4.12 Pooling estimates of covariance components

Information related to pooling of covariance components using a (penalized) maximum likelihood approach is specified in a block entry, beginning with a line containing the code POOL and ending with a line beginning with END. Within the block, the following directives are recognized:
PSEUPED followed (space separated) by a three letter code specifying the assumed pedigree structure and values on sizes or numbers of families. The following pseudo-pedigree codes are available:
PHS denotes a simple balanced paternal half-sib design. Optionally, two integer numbers giving the numbers of sires and the number of progeny per sire, respectively, can be given (space separated, on the same line) after the code. If not given, default values of 10  and 4  are used.

This option is suitable for a simple animal model only. WOMBAT will check that a) there is only one random effect fitted, and b) that this has covariance option NRM. If the MINPAR option (see below) is used, WOMBAT cannot perform these checks; hence the DIRADD code together with the name of the random effect, as described below, need to be given.
HFS implies a balanced hierarchical full-sib design comprised of s  sires, d  dams per sire and n  progeny per dam. Assumed values for s  , d  and n  can be given (space-separated) on the same line. If omitted, default values of s = 10  , d = 5  and n = 4  are used. This option is suitable for a simple animal model or a model fitting maternal permanent environmental effects in addition. Again, if the MINPAR option is used, codes DIRADD and MATPE need to be given in addition.
BON selects a design comprising 8 individual per family in two generations, due to Bondari et al. [4]. Optionally, this code can be followed (space separated) by the number of such families (integer); if not given, a default value of 2 is used. For this design, expectations of covariances between relatives due to direct and maternal effects are available. In order for WOMBAT to ‘know’ which random effect has which structure, additional information is required. This should comprise one additional line per random effect fitted, with each line consisting of a keyword specifying the type of random effect followed (space separated) by the name of the effect as specified in the model of analysis. Keywords recognized are DIRADD for direct additive genetic, MATADD for maternal genetic and MATPE for maternal permanent environmental effects.

EXAMPLE:

MODEL
   RAN     animal  NRM
   RAN     gmdam   NRM
   RAN     pedam   IDE
   trait   bwgt    1
   trait   wwgt    2
   trait   ywgt    3
END MOD
VAR animal   3  NOS
VAR gmdam    3  NOS
VAR pedam    3  NOS
VAR residual 3  NOS
POOL
   PSEUPED    BON  10
      DIRADD  animal
      MATADD  gmdam
      MATPE   pedam
   SMALL      0.0010
   DELTAL     0.0010
END

USR Other designs and models are readily accommodated by specifying the expectations of covariances between family members for each random effect fitted (excluding residuals - this is set automatically to be an identity matrix). This is selected by the USR option which must be followed (space separated) by an integer specifying the family size. Optionally, a second number giving the number of families is recognized (a default value of 2 is used if not given). Then the upper triangle of the matrix of coefficients in the expectation of covariances has to be given for each random effect fitted. For multiple random effects, these have to be given in the same order in which they have been specified in the VAR statements in parameter file, and the matrix for each effect has to begin on a new line.

EXAMPLE:

MODEL
  RAN  animal  NRM
    ...
END MOD
VAR   animal   4  NOSTART
VAR   residual 4  NOSTART
POOL
   PSEUPED  USR  5
   1.0  0.50  0.50  0.50  0.50
   1.0  0.25  0.25  0.25
   1.0  0.25  0.25
   1.0  0.25
   1.0
END

This shows the coefficients for direct additive genetic effects for a family comprising a sire (individual 1) with four progeny from unrelated dams.



SMALL followed (space separated) by a real number which gives the lower limit (≤ 1  ) for smallest eigenvalue allowed in the combined matrices. If this is not specified, a default value of 0.0001  is used.
DELTAL followed (space separated) by a real number specifying the convergence criterion: if the increase in log likelihood between iterates falls below this value, convergence is assumed to have been achieved. If not specified a stringent default of 0.00005  is used.
MINPAR The default assumption for the parameter file used is that it is a ‘full’ parameter file as for a corresponding, multivariate analysis. However, the information on pedigree and data file and their layout are not actually required. Hence the option MINPAR (on a line by itself) is provided which allows use of a ‘minimum’ parameter file, switching off some of the consistency checks otherwise carried out. If used, MINPAR must be the first entry within the POOL block.

The minimum information to be given in the parameter file must comprise:

1.
The ANALysis statement
2.
A VAR line for each covariance matrix, together with the NOSTART option telling WOMBAT not to expect a matrix of starting values.
3.
The POOL block, including statements showing which random effect represents which type of genetic or non-genetic effect.

EXAMPLE:

ANAL MUV 14
VAR  animal    14  NOSTART
VAR  residual  14  NOSTART
POOL
   MINPAR
   SMALL  0.001d0
   PSEUPED  hfs  100  10  4
     DIRADD  animal
END

SINGLE The default form of input for results from part analysis is to read estimates from separate files, in the form of output generated by WOMBAT when carrying out multiple analyses of subsets of traits. Alternatively, all information can be given in a single file. This is selected by the option SINGLE, followed (space separated) by the name of the input file (same line). The layout required for this file is described in 6.6.4.2.
PENALTY followed (space separated) by a code word(s) defining the type and strength of penalty to be applied. Codes for the penalty type recognised are:
CANEIG selects a penalty on the canonical eigenvalues. This has to be followed (space separated) by either ORG or LOG specifying a penalty on eigenvalues on the original scale or transformed to logarithmic scale (no default).
COVARM specifies shrinkage of a covariance matrix towards a given target.
CORREL chooses shrinkage of a correlation matrix towards a given target correlation matrix.

Either COVARM or CORREL can be followed (space separated) by the keyword MAKETAR. If this is given, WOMBAT determines the shrinkage target as the phenotypic covariance (or correlation) matrix obtained by summing estimates of covariances for all sources of variation from the preceding, unpenalized analysis. If this is not given, the upper triangle of the target matrix is expected to be read from a file with the standard name PenTargetMatrix; see 6.6.7.1.

The last entry on the line relates to the tuning factor(s) to be used: If a single penalized analysis is to be carried out, the corresponding tuning factor should be given (real value). To specify multiple penalized analyses, specify the number of separate tuning factors multiplied by − 1  as an integer value (e.g. -3 means 3 analyses), and list the corresponding tuning factors space separated on the next line.

EXAMPLE:

1.
Shrink all correlation matrices towards the phenotypic correlation matrix using a single tuning factor of 0.1  , and calculate the shrinkage target from unpenalized results.
PENALTY CORREL  MAKETAR  0.1
2.
Shrink canonical eigenvalues on the logarithm scale towards their mean, using 5 different tuning factors
PENALTY CANEIG LOG  -5
0.01  0.1  0.5  1.0  2.0