8 5.9 5.4 8.3 7.8 4.5 6.4 6.5 4.3 3.0 4.3 7 23 1 0 8.6 3.6 7.4 5.1 3.5 7.3 4.7 3.7 6.7 6.0 4.8 3.4 4.0 8 25 1 1 6.0 4.1 5.3 4.7 3.5 5.3 5.3 8.0 6.5 3.9 4.7 5.3 4.0 9 26 1 1 6.4 3.6 6.6 6.1 4.0 3.9 5.3 7.1 6.1 3.7 5.6 6.6 3.9 10 28 1 1 7.0 3.3 5.4 5.5 2.6 4.8 4.2 9.0 6.5 5.9 4.3 5.2 3.7 11 34 1 1 8.0 3.3 6.1 5.7 5.5 4.6 4.7 8.7 5.9 3.8 4.7 6.6 4.2 12 39 1 1 8.2 3.6 3.9 6.2 5.8 4.9 5.0 9.0 5.2 7.1 4.7 6.9 4.5 13 40 1 1 6.1 4.9 3.0 4.8 5.1 3.9 6.4 8.2 5.1 6.8 4.5 4.9 3.2 14 41 1 1 8.3 3.4 3.3 5.5 3.1 4.6 5.2 9.1 4.1 1.7 4.6 5.8 3.9 15 42 1 0 9.4 3.8 4.7 5.4 3.8 6.5 4.9 8.5 4.9 6.2 4.1 4.5 4.1 16 51 1 1 6.7 3.6 5.9 4.2 3.4 4.7 4.8 7.2 5.7 5.3 4.0 3.6 2.8 17 55 1 1 8.0 3.3 3.8 5.8 3.2 4.6 4.7 8.7 5.3 4.2 4.9 6.6 4.2 18 57 1 1 7.4 5.1 4.8 7.7 4.5 7.2 6.9 9.6 6.4 7.4 5.7 6.5 5.5 19 59 1 0 10.0 4.3 5.3 3.7 4.2 5.4 4.5 3.8 6.7 4.5 3.7 4.0 3.5 20 62 1 1 8.4 3.8 6.7 5.0 4.5 4.7 5.9 6.7 5.1 4.2 2.7 5.0 3.6 2166 1 1 5.7 3.8 3.5 6.7 5.4 5.7 6.0 8.2 5.4 5.0 4.7 7.3 5.2 SAS Output 2270 1 1 6.4 3.6 2.7 5.3 3.9 3.9 5.3 7.1 5.2 5.5 4.7 6.6 3.9 23 71 1 1 9.1 4.5 6.1 5.9 6.3 5.3 7.1 8.4 7.1 5.7 5.4 6.1 4.4 24 73 1 1 6.5 4.3 2.7 6.6 6.5 6.3 6.0 8.7 4.7 6.3 4.6 5.6 4.6 25 74 1 1 9.9 3.7 7.5 4.7 5.6 7.0 6.7 6.8 7.2 4.6 4.1 3.4 3.9 26 75 1 1 8.5 3.9 5.3 5.5 5.0 4.9 6.0 6.8 5.7 3.6 4.4 5.1 3.7 27 76 1 0 9.9 3.0 6.8 5.0 5.4 5.9 4.8 4.9 7.3 7.6 3.1 4.3 3.8 28 77 1 1 7.6 3.6 7.6 4.6 4.7 4.6 5.0 7.4 8.1 6.6 4.5 5.8 3.9 29 78 1 0 9.4 3.8 7.0 6.2 4.7 6.5 4.9 8.5 7.3 2.4 4.3 4.5 4.1 30 82 1 0 8.7 3.2 6.4 4.9 2.4 6.8 4.6 6.8 6.3 5.1 4.3 3.7 4.0 31 83 1 0 8.6 2.9 5.8 3.9 2.9 5.6 4.0 6.3 6.1 4.0 2.7 3.0 3.0 32 84 1 1 6.4 3.2 6.7 3.6 2.2 2.9 5.0 8.4 7.3 6.5 2.0 3.7 1.6 33 85 1 0 7.7 2.6 6.7 6.6 1.9 7.2 4.3 5.9 6.5 4.1 4.7 3.9 4.3 34 89 1 0 9.1 3.6 5.5 5.4 4.2 6.2 4.6 8.3 6.5 4.1 4.6 4.3 3.9 35 90 1 1 5.5 5.5 7.7 7.0 5.6 5.7 8.2 6.3 7.4 4.9 5.5 6.7 4.9 36 91 1 0 9.1 3.7 7.0 4.1 4.4 6.3 5.4 7.3 7.5 4.6 4.4 3.0 3.3 37 93 1 0 9.2 3.9 4.6 5.3 4.2 8.4 4.8 7.1 6.2 6.6 4.4 2.6 4.2 38 96 1 0 8.6 4.8 5.6 5.3 2.3 6.0 5.7 6.7 5.8 3.6 4.9 3.6 3.6 39 97 1 1 7.4 3.4 2.6 5.0 4.1 4.4 4.8 7.2 4.5 6.4 4.2 5.6 3.7 40 99 1 1 7.8 4.9 5.8 5.3 5.2 5.3 7.1 7.9 6.0 5.7 4.3 4.9 3.9 Chapter 6 Logistic Regression Example The LOGISTIC Procedure Model Information Data Set WORK.HBAT60 Response VariableX4X4 – Region Number of Response Levels 2 Modelbinary logit Optimization TechniqueFisher’s scoring Number of Observations Read60 Number of Observations Used 60 Response Profile Ordered Value X4 Total Frequency 1 0 26 2 1 34 Probability modeled is X4=0. Stepwise Selection Procedure Step 0. Intercept entered: Note: Most typical Regression Diagnostics, e.g., Proc REG and GLM are also availble in Proc Logistic. SAS Output Model Convergence Status Convergence criterion (GCONV=1E – 8) satisfied.- 2 Log L= 82.108 Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi – Square Pr > ChiSq Intercept 1 – 0.2683 0.2605 1.0603 0.3031 Residual Chi – Square Test Chi – Square DF Pr > ChiSq 42.3497 13 <.0001 Analysis of Effects Eligible for Entry Effect DF Score Chi – Square Pr > ChiSq X6 1 11.9251 0.0006 X7 1 2.0517 0.1520 X8 1 1.6089 0.2046 X9 1 0.8656 0.3522 X10 1 0.7914 0.3737 X11 1 18.3231 <.0001 X12 1 8.6217 0.0033 X13 1 21.3297 <.0001 X14 1 0.4654 0.4951 X15 1 0.6138 0.4333 X16 1 0.0899 0.7644 X17 1 21.2035 <.0001 X18 1 0.1567 0.6922 Step 1. Effect X13 entered: Model Convergence Status Convergence criterion (GCONV=1E – 8) satisfied. Model Fit Statistics Criterion Intercept Only Intercept and Covariates AIC 84.108 60.971 SC 86.202 65.160 – 2 Log L 82.108 56.971 R – Square 0.3423Max – rescaled R – Square 0.4591 Testing Global Null Hypothesis: BETA=0 Test Chi – Square DF Pr > ChiSq Likelihood Ratio 25.1363 1 <.0001 SAS Output Score21.3297 1 <.0001 Wald15.4710 1 <.0001 Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi – Square Pr > ChiSq Intercept 1 7.0082 1.8360 14.5698 0.0001 X13 1 – 1.1287 0.2870 15.4710 <.0001 Odds Ratio Estimates Effect Point Estimate 95% Wald Confidence Limits X13 0.323 0.184 0.568 Association of Predicted Probabilities and Observed Responses Percent Concordant 84.5Somers’ D 0.699 Percent Discordant 14.6Gamma 0.705 Percent Tied 0.9Tau- a 0.349 Pairs 884c 0.850 Residual Chi – Square Test Chi – Square DF Pr > ChiSq 27.3151 12 0.0070 Analysis of Effects Eligible for Removal Effect DF Wald Chi – Square Pr > ChiSq X13 1 15.4710 <.0001 Note: No effects for the model in Step 1 are removed. Analysis of Effects Eligiblefor Entry Effect DF Score Chi – Square Pr > ChiSq X6 1 4.8588 0.0275 X7 1 0.1321 0.7163 X8 1 0.0074 0.9315 X9 1 1.3794 0.2402 X10 1 0.1293 0.7192 X11 1 6.1543 0.0131 X12 1 2.7452 0.0975 X14 1 0.6395 0.4239 X15 1 0.3442 0.5574 X16 1 2.5284 0.1118 X17 1 13.7231 0.0002 X18 1 1.2061 0.2721 Statistical Measures. The first statistical measure is the chi-square test for the change in the -2LL value from the base model, which is comparable to the overall F test in multiple regression. Smaller values of the -2LL measure indicate better model fit, and the statistical test is available for assessing the difference between both the base model and other proposed models (in a stepwise procedure, this test is always based on improvement from the prior step). SAS Output Step 2. Effect X17 entered:Model Convergence Status Convergence criterion (GCONV=1E – 8) satisfied. Model Fit Statistics Criterion Intercept Only Intercept and Covariates AIC 84.108 45.960 SC 86.202 52.243 – 2 Log L 82.108 39.960 R – Square 0.5046Max – rescaled R – Square 0.6769 Testing Global Null Hypothesis: BETA=0 Test Chi – Square DF Pr > ChiSq Likelihood Ratio 42.1477 2 <.0001 Score 31.3228 2 <.0001 Wald 14.1772 2 0.0008 Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi – Square Pr > ChiSq Intercept 1 14.1917 3.7123 14.6143 0.0001 X13 1 – 1.0791 0.3574 9.1148 0.0025 X17 1 – 1.8439 0.6388 8.3314 0.0039 Odds Ratio Estimates Effect Point Estimate 95% Wald Confidence Limits X13 0.340 0.169 0.685 X17 0.158 0.045 0.553 Association of Predicted Probabilities and Observed Responses Percent Concordant 92.1Somers’ D 0.843 Percent Discordant 7.8Gamma 0.844 Percent Tied 0.1Tau- a 0.421 Pairs 884c 0.921 Residual Chi – Square Test Chi – Square DF Pr > ChiSq 20.2161 11 0.0425 Analysis of Effects Eligible for Removal Effect DF Wald Chi – Square Pr > ChiSq X13 1 9.1148 0.0025 X17 1 8.3314 0.0039 Model converges and -2 Log L (analogous to Global F-test in PROC REG) shows that overall model is significant (see Residual Chi-square test p< .0001). The “Analysis of Maximum Likelihood Estimates” table summarizes information regarding the independent variables including parameter estimates, variability, and significance. The “Odds Ratio Estimates” table summarizes the significant independent variables and indicates their associated odds ratios and confidence limits. Criterion – Underneath are various measurements used to assess the model fit. The first two, Akaike Information Criterion (AIC) and Schwarz Criterion (SC) are deviants of negative two times the Log-Likelihood (-2 Log L). AIC and SC penalize the log-likelihood by the number of predictors in the model. AIC – This is the Akaike Information Criterion. It is calculated as AIC = -2 Log L + 2((k-1) + s), where k is the number of levels of the dependent variable and s is the number of predictors in the model. AIC is used for the comparison of nonnested models on the same sample. Ultimately, the model with the smallest AIC is considered the best, although the AIC value itself is not meaningful. SC – This is the Schwarz Criterion. It is defined as – 2 Log L + ((k-1) + s)*log(Σ fi), where fi’s are the frequency values of the ith observation, and k and s were defined previously. Like AIC, SC penalizes for the number of predictors in the model and the smallest SC is most desirable and the value itself is not meaningful.. -2 Log L – This is negative two times the log- likelihood. The -2 Log L is used in hypothesis tests for nested models and the value in itself is not meaningful. Intercept Only – This column refers to the respective criterion statistics with no predictors in the model, i.e., just the response variable. Intercept and Covariates – This column corresponds to the respective criterion statistics for the fitted model. A fitted model includes all independent variables and the intercept. We can compare the values in this column with the criteria corresponding Intercept Only value to assess model fit/significance. Test – These are three asymptotically equivalent Chi-Square tests. They test against the null hypothesis that at least one of the predictors’ regression coefficient is not equal to zero in the model. The difference between them are where on the log-likelihood function they are evaluated. Likelihood Ratio – This is the Likelihood Ratio (LR) Chi-Square test that at least one of the predictors’ regression coefficient is not equal to zero in the model. The LR Chi-Square statistic can be calculated by -2 Log L(null model) – 2 Log L(fitted model) = , where L(null model) refers to the Intercept Only model and L(fitted model) refers to the Intercept and Covariates model. Score – This is the Score Chi-Square Test that at least one of the predictors’ regression coefficient is not equal to zero in the model. Wald – This is the Wald Chi-Square Test that at least one of the predictors’ regression coefficient is not equal to zero in the model. Effect – the predictor variables that are interpreted in terms of odds ratios. Point Estimate – the odds ratio corresponding to Effect. The odds ratio is obtained by exponentiating the Estimate, exp[Estimate]. The difference in the log of two odds is equal to the log of the ratio of these two odds. The log of the ratio of two odds is the log odds ratio. Hence, the interpretation of Estimate–the coefficient was interpreted as the difference in log-odds–could also be done in terms of log-odds ratio. When the Estimate is exponentiated, the log-odds ratio becomes the odds ratio. We can interpret the odds ratio as follows: for a one unit change in the predictor variable, the odds ratio for a positive outcome is expected to change by the respective coefficient, given the other variables in the model are held constant. 95% Wald Confidence Limits – This is the Wald Confidence Interval (CI) of an individual odds ratio, given the other predictors are in the model. For a given predictor variable with a level of 95% confidence, we’d say that we are 95% confident that upon repeated trials, 95% of the CI’s would include the “true” population odds ratio. The CI is equivalent to the Chi-Square test statistic: if the CI includes one, we’d fail to reject the null hypothesis that a particular regression coefficient equals zero and the odds ratio equals one, given the other predictors are in the model. An advantage of a CI is that it is illustrative; it provides information on where the “true” parameter may lie and the precision of the point estimate for the odds ratio.Percent Concordant – A pair of observations with different observed responses is said to be concordant if the observation with the lower ordered response value (= 0) has a lower predicted mean score than the observation with the higher ordered response value (= 1). Percent Discordant – If the observation with the lower ordered response value has a higher predicted mean score than the observation with the higher ordered response value, then the pair is discordant. Percent Tied – If a pair of observations with different responses is neither concordant nor discordant, it is a tie. Pairs – This is the total number of distinct pairs in which one case has an observed outcome different from the other member of the pair. SAS Output Note:No effects for the model in Step 2 are removed. Analysis of Effects Eligiblefor Entry Effect DF Score Chi – Square Pr > ChiSq X6 1 0.6561 0.4179 X7 1 3.5008 0.0613 X8 1 0.0063 0.9369 X9 1 0.6926 0.4053 X10 1 0.0914 0.7624 X11 1 3.4094 0.0648 X12 1 0.8492 0.3568 X14 1 2.3269 0.1272 X15 1 0.0257 0.8727 X16 1 0.0103 0.9192 X18 1 2.9074 0.0882 Note: No (additional) effects met the 0.05 significance level for entry into the model. Summary of Stepwise Selection Step Effect DFNumber In Score Chi – Square Wald Chi – Square Pr > ChiSqVariable Label Entered Removed 1 X13 1 1 21.3297 <.0001 X13 – Competitive Pricing 2 X17 1 2 13.7231 0.0002X17 – Price Flexibility Partition for the Hosmer and Lemeshow Test Group Total X4 = 0 X4 = 1 Observed Expected Observed Expected 1 6 0 0.01 6 5.99 2 7 0 0.11 7 6.89 3 6 0 0.16 6 5.84 4 6 1 0.39 5 5.61 5 6 1 2.41 5 3.59 6 6 6 3.61 0 2.39 7 6 4 4.12 2 1.88 8 6 5 5.03 1 0.97 9 6 4 5.34 2 0.66 10 5 5 4.81 0 0.19 Hosmer and Lemeshow Goodness- of – Fit Test Chi – Square DF Pr > ChiSq 9.9225 8 0.2705 The “Hosmer and Lemeshow Goodness of Fit Test” indicates the quality of model fit. If the associated p-value is significant (p<0.05) this would be an indication that we need to rethink our analytic strategy.Somers’ D – Somer’s D is used to determine the strength and direction of relation between pairs of variables. Its values range from -1.0 (all pairs disagree) to 1.0 (all pairs agree). It is defined as (nc-nd)/t where nc is the number of pairs that are concordant, nd the number of pairs that are discordant, and t is the number of total number of pairs with different responses. In our example, it equals the difference between the percent concordant and the percent discordant divided by 100 Gamma – The Goodman-Kruskal Gamma method does not penalize for ties on either variable. Its values range from -1.0 (no association) to 1.0 (perfect association). Because it does not penalize for ties, its value will generally be greater than the values for Somer’s D. Tau-a – Kendall’s Tau-a is a modification of Somer’s D that takes into the account the difference between the number of possible paired observations and the number of paired observations with a different response. It is defined to be the ratio of the difference between the number of concordant pairs and the number of discordant pairs to the number of possible pairs (2(nc-nd)/(N(N-1)). Usually Tau-a is much smaller than Somer’s D since there would be many paired observations with the same response. c – is equivalent to the well known measure ROC. c ranges from 0.5 to 1, where 0.5 corresponds to the model randomly predicting the response, and a 1 corresponds to the model perfectly discriminating the response. A Receiver Operating Characteristic Curve (ROC) is a standard technique for summarizing classifier performance over a range of trade-offs between true positive (TP) and false positive (FP) error rates. SAS Output Classification Table Prob Level Correct Incorrect Percentages Event Non – Event Event Non – Event Correct Sensi- tivity Speci- ficity False POS False NEG 0.000 26 0 34 0 43.3 100.0 0.0 56.7 . 0.100 25 24 10 1 81.7 96.2 70.6 28.6 4.0 0.200 25 24 10 1 81.7 96.2 70.6 28.6 4.0 0.300 25 25 9 1 83.3 96.2 73.5 26.5 3.8 0.400 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.500 24 28 6 2 86.7 92.3 82.4 20.0 6.7 0.600 20 28 6 6 80.0 76.9 82.4 23.1 17.6 0.700 16 31 3 10 78.3 61.5 91.2 15.8 24.4 0.800 13 31 3 13 73.3 50.0 91.2 18.8 29.5 0.900 6 33 1 20 65.0 23.1 97.1 14.3 37.7 1.000 0 34 0 26 56.7 0.0 100.0 . 43.3 Chapter 6 Logistic Regression Example The LOGISTIC Procedure Model Information Data Set WORK.HBAT60 Response Variable X4X4 – Region Number of Response Levels 2 Modelbinary logit Optimization Technique Fisher’s scoring Number of Observations Read 60 Number of Observations Used 60 Response Profile Ordered Value X4 Total Frequency 1 0 26 2 1 34 Probability modeled is X4=0. Model Convergence Status Convergence criterion (GCONV=1E – 8) satisfied. Model Fit Statistics Criterion Intercept Only Intercept and Covariates AIC 84.108 45.960 SC 86.202 52.243 – 2 Log L 82.108 39.960 SAS Output R – Square0.5046Max – rescaled R – Square 0.6769 Testing Global Null Hypothesis: BETA=0 Test Chi – Square DF Pr > ChiSq Likelihood Ratio 42.1477 2 <.0001 Score 31.3228 2 <.0001 Wald 14.1772 2 0.0008 Analysis of Maximum Likelihood Estimates Parameter DF Estimate Standard Error Wald Chi – Square Pr > ChiSq Intercept 1 14.1917 3.7123 14.6143 0.0001 X13 1 – 1.0791 0.3574 9.1148 0.0025 X17 1 – 1.8439 0.6388 8.3314 0.0039 Odds Ratio Estimates Effect Point Estimate 95% Wald Confidence Limits X13 0.340 0.169 0.685 X17 0.158 0.045 0.553 Association of Predicted Probabilities and Observed Responses Percent Concordant 92.1Somers’ D 0.843 Percent Discordant 7.8Gamma 0.844 Percent Tied 0.1Tau- a 0.421 Pairs 884c 0.921 Partition for the Hosmer and Lemeshow Test Group Total X4 = 0 X4 = 1 Observed Expected Observed Expected 1 6 0 0.01 6 5.99 2 7 0 0.11 7 6.89 3 6 0 0.16 6 5.84 4 6 1 0.39 5 5.61 5 6 1 2.41 5 3.59 6 6 6 3.61 0 2.39 7 6 4 4.12 2 1.88 8 6 5 5.03 1 0.97 9 6 4 5.34 2 0.66 10 5 5 4.81 0 0.19 Hosmer and Lemeshow Goodness- of – Fit Test Chi – Square DF Pr > ChiSq 9.9225 8 0.2705 Classification Table Correct Incorrect Percentages Chi-Square, DF and Pr > ChiSq – These are the Chi-Square test statistic, Degrees of Freedom (DF) and associated p-value (PR>ChiSq) corresponding to the specific test that all of the predictors are simultaneously equal to zero. We are testing the probability (PR>ChiSq) of observing a Chi-Square statistic as extreme as, or more so, than the observed one under the null hypothesis; the null hypothesis is that all of the regression coefficients in the model are equal to zero. The DF defines the distribution of the Chi- Square test statistics and is defined by the number of predictors in the model. Typically, PR>ChiSq is compared to a specified alpha level, our willingness to accept a type I error, which is often set at 0.05 or 0.01. The small p-value from the all three tests would lead us to conclude that at least one of the regression coefficients in the model is not equal to zero. Parameter – Underneath are the predictor variables in the model and the intercept. DF – This column gives the degrees of freedom corresponding to the Parameter. Each Parameter estimated in the model requires one DF and defines the Chi-Square distribution to test whether the individual regression coefficient is zero, given the other variables are in the model. Estimate – These are the binary logit regression estimates for the Parameters in the model. The logistic regression model models the log odds of a positive response (probability modeled is honcomp=1) as a linear combination the predictor variables. We can interpret the parameter estimates as follows: for a one unit change in the predictor variable, the difference in log-odds for a positive outcome is expected to change by the respective coefficient, given the other variables in the model are held constant. Intercept – This is the logistic regression estimate when all variables in the model are evaluated at zero. Standard Error – These are the standard errors of the individual regression coefficients. They are used in both the 95% Wald Confidence Limits and the Chi-Square test statistic. SAS Output ProbLevel Event Non -Event Event Non -Event Correct Sensi-tivity Speci-ficity FalsePOS FalseNEG 0.400 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.410 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.420 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.430 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.440 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.450 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.460 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.470 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.480 25 27 7 1 86.7 96.2 79.4 21.9 3.6 0.490 24 28 6 2 86.7 92.3 82.4 20.0 6.7 0.500 24 28 6 2 86.7 92.3 82.4 20.0 6.7 0.510 24 28 6 2 86.7 92.3 82.4 20.0 6.7 0.520 24 28 6 2 86.7 92.3 82.4 20.0 6.7 0.530 22 28 6 4 83.3 84.6 82.4 21.4 12.5 0.540 22 28 6 4 83.3 84.6 82.4 21.4 12.5 0.550 22 28 6 4 83.3 84.6 82.4 21.4 12.5 0.560 22 28 6 4 83.3 84.6 82.4 21.4 12.5 0.570 22 28 6 4 83.3 84.6 82.4 21.4 12.5 0.580 20 28 6 6 80.0 76.9 82.4 23.1 17.6 0.590 20 28 6 6 80.0 76.9 82.4 23.1 17.6 0.600 20 28 6 6 80.0 76.9 82.4 23.1 17.6 Chapter 6 Logistic Regression Example Obs X4 X13 X17 F_X4 I_X4 P_0 P_1 1 1 6.8 5.1 1 1 0.07241 0.92759 2 0 5.3 4.3 0 0 0.63264 0.36736 3 1 4.5 4.0 1 0 0.87653 0.12347 4 1 8.8 4.1 1 1 0.05393 0.94607 5 0 6.8 3.5 0 0 0.59870 0.40130 6 1 8.5 4.7 1 1 0.02540 0.97460 7 1 8.9 4.2 1 1 0.04082 0.95918 8 1 6.9 6.3 1 1 0.00761 0.99239 9 1 8.4 5.8 1 1 0.00381 0.99619 10 0 6.8 3.7 0 0 0.50781 0.49219 11 1 8.2 4.9 1 1 0.02431 0.97569 12 0 7.1 2.6 0 0 0.85016 0.14984 13 1 8.8 6.2 1 1 0.00119 0.99881 14 0 4.9 3.9 0 0 0.84719 0.15281 15 1 6.2 6.2 1 1 0.01924 0.98076 16 1 8.4 6.1 1 1 0.00219 0.99781 17 1 6.2 4.4 1 1 0.35159 0.64841 SAS Output 180 4.8 3.8 0 0 0.88133 0.11867 190 4.8 3.8 0 0 0.88133 0.11867 20 1 7.7 5.5 1 1 0.01394 0.98606 21 0 5.2 2.7 0 0 0.97345 0.02655 22 1 6.6 3.5 1 0 0.64928 0.35072 23 1 9.2 4.5 1 1 0.01740 0.98260 24 1 8.4 4.3 1 1 0.05723 0.94277 25 0 5.6 2.9 0 0 0.94275 0.05725 26 0 6.8 3.5 0 0 0.59870 0.40130 27 1 7.7 4.6 1 1 0.06917 0.93083 28 0 7.4 4.6 0 1 0.09315 0.90685 29 1 5.9 6.3 1 1 0.02206 0.97794 30 0 5.2 4.2 0 0 0.69759 0.30241 31 1 8.4 5.8 1 1 0.00381 0.99619 32 0 3.8 4.0 0 0 0.93793 0.06207 33 1 8.2 7.3 1 1 0.00030 0.99970 34 1 6.8 3.4 1 0 0.64209 0.35791 35 0 4.7 4.2 0 0 0.79825 0.20175 36 0 5.3 4.3 0 0 0.63264 0.36736 37 1 6.3 4.6 1 1 0.25186 0.74814 38 0 5.2 2.7 0 0 0.97345 0.02655 39 0 7.4 3.2 0 0 0.57585 0.42415 40 0 4.4 3.9 0 0 0.90485 0.09515 41 1 5.4 4.9 1 1 0.33833 0.66167 42 0 4.9 3.9 0 0 0.84719 0.15281 43 0 5.8 3.7 0 0 0.75219 0.24781 44 1 6.2 3.1 1 0 0.85632 0.14368 45 1 7.2 3.6 1 1 0.44621 0.55379 46 1 6.2 6.2 1 1 0.01924 0.98076 47 1 6.0 5.9 1 1 0.04062 0.95938 48 1 7.6 5.4 1 1 0.01858 0.98142 49 0 5.0 2.6 0 0 0.98205 0.01795 50 0 4.6 4.1 0 0 0.84127 0.15873 51 1 7.8 3.1 1 0 0.51462 0.48538 52 0 4.9 4.3 0 0 0.72615 0.27385 53 1 7.6 5.4 1 1 0.01858 0.98142 54 1 8.2 4.8 1 1 0.02909 0.97091 55 0 5.9 3.9 0 0 0.65332 0.34668 56 1 9.9 4.0 1 1 0.02049 0.97951 57 1 4.6 4.1 1 0 0.84127 0.15873 58 0 7.4 3.2 0 0 0.57585 0.42415 59 0 5.6 2.9 0 0 0.94275 0.05725 60 1 9.7 5.4 1 1 0.00196 0.99804 SAS Output Chapter 6 Logistic Regression Example The FREQ Procedure Frequency Percent Row Pct Col Pct Table of F_X4 by I_X4 F_X4(From: X4) I_X4(Into: X4) 0 1 Total 0 25 41.67 96.15 80.65 1 1.67 3.85 3.45 26 43.33 1 6 10.00 17.65 19.35 28 46.67 82.35 96.55 34 56.67 Total 31 51.67 29 48.33 60 100.00 Chapter 6 Logistic Regression Example Obs X4 X13 X17 F_X4 I_X4 P_0 P_1 1 1 9.3 6.1 1 1 0.00083 0.99917 2 0 7.6 4.5 0 1 0.09053 0.90947 3 1 8.4 5.8 1 1 0.00381 0.99619 4 1 9.1 6.0 1 1 0.00124 0.99876 5 1 8.4 4.9 1 1 0.01968 0.98032 6 1 4.5 3.0 1 0 0.97820 0.02180 7 0 3.7 3.4 0 0 0.98073 0.01927 8 1 8.0 5.3 1 1 0.01457 0.98543 9 1 7.1 6.6 1 1 0.00354 0.99646 10 1 9.0 5.2 1 1 0.00601 0.99399 11 1 8.7 6.6 1 1 0.00063 0.99937 12 1 9.0 6.9 1 1 0.00026 0.99974 13 1 8.2 4.9 1 1 0.02431 0.97569 14 1 9.1 5.8 1 1 0.00179 0.99821 15 0 8.5 4.5 0 1 0.03632 0.96368 16 1 7.2 3.6 1 1 0.44621 0.55379 17 1 8.7 6.6 1 1 0.00063 0.99937 18 1 9.6 6.5 1 1 0.00029 0.99971 19 0 3.8 4.0 0 0 0.93793 0.06207 20 1 6.7 5.0 1 1 0.09467 0.90533 21 1 8.2 7.3 1 1 0.00030 0.99970 22 1 7.1 6.6 1 1 0.00354 0.99646 23 1 8.4 6.1 1 1 0.00219 0.99781 24 1 8.7 5.6 1 1 0.00398 0.99602 25 1 6.8 3.4 1 0 0.64209 0.35791 Original Data Classifications Crosstabulations SAS Output 261 6.8 5.1 1 1 0.07241 0.92759 270 4.9 4.3 0 0 0.72615 0.27385 28 1 7.4 5.8 1 1 0.01111 0.98889 29 0 8.5 4.5 0 1 0.03632 0.96368 30 0 6.8 3.7 0 0 0.50781 0.49219 31 0 6.3 3.0 0 0 0.86548 0.13452 32 1 8.4 3.7 1 1 0.15508 0.84492 33 0 5.9 3.9 0 0 0.65332 0.34668 34 0 8.3 4.3 0 1 0.06334 0.93666 35 1 6.3 6.7 1 1 0.00696 0.99304 36 0 7.3 3.0 0 0 0.68621 0.31379 37 0 7.1 2.6 0 0 0.85016 0.14984 38 0 6.7 3.6 0 0 0.58019 0.41981 39 1 7.2 5.6 1 1 0.01977 0.98023 40 1 7.9 4.9 1 1 0.03329 0.96671 Chapter 6 Logistic Regression Example The FREQ Procedure Frequency Percent Row Pct Col Pct Table of F_X4 by I_X4 F_X4(From: X4) I_X4(Into: X4) 0 1 Total 0 9 22.50 69.23 81.82 4 10.00 30.77 13.79 13 32.50 1 2 5.00 7.41 18.18 25 62.50 92.59 86.21 27 67.50 Total 11 27.50 29 72.50 40 100.00 Holdout Sample Classifications Crosstabulations

SAS LOGISTIC

SAS HATCO Logistic Regression – HW 2-Group Analysis: Perform a Logistic Regression Analysis paralleling the HBAT analysis performed in the textbook and in class, but now use the HATCO dataset (see HATCO_Split60 data file). Model: X11 = X1 – X7 Split 60-40 Variable: Split60 Stepwise Analysis Classification and Prediction Analysis including Holdout Sample Fully Discuss the Analyses and Solutions Provide a brief 2-3 page management summary including key selected results along with the SAS program and output as an appendix.

SAS LOGISTIC

1 1 4.1 0.6 6.9 4.7 2.4 2.3 5.2 0 32 4.2 1 0 1 1 2 1 1.8 3 6.3 6.6 2.5 4 8.4 1 43 4.3 0 1 0 1 3 0 3.4 5.2 5.7 6 4.3 2.7 8.2 1 48 5.2 0 1 1 2 4 0 2.7 1 7.1 5.9 1.8 2.3 7.8 1 32 3.9 0 1 1 1 5 1 6 0.9 9.6 7.8 3.4 4.6 4.5 0 58 6.8 1 0 1 3 6 1 1.9 3.3 7.9 4.8 2.6 1.9 9.7 1 45 4.4 0 1 1 2 7 1 4.6 2.4 9.5 6.6 3.5 4.5 7.6 0 46 5.8 1 0 1 1 8 1 1.3 4.2 6.2 5.1 2.8 2.2 6.9 1 44 4.3 0 1 0 2 9 0 5.5 1.6 9.4 4.7 3.5 3 7.6 0 63 5.4 1 0 1 3 10 0 4 3.5 6.5 6 3.7 3.2 8.7 1 54 5.4 0 1 0 2 11 1 2.4 1.6 8.8 4.8 2 2.8 5.8 0 32 4.3 1 0 0 1 12 1 3.9 2.2 9.1 4.6 3 2.5 8.3 0 47 5 1 0 1 2 13 1 2.8 1.4 8.1 3.8 2.1 1.4 6.6 1 39 4.4 0 1 0 1 14 1 3.7 1.5 8.6 5.7 2.7 3.7 6.7 0 38 5 1 0 1 1 15 1 4.7 1.3 9.9 6.7 3 2.6 6.8 0 54 5.9 1 0 0 3 16 0 3.4 2 9.7 4.7 2.7 1.7 4.8 0 49 4.7 1 0 0 3 17 1 3.2 4.1 5.7 5.1 3.6 2.9 6.2 0 38 4.4 1 1 1 2 18 0 4.9 1.8 7.7 4.3 3.4 1.5 5.9 0 40 5.6 1 0 0 2 19 0 5.3 1.4 9.7 6.1 3.3 3.9 6.8 0 54 5.9 1 0 1 3 20 1 4.7 1.3 9.9 6.7 3 2.6 6.8 0 55 6 1 0 0 3 21 0 3.3 0.9 8.6 4 2.1 1.8 6.3 0 41 4.5 1 0 0 2 22 0 3.4 0.4 8.3 2.5 1.2 1.7 5.2 0 35 3.3 1 0 0 1 23 1 3 4 9.1 7.1 3.5 3.4 8.4 0 55 5.2 1 1 0 3 24 1 2.4 1.5 6.7 4.8 1.9 2.5 7.2 1 36 3.7 0 1 0 1 25 1 5.1 1.4 8.7 4.8 3.3 2.6 3.8 0 49 4.9 1 0 0 2 26 1 4.6 2.1 7.9 5.8 3.4 2.8 4.7 0 49 5.9 1 0 1 3 27 0 2.4 1.5 6.6 4.8 1.9 2.5 7.2 1 36 3.7 0 1 0 1 28 1 5.2 1.3 9.7 6.1 3.2 3.9 6.7 0 54 5.8 1 0 1 3 29 1 3.5 2.8 9.9 3.5 3.1 1.7 5.4 0 49 5.4 1 0 1 3 30 0 4.1 3.7 5.9 5.5 3.9 3 8.4 1 46 5.1 0 1 0 2 31 1 3 3.2 6 5.3 3.1 3 8 1 43 3.3 0 1 0 1 32 1 2.8 3.8 8.9 6.9 3.3 3.2 8.2 0 53 5 1 1 0 3 33 1 5.2 2 9.3 5.9 3.7 2.4 4.6 0 60 6.1 1 0 0 3 34 0 3.4 3.7 6.4 5.7 3.5 3.4 8.4 1 47 3.8 0 1 0 1 35 0 2.4 1 7.7 3.4 1.7 1.1 6.2 1 35 4.1 0 1 0 1 36 1 1.8 3.3 7.5 4.5 2.5 2.4 7.6 1 39 3.6 0 1 1 1 37 0 3.6 4 5.8 5.8 3.7 2.5 9.3 1 44 4.8 0 1 1 2 38 0 4 0.9 9.1 5.4 2.4 2.6 7.3 0 46 5.1 1 0 1 3 39 1 0 2.1 6.9 5.4 1.1 2.6 8.9 1 29 3.9 0 1 1 1 40 0 2.4 2 6.4 4.5 2.1 2.2 8.8 1 28 3.3 0 1 1 1 41 0 1.9 3.4 7.6 4.6 2.6 2.5 7.7 1 40 3.7 0 1 1 1 42 1 5.9 0.9 9.6 7.8 3.4 4.6 4.5 0 58 6.7 1 0 1 3 43 1 4.9 2.3 9.3 4.5 3.6 1.3 6.2 0 53 5.9 1 0 0 3 44 0 5 1.3 8.6 4.7 3.1 2.5 3.7 0 48 4.8 1 0 0 2 45 1 2 2.6 6.5 3.7 2.4 1.7 8.5 1 38 3.2 0 1 1 1 46 0 5 2.5 9.4 4.6 3.7 1.4 6.3 0 54 6 1 0 0 3 47 1 3.1 1.9 10 4.5 2.6 3.2 3.8 0 55 4.9 1 0 1 3 48 1 3.4 3.9 5.6 5.6 3.6 2.3 9.1 1 43 4.7 0 1 1 2 49 1 5.8 0.2 8.8 4.5 3 2.4 6.7 0 57 4.9 1 0 1 3 50 1 5.4 2.1 8 3 3.8 1.4 5.2 0 53 3.8 1 0 1 3 51 1 3.7 0.7 8.2 6 2.1 2.5 5.2 0 41 5 1 0 0 2 52 1 2.6 4.8 8.2 5 3.6 2.5 9 1 53 5.2 0 1 1 2 53 1 4.5 4.1 6.3 5.9 4.3 3.4 8.8 1 50 5.5 0 1 0 2 54 1 2.8 2.4 6.7 4.9 2.5 2.6 9.2 1 32 3.7 0 1 1 1 55 0 3.8 0.8 8.7 2.9 1.6 2.1 5.6 0 39 3.7 1 0 0 1 56 0 2.9 2.6 7.7 7 2.8 3.6 7.7 0 47 4.2 1 1 1 2 57 0 4.9 4.4 7.4 6.9 4.6 4 9.6 1 62 6.2 0 1 0 2 58 1 5.4 2.5 9.6 5.5 4 3 7.7 0 65 6 1 0 0 3 59 1 4.3 1.8 7.6 5.4 3.1 2.5 4.4 0 46 5.6 1 0 1 3 60 0 2.3 4.5 8 4.7 3.3 2.2 8.7 1 50 5 0 1 1 2 61 1 3.1 1.9 9.9 4.5 2.6 3.1 3.8 0 54 4.8 1 0 1 3 62 0 5.1 1.9 9.2 5.8 3.6 2.3 4.5 0 60 6.1 1 0 0 3 63 0 4.1 1.1 9.3 5.5 2.5 2.7 7.4 0 47 5.3 1 0 1 3 64 0 3 3.8 5.5 4.9 3.4 2.6 6 0 36 4.2 1 1 1 2 65 1 1.1 2 7.2 4.7 1.6 3.2 10 1 40 3.4 0 1 1 1 66 0 3.7 1.4 9 4.5 2.6 2.3 6.8 0 45 4.9 1 0 0 2 67 1 4.2 2.5 9.2 6.2 3.3 3.9 7.3 0 59 6 1 0 0 3 68 1 1.6 4.5 6.4 5.3 3 2.5 7.1 1 46 4.5 0 1 0 2 69 0 5.3 1.7 8.5 3.7 3.5 1.9 4.8 0 58 4.3 1 0 0 3 70 1 2.3 3.7 8.3 5.2 3 2.3 9.1 1 49 4.8 0 1 1 2 71 1 3.6 5.4 5.9 6.2 4.5 2.9 8.4 1 50 5.4 0 1 1 2 72 1 5.6 2.2 8.2 3.1 4 1.6 5.3 0 55 3.9 1 0 1 3 73 1 3.6 2.2 9.9 4.8 2.9 1.9 4.9 0 51 4.9 1 0 0 3 74 0 5.2 1.3 9.1 4.5 3.3 2.7 7.3 0 60 5.1 1 0 1 3 75 0 3 2 6.6 6.6 2.4 2.7 8.2 1 41 4.1 0 1 0 1 76 0 4.2 2.4 9.4 4.9 3.2 2.7 8.5 0 49 5.2 1 0 1 2 77 0 3.8 0.8 8.3 6.1 2.2 2.6 5.3 0 42 5.1 1 0 0 2 78 0 3.3 2.6 9.7 3.3 2.9 1.5 5.2 0 47 5.1 1 0 1 3 79 1 1 1.9 7.1 4.5 1.5 3.1 9.9 1 39 3.3 0 1 1 1 80 1 4.5 1.6 8.7 4.6 3.1 2.1 6.8 0 56 5.1 1 0 0 3 81 1 5.5 1.8 8.7 3.8 3.6 2.1 4.9 0 59 4.5 1 0 0 3 82 1 3.4 4.6 5.5 8.2 4 4.4 6.3 0 47 5.6 1 1 1 2 83 0 1.6 2.8 6.1 6.4 2.3 3.8 8.2 1 41 4.1 0 1 0 1 84 1 2.3 3.7 7.6 5 3 2.5 7.4 0 37 4.4 1 1 0 1 85 0 2.6 3 8.5 6 2.8 2.8 6.8 1 53 5.6 0 1 0 2 86 1 2.5 3.1 7 4.2 2.8 2.2 9 1 43 3.7 0 1 1 1 87 0 2.4 2.9 8.4 5.9 2.7 2.7 6.7 1 51 5.5 0 1 0 2 88 1 2.1 3.5 7.4 4.8 2.8 2.3 7.2 0 36 4.3 1 1 0 1 89 1 2.9 1.2 7.3 6.1 2 2.5 8 1 34 4 0 1 1 1 90 1 4.3 2.5 9.3 6.3 3.4 4 7.4 0 60 6.1 1 0 0 3 91 0 3 2.8 7.8 7.1 3 3.8 7.9 0 49 4.4 1 1 1 2 92 1 4.8 1.7 7.6 4.2 3.3 1.4 5.8 0 39 5.5 1 0 0 2 93 1 3.1 4.2 5.1 7.8 3.6 4 5.9 0 43 5.2 1 1 1 2 94 0 1.9 2.7 5 4.9 2.2 2.5 8.2 1 36 3.6 0 1 0 1 95 1 4 0.5 6.7 4.5 2.2 2.1 5 0 31 4 1 0 1 1 96 1 0.6 1.6 6.4 5 0.7 2.1 8.4 1 25 3.4 0 1 1 1 97 1 6.1 0.5 9.2 4.8 3.3 2.8 7.1 0 60 5.2 1 0 1 3 98 0 2 2.8 5.2 5 2.4 2.7 8.4 1 38 3.7 0 1 0 1 99 1 3.1 2.2 6.7 6.8 2.6 2.9 8.4 1 42 4.3 0 1 0 1 100 0 2.5 1.8 9 5 2.2 3 6 0 33 4.4 1 0 0 1

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