DIALLEL procedure

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


Options

PRINT = strings
Controls printed output (data, vrwr, regression, aov, means); 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
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.

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.

   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

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.