### Example 59.6 Line-Source Sprinkler Irrigation

These data appear in Hanks et al. (1980); Johnson, Chaudhuri, and Kanemasu (1983); Stroup (1989b). Three cultivars (Cult) of winter wheat are randomly assigned to rectangular plots within each of three blocks (Block). The nine plots are located side by side, and a line-source sprinkler is placed through the middle. Each plot is subdivided into twelve subplots—six to the north of the line source, six to the south (Dir). The two plots closest to the line source represent the maximum irrigation level (Irrig=6), the two next-closest plots represent the next-highest level (Irrig=5), and so forth.

This example is a case where both and can be modeled. One of Stroup’s models specifies a diagonal containing the variance components for Block, Block*Dir, and Block*Irrig, and a Toeplitz with four bands. The SAS statements to fit this model and carry out some further analyses follow.

Caution: This analysis can require considerable CPU time.

data line;
length Cult$8; input Block Cult$ @;
row = _n_;
do Sbplt=1 to 12;
if Sbplt le 6 then do;
Irrig = Sbplt;
Dir = 'North';
end; else do;
Irrig = 13 - Sbplt;
Dir = 'South';
end;
input Y @; output;
end;
datalines;
1 Luke     2.4 2.7 5.6 7.5 7.9 7.1 6.1 7.3 7.4 6.7 3.8 1.8
1 Nugaines 2.2 2.2 4.3 6.3 7.9 7.1 6.2 5.3 5.3 5.2 5.4 2.9
1 Bridger  2.9 3.2 5.1 6.9 6.1 7.5 5.6 6.5 6.6 5.3 4.1 3.1
2 Nugaines 2.4 2.2 4.0 5.8 6.1 6.2 7.0 6.4 6.7 6.4 3.7 2.2
2 Bridger  2.6 3.1 5.7 6.4 7.7 6.8 6.3 6.2 6.6 6.5 4.2 2.7
2 Luke     2.2 2.7 4.3 6.9 6.8 8.0 6.5 7.3 5.9 6.6 3.0 2.0
3 Nugaines 1.8 1.9 3.7 4.9 5.4 5.1 5.7 5.0 5.6 5.1 4.2 2.2
3 Luke     2.1 2.3 3.7 5.8 6.3 6.3 6.5 5.7 5.8 4.5 2.7 2.3
3 Bridger  2.7 2.8 4.0 5.0 5.2 5.2 5.9 6.1 6.0 4.3 3.1 3.1
;


proc mixed;
class Block Cult Dir Irrig;
model Y = Cult|Dir|Irrig@2;
random Block Block*Dir Block*Irrig;
repeated / type=toep(4) sub=Block*Cult r;
lsmeans Cult|Irrig;
estimate 'Bridger vs Luke' Cult 1 -1 0;
estimate 'Linear Irrig' Irrig -5 -3 -1 1 3 5;
estimate 'B vs L x Linear Irrig' Cult*Irrig
-5 -3 -1 1 3 5 5 3 1 -1 -3 -5;
run;


The preceding statements use the bar operator ( | ) and the at sign (@) to specify all two-factor interactions between Cult, Dir, and Irrig as fixed effects.

The RANDOM statement sets up the and matrices corresponding to the random effects Block, Block*Dir, and Block*Irrig.

In the REPEATED statement, the TYPE=TOEP(4) option sets up the blocks of the matrix to be Toeplitz with four bands below and including the main diagonal. The subject effect is Block*Cult, and it produces nine 1212 blocks. The R option requests that the first block of be displayed.

Least squares means (LSMEANS) are requested for Cult, Irrig, and Cult*Irrig, and a few ESTIMATE statements are specified to illustrate some linear combinations of the fixed effects.

The results from this analysis are shown in Output 59.6.1.

The Covariance Structures row in Output 59.6.1 reveals the two different structures assumed for and .

Output 59.6.1: Model Information in Line-Source Sprinkler Analysis

The Mixed Procedure

Model Information
Data Set WORK.LINE
Dependent Variable Y
Covariance Structures Variance Components, Toeplitz
Subject Effect Block*Cult
Estimation Method REML
Residual Variance Method Profile
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Containment

The levels of each classification variable are listed as a single string in the Values column, regardless of whether the levels are numeric or character (Output 59.6.2).

Output 59.6.2: Class Level Information

Class Level Information
Class Levels Values
Block 3 1 2 3
Cult 3 Bridger Luke Nugaines
Dir 2 North South
Irrig 6 1 2 3 4 5 6

Even though there is a SUBJECT= effect in the REPEATED statement, the analysis considers all of the data to be from one subject because there is no corresponding SUBJECT= effect in the RANDOM statement (Output 59.6.3).

Output 59.6.3: Model Dimensions and Number of Observations

Dimensions
Covariance Parameters 7
Columns in X 48
Columns in Z 27
Subjects 1
Max Obs Per Subject 108

Number of Observations
Number of Observations Used 108
Number of Observations Not Used 0

The Newton-Raphson algorithm converges successfully in seven iterations (Output 59.6.4).

Output 59.6.4: Iteration History and Convergence Status

Iteration History
Iteration Evaluations -2 Res Log Like Criterion
0 1 226.25427252
1 4 187.99336173 .
2 3 186.62579299 0.10431081
3 1 184.38218213 0.04807260
4 1 183.41836853 0.00886548
5 1 183.25111475 0.00075353
6 1 183.23809997 0.00000748
7 1 183.23797748 0.00000000

 Convergence criteria met.

The first block of the estimated matrix has the TOEP(4) structure, and the observations that are three plots apart exhibit a negative correlation (Output 59.6.5).

Output 59.6.5: Estimated R Matrix for the First Subject

Estimated R Matrix for Block*Cult 1 Bridger
Row Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9 Col10 Col11 Col12
1 0.2850 0.007986 0.001452 -0.09253
2 0.007986 0.2850 0.007986 0.001452 -0.09253
3 0.001452 0.007986 0.2850 0.007986 0.001452 -0.09253
4 -0.09253 0.001452 0.007986 0.2850 0.007986 0.001452 -0.09253
5   -0.09253 0.001452 0.007986 0.2850 0.007986 0.001452 -0.09253
6     -0.09253 0.001452 0.007986 0.2850 0.007986 0.001452 -0.09253
7       -0.09253 0.001452 0.007986 0.2850 0.007986 0.001452 -0.09253
8         -0.09253 0.001452 0.007986 0.2850 0.007986 0.001452 -0.09253
9           -0.09253 0.001452 0.007986 0.2850 0.007986 0.001452 -0.09253
10             -0.09253 0.001452 0.007986 0.2850 0.007986 0.001452
11               -0.09253 0.001452 0.007986 0.2850 0.007986
12                 -0.09253 0.001452 0.007986 0.2850

Output 59.6.6 lists the estimated covariance parameters from both and . The first three are the variance components making up the diagonal , and the final four make up the Toeplitz structure in the blocks of . The Residual row corresponds to the variance of the Toeplitz structure, and it represents the parameter profiled out during the optimization process.

Output 59.6.6: Estimated Covariance Parameters

Covariance Parameter Estimates
Cov Parm Subject Estimate
Block   0.2194
Block*Dir   0.01768
Block*Irrig   0.03539
TOEP(2) Block*Cult 0.007986
TOEP(3) Block*Cult 0.001452
TOEP(4) Block*Cult -0.09253
Residual   0.2850

The –2 Res Log Likelihood value in Output 59.6.7 is the same as the final value listed in the Iteration History table (Output 59.6.4).

Output 59.6.7: Fit Statistics Based on the Residual Log Likelihood

Fit Statistics
-2 Res Log Likelihood 183.2
AIC (smaller is better) 197.2
AICC (smaller is better) 198.8
BIC (smaller is better) 190.9

Every fixed effect except for Dir and Cult*Irrig is significant at the 5% level (Output 59.6.8).

Output 59.6.8: Tests for Fixed Effects

Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
Cult 2 68 7.98 0.0008
Dir 1 2 3.95 0.1852
Cult*Dir 2 68 3.44 0.0379
Irrig 5 10 102.60 <.0001
Cult*Irrig 10 68 1.91 0.0580
Dir*Irrig 5 68 6.12 <.0001

The Estimates table lists the results from the various linear combinations of fixed effects specified in the ESTIMATE statements (Output 59.6.9). Bridger is not significantly different from Luke, and Irrig possesses a strong linear component. This strength appears to be influencing the significance of the interaction.

Output 59.6.9: Estimates

Estimates
Label Estimate Standard Error DF t Value Pr > |t|
Bridger vs Luke -0.03889 0.09524 68 -0.41 0.6843
Linear Irrig 30.6444 1.4412 10 21.26 <.0001
B vs L x Linear Irrig -9.8667 2.7400 68 -3.60 0.0006

The least squares means shown in Output 59.6.10 are useful in comparing the levels of the various fixed effects. For example, it appears that irrigation levels 5 and 6 have virtually the same effect.

Output 59.6.10: Least Squares Means for Cult, Irrig, and Their Interaction

Least Squares Means
Effect Cult Irrig Estimate Standard Error DF t Value Pr > |t|
Cult Bridger   5.0306 0.2874 68 17.51 <.0001
Cult Luke   5.0694 0.2874 68 17.64 <.0001
Cult Nugaines   4.7222 0.2874 68 16.43 <.0001
Irrig   1 2.4222 0.3220 10 7.52 <.0001
Irrig   2 3.1833 0.3220 10 9.88 <.0001
Irrig   3 5.0556 0.3220 10 15.70 <.0001
Irrig   4 6.1889 0.3220 10 19.22 <.0001
Irrig   5 6.4000 0.3140 10 20.38 <.0001
Irrig   6 6.3944 0.3227 10 19.81 <.0001
Cult*Irrig Bridger 1 2.8500 0.3679 68 7.75 <.0001
Cult*Irrig Bridger 2 3.4167 0.3679 68 9.29 <.0001
Cult*Irrig Bridger 3 5.1500 0.3679 68 14.00 <.0001
Cult*Irrig Bridger 4 6.2500 0.3679 68 16.99 <.0001
Cult*Irrig Bridger 5 6.3000 0.3463 68 18.19 <.0001
Cult*Irrig Bridger 6 6.2167 0.3697 68 16.81 <.0001
Cult*Irrig Luke 1 2.1333 0.3679 68 5.80 <.0001
Cult*Irrig Luke 2 2.8667 0.3679 68 7.79 <.0001
Cult*Irrig Luke 3 5.2333 0.3679 68 14.22 <.0001
Cult*Irrig Luke 4 6.5500 0.3679 68 17.80 <.0001
Cult*Irrig Luke 5 6.8833 0.3463 68 19.87 <.0001
Cult*Irrig Luke 6 6.7500 0.3697 68 18.26 <.0001
Cult*Irrig Nugaines 1 2.2833 0.3679 68 6.21 <.0001
Cult*Irrig Nugaines 2 3.2667 0.3679 68 8.88 <.0001
Cult*Irrig Nugaines 3 4.7833 0.3679 68 13.00 <.0001
Cult*Irrig Nugaines 4 5.7667 0.3679 68 15.67 <.0001
Cult*Irrig Nugaines 5 6.0167 0.3463 68 17.37 <.0001
Cult*Irrig Nugaines 6 6.2167 0.3697 68 16.81 <.0001

An interesting exercise is to fit other variance-covariance models to these data and to compare them to this one by using likelihood ratio tests, Akaike’s information criterion, or Schwarz’s Bayesian information criterion. In particular, some spatial models are worth investigating (Marx and Thompson, 1987; Zimmerman and Harville, 1991). The following is one example of spatial model statements:

proc mixed;
class Block Cult Dir Irrig;
model Y = Cult|Dir|Irrig@2;
repeated / type=sp(pow)(Row Sbplt) sub=intercept;
run;


The TYPE=SP(POW)(Row Sbplt) option in the REPEATED statement requests the spatial power structure, with the two defining coordinate variables being Row and Sbplt. The SUBJECT=INTERCEPT option indicates that the entire data set is to be considered as one subject, thereby modeling as a dense 108108 covariance matrix. See Wolfinger (1993) for further discussion of this example and additional analyses.