Page 236 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

The principles required to apply a linear model for the analysis of categorical traits
are the same as discussed in the previous chapters; therefore, the main focus of this
chapter is on threshold models, assuming a normal distribution for the liability.
Cameron (1997) illustrated the analysis of a binary trait with a threshold model using
a logit function. In this chapter, sample data used for the illustration with the threshold
model have also been analysed with a linear model for the purposes of comparison.

13.2 The Threshold Model

13.2.1 Defining some functions of the normal distribution

The use of the threshold model involves the use of some functions of the normal
distribution and these are briefly defined. Assume the number of lambs born alive to
ewes in the breeding season is scored using four categories. The distribution of liabil-
ity for the number of lambs born alive with three thresholds (t ) can be illustrated as
j
in Fig. 13.1, where N is the number of ewes with the jth number of lambs and are
j
those exceeding the threshold point t , when j > 1 and j ≤ m −1.
j−1
With the assumption that the liability (l) is normally distributed (l ~ N(0,1)), the
height of the normal curve at t (f(t )) is:
j j
−
2
f( ) = exp( 0.5 ) / 2 p (13.1)
t
t
j
j
For instance, given that t = 0.779, then f(0.779) = 0.2945.
j
The function F() is the standard cumulative distribution function of the normal
distribution. Thus F(k) or F gives the areas under the normal curve up to and
k
including the kth category. Given that there are m categories, then F = 1 when the
k
kth category equals m. For a variable x, for instance, drawn from a normal distribu-
tion, the value F can be computed, using a subroutine from the IMSL (1980) library.
x
Thus if x = 0.560, then F(0.560) = 0.7123.
P(k) defines the probability of a response being observed in category k assum-
ing a normal distribution. This is also the same probability that a response is
between the thresholds defined by category k. Thus P(k) or P may be calculated as
k

N 1 N 2 N 3 N 4
t 1 t 2 t 3
Fig. 13.1. The distribution of liability for number of lambs born alive with four categories
and three thresholds.

220 Chapter 13