2.5 Progeny trials in incomplete blocks

It is often difficult to find a sufficiently large homogeneous area to test the number C of tenera crosses or progenies from dura mothers and pisifera fathers in a Completely Randomized Design. The experimental field has often only homogeneous parts which are so large that they can only contain a part of the crosses; in such a homogeneous part ( block) of the experimental field the crosses can be compared under the same conditions. To take care of the heterogeneous growing conditions in an experimental field one can use a Randomized Incomplete Block design. If all the progenies are present in a block with size k, it is called a complete block; block size k = C. But often the block (homogeneous part of the experimental field) is not large enough to contain all the progenies; block size k<C . In that case an incomplete block design is used. The well known incomplete block designs, such as balanced incomplete block designs (BIBD), partially balanced incomplete block designs with two associate classes (PBIBD with 2 associate classes), lattices (for the case of C=k*k) and rectangular lattices (for the case of C=k*(k+1) ) can be found in the book of Cochran and Cox (1957). Often the number of tested progenies C does not fit with the above mentioned incomplete block designs as given in the book of Cochran and Cox (1957). An extension of the incomplete block designs is given by Patterson, Williams and Hunter (1978). They introduced the so-called alpha-designs. For many combinations of progenies C and block sizes k they give a procedure to construct these alpha-designs.

All these above mentioned designs are connected. A block design is called connected if for each pair (A_i,A_j) of the C progenies A1,...,AC, there is a chain, A_i=A₍₁₎,...,A_(m)=A_j, in which each two adjacent links of the chain occur together in the same block. The block design is otherwise called disconnected. In a connected block design one can estimate all differences between the progenies.

But later on we also want to estimate, from the yield of the C tenera progenies, the General Combining Abilities of the A dura mothers and the B pisifera fathers. Therefore we must use a connected crossing design for the dura and the pisifera.

The model for an incomplete block design with C progenies and NB incomplete blocks is such that the expected yield E_(ygh) of a tenera offspring T_g (g=1,...,C) of a dura mother D_i (i=1,...,A) and a pisifera father P_j (j=1,...,B), which is allotted to a plot in an incomplete block Bl_h (h=1,...,NB), can be described as the sum of a general constant f , an effect t _g of the tenera T_g and an effect d _h of the block Bl_h, hence

for g=1,...,C and h=1,...,NB .

The yield y_gh of the progeny T_g in the block Bl_h can be described as y_gh = E(y_gh) + e_gh , where e_gh is the environmental effect or plot error with expectation E(e_gh) = 0 and variance Var(e_gh) = s ², these errors are uncorrelated. Because we have allotted the plots of a block at random to the progenies, which must be tested in this block according to the design, this assumption of uncorrelated errors is reasonable.

The model described for y_gh is an additive model of the tenera effects and the block effects. In section 2.2.1 we have already described how the parameters of an additive model can be estimated with the Least Squares Method. To estimate these parameters we must solve the so-called Normal Equations. See Appendix 4 for the analysis of an incomplete block design.

Good statistical packages such as SAS, SPSS, SYSTAT, BMDP or GENSTAT can give us a solution of the Normal Equations; an estimate for the variance s ² follows from the ANOVA table as Mean Square Residual (or Error).

To calculate the estimates of the General Combining Abilities of the A dura and the B pisifera, which are used to derive the C crosses, we construct a two-way table with A rows for the dura and with B columns for the pisifera. In a cell D_ixP_j we insert the Least Squares Mean for D_ixP_j, according to the number of plots of this cross in the experimental field. This two-way table is then analyzed according to an additive model for the dura and pisifera effects, as has been described in section 2.2.1. The Least Squares estimates for m , for the GCA dura effect a _i and for the GCA pisifera effect b _j are m, a_i and b_j respectively. The estimate for the expected value of a cross D_ixP_j according to the additive model of the GCA values is then

m + a_i + b_j .

This procedure gives sufficiently accurate General Combining Abilities estimates to rank the dura and the pisifera.

Note that this procedure is an approximative procedure to estimate the General Combining Abilities, using a randomized incomplete block design to compare the C crosses. A more elaborate analysis would need a three-way table analysis according to blocks, dura and pisifera. But the two-step analysis gives results in a very good approximation; in the case of a completely randomized design or a randomized complete block design the two-step procedure gives us the correct estimates.

Furthermore, with a large crossing trial in an incomplete block design one can get difficulties with the size of the classifications to analyze it directly as a three-way classification with a statistical package.

In the case of disconnected crossing schemes, one can always analyze the connected parts of the crossing schemes separately. In each connected part insert the Least Squares Means of the tenera offsprings. The ranking of the dura and pisifera parents belonging to the connected parts of this crossing scheme can then be given.

To calculate the estimates of the Specific Combining Abilities we must calculate the difference between the Least Squares Mean of a tenera Tg and the estimate of the expected value of this cross according to the additive model of GCA values m + a_i + b_j , hence

SCA(T_g) = LSM(T_g) - (m + a_i + b_j ).

If we may assume that the errors (and hence the yields) are Normally distributed, then we can test whether an additive model for GCA values is reasonable, otherwise stated that the SCA values are equal.

 To test the null-hypothesis "The SCA values are equal " we use the test-statistic

The sum of squares for the SCA values, SS(SCA), can be calculated as the sum of the squared SCA values. The degrees of freedom of this SS(SCA) is df(SCA)=C-(A+B-1), where C=number of the tenera crossings in the connected crossing scheme, A=number of dura and B=number of pisifera. If F >F(a %) the null-hypothesis is rejected at significance level a %, where F(a %) is the right-sided a %-point of the F-distribution with df(SCA) and df(res) degrees of freedom.

EXAMPLE 4

Let us consider the case that we have made C=10 connected tenera crosses derived from A=5 dura mothers and B=5 pisifera fathers. In the following table the crossing scheme is given; a dot (.) indicates a cross which has not been made.

Suppose that the experimental field is very heterogeneous, and that we can only find homogeneous parts (blocks) of maximal four plots. We want to have each progeny tested on four plots. An alpha-design with block size 4, 3 and 3 with four replications has been used. Hence there were a total of NB=12 blocks, where the blocks 1, 2 and 3 form one super-block or replication; further blocks {4, 5 and 6}, blocks {7, 8 and 9} and blocks {10, 11 and 12} form other replications.

The Normal Equations are as follows.

The first Normal Equation (1) is:

 40*f + å _g4*t_g + 4*d₁ + 3*d₂ + 3*d₃ +3*d₄ + 3*d₅ + 4*d₆ +3*d₇ + 4*d₈ +3*d₉ + 3*d₁₀ +4*d₁₁ +3*d₁₂ = 263.62

the first equation of (2) is:

etc.

the first equation of (3) is:

etc.

A solution of these Normal Equations is

Hence the Least Squares Means for the tenera offsprings are:

LSM(T₁)=6.5583, LSM(T₂)=6.1765, LSM(T₃)=6.3328, LSM(T₄)=6.4892, LSM(T₅)=6.7760, LSM(T₆)=7.1923, LSM(T₇)=6.6382, LSM(T₈)=6.2922, LSM(T₉)=6.3096, LSM(T₁₀)=6.9567 .

 The residual sum of squares is SS(res) =

å _{gå h} y_gh² – [f*263.62+t₁*26.41+ . . . +d₁*29.9 + . . . ]

=1757.5708 - 17448.4275 = 9.1433

with degrees of freedom

df(res)=40-(10+12-1) =19 and hence

s ² = 9.1433/19= 0.4812 .

To estimate the General Combining Abilities of the dura and pisifera parents we make a two-way table for the dura D_i and pisifera P_j. In each cell of a realized cross the Least Squares Mean is inserted as many times as there are plots for that cross in the Incomplete Block Design; in this example this is four times. In the table we indicate the value only once.

The Least Squares estimates for the additive model of dura and pisifera effects are:

m=6.3588, a₁=0.5487, a₂=0.0685,

a₃=0.1266, a₄=0.4444, a₅=0.0000,

b₁=-0.3000, b₂=-0.0454, b₃=0.3398,

b₄=-0.1158, b₅=0.0000 .

The estimate of the Specific Combining Ability of T₁ = D₁xP₁ is LSM(T₁)-(m+a₁+b₁)

= 6.5583 -[6.3588 + 0.5487 +(-0.3000)] = -0.0492 .

In the following table the estimates of the Specific Combining Abilities (SCA) are given; each SCA value must be repeated four times but in the table only a SCA value is given once.

The sum of squares for the Specific Combining Abilities (SCA) is the sum of all the squares of the SCA values (note that each entry in the table must be replicated four times), SS(SCA)= 4*10*(0.0492)² = 0.0968 . The degrees of freedom for this SS(SCA) is df(SCA) = 10 - ( 5 + 5 -1 ) =1 .

The test-statistic to test the null-hypothesis "The SCA values are equal" is

The 5% right sided significance point of the F-distribution with 1 and 19 degrees of freedom is F(5%)=4.38, hence the null hypothesis is not rejected because F= 0.096 <F(5%) = 4.38.