Page 237 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

P(k) = F(k) − F(k − 1) with F(k − 1) = 0, when k = 1; or expressed in terms of thresh-
olds defining the category k, P = Ft − Ft . For instance in Fig. 13.1, the
k k (k−1)
probability of response in the k category (P ) can be computed as:
k
P = F(t ) (13.2)
1 1
P = F(t ) − F(t ) (13.3)
2 2 1
P = F(t ) − F(t ) and
3 3 2
P = 1 − F(t )
4 3
13.2.2 Data organization and the threshold model

Usually, the data are organized into an s by m contingency table (Table 13.1), where
the s rows represent individuals or herd–year subclasses of effects, such as herd, and
the m columns indicate ordered categories of response. If the rows represent individu-
als, then all n will be zero except one and the n = 1, for j = 1,..., s.
jk j.
The linear model for the analysis of the liability is:
y = Xb + Zu + e

where y is the vector of liability on a normal scale, b and u are vectors of fixed and
random (sire or animal) effects, respectively, and X and Z are incidence matrices
relating data to fixed effects and responses effects, respectively. Since y is not
observed, it is not possible to solve for u using the usual MME.
Given that H′ = [t′, b′, u′], where t is the vector for the threshold effects, Gianola
ˆ
and Foulley (1983) proceeded to find the estimator Hthat maximizes the log of the

posterior density L(H). The resulting set of equations involved in the differentiation
were not linear with respect to H. They therefore provided the following non-linear
iterative system of equations based on the first and second derivatives, assuming a
normal distribution to obtain solutions for Dt, Db and Du:
⎡ Q L X L Z ⎤ ⎡ t Δ ⎤ ⎡p ⎤
′
′
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
′
′
′
′
⎢
⎢ XL X WX X WZ ⎥ ⎢ Δ b = Xv ⎥ (13.4)
⎥
⎢ ′ ′ ′ −1 −1⎥ ⎢ Δ u⎥ ⎢ −1 −1 ⎥
+
⎣ ZL Z WX Z WZ A G ⎦ ⎣ ⎦ ⎦ ⎣ ′ Zv − A G u ⎦
Table 13.1. Ordered categorical data arranged as an s by m contingency table.
Categories a
Subclasses 1 2 … k … m Totals b
1 n n … n … n n
11 12 1k 1m 1.
2 n 21 n 22 … n 2k … n 2m n 2.

j n j1 n j2 … n jk … n jm n j.

s n n … n … n n
s1 s2 sk sm s.
a n = number of counts in category k of response in row j.
jk
j. ∑
b m
n = k=1 n jk
Analysis of Ordered Categorical Traits 221