Evaluating mating designs

2.3 Evaluating mating designs

Assume that we want to make C crosses derived from A dura and B pisifera. If C=A*B then we have only one mating design, a complete diallel crossing scheme. In an incomplete diallel where the number of crosses C is less than A*B there are many possible mating designs. As we have already explained in section 2.2.1, the entire set of A dura and B pisifera can only be compared on the basis of the GCA values if the crossing scheme is connected. A necessary condition for a connected scheme is that C³ A+B-1, but this condition is not sufficient. For a given number of crosses C in an incomplete diallel scheme, where A+B-1 £ C <A*B, the problem is now to find a good connected mating design. The choice between several connected mating designs can best be based on the standard error of the estimator for the difference in the GCA value of all the dura pairs and the pisifera pairs. The standard error of the estimator for the difference in the GCA value between two dura parents or pisifera parents P_i and P_j is S_ij*s , where s is the residual standard deviation and the value of S_ij depends solely on the mating scheme. The value of s depends on the studied trait, the variation between the plots in the experimental field and the plot size.

As we have already explained in section 2.2.1 (see Example 2), the GCA values are estimated by solving the Normal Equations M * p=t, where M is the matrix of the coefficients in the Normal Equations, p is the column of parameters from the linear additive model of the dura and pisifera genetic effects, and t is the column of the totals in the right-hand side of the Normal Equations. A solution of these Normal Equations can be written as p=M^-* t, where M^- is a generalized inversion of M. In other words it fulfills the condition that M* M^- * M = M.

A difference of the GCA values between two parents P_i and P_j will be estimated as the difference between the estimates of the parameters of these parents P_i and P_j and this is the same for each solution of the Normal Equations. This difference is a linear combination of the parameter-estimates and can be written in matrix notation as c’*p; the variance of c’*p is given by (c’*M^-*c)*s ². The standard error of this estimator c’*p is the square root of the variance of c’*p, hence

From a mating design follows the matrix M and hence the Sij depends solely on the mating design.

For complete crossing schemes (as a complete diallel) with A dura and B pisifera (each cross occurs on r plots) the standard error of the estimator of the difference between the GCA values of the dura parents is the same for all pairs of dura and S_ij is ; also the standard error of the estimator of the difference between the GCA values of the pisifera parents is the same for all pairs of pisifera and S_ijis .

For incomplete mating designs we can get the same standard error for the estimator of the difference between the GCA values of the dura parents if these dura are balanced over the pisifera. This means that each pisifera has the same number of k (<A) dura and each pair of dura occurs the same number of times together with a pisifera. In this case we have an incomplete balanced mating design.

For incomplete unbalanced mating designs the standard error of the estimator of the differences in GCA values varies across the parents. The quality of such mating designs can be measured by the average and range of the standard errors of the estimator of the differences between the GCA values of a pair of dura parents or of a pair of pisifera parents. As shown above, such quality evaluation can solely be based on S_ijvalues.

To find a good mating design one can search for balanced or partially balanced incomplete mating designs. For such incomplete mating designs one can use the incomplete block designs (see section 2.5). In such incomplete block designs there must be compared v treatments in blocks of sizes of k plots, where the block size k <v. Well known incomplete block designs are lattices where v = k*k or rectangular lattices where v = k*(k+1). (See Cochran & Cox, 1957). To extend the possibilities for v unequal to k*k or k*(k+1) there are the so called alpha-designs (see Patterson, Williams and Hunter, 1978). To use such an incomplete block design the role of the treatments is played by the dura and the role of the incomplete blocks is played by the pisifera. So we must look for incomplete block designs with A treatments and B blocks. The block size k is then chosen as C/B, where C is the number of crosses used. If there is no incomplete block design which fits the requirements, we can always start from a smaller design and add some extra treatments (=dura) to the blocks (=pisifera).

As an example we give here some mating designs involving C=40 crosses among A=20 dura and B=10 pisifera. In these designs each dura must be crossed with two pisifera; furthermore each pisifera must be crossed with four dura.

Two designs (I and II) were solely chosen intuitively on the basis of symmetry by two experienced oil palm breeders and the last design (III) is an alpha-design. A realized cross is indicated by an asterisk (*).

Design I
		Pisifera
		1	2	3	4	5	6	7	8	9	10

Dura	1	*									*
	2	*									*
	3	*	*
	4		*							*
	5		*	*
	6			*					*
	7			*	*
	8				*			*
	9				*	*
	10					*	*
	11					*	*
	12					*	*
	13						*	*
	14				*			*
	15							*	*
	16			*					*
	17								*	*
	18		*							*
	19									*	*
	20	*									*

Design II
		Pisifera
		1	2	3	4	5	6	7	8	9	10

Dura	1	*									*
	2	*					*
	3	*	*
	4		*					*
	5		*	*
	6			*					*
	7			*	*
	8				*					*
	9				*	*
	10					*					*
	11					*	*
	12	*					*
	13						*	*
	14		*					*
	15							*	*
	16			*					*
	17								*	*
	18				*					*
	19									*	*
	20					*					*

Design III
		Pisifera
		1	2	3	4	5	6	7	8	9	10

Dura	1	*					*
	2		*					*
	3			*					*
	4				*					*
	5					*					*
	6	*									*
	7		*				*
	8			*				*
	9				*				*
	10					*				*
	11	*									*
	12		*							*
	13			*			*
	14				*			*
	15					*			*
	16	*							*
	17		*							*
	18			*							*
	19				*		*
	20					*		*

For each design one can calculate beforehand the average of the S_ij-values of the standard errors of the estimator of the differences between the GCA values of the dura and pisifera parents as well as their range.

In the following table the minimum, maximum and average of the standard errors of the estimator of the difference between GCA values of pairs of dura and pisifera parents ( S_ij*s ), divided by s , for the mating Designs I, II (constructed by the experienced oil palm breeders) and III (based on an alpha-design) are given.

mating design	pairs of dura			pairs of pisifera
mating design	min	max	average	min	max	average

I	1	2.236	1.561	0.765	2.072	1.417
II	1	1.483	1.313	0.841	1.25	1.125
III	1.125	1.291	1.214	0.949	1.08	1.001

From the table it is clear that design III (the alpha-design), which has the smallest average value for Sij for the dura and the pisifera pairs, and moreover has the smallest range (max - min) for Sij, is the mating design which must be preferred.

Hence it is worthwhile to use an alpha-design for a mating design and be careful to rely too much on "experience"!