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The MIXED Procedure

Example 41.4: Known G and R

This animal breeding example from Henderson (1984, p. 53) 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 G and R are known.

G= [ 2 & 1 & 1 & 2 & 1 & 1 \ 1 & 2 & .5 & 1 & 2 & .5 \ 1 & .5 & 2 & 1 & .5 & 2 \... ... & 3 & 1.5 & 1.5 \ 1 & 2 & .5 & 1.5 & 3 & .75 \ 1 & .5 & 2 & 1.5 & .75 & 3 \ ]

R= [ 4 & 0 & 0 & 1 & 0 & 0 \ 0 & 4 & 0 & 0 & 1 & 0 \ 0 & 0 & 4 & 0 & 0 & 1 \ 1 & 0 & 0 & 5 & 0 & 0 \ 0 & 1 & 0 & 0 & 5 & 0 \ 0 & 0 & 1 & 0 & 0 & 5 \ ]

In order to read G into PROC MIXED 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 using Row, Col, and Value variables, although this would require 21 observations instead of 6 for this example.

The PROC MIXED code is 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 X 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 using the S and OUTP= options, respectively.

The random effect Trait*Animal leads to a Z matrix with six columns, the first five corresponding to the identity matrix and the last consisting of 0s. An unstructured G matrix is specified using the TYPE=UN option, and it is read into PROC MIXED from a SAS data set using the GDATA=G specification. The G and GI options request the display of G and G-1, respectively. The S option requests that the random-effects solution vector be displayed.

Note that the preceding R matrix is block diagonal if the data are sorted by animals. The REPEATED statement exploits this fact by requesting R to have unstructured 2×2 blocks corresponding to animals, which are the subjects. The R and RI options request that the estimated 2×2 blocks for the first animal and its inverse be displayed. The PARMS statement lists the parameters of this 2×2 matrix. Note that the parameters from G are not specified in the PARMS statement because they have already been assigned 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 41.4.1.

Output 41.4.1: Known G and R

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


The "Unstructured" covariance structure applies to both G and R here.

The Mixed Procedure

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


The levels of Trait and Animal have been specified correctly.

The Mixed Procedure

Dimensions
Covariance Parameters 3
Columns in X 2
Columns in Z 6
Subjects 1
Max Obs Per Subject 6
Observations Used 5
Observations Not Used 1
Total Observations 6


The three covariance parameters indicated here correspond to those from the R matrix. Those from G are considered fixed and known because of the GDATA= option.

The Mixed Procedure

Parameter Search
CovP1 CovP2 CovP3 Res Log Like -2 Res Log Like
4.0000 1.0000 5.0000 -7.3731 14.7463


The preceding table results from the PARMS statement.

The Mixed Procedure

Estimated R Matrix for Subject 1
Row Col1 Col2
1 4.0000 1.0000
2 1.0000 5.0000


The block of R corresponding to the first animal is shown in the "Estimated R Matrix" table.

The Mixed Procedure

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


The inverse of the block of R corresponding to the first animal is shown in the preceding table.

The Mixed Procedure

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


The preceding table lists the G matrix as specified in the GDATA= data set.

The Mixed Procedure

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 preceding table lists G-1. The blank values correspond to zeros.

The Mixed Procedure

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 parameters from R are listed again.

The Mixed Procedure

Fit Statistics
Res Log Likelihood -7.4
Akaike's Information Criterion -10.4
Schwarz's Bayesian Criterion -10.1
-2 Res Log Likelihood 14.7


You can use this model-fitting information to compare this model with others.

The Mixed Procedure

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 coefficients of the mixed model equations agree with Henderson (1984, p. 55).

The Mixed Procedure

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 solution to the mixed model equations also matches that given by Henderson (1984, p. 55).

The Mixed Procedure

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


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

The Mixed Procedure

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 Estimate column 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.

The Mixed Procedure

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.

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 3.11701 0 0.05 . . .


The preceding table contains the predicted values of the observations based on the trait and breeding value estimates, that is, the fixed and random effects. 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. Refer to Henderson (1984) for information on predicting future records. The L95 and U95 columns usually contain confidence limits for the predicted values; they are missing here because the random-effects parameter degrees of freedom equals 0.

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