Appendix 4

Analysis of an incomplete block design

The model for an incomplete block design with C progenies and NB incomplete blocks is such that the expected yield E(y_gh) 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

E(y_gh) = f + t _g + d _h

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. Let the Least Squares estimates be denoted by f for f , t_g for t _g and d_h for d _h. Let further n_gh be 1 if progeny T_g is present in block Bl_h and n_gh be 0 if progeny T_g is not present in block Bl_h.

The Normal Equations are then:

n.. * f + S _g n_g. * t_g + S _h n._h * d_h = y.. (1)

n_g. * f + n_g. * t_g + S _h n_gh * d_h = y_g.

for g=1,..., (2)

n._h * f + S _g n_gh * t_g + n._h * d_h = y._h

for h=1,...,NB (3)

Note that these equations are not independent. Equation (1) is equal to the sum of the equations of (2); also, equation (1) is equal to the sum of the equations of (3). Hence there are two linear dependencies between the Normal Equations. We will choose as a solution of the Normal Equations that solution where we take t_C=0 and d_NB=0.

The Least Squares Mean of an offspring T_g is defined as

f + t_g + S _h d_h /NB

and this is the same for every solution of the Normal Equations.

The estimate for the variance s ² is s² = SS(res)/df(res) , where the residual sum of squares SS(res) is calculated as

SS(res) = S _{gS h} y_gh² -[ f*y.. + S _g t_g * y_g. + S _h d_h * y._h ]

with degrees of freedom

df(res) = n.. - [ C + NB -1 ].