Half-sib analysis - Animal model

Introduction

In this section we will illustrate the use of .ped files to define the genetic pedigree structure between animals. This is an alternate method of estimating additive genetic variance for these data. The variance matrix of the animals (sires, dams and lambs) for which we only have data on lambs is given by Var(u_A = Σ_A cross A^-1 where A^-1 is the inverse of the genetic relationship matrix. There are a total of 10696 = 92 + 3561 + 7043 animals in the pedigree. The ASReml input file is presented below. Note that this model is not equivalent to the sire/dam/litter model with respect to the animal/litter components for gfw, fd and fat.

 Multivariate Animal model
   tag  !P
   sire
   dam  !P
   grp    49
   sex
   brr     4
   litter 4871
   age       wwt   !m0 ywt   !m0    # !M0 identifies missing values
   gfw   !m0 fdm   !m0 fat   !m0
 coop.fmt  # read pedigree from first three fields
 coop.fmt   !DOPATH $1 !CONTINUE !MAXIT 20 !STEP 0.01
   #   $1 allows selection of PATH as a command line argument
 !PATH 3
  !EXTRA 4     #  Force 4 more iterations after convergence criterion met
 !PATH
 wwt ywt gfw fdm fat ~ Trait Tr.age Tr.brr Tr.sex Tr.age.sex,
           !r Tr.tag ,
          at(Tr,1).dam, at(Tr,2).dam, -at(Tr,3).dam .003,
          at(Tr,1).lit, at(Tr,2).lit, at(Tr,3).lit, at(Tr,4).lit,
          at(Trait,1).age.grp .0024,
          at(Trait,2).age.grp .0019,
          at(Trait,4).age.grp .0020,
          at(Trait,5).age.grp .00026,
          at(Trait,1).sex.grp .93,
          at(Trait,2).sex.grp 16.0,
          at(Trait,3).sex.grp .28,
          at(Trait,5).sex.grp 1.18,
  !f Tr.grp

 1 2 3    # One multivariate R structure, 3 G structures

 0 0 0    # No structure across lamb records
          # First zero lets ASReml count te number of records
 Tr 0 US                  #General structure across traits
 7.66
 5.33 13
 .18 .66 .10
 .78 2.1 .27 3.2
 .73 2.02 .08 .20 1.44

 Tr.tag 2                  # Direct animal effects.

 !PATH 2
 Tr 0 FA1 !GP
    0.5 0.5 -.01 -.01 0.1
 2.4 5.2 0.06 .8 .14

 !PATH 3
 Tr 0 US
 2.4800
  2.8 6.4
 0.0128 0.03 0.06
 -.1 -.22 -.0011 0.72
  0.24 0.55     0.0026 -0.0202 0.14
 !PATH
 tag

 at(Tr,1).dam 2                  # Maternal effects.
 !PATH 2
 2 0 CORGH !GFU
 .99
 1.6 2.54
 !PATH 3
 2 0 US !GU
 1.1 .58 .31
 !PATH
 dam

 at(Tr,1).lit 2                  # Litter effects.
 !PATH 2
 4 0 FA1 !GP                     # Factor Analytic
 .5 .5 .01 .1 .01
 4.95 4.63 0.037 0.941 0.102
 !PATH 3
 4 0 US                          # Unstructured
  5.073
   3.545      3.914
  0.1274     0.08909 0.02865
  0.07277 0.05090 0.001829  1.019

 !PATH
 lit

The term Tr.tag now replaces the sire (and part of dam) terms in the half-sib analysis. This analysis uses information from both sires and dams to estimate additive genetic variance. The dam variance component is this analysis only estimates the maternal variance component. It is only significant for the weaning and yearling weights. The litter variation remains unchanged. The ASReml input file again consists of several parts, which progressively build up to fitting unstructured variance models to Tr.tag, Tr.dam, Tr.litter and error. A portion of the output file is

    tag  !P
    dam  !P
    age       wwt   !m0 ywt   !m0
    gfw   !m0 fdm   !m0 fat   !m0
  A-inverse retrieved from ainverse.bin
  PEDIGREE [pcoop.fmt ] has    10696 identities,   29474 Non zero elements
  QUALIFIERS: !CONTINUE !MAXIT 20  !STEP 0.01
  QUALIFIERS: !EXTRA 4
  QUALIFIER: !DOPATH    3 is active
  Reading pcoop.fmt  FREE FORMAT skipping     0 lines

  Multivariate analysis of wwt            ywt            gfw            fdm

  Multivariate analysis of fat
  Using     7043 records of    7043 read
   Model term                  Size #miss #zero   MinNon0    Mean      MaxNon0
    1 tag                 !P  10696     0     0   3.000     5380.     0.1070E+05
    2 sire                              0     0  1.000      48.06      92.00
    3 dam                 !P  10696     0     0   1.000     5197.     0.1070E+05
       :
  Forming   95033 equations:  40 dense.
  Initial updates will be shrunk by factor    0.010
  Restarting iteration from previous solution
  Notice: LogL values are reported relative to a base of      -20000.00
  NOTICE:     76 singularities detected in design matrix.
    1 LogL=-1437.10     S2=  1.0000      35006 df    :   2 components constrained
    2 LogL=-1436.87     S2=  1.0000      35006 df    :   3 components constrained
    3 LogL=-1434.97     S2=  1.0000      35006 df    :   2 components constrained
    4 LogL=-1430.73     S2=  1.0000      35006 df    :   2 components constrained
    5 LogL=-1424.71     S2=  1.0000      35006 df    :   1 components constrained
    6 LogL=-1417.98     S2=  1.0000      35006 df    :   1 components constrained
    7 LogL=-1417.77     S2=  1.0000      35006 df    :   1 components constrained
    8 LogL=-1417.62     S2=  1.0000      35006 df    :   1 components constrained
    9 LogL=-1417.28     S2=  1.0000      35006 df
   10 LogL=-1417.23     S2=  1.0000      35006 df
       :
   16 LogL=-1417.23     S2=  1.0000      35006 df

  Source                Model  terms     Gamma     Component    Comp/SE   % C
  at(Trait,1).age.grp      49     49  0.132682E-02  0.132682E-02   2.02   0 P
  at(Trait,2).age.grp      49     49  0.908220E-03  0.908220E-03   1.15   0 P
  at(Trait,4).age.grp      49     49  0.175614E-02  0.175614E-02   1.13   0 P
  at(Trait,5).age.grp      49     49  0.223617E-03  0.223617E-03   1.73   0 P
  at(Trait,1).sex.grp      49     49  0.902586      0.902586       2.88   0 P
  at(Trait,2).sex.grp      49     49   15.3623       15.3623       3.50   0 P
  at(Trait,3).sex.grp      49     49  0.280673      0.280673       3.71   0 P
  at(Trait,5).sex.grp      49     49   1.42136       1.42136       1.80   0 P
  Residual            UnStru   1   1   7.47555       7.47555      13.86   0 U
   :
 Covariance/Variance/Correlation Matrix UnStructured Residual
   7.476     0.4918     0.1339     0.1875     0.1333
   4.768      12.57     0.4381     0.3425     0.3938
  0.1189     0.5049     0.1056     0.4864     0.1298
  0.9377      2.221     0.2891      3.345     0.1171
  0.4208      1.612     0.4869E-01 0.2473      1.333

  Covariance/Variance/Correlation Matrix UnStructured Tr.tag
   3.898     0.8164     0.5763     0.3899E-01 0.6148
   4.877      9.154     0.3689    -0.1849     0.7217
  0.3029     0.2971     0.7085E-01-0.2415E-01 0.3041
  0.6021E-01-0.4375    -0.5027E-02 0.6117    -0.4672
  0.6154      1.107     0.4104E-01-0.1853     0.2570

  Covariance/Variance/Correlation Matrix UnStructured at(Tr,1).dam
  0.9988     0.7024
  0.5881    -0.7018

  Covariance/Variance/Correlation Matrix UnStructured at(Tr,1).lit
   3.714     0.5511     0.1635    -0.6157E-01
   2.019      3.614     0.5176    -0.4380
  0.4506E-01 0.1407     0.2045E-01-0.3338
 -0.1021    -0.7166    -0.4108E-01 0.7407

                                    Wald F statistics
      Source of Variation           NumDF              F-inc
   15 Tr.age                            5              99.16
   16 Tr.brr                           15             116.52
   17 Tr.sex                            5              59.94
   19 Tr.age.sex                        4               5.10

There is no guarantee that unstructured variance component matrices will be positive definite unless !GP qualifier is set. This example highlights this issue. We used the !GU qualifier on the maternal component to obtain the matrix

  0.9988  0.5881
  0.5881 -0.7018

ASReml reports the correlation as 0.7024 which it obtains by ignoring the sign in -0.7018. This is the maternal component for ywt. Since it is entirely reasonable to expect maternal influences on growth to have dissipated at 12 months of age, it would be reasonable to refit the model omitting at(Tr,2).dam and changing the dimension of the G structure.

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