Example 58.4 Known G and R

This animal breeding example from Henderson (1984, p. 48) considers multiple traits. The data are artificial and consist of measurements of two traits on three animals, but the second trait of the third animal is missing. Assuming an additive genetic model, you can use PROC MIXED to predict the breeding value of both traits on all three animals and also to predict the second trait of the third animal. The data are as follows:

```data h;
input Trait Animal Y;
datalines;
1 1 6
1 2 8
1 3 7
2 1 9
2 2 5
2 3 .
;
```

Both and are known.

In order to read into PROC MIXED by using the GDATA= option in the RANDOM statement, perform the following DATA step:

```data g;
input Row Col1-Col6;
datalines;
1  2  1  1  2   1     1
2  1  2 .5  1   2    .5
3  1 .5  2  1    .5  2
4  2  1  1  3   1.5  1.5
5  1  2 .5  1.5 3     .75
6  1 .5  2  1.5  .75 3
;
```

The preceding data are in the dense representation for a GDATA= data set. You can also construct a data set with the sparse representation by using Row, Col, and Value variables, although this would require 21 observations instead of 6 for this example.

The PROC MIXED statements are as follows:

```proc mixed data=h mmeq mmeqsol;
class Trait Animal;
model Y = Trait / noint s outp=predicted;
random Trait*Animal / type=un gdata=g g gi s;
repeated / type=un sub=Animal r ri;
parms (4) (1) (5) / noiter;
run;
proc print data=predicted;
run;
```

The MMEQ and MMEQSOL options request the mixed model equations and their solution. The variables Trait and Animal are classification variables, and Trait defines the entire matrix for the fixed-effects portion of the model, since the intercept is omitted with the NOINT option. The fixed-effects solution vector and predicted values are also requested by using the S and OUTP= options, respectively.

The random effect Trait*Animal leads to a matrix with six columns, the first five corresponding to the identity matrix and the last consisting of 0s. An unstructured matrix is specified by using the TYPE=UN option, and it is read into PROC MIXED from a SAS data set by using the GDATA=G specification. The G and GI options request the display of and , respectively. The S option requests that the random-effects solution vector be displayed.

Note that the preceding matrix is block diagonal if the data are sorted by animals. The REPEATED statement exploits this fact by requesting to have unstructured 22 blocks corresponding to animals, which are the subjects. The R and RI options request that the estimated 22 blocks for the first animal and its inverse be displayed. The PARMS statement lists the parameters of this 22 matrix. Note that the parameters from are not specified in the PARMS statement because they have already been assigned by using the GDATA= option in the RANDOM statement. The NOITER option prevents PROC MIXED from computing residual (restricted) maximum likelihood estimates; instead, the known values are used for inferences.

The results from this analysis are shown in Output 58.4.1Output 58.4.12.

The "Unstructured" covariance structure (Output 58.4.1) applies to both and here. The levels of Trait and Animal have been specified correctly.

Output 58.4.1 Model and Class Level Information
The Mixed Procedure

Model Information
Data Set WORK.H
Dependent Variable Y
Covariance Structure Unstructured
Subject Effect Animal
Estimation Method REML
Residual Variance Method None
Fixed Effects SE Method Model-Based
Degrees of Freedom Method Containment

Class Level Information
Class Levels Values
Trait 2 1 2
Animal 3 1 2 3

The three covariance parameters indicated in Output 58.4.2 correspond to those from the matrix. Those from are considered fixed and known because of the GDATA= option.

Output 58.4.2 Model Dimensions and Number of Observations
Dimensions
Covariance Parameters 3
Columns in X 2
Columns in Z 6
Subjects 1
Max Obs Per Subject 6

Number of Observations
Number of Observations Used 5
Number of Observations Not Used 1

Because starting values for the covariance parameters are specified in the PARMS statement, the MIXED procedure prints the residual (restricted) log likelihood at the starting values. Because of the NOITER option in the PARMS statement, this is also the final log likelihood in this analysis (Output 58.4.3).

Output 58.4.3 REML Log Likelihood
Parameter Search
CovP1 CovP2 CovP3 Res Log Like -2 Res Log Like
4.0000 1.0000 5.0000 -7.3731 14.7463

The block of corresponding to the first animal and the inverse of this block are shown in Output 58.4.4.

Output 58.4.4 Inverse R Matrix
Estimated R Matrix for Animal 1
Row Col1 Col2
1 4.0000 1.0000
2 1.0000 5.0000

Estimated Inv(R) Matrix for
Animal 1
Row Col1 Col2
1 0.2632 -0.05263
2 -0.05263 0.2105

The matrix as specified in the GDATA= data set and its inverse are shown in Output 58.4.5 and Output 58.4.6.

Output 58.4.5 G Matrix
Estimated G Matrix
Row Effect Trait Animal Col1 Col2 Col3 Col4 Col5 Col6
1 Trait*Animal 1 1 2.0000 1.0000 1.0000 2.0000 1.0000 1.0000
2 Trait*Animal 1 2 1.0000 2.0000 0.5000 1.0000 2.0000 0.5000
3 Trait*Animal 1 3 1.0000 0.5000 2.0000 1.0000 0.5000 2.0000
4 Trait*Animal 2 1 2.0000 1.0000 1.0000 3.0000 1.5000 1.5000
5 Trait*Animal 2 2 1.0000 2.0000 0.5000 1.5000 3.0000 0.7500
6 Trait*Animal 2 3 1.0000 0.5000 2.0000 1.5000 0.7500 3.0000

Output 58.4.6 Inverse G Matrix
Estimated Inv(G) Matrix
Row Effect Trait Animal Col1 Col2 Col3 Col4 Col5 Col6
1 Trait*Animal 1 1 2.5000 -1.0000 -1.0000 -1.6667 0.6667 0.6667
2 Trait*Animal 1 2 -1.0000 2.0000   0.6667 -1.3333
3 Trait*Animal 1 3 -1.0000   2.0000 0.6667   -1.3333
4 Trait*Animal 2 1 -1.6667 0.6667 0.6667 1.6667 -0.6667 -0.6667
5 Trait*Animal 2 2 0.6667 -1.3333   -0.6667 1.3333
6 Trait*Animal 2 3 0.6667   -1.3333 -0.6667   1.3333

The table of covariance parameter estimates in Output 58.4.7 displays only the parameters in . Because of the GDATA= option in the RANDOM statement, the G-side parameters do not participate in the parameter estimation process. Because of the NOITER option in the PARMS statement, however, the R-side parameters in this output are identical to their starting values.

Output 58.4.7 R-Side Covariance Parameters
Covariance Parameter Estimates
Cov Parm Subject Estimate
UN(1,1) Animal 4.0000
UN(2,1) Animal 1.0000
UN(2,2) Animal 5.0000

The coefficients of the mixed model equations in Output 58.4.8 agree with Henderson (1984, p. 55). Recall from Output 58.4.1 that there are 2 columns in and 6 columns in . The first 8 columns of the mixed model equations correspond to the and components. Column 9 represents the Y border.

Output 58.4.8 Mixed Model Equations with Y Border
Mixed Model Equations
Row Effect Trait Animal Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9
1 Trait 1   0.7763 -0.1053 0.2632 0.2632 0.2500 -0.05263 -0.05263   4.6974
2 Trait 2   -0.1053 0.4211 -0.05263 -0.05263   0.2105 0.2105   2.2105
3 Trait*Animal 1 1 0.2632 -0.05263 2.7632 -1.0000 -1.0000 -1.7193 0.6667 0.6667 1.1053
4 Trait*Animal 1 2 0.2632 -0.05263 -1.0000 2.2632   0.6667 -1.3860   1.8421
5 Trait*Animal 1 3 0.2500   -1.0000   2.2500 0.6667   -1.3333 1.7500
6 Trait*Animal 2 1 -0.05263 0.2105 -1.7193 0.6667 0.6667 1.8772 -0.6667 -0.6667 1.5789
7 Trait*Animal 2 2 -0.05263 0.2105 0.6667 -1.3860   -0.6667 1.5439   0.6316
8 Trait*Animal 2 3     0.6667   -1.3333 -0.6667   1.3333

The solution to the mixed model equations also matches that given by Henderson (1984, p. 55). After solving the augmented mixed model equations, you can find the solutions for fixed and random effects in the last column (Output 58.4.9).

Output 58.4.9 Solutions of the Mixed Model Equations with Y Border
Mixed Model Equations Solution
Row Effect Trait Animal Col1 Col2 Col3 Col4 Col5 Col6 Col7 Col8 Col9
1 Trait 1   2.5508 1.5685 -1.3047 -1.1775 -1.1701 -1.3002 -1.1821 -1.1678 6.9909
2 Trait 2   1.5685 4.5539 -1.4112 -1.3534 -0.9410 -2.1592 -2.1055 -1.3149 6.9959
3 Trait*Animal 1 1 -1.3047 -1.4112 1.8282 1.0652 1.0206 1.8010 1.0925 1.0070 0.05450
4 Trait*Animal 1 2 -1.1775 -1.3534 1.0652 1.7589 0.7085 1.0900 1.7341 0.7209 -0.04955
5 Trait*Animal 1 3 -1.1701 -0.9410 1.0206 0.7085 1.7812 1.0095 0.7197 1.7756 0.02230
6 Trait*Animal 2 1 -1.3002 -2.1592 1.8010 1.0900 1.0095 2.7518 1.6392 1.4849 0.2651
7 Trait*Animal 2 2 -1.1821 -2.1055 1.0925 1.7341 0.7197 1.6392 2.6874 0.9930 -0.2601
8 Trait*Animal 2 3 -1.1678 -1.3149 1.0070 0.7209 1.7756 1.4849 0.9930 2.7645 0.1276

The solutions for the fixed and random effects in Output 58.4.10 correspond to the last column in Output 58.4.9. Note that the standard errors for the fixed effects and the prediction standard errors for the random effects are the square root values of the diagonal entries in the solution of the mixed model equations (Output 58.4.9).

Output 58.4.10 Solutions for Fixed and Random Effects
Solution for Fixed Effects
Effect Trait Estimate Standard Error DF t Value Pr > |t|
Trait 1 6.9909 1.5971 3 4.38 0.0221
Trait 2 6.9959 2.1340 3 3.28 0.0465

Solution for Random Effects
Effect Trait Animal Estimate Std Err Pred DF t Value Pr > |t|
Trait*Animal 1 1 0.05450 1.3521 0 0.04 .
Trait*Animal 1 2 -0.04955 1.3262 0 -0.04 .
Trait*Animal 1 3 0.02230 1.3346 0 0.02 .
Trait*Animal 2 1 0.2651 1.6589 0 0.16 .
Trait*Animal 2 2 -0.2601 1.6393 0 -0.16 .
Trait*Animal 2 3 0.1276 1.6627 0 0.08 .

The estimates for the two traits are nearly identical, but the standard error of the second trait is larger because of the missing observation.

The Estimate column in the "Solution for Random Effects" table lists the best linear unbiased predictions (BLUPs) of the breeding values of both traits for all three animals. The p-values are missing because the default containment method for computing degrees of freedom results in zero degrees of freedom for the random effects parameter tests.

Output 58.4.11 Significance Test Comparing Traits
Type 3 Tests of Fixed Effects
Effect Num DF Den DF F Value Pr > F
Trait 2 3 10.59 0.0437

The two estimated traits are significantly different from zero at the 5% level (Output 58.4.11).

Output 58.4.12 displays the predicted values of the observations based on the trait and breeding value estimates—that is, the fixed and random effects.

Output 58.4.12 Predicted Observations
Obs Trait Animal Y Pred StdErrPred DF Alpha Lower Upper Resid
1 1 1 6 7.04542 1.33027 0 0.05 . . -1.04542
2 1 2 8 6.94137 1.39806 0 0.05 . . 1.05863
3 1 3 7 7.01321 1.41129 0 0.05 . . -0.01321
4 2 1 9 7.26094 1.72839 0 0.05 . . 1.73906
5 2 2 5 6.73576 1.74077 0 0.05 . . -1.73576
6 2 3 . 7.12015 2.99088 0 0.05 . . .

The predicted values are not the predictions of future records in the sense that they do not contain a component corresponding to a new observational error. See Henderson (1984) for information about predicting future records. The Lower and Upper columns usually contain confidence limits for the predicted values; they are missing here because the random-effects parameter degrees of freedom equals 0.