As mentioned in Section 1.13.4 significance tests concerning the treatment effects are performed as approximate F test by substituting ρ for ρ in the coefficient matrixAof (1.60), using any of the previously described methods of estimating ρ. Such tests are mostly conveniently performed by choosing any option in SAS PROC MIXED.
An exact test, based, however, on the assumption of normality, for H0: τ1=τ2= · · · =τt
against
H1: not allτi equal
was developed by Cohen and Sackrowitz (1989). We shall give a brief descrip- tion here, but the reader should refer to the original article for details. The test is based on invariance properties and utilizes information from the intrablock analysis (as described in Sections 2.4.1 and 1.3) and the interblock analysis (as described for the general case in Section 1.7.2) by combining theP values from
the two respective independent tests. In order to define the test we establish the following notation:
τ =solution to the intrablock normal equations, (2.13) τ∗=solution to the interblock normal equations, (1.48),
withL=I
O=matrix oft−1 orthonormal contrasts for treatment effects U1=Oτ
U2=Oτ∗ U12=U1U1 U22=U2U2
V1=U1/U1 V2=U2/U2
R=V1V2
S2=SS(Error from intrablock analysis) S∗2=SS(Error from interblock analysis)
= b j=1
Bj−kµ∗− t
i=1
nijτi∗
2
T1=S2+λt k U12 T2=S∗2+r−λ
k U22 a=min(T1/T2,1) γ =1/(a+1)
P =P value for testingH0using intrablock analysis
P∗=P value for testingH0using interblock analysis, based on F statistic
F∗= (b−t)(r−λ) k(t−1)
U22
S∗2 with t−1 andb−t d.f.
Z1= −nP Z2= −nP∗
z=γ Z1+(1−γ )Z2
Then, forγ = 12, theP value for the exact test is given by P∗∗=
γ e−z/γ−(1−γ )e−z/(1−γ )
/[(2γ −1)(1+R)]
and for γ = 12 by (2z+1)e−2z!
/(1+R). Cohen and Sackrowitz (1989) per- formed some power simulations and showed that the exact test is more powerful than the usual (approximate) F test. For a more general discussion see also Mathew, Sinha, and Zhou (1993) and Zhou and Mathew (1993).
We shall illustrate the test by using an example from Lentner and Bishop (1993).
Example 2.2 An experiment is concerned with studying the effect of six diets on weight gain of rabbits using 10 litters of three rabbits each. The data are given in Table 2.1A. The parameters of the BIB design aret =6, b=10, k= 3, r =5, λ=2.
The intrablock analysis using SAS PROC GLM is presented in Table 2.1A yieldingF =3.16 andP =0.0382. In addition, Table 2.1A gives the SAS solu- tions to the normal equations, which are then used to compute the set of five orthonormal contrasts based on
−5/√
70 −3/√
70 −1/√
70 1/√
70 3/√
70 5/√
70 5/√
84 −1/√
84 −4/√
84 −4/√
84 −1/√
84 5/√ 84
−5/√
180 7/√
180 4/√
180 −4/√
180 −7/√
180 5/√ 180 1/√
28 −3/√
28 2/√
28 2/√
28 −3/√
28 1/√ 28
−1/√
252 5/√
252 −10/√
252 10/√
252 −5/√
252 1/√ 252
The linear, quadratic, cubic, quartic, and quintic parameter estimates represent the elements ofU1, yieldingU12=39.6817.
A solution to the interblock normal equations [usingL=I in (1.48)] is given in Table 2.1B.
The estimatesτ∗are then used to obtainU2andU22=537.5284. We also obtain from Table 2.1BF∗=2.23 and P∗=0.2282.
UsingS2=150.7728 from Table 2.1A andS∗2=577.9133 from Table 2.1B, all other values needed for the test can be computed. Withγ =0.7828 we obtain Z=2.8768, and finallyP∗∗=0.03. This result is in good agreement with theP value of 0.0336 obtained from the combined analysis using SAS PROC MIXED
with the REML option (see Table 2.1A).
Table 2.1A Data, Intrablock, and Combined Analysis for BIB Design (t=6,b=10,r=5,k=3, LAMBDA=2)
options nodate pageno=1;
data rabbit1;
input B T Y @@;
datalines;
1 6 42.2 1 2 32.6 1 3 35.2 2 3 40.9 2 1 40.1 2 2 38.1 3 3 34.6 3 6 34.3 3 4 37.5 4 1 44.9 4 5 40.8 4 3 43.9 5 5 32.0 5 3 40.9 5 4 37.3 6 2 37.3 6 6 42.8 6 5 40.5 7 4 37.9 7 1 45.2 7 2 40.6 8 1 44.0 8 5 38.5 8 6 51.9 9 4 27.5 9 2 30.6 9 5 20.6 10 6 41.7 10 4 42.3 10 1 37.3
; run;
proc print data=rabbit1;
title1 'TABLE 2.1A';
title2 'DATA FOR BIB DESIGN';
title3 '(t=6, b=10, r=5, k=3, LAMBDA=2)';
run;
proc glm data=rabbit1;
class B T;
model Y=B T/solution;
estimate 'linear' T -5 -3 -1 1 3 5/divisor=8.3667;
estimate 'quad' T 5 -1 -4 -4 -1 5/divisor=9.1652;
estimate 'cubic' T -5 7 4 -4 -7 5/divisor=13.4164;
estimate 'quartic' T 1 -3 2 2 -3 1/divisor=5.2915;
estimate 'quintic' T -1 5 -10 10 -5 1/divisor=15.8745;
title2 'INTRA-BLOCK ANALYSIS';
title3 'WITH ORTHONORMAL CONTRASTS';
run;
proc mixed data=rabbit1;
class B T;
model Y=T/solution;
random B;
lsmeans T;
title2 'COMBINED INTRA- AND INTER-BLOCK ANALYSIS';
title3 '(USING METHOD DESCRIBED IN SECTION 1.8)';
run;
Table 2.1A (Continued)
Obs B T Y
1 1 6 42.2
2 1 2 32.6
3 1 3 35.2
4 2 3 40.9
5 2 1 40.1
6 2 2 38.1
7 3 3 34.6
8 3 6 34.3
9 3 4 37.5
10 4 1 44.9
11 4 5 40.8
12 4 3 43.9
13 5 5 32.0
14 5 3 40.9
15 5 4 37.3
16 6 2 37.3
17 6 6 42.8
18 6 5 40.5
19 7 4 37.9
20 7 1 45.2
21 7 2 40.6
22 8 1 44.0
23 8 5 38.5
24 8 6 51.9
25 9 4 27.5
26 9 2 30.6
27 9 5 20.6
28 10 6 41.7
29 10 4 42.3
30 10 1 37.3
The GLM Procedure Class Level Information Class Levels Values
B 10 1 2 3 4 5 6 7 8 9 10
T 6 1 2 3 4 5 6
Table 2.1A (Continued)
Number of observations 30 Dependent Variable: Y
Sum of Mean
Source DF Squares Square F Value Pr > F Model 14 889.113889 63.508135 6.32 0.0005 Error 15 150.772778 10.051519
Corrected
Total 29 1039.886667
R-Square Coeff Var Root MSE Y Mean 0.855010 8.241975 3.170413 38.46667
Source DF Type I SS Mean Square F Value Pr > F
B 9 730.3866667 81.1540741 8.07 0.0002
T 5 158.7272222 31.7454444 3.16 0.0382
Source DF Type III SS Mean Square F Value Pr > F
B 9 595.7352222 66.1928025 6.59 0.0008
T 5 158.7272222 31.7454444 3.16 0.0382
Standard
Parameter Estimate Error t Value Pr > |t|
linear 0.68326421 1.58518760 0.43 0.6726
quad 2.35674071 1.58519809 1.49 0.1578
cubic 3.14664639 1.58520742 1.99 0.0657
quartic 4.74975590 1.58520728 3.00 0.0090 quintic 1.09504761 1.58520728 0.69 0.5002
Standard
Parameter Estimate Error t Value Pr > |t|
Intercept 42.61111111 B 2.24182052 19.01 <.0001 B 1 -3.29722222 B 2.79604144 -1.18 0.2567
B 2 0.83611111 B 2.79604144 0.30 0.7690
Table 2.1A (Continued)
Standard
Parameter Estimate Error t Value Pr > |t|
B 3 -5.10000000 B 2.69433295 -1.89 0.0778
B 4 5.49722222 B 2.79604144 1.97 0.0681
B 5 -0.99166667 B 2.79604144 -0.35 0.7278
B 6 2.11111111 B 2.79604144 0.76 0.4619
B 7 2.48055556 B 2.69433295 0.92 0.3718
B 8 6.13055556 B 2.69433295 2.28 0.0380
B 9 -10.77777778 B 2.79604144 -3.85 0.0016
B 10 0.00000000 B . . .
T 1 -3.30000000 B 2.24182052 -1.47 0.1617 T 2 -5.04166667 B 2.24182052 -2.25 0.0400 T 3 -2.90000000 B 2.24182052 -1.29 0.2154 T 4 -3.23333333 B 2.24182052 -1.44 0.1698 T 5 -8.52500000 B 2.24182052 -3.80 0.0017
T 6 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.
COMBINED INTRA- AND INTERBLOCK ANALYSIS (USING METHOD DESCRIBED IN SECTION 1.8)
The Mixed Procedure Model Information
Data Set WORK.RABBIT1
Dependent Variable Y
Covariance Structure Variance Components Estimation Method REML
Residual Variance Method Profile Fixed Effects SE Method Model-Based Degrees of Freedom Method Containment
Class Level Information Class Levels Values
B 10 1 2 3 4 5 6 7 8 9 10
T 6 1 2 3 4 5 6
Table 2.1A (Continued)
Dimensions
Covariance Parameters 2
Columns in X 7
Columns in Z 10
Subjects 1
Max Obs Per Subject 30 Observations Used 30 Observations Not Used 0 Total Observations 30
Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 160.26213715
1 2 150.36288495 0.00010765
2 1 150.35691057 0.00000054
3 1 150.35688183 0.00000000
Convergence criteria met.
Covariance Parameter Estimates
Cov Parm Estimate
B 21.6953
Residual 10.0840
Fit Statistics
-2 Res Log Likelihood 150.4 AIC (smaller is better) 154.4 AICC (smaller is better) 154.9 BIC (smaller is better) 155.0
Solution for Fixed Effects Standard
Effect T Estimate Error DF t Value Pr > |t|
Intercept 42.3454 2.1303 9 19.88 <.0001
T 1 -2.8100 2.2087 15 -1.27 0.2227
Table 2.1A (Continued)
Solution for Fixed Effects Standard
Effect T Estimate Error DF t Value Pr > |t|
T 2 -5.3172 2.2087 15 -2.41 0.0294
T 3 -2.9941 2.2087 15 -1.36 0.1953
T 4 -3.6952 2.2087 15 -1.67 0.1150
T 5 -8.4560 2.2087 15 -3.83 0.0016
T 6 0 . . . .
Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
T 5 15 3.28 0.0336
Least Squares Means Standard
Effect T Estimate Error DF t Value Pr > |t|
T 1 39.5354 2.1303 15 18.56 <.0001
T 2 37.0282 2.1303 15 17.38 <.0001
T 3 39.3513 2.1303 15 18.47 <.0001
T 4 38.6502 2.1303 15 18.14 <.0001
T 5 33.8894 2.1303 15 15.91 <.0001
T 6 42.3454 2.1303 15 19.88 <.0001
Table 2.1B Data of Block Totals and Interblock Analysis options nodate pageno=1;
data rabbit2;
input y x1 x2 x3 x4 x5 x6;
datalines;
110.0 0 1 1 0 0 1 119.1 1 1 1 0 0 0 106.4 0 0 1 1 0 1 129.6 1 0 1 0 1 0 110.2 0 0 1 1 1 0 120.6 0 1 0 0 1 1 123.7 1 1 0 1 0 0 134.4 1 0 0 0 1 1
Table 2.1B (Continued) 78.7 0 1 0 1 1 0 121.3 1 0 0 1 0 1
; run;
proc print data=rabbit2;
title1 'TABLE 2.1 B';
title2 'DATA OF BLOCK TOTALS';
proc glm data=rabbit2;
model y=x1 x2 x3 x4 x5 x6;
title2 'INTER-BLOCK ANALYSIS';
run;
Obs y x1 x2 x3 x4 x5 x6
1 110.0 0 1 1 0 0 1
2 119.1 1 1 1 0 0 0
3 106.4 0 0 1 1 0 1
4 129.6 1 0 1 0 1 0
5 110.2 0 0 1 1 1 0
6 120.6 0 1 0 0 1 1
7 123.7 1 1 0 1 0 0
8 134.4 1 0 0 0 1 1
9 78.7 0 1 0 1 1 0
10 121.3 1 0 0 1 0 1
INTERBLOCK ANALYSIS The GLM Procedure Number of observations 10
The GLM Procedure Dependent Variable: y
Sum of Mean
Source DF Squares Square F Value Pr > F Model 5 1613.246667 322.649333 2.23 0.2282 Error 4 577.913333 144.478333
Corrected Total 9 2191.160000
Table 2.1B (Continued)
R-Square Coeff Var Root MSE y Mean 0.736252 10.41587 12.01991 115.4000
Source DF Type I SS Mean Square F Value Pr > F x1 1 1044.484000 1044.484000 7.23 0.0547
x2 1 89.792667 89.792667 0.62 0.4746
x3 1 10.454444 10.454444 0.07 0.8012
x4 1 407.075556 407.075556 2.82 0.1685
x5 1 61.440000 61.440000 0.43 0.5499
x6 0 0.000000 . . .
Standard
Parameter Estimate Error t Value Pr > |t|
Intercept 131.1000000 B 19.38152901 6.76 0.0025
x1 11.8000000 B 9.81421871 1.20 0.2955
x2 -13.5333333 B 9.81421871 -1.38 0.2400
x3 -5.8000000 B 9.81421871 -0.59 0.5863
x4 -17.4666667 B 9.81421871 -1.78 0.1497
x5 -6.4000000 B 9.81421871 -0.65 0.5499
x6 0.0000000 B . . .