Analyses full and half diallel tables with parents (J.F. Potter).

### Options

`PRINT` = string tokens |
Controls printed output (`data` , `vrwr` , `regression` , `aov` , `means` , `griffingaov` ); default `data` , `vrwr` , `regr` , `aov` , `mean` |
---|---|

`LABELS` = text |
Labels for rowcols, one text value for each, column j has the same label as row j, so each value of `LABELS` is the label for a pair of parents, applying to a rowcol; default 1…N, where N is the dimension of each diallel table |

`METHOD` = string token |
Whether to perform full or half diallel analysis (`half` , `full` ); default `full` |

### Parameter

`DATA` = matrices |
Each matrix contains the data for one block in the analysis, half diallel tables are presented as square matrices with the upper triangles and leading diagonals containing the values of interest, the matrices must be of the same size |
---|

### Description

`DIALLEL`

performs analysis of variance of full diallel tables (Hayman 1954) and half diallels (Jones 1965). Work on variance and covariance relationships is also performed (Jinks 1954). The data are specified by the `DATA`

parameter, in a square matrix for every block in the analyses. Half diallel tables are presented as square matrices with the upper triangle and leading diagonal containing the values of interest. The `PRINT`

option controls printed output:

`data` |
data values, |
---|---|

`vrwr` |
variances and covariances of rowcols, |

`regression` |
regression of the variances on the covariances, |

`aov` |
analysis of variance table, |

`means` |
means, |

`griffingaov` |
analysis of variance defined by Griffing (1956), which provides estimates of general combining ability (GCA) and specific combining ability (SCA). |

The `LABELS`

option can give a text to be used for labelling rowcols (called arrays in the literature). The `METHOD`

option specifies whether analysis of full or half diallels is required.

Options: `PRINT`

, `LABELS`

, `METHOD`

.

Parameter: `DATA`

.

### Method

`DIALLEL`

performs analysis of variance of full diallel tables, according to the method of Hayman (1954), and half diallels, according to the method of Jones (1965). A diallel table is a representation of the results of crossing a set of male and female homozygous parents in all possible combinations, including male:female reciprocation in full diallels. `DIALLEL`

expects parent values (selfs) to be present as the leading diagonal of the table (whether a full or half matrix).

The analysis of variance estimates the following genetic components of variation.

*a*: variation between mean effects of each parental line. Genetically this provides a test of additive variation, but also detects dominance if asymmetry present, i.e. if alleles at any one locus are not equally frequent (Hayman 1954).

*b*: variation caused by dominance at some of the loci. This term splits into:

*b1*: if significant this shows that dominance is largely uni-directional;

*b2*: estimates the asymmetry mentioned in *a*;

*b3*: signifies that some dominance is peculiar to individual crosses; If the symmetry condition is met, *b1* and *b3* together give a test of dominance equivalent to *b*.

*c*: variation between average maternal effects of each parental line.

*d*: variation in the reciprocal differences not attributable to *c*.

*t*: total variation.

Components *c* and *d* are reciprocal effects not available in half diallels. In the absence of replication, the *d* term should be used as the error term for testing components *a* to *c* in the full diallel. In the Griffing analysis, *a* corresponds to GCA, and *b* corresponds to SCA.

`DIALLEL`

can also analyse over any number of blocks, in which case block effects are also estimated, and block interactions with the above components can then be used as estimates of error to test the significance of the components.

Variances of rowcols (*Vr*) are compared with the covariance of the rowcols (*Wr*) with the corresponding concurrent parents, using the method of Jinks (1954). This entails the regression of *Wr* on *Vr*, which gives measures of adequacy of the model, average dominance, and the distribution of dominant and recessive genes. The analysis of diallel tables is more fully described by Mather and Jinks (1971).

Many other diallel methods exist, `DIALLEL`

representing quite a complex one, but one which makes fairly limiting assumptions, e.g. only a reference population in Hardy-Weinberg equilibrium with respect to individual loci and linkage equilibrium with respect to all pairs of loci can legitimately be used to estimate the genetic variance components. This means a large population reproducing by panmixia without selection. This and other difficulties such as the need for distinction between ancestral and descendant reference populations are discussed by Wright (1985).

### Action with `RESTRICT`

Restrictions are ignored for text `LABELS`

and are not relevant for `DATA`

, which is of type matrix.

### References

Griffing, B. (1956). Concept of general and specific combining ability in relation to diallel crossing system. *Aust. J. Biol.*, 9, 463-493.

Hayman, B.I. (1954). The Analysis of Variance of Diallel Tables. *Biometrics*, 10, 235-244.

Jones, R.M. (1965). Analysis of Variance of the Half Diallel Table. *Heredity*, 20, 117-121.

Jinks, J.L. (1954). The Analysis of Continuous Variation in a Diallel Cross of Nicotiana rustica Varieties. *Genetics*, 39, 767-788.

Mather, K. & Jinks, J.L. (1971). *Biometrical Genetics*, 249-284. Chapman & Hall Ltd.

Wright, A.J. (1985). Diallel Designs, Analyses, and Reference Populations. *Heredity*, 54, 307-311.

### See also

Procedure: `FDIALLEL`

.

Commands for: Analysis of variance, REML analysis of linear mixed models.

### Example

CAPTION 'DIALLEL example',\ !t('Data from Hayman, B.I. (1954). Biometrics 10, 235-244.',\ 'Two blocks of 8 x 8 full (default setting of option METHOD)',\ 'diallel tables. Analysis over blocks giving block interactions.',\ 'Text provided to label rowcols.'); STYLE=meta,plain TEXT [VALUES=one,two,'3',' 4',Five,'6','7','8'] Parents MATRIX [ROWS=8; COLUMNS=8] Blockdat[1...2] READ [SERIAL=yes] Blockdat[] 276 156 322 250 162 193 222 152 136 166 164 134 102 150 96 90 246 158 416 213 160 222 128 166 318 132 218 272 138 195 108 124 150 124 164 164 156 158 100 114 182 136 204 216 133 174 112 120 174 86 194 142 86 92 58 94 152 128 158 136 126 114 84 142 : 302 178 274 246 140 204 254 154 142 175 136 128 128 174 116 114 242 174 360 178 140 208 160 154 204 138 206 210 130 192 138 176 180 140 156 146 176 192 104 170 186 146 202 222 150 166 136 176 162 100 162 100 98 84 48 142 154 138 140 144 124 112 96 166 : DIALLEL [PRINT=data,vrwr,aov,regression,means,griffingaov; LABELS=Parents]\ Blockdat[] CAPTION !t('Data from Jones, R.M. (1965). Heredity 20, 117-121.',\ 'Single block of half diallel.',\ 'Default rowcol labelling (in this case 1...4)') MATRIX [ROWS=4; COLUMNS=4] Data READ Data 33.9 42 35.6 38.7 0 31 36.7 39.1 0 0 30 34.5 0 0 0 32.8 : DIALLEL [PRINT=data,vrwr,aov,regression,means,griffingaov; METHOD=half] Data