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This page shows how to obtain the results from Kirk Chapter 7 using SAS.
Use data file rb4 (page 167).
data rb4; input y a s; datalines; 3 1 1 4 2 1 4 3 1 3 4 1 2 1 2 4 2 2 4 3 2 5 4 2 2 1 3 3 2 3 3 3 3 6 4 3 3 1 4 3 2 4 3 3 4 5 4 4 1 1 5 2 2 5 4 3 5 7 4 5 3 1 6 3 2 6 6 3 6 6 4 6 4 1 7 4 2 7 5 3 7 10 4 7 6 1 8 5 2 8 5 3 8 8 4 8 ; run;
Table 7.2-1, page 260.
Note: The all option in the table statement display row and column sums.
options formchar='|-*+*+++*+*'; proc tabulate data=rb4; class s a; var y; table s all, sum*y=' '*(a all); run;*--------------+----------------------------------------------------------------* | | Sum | | +---------------------------------------------------+------------+ | | a | | | +------------+------------+------------+------------+ | | | 1 | 2 | 3 | 4 | All | +--------------+------------+------------+------------+------------+------------+ |s | | | | | | +--------------+ | | | | | |1 | 3.00| 4.00| 4.00| 3.00| 14.00| +--------------+------------+------------+------------+------------+------------+ |2 | 2.00| 4.00| 4.00| 5.00| 15.00| +--------------+------------+------------+------------+------------+------------+ |3 | 2.00| 3.00| 3.00| 6.00| 14.00| +--------------+------------+------------+------------+------------+------------+ |4 | 3.00| 3.00| 3.00| 5.00| 14.00| +--------------+------------+------------+------------+------------+------------+ |5 | 1.00| 2.00| 4.00| 7.00| 14.00| +--------------+------------+------------+------------+------------+------------+ |6 | 3.00| 3.00| 6.00| 6.00| 18.00| +--------------+------------+------------+------------+------------+------------+ |7 | 4.00| 4.00| 5.00| 10.00| 23.00| +--------------+------------+------------+------------+------------+------------+ |8 | 6.00| 5.00| 5.00| 8.00| 24.00| +--------------+------------+------------+------------+------------+------------+ |All | 24.00| 28.00| 34.00| 50.00| 136.00| *--------------+------------+------------+------------+------------+------------*
Table 7.2-2, page 261.
proc glm data=rb4; class a s; model y = a s; run;The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 10 80.5000000 8.0500000 5.73 0.0004 Error 21 29.5000000 1.4047619 Corrected Total 31 110.0000000 R-Square Coeff Var Root MSE y Mean 0.731818 27.88768 1.185227 4.250000 Source DF Type I SS Mean Square F Value Pr > F a 3 49.00000000 16.33333333 11.63 0.0001 s 7 31.50000000 4.50000000 3.20 0.0180 Source DF Type III SS Mean Square F Value Pr > F a 3 49.00000000 16.33333333 11.63 0.0001 s 7 31.50000000 4.50000000 3.20 0.0180
Table 7.3-2, page 271.
Note: The process of obtaining the Tukey's test for additivity takes several steps.
proc means mean data=rb4; /* obtain grand mean */ var y; run;Analysis Variable : y Mean ------------ 4.2500000 ------------ proc glm data=rb4; class a s; model y = a s; output out=rb4out predicted=yhat; /* save the predicted scores yhat */ run; [output omitted] data rb4out; set rb4out; ybar = 4.25; /* the grand mean */ ystar = (yhat-ybar)**2; run; proc glm data=rb4out; class a s; model y = a s ystar / ss3; /* use sums of squares type 3 */ /* the test for ystar is Tukeys test for additivity' */ run; The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 11 82.2738905 7.4794446 5.40 0.0006 Error 20 27.7261095 1.3863055 Corrected Total 31 110.0000000 R-Square Coeff Var Root MSE y Mean 0.747944 27.70388 1.177415 4.250000 Source DF Type III SS Mean Square F Value Pr > F a 3 35.72642695 11.90880898 8.59 0.0007 s 7 23.76793627 3.39541947 2.45 0.0549 ystar 1 1.77389051 1.77389051 1.28 0.2714
Figure 7.3-1, page 272.
proc glm data=rb4; class a s; model y = a s; output out=rb4out2 predicted=p rstudent=r ; run;[output omitted] symbol1 v=circle; proc gplot data=rb4out2; plot r*p; run;
Various computations on compound symmetry, pages 274-282.
Note 1: This analysis requires that the data be organized in its wide form, with one line per subject.
Note 2: Proc corr is used with the cov option to display the covariance matrix. The nocorr and nosimple options suppress the correlation matrix and descriptive statistics.
data rb4wide; input s y1 y2 y3 y4; datalines; 1 3 4 4 3 2 2 4 4 5 3 2 3 3 6 4 3 3 3 5 5 1 2 4 7 6 3 3 6 6 7 4 4 5 10 8 6 5 5 8 ; run; proc corr cov nocorr nosimple data=rb4wide; var y1 y2 y3 y4; run;The CORR Procedure 4 Variables: y1 y2 y3 y4 Covariance Matrix, DF = 7 y1 y2 y3 y4 y1 2.285714286 1.142857143 0.714285714 1.285714286 y2 1.142857143 0.857142857 0.285714286 0.285714286 y3 0.714285714 0.285714286 1.071428571 0.928571429 y4 1.285714286 0.285714286 0.928571429 4.500000000 proc glm data=rb4wide; model y1 y2 y3 y4 = / nouni; repeated a; run; The GLM Procedure Repeated Measures Analysis of Variance Repeated Measures Level Information Dependent Variable y1 y2 y3 y4 Level of a 1 2 3 4 Manova Test Criteria and Exact F Statistics for the Hypothesis of no a Effect H = Type III SSCP Matrix for a E = Error SSCP Matrix S=1 M=0.5 N=1.5 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.24580793 5.11 3 5 0.0554 Pillai's Trace 0.75419207 5.11 3 5 0.0554 Hotelling-Lawley Trace 3.06821705 5.11 3 5 0.0554 Roy's Greatest Root 3.06821705 5.11 3 5 0.0554 Univariate Tests of Hypotheses for Within Subject Effects Adj Pr > F Source DF Type III SS Mean Square F Value Pr > F G - G H - F a 3 49.00000000 16.33333333 11.63 0.0001 0.0015 0.0003 Error(a) 21 29.50000000 1.40476190 Greenhouse-Geisser Epsilon 0.6195 Huynh-Feldt Epsilon 0.8343
Use data file rb3.
data rb3; input s a y; datalines; 1 1 15 1 2 12 1 3 11 2 2 11 2 3 9 3 1 13 3 2 13 ; run;
Table 7.8-2, page 299.
proc tabulate data=rb3; class s a; var y; table s, sum*y=' '*a; run;*----------------+--------------------------------------* | | Sum | | +--------------------------------------+ | | a | | +------------+------------+------------+ | | 1 | 2 | 3 | +----------------+------------+------------+------------+ |s | | | | +----------------+ | | | |1 | 15.00| 12.00| 11.00| +----------------+------------+------------+------------+ |2 | .| 11.00| 9.00| +----------------+------------+------------+------------+ |3 | 13.00| 13.00| .| *----------------+------------+------------+------------*
Table 7.8-3, page 301.
proc glm data=rb3; class a s; model y = a s / ss3; run;The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 4 19.73333333 4.93333333 4.35 0.1954 Error 2 2.26666667 1.13333333 Corrected Total 6 22.00000000 R-Square Coeff Var Root MSE y Mean 0.896970 8.871511 1.064581 12.00000 Source DF Type III SS Mean Square F Value Pr > F a 2 8.40000000 4.20000000 3.71 0.2125 s 2 3.73333333 1.86666667 1.65 0.3778
Use datafile grb4.
data grb4; input g id a y; datalines; 1 1 1 3 1 2 1 3 1 1 2 4 1 2 2 3 1 1 3 4 1 2 3 3 1 1 4 3 1 2 4 5 2 1 1 2 2 2 1 2 2 1 2 4 2 2 2 3 2 1 3 4 2 2 3 3 2 1 4 5 2 2 4 6 3 1 1 6 3 2 1 3 3 1 2 5 3 2 2 3 3 1 3 5 3 2 3 6 3 1 4 8 3 2 4 6 4 1 1 1 4 2 1 4 4 1 2 2 4 2 2 4 4 1 3 4 4 2 3 5 4 1 4 7 4 2 4 10 ; run;
Tables 7.9-1, page 305.
proc tabulate data=grb4; class g id a; var y; table g*id, sum*y=' '*a; run;*----------------+---------------------------------------------------* | | Sum | | +---------------------------------------------------+ | | a | | +------------+------------+------------+------------+ | | 1 | 2 | 3 | 4 | +-------+--------+------------+------------+------------+------------+ |g |id | | | | | +-------+--------+ | | | | |1 |1 | 3.00| 4.00| 4.00| 3.00| | +--------+------------+------------+------------+------------+ | |2 | 3.00| 3.00| 3.00| 5.00| +-------+--------+------------+------------+------------+------------+ |2 |1 | 2.00| 4.00| 4.00| 5.00| | +--------+------------+------------+------------+------------+ | |2 | 2.00| 3.00| 3.00| 6.00| +-------+--------+------------+------------+------------+------------+ |3 |1 | 6.00| 5.00| 5.00| 8.00| | +--------+------------+------------+------------+------------+ | |2 | 3.00| 3.00| 6.00| 6.00| +-------+--------+------------+------------+------------+------------+ |4 |1 | 1.00| 2.00| 4.00| 7.00| | +--------+------------+------------+------------+------------+ | |2 | 4.00| 4.00| 5.00| 10.00| *-------+--------+------------+------------+------------+------------* proc tabulate data=grb4; class g a; var y; table g all, sum*y=' '*(a all); run; *----------------+----------------------------------------------------------------* | | Sum | | +---------------------------------------------------+------------+ | | a | | | +------------+------------+------------+------------+ | | | 1 | 2 | 3 | 4 | All | +----------------+------------+------------+------------+------------+------------+ |g | | | | | | +----------------+ | | | | | |1 | 6.00| 7.00| 7.00| 8.00| 28.00| +----------------+------------+------------+------------+------------+------------+ |2 | 4.00| 7.00| 7.00| 11.00| 29.00| +----------------+------------+------------+------------+------------+------------+ |3 | 9.00| 8.00| 11.00| 14.00| 42.00| +----------------+------------+------------+------------+------------+------------+ |4 | 5.00| 6.00| 9.00| 17.00| 37.00| +----------------+------------+------------+------------+------------+------------+ |All | 24.00| 28.00| 34.00| 50.00| 136.00| *----------------+------------+------------+------------+------------+------------*
Table 7.9-2, page 306.
Note: To obtain the correct F-ratio for a, a*b is used as the error term.
proc glm data=grb4; class a g id; model y = a g a*g / ss3; test h=a e=a*g; run;The GLM Procedure Dependent Variable: y Sum of Source DF Squares Mean Square F Value Pr > F Model 15 85.0000000 5.6666667 3.63 0.0074 Error 16 25.0000000 1.5625000 Corrected Total 31 110.0000000 R-Square Coeff Var Root MSE y Mean 0.772727 29.41176 1.250000 4.250000 Source DF Type III SS Mean Square F Value Pr > F a 3 49.00000000 16.33333333 10.45 0.0005 g 3 16.75000000 5.58333333 3.57 0.0377 a*g 9 19.25000000 2.13888889 1.37 0.2795 Tests of Hypotheses Using the Type III MS for a*g as an Error Term Source DF Type III SS Mean Square F Value Pr > F a 3 49.00000000 16.33333333 7.64 0.0076
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