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 |
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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 |
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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