options linesize = 80 ; footnote "~john-c/5421/incomplete.sas &sysdate &systime" ; data incomplete ; blocks = 4 ; treatments = 4 ; mu = 6 ; amean = 5 ; bmean = 7 ; cmean = 10 ; dmean = 13 ; sigma = 3 ; reps = 5 ; block = 1 ; pattern = 'ACD'; do j = 1 to reps ; assay = 'A' ; y = mu + amean + sigma*rannor(-1) ; output ; assay = 'C' ; y = mu + cmean + sigma*rannor(-1) ; output ; assay = 'D' ; y = mu + dmean + sigma*rannor(-1) ; output ; end ; block = 2 ; pattern = 'ABC'; do j = 1 to reps ; assay = 'A' ; y = mu + amean + sigma*rannor(-1) ; output ; assay = 'B' ; y = mu + bmean + sigma*rannor(-1) ; output ; assay = 'C' ; y = mu + cmean + sigma*rannor(-1) ; output ; end ; block = 3 ; pattern = 'BCD'; do j = 1 to reps ; assay = 'B' ; y = mu + bmean + sigma*rannor(-1) ; output ; assay = 'C' ; y = mu + cmean + sigma*rannor(-1) ; output ; assay = 'D' ; y = mu + dmean + sigma*rannor(-1) ; output ; end ; block = 4 ; pattern = 'ABD'; do j = 1 to reps ; assay = 'A' ; y = mu + amean + sigma*rannor(-1) ; output ; assay = 'B' ; y = mu + bmean + sigma*rannor(-1) ; output ; assay = 'D' ; y = mu + dmean + sigma*rannor(-1) ; output ; end ; run ; proc freq data = incomplete ; tables block * assay / chisq ; run ; proc glm data = incomplete ; class block assay pattern ; model y = block assay / solution ; title1 'Incomplete block design ...' ; run ; proc glm data = incomplete ; class block assay pattern ; model y = block | assay / solution ; title1 'Incomplete block design ...' ; run ; ================================================================================ The SAS System 1 19:59 Monday, April 25, 2011 The FREQ Procedure Table of block by assay block assay Frequency| Percent | Row Pct | Col Pct |A |B |C |D | Total ---------+--------+--------+--------+--------+ 1 | 5 | 0 | 5 | 5 | 15 | 8.33 | 0.00 | 8.33 | 8.33 | 25.00 | 33.33 | 0.00 | 33.33 | 33.33 | | 33.33 | 0.00 | 33.33 | 33.33 | ---------+--------+--------+--------+--------+ 2 | 5 | 5 | 5 | 0 | 15 | 8.33 | 8.33 | 8.33 | 0.00 | 25.00 | 33.33 | 33.33 | 33.33 | 0.00 | | 33.33 | 33.33 | 33.33 | 0.00 | ---------+--------+--------+--------+--------+ 3 | 0 | 5 | 5 | 5 | 15 | 0.00 | 8.33 | 8.33 | 8.33 | 25.00 | 0.00 | 33.33 | 33.33 | 33.33 | | 0.00 | 33.33 | 33.33 | 33.33 | ---------+--------+--------+--------+--------+ 4 | 5 | 5 | 0 | 5 | 15 | 8.33 | 8.33 | 0.00 | 8.33 | 25.00 | 33.33 | 33.33 | 0.00 | 33.33 | | 33.33 | 33.33 | 0.00 | 33.33 | ---------+--------+--------+--------+--------+ Total 15 15 15 15 60 25.00 25.00 25.00 25.00 100.00 Statistics for Table of block by assay Statistic DF Value Prob ------------------------------------------------------ Chi-Square 9 20.0000 0.0179 Likelihood Ratio Chi-Square 9 34.5218 <.0001 Mantel-Haenszel Chi-Square 1 0.0000 1.0000 Phi Coefficient 0.5774 Contingency Coefficient 0.5000 Cramer's V 0.3333 WARNING: 100% of the cells have expected counts less than 5. Chi-Square may not be a valid test. Sample Size = 60 ================================================================================ Incomplete block design ... 2 19:59 Monday, April 25, 2011 The GLM Procedure Class Level Information Class Levels Values block 4 1 2 3 4 assay 4 A B C D pattern 4 ABC ABD ACD BCD Number of observations 60 ================================================================================ Incomplete block design ... 3 19:59 Monday, April 25, 2011 The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 6 469.1362917 78.1893820 10.78 <.0001 Error 53 384.5241685 7.2551730 Corrected Total 59 853.6604603 R-Square Coeff Var Root MSE y Mean 0.549558 18.49189 2.693543 14.56607 Source DF Type I SS Mean Square F Value Pr > F block 3 72.8195276 24.2731759 3.35 0.0258 assay 3 396.3167641 132.1055880 18.21 <.0001 Source DF Type III SS Mean Square F Value Pr > F block 3 11.9432191 3.9810730 0.55 0.6512 assay 3 396.3167641 132.1055880 18.21 <.0001 Standard Parameter Estimate Error t Value Pr > |t| Intercept 17.67895505 B 0.92002003 19.22 <.0001 block 1 0.14319292 B 1.04320465 0.14 0.8913 block 2 0.77047718 B 1.04320465 0.74 0.4634 block 3 1.16855244 B 1.04320465 1.12 0.2677 block 4 0.00000000 B . . . assay A -6.59225838 B 1.04320465 -6.32 <.0001 assay B -5.91235861 B 1.04320465 -5.67 <.0001 assay C -2.02913490 B 1.04320465 -1.95 0.0571 assay D 0.00000000 B . . . NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable. ================================================================================ Incomplete block design ... 4 19:59 Monday, April 25, 2011 The GLM Procedure Class Level Information Class Levels Values block 4 1 2 3 4 assay 4 A B C D pattern 4 ABC ABD ACD BCD Number of observations 60 The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 11 539.0146686 49.0013335 7.48 <.0001 Error 48 314.6457916 6.5551207 Corrected Total 59 853.6604603 R-Square Coeff Var Root MSE y Mean 0.631416 17.57713 2.560297 14.56607 Source DF Type I SS Mean Square F Value Pr > F block 3 72.8195276 24.2731759 3.70 0.0178 assay 3 396.3167641 132.1055880 20.15 <.0001 block*assay 5 69.8783769 13.9756754 2.13 0.0776 Source DF Type III SS Mean Square F Value Pr > F block 3 11.9432191 3.9810730 0.61 0.6135 assay 3 396.3167641 132.1055880 20.15 <.0001 block*assay 5 69.8783769 13.9756754 2.13 0.0776 Standard Parameter Estimate Error t Value Pr > |t| Intercept 17.70721839 B 1.14499962 15.46 <.0001 block 1 1.42189963 B 1.61927399 0.88 0.3843 block 2 -1.19480022 B 2.28999924 -0.52 0.6042 block 3 -0.19494430 B 1.61927399 -0.12 0.9047 block 4 0.00000000 B . . . assay A -6.36994595 B 1.61927399 -3.93 0.0003 assay B -6.21946106 B 1.61927399 -3.84 0.0004 assay C -0.70820211 B 1.61927399 -0.44 0.6638 assay D 0.00000000 B . . . block*assay 1 A -3.46663443 B 2.28999924 -1.51 0.1366 block*assay 1 C -1.99752095 B 2.28999924 -0.87 0.3874 block*assay 1 D 0.00000000 B . . . block*assay 2 A 3.40147782 B 2.80466483 1.21 0.2311 block*assay 2 B 1.17342160 B 2.80466483 0.42 0.6775 block*assay 2 C 0.00000000 B . . . block*assay 3 B 2.99186988 B 2.28999924 1.31 0.1976 block*assay 3 C 0.00000000 B . . . block*assay 3 D 0.00000000 B . . . block*assay 4 A 0.00000000 B . . . ================================================================================ Incomplete block design ... 6 19:59 Monday, April 25, 2011 The GLM Procedure Dependent Variable: y Standard Parameter Estimate Error t Value Pr > |t| block*assay 4 B 0.00000000 B . . . block*assay 4 D 0.00000000 B . . . NOTE: The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable.