Specialised analyses with WOMBAT

This page provides some hints for non-standard analyses.

WOMBAT can accommodate a factor-analytic model for the structure of covariance matrices of random effects. While there is no explicit option for this structure, it is easily fitted indirectly.

Factor analysis implies a latent model which models a random effect (r) for a set of q traits as the sum of a vector of m common effects (c), weighted by a matrix of factor loadings (F), and a vector of q specific effects (s). i.e. r = F c + s. The common effects are assumed to be iid distributed with variances of unity, while the specific effects are assumed to be uncorrelated but have heterogeneous variances. This gives Var(r) = F F' + D, with D the the diagonal matrix of specific variances.

WOMBAT allows for the first part, i.e. has an option which allows Var(r) to be estimated as F F' with reduced rank m. In addition, WOMBAT incorporates an option for diagonal covariance matrices. Hence a factor-analytic model is readily fitted by simply fitting the corresponding random effect twice: In the first instance with reduced rank covariance matrix, representing the common factors, and in the second instance with diagonal covariance matrix, representing the specific effects.

WOMBAT only allows a particular column in the data file to be used once, i.e. to be mapped to a single effect in the model of analysis. Hence, fitting a factor-analytic structure for a particular effect requires the corresponding code (column) in the data file to be replicated ! If WOMBAT encounters two random effects, listed successively in the .par file, with the names only differing by the first letter (e.g. animal and bnimal in Example 5) and with reduced rank and diagonal covariance matrices, respectively, it will assume that a factor-analytic model has been intended and automatically write out summary statistics for the sum of the two matrices.

COM  Example parameter file for factor analytic model
ANAL MUV  PC 6
PEDS ../carcped.dat
DATA ../carcsix.dat GRP
  TRNOS 1 2 3 4 5 6
  traitno 6
  animal 0
  bnimal 0
  cg     299
  NAMES  cwt  cema  cimf  crby  cp8  crib
END

MODEL
  FIX cg
  RAN animal  nrm
  RAN bnimal  nrm
  tr cwt 1
  tr cema 2
  tr cimf 3
  tr crby 4
  tr cp8  5
  tr crib 6
END MOD

VAR animal 6 1
 177  29 3.1 -11 15 8
 9.8 0.03 -0.2 1.2 1.6
 0.91 -0.76 0.9 0.3
 3.62 -2.9 -1.96
 4.95 2.2
 1.95

VAR bnimal 6 6 DIAG
 50  3  0.3  0.5  1.1  0.8
VAR residual 6 6
 292  39  3.2  -5.9  22.4  11.
 36   0.8  -0.9  4.7 1.2
 1  -0.06  0.15  0.32
 0.73  -0.55 -0.45
 6.9  2.0
 5.1

In some instances we have clones, i.e. subjects which are genetically identical. For instance, monozygotic twins are naturally occuring clones. If these individuals have different identity codes and the same parents, WOMBAT will simply treat them as ordinary full-sibs. Alternatively, if all members of the clone are given the same identity, records on all members of the clone are treated as if they were repeated observations for the same individual. That is not correct either - hence special action needs to be taken.

To model clones correctly, we need to fit the same genetic effects for all members of the clone, but allow for different environmental effects for different individuals. WOMBAT has no in-built option to do this automatically, but can easily be tricked to do so:

• Give all members of the clone the same genetic identity.
• For data with repeated records per trait, assign a different code for permanent environmental effects to each clone member.
• WOMBAT requires the data file to be sorted according to traits within individual. However, it does not require identities to be in ascending or descending order – the program simply checks for a change in identity to determine where the records for a new individual start.
  Hence, the `trick' is to arrange the data file so that members of the same clone do not occur successively (while still keeping records taken on the same individual together and, for multivariate analyses, sorted in ascending trait number). This will cause WOMBAT to assign different residual effects to different individuals and thus model the genetic and environmental covariance structure correctly.

  Example: Individuals "1","2" and "3" belong to a clone given the identity "100",
           "4", "5" and "6" do not belong to a clone
  
    Trait Genetic_Code  Record  Individual
      1     100          10          1
      2     100          99          1
      1       4          12          4
      2       4          88          4
      1     100           9          2
      2     100          94          2
      1       5          13          5
      2       5          87          5
      1     100          11          3
      2     100          77          3
      1       6           7          6

  WOMBAT is likely to refuse running your analysis on grounds that there are too few “repeated” records per individual or that the data file is not sorted correctly. To override these `safety catches', you need to add the special option “clones”, i.e. your parameter file should have a block

  SPECIAL
      CLONES
  END SPECIAL

Alternatively, if animals are almost – but not 100% – identical, the inverse of the numerator relationship matrix can be constructed as described by Oikawa & Yasuda (2009) and fed into WOMBAT as a *.giv file.

[1989, article]

Kennedy, B. W., & Schaeffer, L. R. (1989). Genetic evaluation under an animal model when identical genotypes are represented in a population. J. Anim. Sci., 67, 1946-1955.

[2009, article | www]

Oikawa, T., & Yasuda, K. (2009). Inclusion of genetically identical animals to a numerator relationship matrix and modification of its inverse. Genetics Selection Evolution, 41(1), 25.

[2003, book]

Ruppert, D., Wand, M. P., & Carroll, R. J. (2003). Semiparametric Regression. New York: Cambridge University Press.

[2005, article]

Cantet, R. J. C., Birchmeier, A. N., Cayo, C. A. W., & Fioretti, C. (2005). Semiparametric animal models via penalized splines as alternatives to models with contemporary groups. J. Anim. Sci., 83(11), 2482-2494.