Test-day fat to protein ratio as an indicator trait for sub-clinical ...

distributions of milk yield, F:P and SCS may therefore also be different in different parts
of lactation.
The objectives of this research were to analyze first lactation TD milk yield, F:P
and SCS of Canadian Holsteins cows with a three-trait normal mixture random regression
model, allowing for heterogeneity of distributions in different parts of lactation. Genetic
parameters were estimated on lactational and daily bases, and distributions of data from
IM- and IM+ categories were characterized.
MATERIAL AND METHODS
Data
Canadian Holstein first lactation TD data with calving years from 1988 to 2007
were obtained from the Canadian Dairy Network. A small data set was selected by
random sampling of herds with minimum of 50 cows each, and with minimum of 7 TD
records per cow. This resulted in 214,813 TD milk (kg), F:P and SCS records on 25,950
cows. Each cow had all three traits recorded on a given TD. Days in milk (DIM) were
from 5 to 305. Daily averages with corresponding variances are plotted in Figures 1 - 3
for milk, F:P and SCS, respectively. The number of herd – test-day (HTD) classes was
23,984 and there were 80 levels of region – age – season of calving (RAS). Pedigree
data included 66,244 animals. The same edited data set was previously used by Jamrozik
and Schaeffer (2010) for fitting a two-trait mixture model.
DIM were split into 30 disjoint intervals: 5 – 15 DIM, and 29 intervals each of
length of 10 DIM for days from 16 to 305. Records were relatively well distributed across
DIM intervals as well as days in lactation. Number of TD records in the first DIM
interval was equal to 6731; the remaining intervals included from 5890 to 7610 records.
The 5 th and 305 th DIM had 503 and 508 TD records, respectively; the largest number of
records (805) was for the 228 th day of lactation.
Model
The finite mixture model assigns records of milk, F:P and SCS on a given TD into
one of the two components, depending on the unknown health status of the cows. Group
membership (healthy or infected) of the j-th record in the i-th DIM interval was defined
by a binary vector zi where zij = 0 (1) indicated a record from an IM- (IM+) cow. The
equation for the mixture model can be written following Ødegård et al. (2003) as:
yi = X0i 0 + Mzi X1i 1i + Zi a + Wi p + ei,
where yi = vector of observations (milk, F:P and SCS) in the i-th DIM interval, 0 =
vector of fixed effects common to all records in all DIM intervals, 1i = vector of fixed
effect corresponding to data from infected cows, a = vector of random animal additive
genetic effects, p = vector of random permanent environmental (PE) effects, ei = vector
of random residuals, Mzi = matrix with diagonal elements of zij, and X0i, X1i, Zi, Wi =
known incidence matrices. The fixed effects in 0 were HTD effects and regression on
DIM within RAS levels, and elements of 1i included mean differences between ‘sick’
and ‘healthy’ components for milk, F:P and SCS. Additive genetic and PE effects were
regressions on DIM; all regressions (fixed and random) were modeled with orthogonal
Legendre polynomials of order 4. Residuals of records on different TD were
uncorrelated. The conditional distribution of yi was assumed to be:
yi | 0, 1i, a, p, Ri, zi ~ N[ X0i 0 + Mz X1i 1i + Zi a + Wi p, Ri],
2