Covariance components

4.7 Covariance components

Next, the parameter file needs to specify values for all (co)variance components in the model of analysis. For variance component estimation, these are the starting values used. For simple BLUP analyses and simulation runs, these are the values assumed to be the population values.

The input matrices for full rank analyses must be positive definite, i.e. cannot have eigenvalues less than the operational zero. For reduced rank (PC) analyses, some ‘zero’ but no negative eigenvalues are acceptable, provided the rank (i.e. the number of non-zero eigenvalues) is equal to (or greater than) the number of principal components to be fitted.

4.7.1 Residual covariances

4.7.1.1 ‘Standard’ multivariate analyses

The residual covariance matrix is specified by

1.: A line beginning with the code RESIDUAL (can be abbreviated to RES) or ERROR (can be abbreviated to ERR). This is followed by the dimension of the covariance matrix, $q$ (INTEGER number, space separated). If an analysis type PC has been specified, a second number specifying the rank of the estimated covariance matrix, needs to be given (even if this is equal to $q$ ).
2.: The $q(q+ 1)∕2$ elements of the upper triangle of the residual covariance matrix, given row-wise (i.e. $σ21$ , $σ12$ , $...$ , $σ1q$ , $σ22$ , $σ23$ , $...$ , $σ2q$ ).
These can be given several elements per line (space separated, taking care not to exceed the total line length of 78 characters), or one per line, or a mixture – WOMBAT will attempt to read until it has acquired all $q(q+ 1)∕2$ elements required.

4.7.1.2 Random regression analyses

Again, the residual covariance matrix is specified by

1.

A line beginning with the code RESIDUAL (can be abbreviated to RES) or ERROR (can be abbreviated to ERR). This should be followed by the dimension ( $q$ ) of each residual covariance matrix (INTEGER number), usually equal to the number of traits, and a code indicating what kind of error covariance function is to be fitted. The following codes are recognised :
HOM This code specifies homogeneous error covariances for all values of the control variable.
HET This code specifies heterogeneous error covariances, with the covariance function a step function of the control variable. It should be followed (space separated) by an INTEGER number giving the number of steps.
FUN This code specifies that error covariances change as a smooth function of the control variable. [Not yet implemented !]

2.

One or more lines with the residual covariance matrices, consisting of the $q(q+ 1)∕2$ elements of the upper triangle given row-wise, and, if applicable, additional information.

For HOM only a single covariance matrix needs to be given.
For HET the number of covariance matrices specified must be equal to the number of steps (in the step function, i.e. intervals). Each should begin on a new line and be preceded by two INTEGER values (space separated) indicating the upper and lower limits (inclusive) of the interval of the control variable for which this matrix is applicable.

EXAMPLE: For a univariate RR analysis with values of the control variable ranging from 1 to 5, we want to fit separate residual variances for 1 & 2, 3 & 4 and 5:
      VAR  residual  1  HET  3
        1  2   1.234
        3  4   3.456
        5  5   5.678
For FUN the type of function and its coefficients need to be given, as well as the covariance matrix at the lowest value of the control variable.

4.7.2 Covariances for random effects

Similarly, for each covariance matrix due to random effects to be estimated, a ‘header’ line and the elements of the covariance matrix need to be specified.

1.: A line beginning with the code VARIANCE (can be abbreviated to VAR), followed by the name of the random effect and the dimension of the covariance matrix, $q$ (INTEGER number, space separated). If an analysis type PC has been specified, a second number specifying the rank of the estimated covariance matrix, needs to be given (even if this is equal to $q$ ).
The name can simply the random effects name as specified for the model of analysis. Alternatively, it can be of the form “vn1+vn2” where vn1 and vn2 are names of random effects specified in the MODEL block. This denotes that the two random effects are assumed to be correlated and that their joint covariance matrix is to be estimated¹.
2.: The $q(q+ 1)∕2$ elements of the upper triangle of the covariance matrix, given row-wise (i.e. $σ2 1$ , $σ12$ , $...$ , $σ1q$ , $σ2 2$ , $σ23$ , $...$ , $σ2 q$ ).
Again, these can be given several elements per line (space separated, taking care not to exceed the total line length of 78 characters), or one per line, or a mixture – WOMBAT will attempt to read until it has acquired all $q(q+ 1)∕2$ elements required.

¹Currently implemented for full-rank, ‘standard’ analyses only !

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