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 The MIXED Procedure

## Example 41.6: Line-Source Sprinkler Irrigation

These data appear in Hanks et al. (1980), Johnson, Chaudhuri, and Kanemasu (1983), and 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 G and R can be modeled. One of Stroup's models specifies a diagonal G containing the variance components for Block, Block*Dir, and Block*Irrig, and a Toeplitz R with four bands. The SAS code to fit this model and carry out some further analyses follows.

Caution: This analysis may 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 code uses 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 Z and G 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 R matrix to be Toeplitz with four bands below and including the main diagonal. The subject effect is Block( Cult), and it produces nine 12×12 blocks. The R option requests that the first block of R 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 41.6.1.

Output 41.6.1: Line-Source Sprinkler Irrigation 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 Covariance Structures row reveals the two different structures assumed for G and R.

 The Mixed Procedure

 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

The levels of each class variable are listed as a single string in the Values column, regardless of whether the levels are numeric or character.

 The Mixed Procedure

 Dimensions Covariance Parameters 7 Columns in X 48 Columns in Z 27 Subjects 1 Max Obs Per Subject 108 Observations Used 108 Observations Not Used 0 Total Observations 108

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.

 The Mixed Procedure

 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 Newton-Raphson algorithm converges successfully in seven iterations.

 The Mixed Procedure

 Estimated R Matrix for Subject 1 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

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

 The Mixed Procedure

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

 The Mixed Procedure

 Fit Statistics Res Log Likelihood -91.6 Akaike's Information Criterion -98.6 Schwarz's Bayesian Criterion -95.5 -2 Res Log Likelihood 183.2

The "-2 Res Log Likelihood" value is the same as the final value listed in the "Iteration History" table.

 The Mixed Procedure

 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

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

 The Mixed Procedure

 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 "Estimates" table lists the results from the various linear combinations of fixed effects specified in the ESTIMATE statements. 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.

 The Mixed Procedure

 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

The LS-means 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.

An interesting exercise is to try fitting other variance-covariance models to these data and comparing them to this one 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 code.

```   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 SUB=INTERCEPT option indicates that the entire data set is to be considered as one subject, thereby modeling R as a dense 108×108 covariance matrix. Refer to Wolfinger (1993) for further discussion of this example and additional analyses.

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