* ch18eg2.sas, using data from exercise 18.17; * 1-way ANOVA: residual plots, Hartley's test, * Brown-Forsythe Test, Box-Cox transformations, * Satterthwaite's procedure, and non-parametric test; ; options ls=76 nodate; ; data WindSpd; do A=1 to 4; * A = speed; do j=1 to 16; drop j; input y @@; * y = number of breaks; output; if (y ne .); end; end; lines; 4 3 2 3 4 4 3 6 5 4 2 4 4 2 3 4 7 6 4 6 7 2 9 5 5 9 3 8 6 4 7 6 12 6 14 12 10 9 12 17 7 6 12 11 6 13 10 14 17 15 7 20 13 11 16 25 11 24 18 21 16 19 9 23 ; * Fit the ANOVA models and save info for residual plots; proc glm noprint; class A; model y = A; output out=B r=e p=yhat; title 'Chapter 18 example using data of exercise 18.17.'; proc rank normal=Blom; var e; ranks nscore; ; proc print; title2 'Data, fitted values, residuals, and normal scores.'; ; * Residual plots; proc plot; plot e*yhat / vpos=19 vref=0; plot e*nscore / vpos=19 vref=0 href=0; title2 'Residual Plots.'; ; * Non-parametric Test (using ranks); proc rank data=WindSpd; var y; ranks rank; title2 'Non-parametric Test (F statistics using ranks).'; proc print; proc plot; plot rank*A / vpos=19; proc glm; class A; model rank = A; means A; ; * Hartley's test, and Box-Cox transformation; proc sort data=WindSpd; by A; proc means noprint data=WindSpd; by A; var y; output out=C mean=ybar var=s2; data C; set C; ln_ybar=log(ybar); ln_s2=log(s2); proc print; title2 'Conduct Hartleys Test, and Determine a Box-Cox Transformation.'; proc plot; plot ln_s2*ln_ybar / vpos=19; ; * Based on the plot, use h(y)=ln(y); data WindSpd; set WindSpd; ln_y = log(y); proc glm; class A; model ln_y = A; means A / Tukey alpha=0.01; output out=B r=e p=lnyhat; title2 'Analysis of ln(y). (How to interpret the results?)'; proc rank normal=Blom; var e; ranks nscore; ; * Residual plots; proc plot; plot e*lnyhat / vpos=19 vref=0; plot e*nscore / vpos=19 vref=0 href=0; ; proc corr nosimple; var e nscore; ; * Brown-Forsythe Test; data WindSpd; set Windspd; * make this the default data set; title2 'Brown-Forsythe Test.'; proc means noprint; by A; var y; output out=D median=median; data dij; merge WindSpd D; by A; d=abs(y-median); proc print; proc glm; class A; model d = A; ; * Use Satterthwaite's method for comparing means; data A12; set WindSpd; if A=1 or A=2; proc ttest ci=none; var y; class A; * no CI for sigma; title2 'Compare levels 1 and 2 of A using Satterthwaites approximation.'; ; data A13; set WindSpd; if A=1 or A=3; proc ttest ci=none; var y; class A; title2 'Compare levels 1 and 3 of A using Satterthwaites approximation.'; ; data A14; set WindSpd; if A=1 or A=4; proc ttest ci=none; var y; class A; title2 'Compare levels 1 and 4 of A using Satterthwaites approximation.'; ; data A23; set WindSpd; if A=2 or A=3; proc ttest ci=none; var y; class A; title2 'Compare levels 2 and 3 of A using Satterthwaites approximation.'; ; data A24; set WindSpd; if A=2 or A=4; proc ttest ci=none; var y; class A; title2 'Compare levels 2 and 4 of A using Satterthwaites approximation.'; ; data A34; set WindSpd; if A=3 or A=4; proc ttest ci=none; var y; class A; title2 'Compare levels 3 and 4 of A using Satterthwaites approximation.'; Chapter 18 example using data of exercise 18.17. 1 Data, fitted values, residuals, and normal scores. Obs A y e yhat nscore 1 1 4 0.4375 3.5625 0.19714 2 1 3 -0.5625 3.5625 -0.27725 3 1 2 -1.5625 3.5625 -0.67244 4 1 3 -0.5625 3.5625 -0.27725 5 1 4 0.4375 3.5625 0.19714 6 1 4 0.4375 3.5625 0.19714 7 1 3 -0.5625 3.5625 -0.27725 8 1 6 2.4375 3.5625 0.94222 9 1 5 1.4375 3.5625 0.72138 10 1 4 0.4375 3.5625 0.19714 11 1 2 -1.5625 3.5625 -0.67244 12 1 4 0.4375 3.5625 0.19714 13 1 4 0.4375 3.5625 0.19714 14 1 2 -1.5625 3.5625 -0.67244 15 1 3 -0.5625 3.5625 -0.27725 16 1 4 0.4375 3.5625 0.19714 17 2 7 1.1250 5.8750 0.42208 18 2 6 0.1250 5.8750 -0.07818 19 2 4 -1.8750 5.8750 -0.88300 20 2 6 0.1250 5.8750 -0.07818 21 2 7 1.1250 5.8750 0.42208 22 2 2 -3.8750 5.8750 -1.18163 23 2 9 3.1250 5.8750 1.07192 24 2 5 -0.8750 5.8750 -0.53126 25 2 5 -0.8750 5.8750 -0.53126 26 2 9 3.1250 5.8750 1.07192 27 2 3 -2.8750 5.8750 -0.97263 28 2 8 2.1250 5.8750 0.79929 29 2 6 0.1250 5.8750 -0.07818 30 2 4 -1.8750 5.8750 -0.88300 31 2 7 1.1250 5.8750 0.42208 32 2 6 0.1250 5.8750 -0.07818 33 3 12 1.3125 10.6875 0.57738 34 3 6 -4.6875 10.6875 -1.36040 35 3 14 3.3125 10.6875 1.22282 36 3 12 1.3125 10.6875 0.57738 37 3 10 -0.6875 10.6875 -0.44335 38 3 9 -1.6875 10.6875 -0.79929 39 3 12 1.3125 10.6875 0.57738 40 3 17 6.3125 10.6875 1.58555 41 3 7 -3.6875 10.6875 -1.10656 42 3 6 -4.6875 10.6875 -1.36040 43 3 12 1.3125 10.6875 0.57738 44 3 11 0.3125 10.6875 0.01951 45 3 6 -4.6875 10.6875 -1.36040 46 3 13 2.3125 10.6875 0.85420 47 3 10 -0.6875 10.6875 -0.44335 48 3 14 3.3125 10.6875 1.22282 49 4 17 0.4375 16.5625 0.19714 50 4 15 -1.5625 16.5625 -0.67244 51 4 7 -9.5625 16.5625 -2.33669 52 4 20 3.4375 16.5625 1.35601 53 4 13 -3.5625 16.5625 -1.03727 54 4 11 -5.5625 16.5625 -1.66320 55 4 16 -0.5625 16.5625 -0.27725 Chapter 18 example using data of exercise 18.17. 2 Data, fitted values, residuals, and normal scores. Obs A y e yhat nscore 56 4 25 8.4375 16.5625 2.33669 57 4 11 -5.5625 16.5625 -1.66320 58 4 24 7.4375 16.5625 1.95500 59 4 18 1.4375 16.5625 0.72138 60 4 21 4.4375 16.5625 1.46117 61 4 16 -0.5625 16.5625 -0.27725 62 4 19 2.4375 16.5625 0.94222 63 4 9 -7.5625 16.5625 -1.95500 64 4 23 6.4375 16.5625 1.74084 Chapter 18 example using data of exercise 18.17. 3 Residual Plots. Plot of e*yhat. Legend: A = 1 obs, B = 2 obs, etc. 10 + | A | A | A A e | | A | B B A | A A A A | A C D A 0 +--------G---------D------------------A----------------------A-------- | G B B C | B A | B A A | C | B | | A | -10 + A ---+-------+-------+-------+-------+-------+-------+-------+-------+-- 2 4 6 8 10 12 14 16 18 yhat Chapter 18 example using data of exercise 18.17. 4 Residual Plots. Plot of e*nscore. Legend: A = 1 obs, B = 2 obs, etc. 10 + | | | A | | A | | A A e | | | | A | | BB A | | BB | | CD B 0 +----------------------------------DA-H--------------------------------- | DBB F | | BA | | AAB | | C | | B | | | | A | | | -10 + A | ---+----------+----------+----------+----------+----------+----------+-- -3 -2 -1 0 1 2 3 Rank for Variable e Chapter 18 example using data of exercise 18.17. 5 Non-parametric Test (F statistics using ranks). Obs A y rank 1 1 4 14.0 2 1 3 7.0 3 1 2 2.5 4 1 3 7.0 5 1 4 14.0 6 1 4 14.0 7 1 3 7.0 8 1 6 25.5 9 1 5 20.0 10 1 4 14.0 11 1 2 2.5 12 1 4 14.0 13 1 4 14.0 14 1 2 2.5 15 1 3 7.0 16 1 4 14.0 17 2 7 32.0 18 2 6 25.5 19 2 4 14.0 20 2 6 25.5 21 2 7 32.0 22 2 2 2.5 23 2 9 37.5 24 2 5 20.0 25 2 5 20.0 26 2 9 37.5 27 2 3 7.0 28 2 8 35.0 29 2 6 25.5 30 2 4 14.0 31 2 7 32.0 32 2 6 25.5 33 3 12 46.5 34 3 6 25.5 35 3 14 51.5 36 3 12 46.5 37 3 10 40.5 38 3 9 37.5 39 3 12 46.5 40 3 17 56.5 41 3 7 32.0 42 3 6 25.5 43 3 12 46.5 44 3 11 43.0 45 3 6 25.5 46 3 13 49.5 47 3 10 40.5 48 3 14 51.5 49 4 17 56.5 50 4 15 53.0 51 4 7 32.0 52 4 20 60.0 53 4 13 49.5 54 4 11 43.0 55 4 16 54.5 Chapter 18 example using data of exercise 18.17. 6 Non-parametric Test (F statistics using ranks). Obs A y rank 56 4 25 64.0 57 4 11 43.0 58 4 24 63.0 59 4 18 58.0 60 4 21 61.0 61 4 16 54.5 62 4 19 59.0 63 4 9 37.5 64 4 23 62.0 Chapter 18 example using data of exercise 18.17. 7 Non-parametric Test (F statistics using ranks). Plot of rank*A. Legend: A = 1 obs, B = 2 obs, etc. R | a 80 + n | k | | B f 60 + E o | A D r | C A | E B V 40 + B C A a | A r | C A A i | A D C a 20 + A B b | G B l | e | G B 0 + y | ---+-----------------+-----------------+-----------------+-- 1 2 3 4 A Chapter 18 example using data of exercise 18.17. 8 Non-parametric Test (F statistics using ranks). The GLM Procedure Class Level Information Class Levels Values A 4 1 2 3 4 Number of Observations Read 64 Number of Observations Used 64 Chapter 18 example using data of exercise 18.17. 9 Non-parametric Test (F statistics using ranks). The GLM Procedure Dependent Variable: rank Rank for Variable y Sum of Source DF Squares Mean Square F Value Pr > F Model 3 16539.15625 5513.05208 64.14 <.0001 Error 60 5157.34375 85.95573 Corrected Total 63 21696.50000 R-Square Coeff Var Root MSE rank Mean 0.762296 28.52687 9.271231 32.50000 Chapter 18 example using data of exercise 18.17. 10 Non-parametric Test (F statistics using ranks). The GLM Procedure Level of -------------rank------------ A N Mean Std Dev 1 16 11.1875000 6.5035247 2 16 24.0937500 10.5502271 3 16 41.5625000 9.8908628 4 16 53.1562500 9.6120042 Chapter 18 example using data of exercise 18.17. 11 Conduct Hartleys Test, and Determine a Box-Cox Transformation. Obs A _TYPE_ _FREQ_ ybar s2 ln_ybar ln_s2 1 1 0 16 3.5625 1.1958 1.27046 0.17884 2 2 0 16 5.8750 3.9833 1.77071 1.38212 3 3 0 16 10.6875 10.4958 2.36907 2.35098 4 4 0 16 16.5625 28.9292 2.80714 3.36485 Chapter 18 example using data of exercise 18.17. 12 Conduct Hartleys Test, and Determine a Box-Cox Transformation. Plot of ln_s2*ln_ybar. Legend: A = 1 obs, B = 2 obs, etc. ln_s2 | 4 + | | | A 3 + | | | A 2 + | | A | 1 + | | | A 0 + | ---+-------------+-------------+-------------+-------------+-- 1.0 1.5 2.0 2.5 3.0 ln_ybar Chapter 18 example using data of exercise 18.17. 13 Analysis of ln(y). (How to interpret the results?) The GLM Procedure Class Level Information Class Levels Values A 4 1 2 3 4 Number of Observations Read 64 Number of Observations Used 64 Chapter 18 example using data of exercise 18.17. 14 Analysis of ln(y). (How to interpret the results?) The GLM Procedure Dependent Variable: ln_y Sum of Source DF Squares Mean Square F Value Pr > F Model 3 21.69169895 7.23056632 56.78 <.0001 Error 60 7.64042176 0.12734036 Corrected Total 63 29.33212071 R-Square Coeff Var Root MSE ln_y Mean 0.739520 17.84571 0.356848 1.999628 Chapter 18 example using data of exercise 18.17. 15 Analysis of ln(y). (How to interpret the results?) The GLM Procedure Tukey's Studentized Range (HSD) Test for ln_y NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.01 Error Degrees of Freedom 60 Error Mean Square 0.12734 Critical Value of Studentized Range 4.59444 Minimum Significant Difference 0.4099 Means with the same letter are not significantly different. Tukey Grouping Mean N A A 2.7499 16 4 B 2.3211 16 3 C 1.7039 16 2 D 1.2237 16 1 Chapter 18 example using data of exercise 18.17. 16 Analysis of ln(y). (How to interpret the results?) Plot of e*lnyhat. Legend: A = 1 obs, B = 2 obs, etc. e | 1.0 + | | | A 0.5 + B A A | A A B B | C A C | G D E B 0.0 +---------------------------------------B-----------C--------- | D B A A | | B A B -0.5 + C C A | A | A | -1.0 + A | ---+-------------+-------------+-------------+-------------+-- 1.0 1.5 2.0 2.5 3.0 lnyhat Chapter 18 example using data of exercise 18.17. 17 Analysis of ln(y). (How to interpret the results?) Plot of e*nscore. Legend: A = 1 obs, B = 2 obs, etc. e | | 1.0 + | | | | | | | A 0.5 + | A B A | | BAAAA | | ACBA | AADA G D 0.0 +------------------------------ABB-+----------------------------------- | A DAB | | | | ABB | -0.5 + A C C | | A | | A | | | -1.0 + A | | | --+----------+----------+----------+----------+----------+----------+-- -3 -2 -1 0 1 2 3 Rank for Variable e Chapter 18 example using data of exercise 18.17. 18 Analysis of ln(y). (How to interpret the results?) The CORR Procedure 2 Variables: e nscore Pearson Correlation Coefficients, N = 64 Prob > |r| under H0: Rho=0 e nscore e 1.00000 0.97166 <.0001 nscore 0.97166 1.00000 Rank for Variable e <.0001 Chapter 18 example using data of exercise 18.17. 19 Brown-Forsythe Test. Obs A y ln_y _TYPE_ _FREQ_ median d 1 1 4 1.38629 0 16 4.0 0.0 2 1 3 1.09861 0 16 4.0 1.0 3 1 2 0.69315 0 16 4.0 2.0 4 1 3 1.09861 0 16 4.0 1.0 5 1 4 1.38629 0 16 4.0 0.0 6 1 4 1.38629 0 16 4.0 0.0 7 1 3 1.09861 0 16 4.0 1.0 8 1 6 1.79176 0 16 4.0 2.0 9 1 5 1.60944 0 16 4.0 1.0 10 1 4 1.38629 0 16 4.0 0.0 11 1 2 0.69315 0 16 4.0 2.0 12 1 4 1.38629 0 16 4.0 0.0 13 1 4 1.38629 0 16 4.0 0.0 14 1 2 0.69315 0 16 4.0 2.0 15 1 3 1.09861 0 16 4.0 1.0 16 1 4 1.38629 0 16 4.0 0.0 17 2 7 1.94591 0 16 6.0 1.0 18 2 6 1.79176 0 16 6.0 0.0 19 2 4 1.38629 0 16 6.0 2.0 20 2 6 1.79176 0 16 6.0 0.0 21 2 7 1.94591 0 16 6.0 1.0 22 2 2 0.69315 0 16 6.0 4.0 23 2 9 2.19722 0 16 6.0 3.0 24 2 5 1.60944 0 16 6.0 1.0 25 2 5 1.60944 0 16 6.0 1.0 26 2 9 2.19722 0 16 6.0 3.0 27 2 3 1.09861 0 16 6.0 3.0 28 2 8 2.07944 0 16 6.0 2.0 29 2 6 1.79176 0 16 6.0 0.0 30 2 4 1.38629 0 16 6.0 2.0 31 2 7 1.94591 0 16 6.0 1.0 32 2 6 1.79176 0 16 6.0 0.0 33 3 12 2.48491 0 16 11.5 0.5 34 3 6 1.79176 0 16 11.5 5.5 35 3 14 2.63906 0 16 11.5 2.5 36 3 12 2.48491 0 16 11.5 0.5 37 3 10 2.30259 0 16 11.5 1.5 38 3 9 2.19722 0 16 11.5 2.5 39 3 12 2.48491 0 16 11.5 0.5 40 3 17 2.83321 0 16 11.5 5.5 41 3 7 1.94591 0 16 11.5 4.5 42 3 6 1.79176 0 16 11.5 5.5 43 3 12 2.48491 0 16 11.5 0.5 44 3 11 2.39790 0 16 11.5 0.5 45 3 6 1.79176 0 16 11.5 5.5 46 3 13 2.56495 0 16 11.5 1.5 47 3 10 2.30259 0 16 11.5 1.5 48 3 14 2.63906 0 16 11.5 2.5 49 4 17 2.83321 0 16 16.5 0.5 50 4 15 2.70805 0 16 16.5 1.5 51 4 7 1.94591 0 16 16.5 9.5 52 4 20 2.99573 0 16 16.5 3.5 53 4 13 2.56495 0 16 16.5 3.5 54 4 11 2.39790 0 16 16.5 5.5 55 4 16 2.77259 0 16 16.5 0.5 Chapter 18 example using data of exercise 18.17. 20 Brown-Forsythe Test. Obs A y ln_y _TYPE_ _FREQ_ median d 56 4 25 3.21888 0 16 16.5 8.5 57 4 11 2.39790 0 16 16.5 5.5 58 4 24 3.17805 0 16 16.5 7.5 59 4 18 2.89037 0 16 16.5 1.5 60 4 21 3.04452 0 16 16.5 4.5 61 4 16 2.77259 0 16 16.5 0.5 62 4 19 2.94444 0 16 16.5 2.5 63 4 9 2.19722 0 16 16.5 7.5 64 4 23 3.13549 0 16 16.5 6.5 Chapter 18 example using data of exercise 18.17. 21 Brown-Forsythe Test. The GLM Procedure Class Level Information Class Levels Values A 4 1 2 3 4 Number of Observations Read 64 Number of Observations Used 64 Chapter 18 example using data of exercise 18.17. 22 Brown-Forsythe Test. The GLM Procedure Dependent Variable: d Sum of Source DF Squares Mean Square F Value Pr > F Model 3 111.5468750 37.1822917 9.54 <.0001 Error 60 233.8125000 3.8968750 Corrected Total 63 345.3593750 R-Square Coeff Var Root MSE d Mean 0.322988 85.94505 1.974050 2.296875 Chapter 18 example using data of exercise 18.17. 23 Compare levels 1 and 2 of A using Satterthwaites approximation. The TTEST Procedure Variable: y A N Mean Std Dev Std Err Minimum Maximum 1 16 3.5625 1.0935 0.2734 2.0000 6.0000 2 16 5.8750 1.9958 0.4990 2.0000 9.0000 Diff (1-2) -2.3125 1.6092 0.5689 A Method Mean 95% CL Mean 1 3.5625 2.9798 4.1452 2 5.8750 4.8115 6.9385 Diff (1-2) Pooled -2.3125 -3.4744 -1.1506 Diff (1-2) Satterthwaite -2.3125 -3.4887 -1.1363 Method Variances DF t Value Pr > |t| Pooled Equal 30 -4.06 0.0003 Satterthwaite Unequal 23.262 -4.06 0.0005 Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 15 15 3.33 0.0258 Chapter 18 example using data of exercise 18.17. 24 Compare levels 1 and 3 of A using Satterthwaites approximation. The TTEST Procedure Variable: y A N Mean Std Dev Std Err Minimum Maximum 1 16 3.5625 1.0935 0.2734 2.0000 6.0000 3 16 10.6875 3.2397 0.8099 6.0000 17.0000 Diff (1-2) -7.1250 2.4178 0.8548 A Method Mean 95% CL Mean 1 3.5625 2.9798 4.1452 3 10.6875 8.9612 12.4138 Diff (1-2) Pooled -7.1250 -8.8708 -5.3792 Diff (1-2) Satterthwaite -7.1250 -8.9183 -5.3317 Method Variances DF t Value Pr > |t| Pooled Equal 30 -8.34 <.0001 Satterthwaite Unequal 18.374 -8.34 <.0001 Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 15 15 8.78 0.0001 Chapter 18 example using data of exercise 18.17. 25 Compare levels 1 and 4 of A using Satterthwaites approximation. The TTEST Procedure Variable: y A N Mean Std Dev Std Err Minimum Maximum 1 16 3.5625 1.0935 0.2734 2.0000 6.0000 4 16 16.5625 5.3786 1.3446 7.0000 25.0000 Diff (1-2) -13.0000 3.8810 1.3722 A Method Mean 95% CL Mean 1 3.5625 2.9798 4.1452 4 16.5625 13.6965 19.4285 Diff (1-2) Pooled -13.0000 -15.8023 -10.1977 Diff (1-2) Satterthwaite -13.0000 -15.9054 -10.0946 Method Variances DF t Value Pr > |t| Pooled Equal 30 -9.47 <.0001 Satterthwaite Unequal 16.238 -9.47 <.0001 Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 15 15 24.19 <.0001 Chapter 18 example using data of exercise 18.17. 26 Compare levels 2 and 3 of A using Satterthwaites approximation. The TTEST Procedure Variable: y A N Mean Std Dev Std Err Minimum Maximum 2 16 5.8750 1.9958 0.4990 2.0000 9.0000 3 16 10.6875 3.2397 0.8099 6.0000 17.0000 Diff (1-2) -4.8125 2.6906 0.9513 A Method Mean 95% CL Mean 2 5.8750 4.8115 6.9385 3 10.6875 8.9612 12.4138 Diff (1-2) Pooled -4.8125 -6.7553 -2.8697 Diff (1-2) Satterthwaite -4.8125 -6.7719 -2.8531 Method Variances DF t Value Pr > |t| Pooled Equal 30 -5.06 <.0001 Satterthwaite Unequal 24.952 -5.06 <.0001 Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 15 15 2.63 0.0701 Chapter 18 example using data of exercise 18.17. 27 Compare levels 2 and 4 of A using Satterthwaites approximation. The TTEST Procedure Variable: y A N Mean Std Dev Std Err Minimum Maximum 2 16 5.8750 1.9958 0.4990 2.0000 9.0000 4 16 16.5625 5.3786 1.3446 7.0000 25.0000 Diff (1-2) -10.6875 4.0566 1.4342 A Method Mean 95% CL Mean 2 5.8750 4.8115 6.9385 4 16.5625 13.6965 19.4285 Diff (1-2) Pooled -10.6875 -13.6166 -7.7584 Diff (1-2) Satterthwaite -10.6875 -13.6888 -7.6862 Method Variances DF t Value Pr > |t| Pooled Equal 30 -7.45 <.0001 Satterthwaite Unequal 19.054 -7.45 <.0001 Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 15 15 7.26 0.0004 Chapter 18 example using data of exercise 18.17. 28 Compare levels 3 and 4 of A using Satterthwaites approximation. The TTEST Procedure Variable: y A N Mean Std Dev Std Err Minimum Maximum 3 16 10.6875 3.2397 0.8099 6.0000 17.0000 4 16 16.5625 5.3786 1.3446 7.0000 25.0000 Diff (1-2) -5.8750 4.4399 1.5697 A Method Mean 95% CL Mean 3 10.6875 8.9612 12.4138 4 16.5625 13.6965 19.4285 Diff (1-2) Pooled -5.8750 -9.0808 -2.6692 Diff (1-2) Satterthwaite -5.8750 -9.1105 -2.6395 Method Variances DF t Value Pr > |t| Pooled Equal 30 -3.74 0.0008 Satterthwaite Unequal 24.618 -3.74 0.0010 Equality of Variances Method Num DF Den DF F Value Pr > F Folded F 15 15 2.76 0.0584