Example 56.6 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 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 56.6.1.

The "Covariance Structures" row in Output 56.6.1 reveals the two different structures assumed for and .

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 56.6.2).

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 56.6.3).

The Newton-Raphson algorithm converges successfully in seven iterations (Output 56.6.4).

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 56.6.5).

Output 56.6.6 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.

The " Res Log Likelihood" value in Output 56.6.7 is the same as the final value listed in the "Iteration History" table (Output 56.6.4).

Every fixed effect except for Dir and Cult*Irrig is significant at the 5% level (Output 56.6.8).

The "Estimates" table lists the results from the various linear combinations of fixed effects specified in the ESTIMATE statements (Output 56.6.9). 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.

The least squares means shown in Output 56.6.10 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.

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.