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