Calculates generalized heritability for a random term in a `REML`

analysis (R.W. Payne).

### Options

`PRINT` = string tokens |
Controls printed output (`heritability` ); default `heri` |
---|---|

`SAVE` = REML save structure |
Save structure of the analysis from which to calculate the heritabilities; default uses the most recent `REML` analysis |

### Parameters

`TERMS` = formula |
Random terms whose heritabilities are to be calculated |
---|---|

`HERITABILITY` = scalar or variate |
Saves the heritabilities |

`EXIT` = scalar or variate |
Exit status for the calculations: one if unsuccessful, otherwise zero |

### Description

`VHERITABILITY`

can be used to calculate the generalized heritability for random terms in a `REML`

analysis, using the definition of Cullis, Smith & Coombes (2006). This provides a natural extension of the standard concept of heritability, which was defined in the context of conventional designs like complete randomized block designs, to more complicated analyses like those with spatial correlation models (see `VSTRUCTURE`

).

Heritability measures the proportion of variance that is attributable to the effects of the term. It is often used by plant breeders to assess the proportion of the variance of a phenotypic trait that is attributable to the effects of genotypes, thus providing an indication of potential benefits of selection. Technically, `VHERITABILITY`

provides a broad-sense measure of heritability, on a mean-line basis, that comprises the sum of additive, dominance and epistatic effects. For more details see Falconer & Mackay (1996) or Piepho & Möhring (2007).

By default, the heritabilities are usually calculated from the most recent `REML`

analysis. However, you can use the `SAVE`

parameter to specify the save structure from an earlier analysis.

The `TERMS`

parameter supplies a model formula to specify the terms whose heritabilities are to be calculated. These must all be in the random model of the analysis.

The `HERITABILITY`

parameter allows you to save the heritabilities, in a scalar if there is a single term, or in a variate if there are several. Similarly the `EXIT`

parameter can save a scalar or variate indicating whether each heritability was calculated successfully (zero for success and one for failure). Possible reasons for failures may include the fact that the term was not in the random model, or that it has a negative variance component.

By default, the heritabilities are printed, but you can set option `PRINT=*`

to suppress this.

Options: `PRINT`

, `SAVE`

.

Parameters: `TERMS`

, `HERITABILITY`

, `EXIT`

.

### Method

Cullis, Smith & Coombes (2006) define the heritability as

1 – A_{tt} / (2 *γ _{g}*

^{2})

where *γ _{g}*

^{2}is the variance component of the term divided by the residual variance, and A

_{tt}is the average variance for differences between the effects of the term divided by the residual variance. This can be simplified, by cancelling out the residual variance, to become

1 – V_{tt} / (2 *σ _{g}*

^{2})

where *σ _{g}*

^{2}is the variance component of the term, and V

_{tt}is the average variance for differences between the effects of the term. These are obtained from

`VKEEP`

and `VPREDICT`

, respectively.### References

Cullis, B.R., Smith, A. & Coombes, N. (2006). On the design of early generation variety trials with correlated data. *JABES*, 11, 381–393.

Falconer, D. S. & Mackay, T. (1996). *Introduction to Quantitative Genetics, 4th ed.* Longman, Harlow.

Piepho, H-P & Möhring, J. (2007). Computing heritability and selection response from unbalanced plant breeding trials. Genetics, 177, 1881-1888.

### See also

Commands for: REML analysis of linear mixed models.

### Example

CAPTION 'VHERITABILITY example',\ !t('Yield trial of oats in an alpha design. John & Williams',\ '(1995, Cyclic & Computer Generated Designs, 2nd ed., p.146);',\ 'also see Piepho & Mohring (2007, Genetics, 107, 1881-8).');\ STYLE=meta,plain SPLOAD [PRINT=*] '%gendir%/Examples/VHER-1.gsh' VCOMPONENTS replicate/block+variety REML yield VHERITABILITY variety