R . Rupp, D. Boichard Livestock Production Science 62 2000 169 –180
171
traits. McDaniel 1993 found that regressions of SCC information and to occur from 5 to 35 days
absence or presence of mastitis cases in first lactation after calving. Mastitis events were declared by
cows on PTA for sire somatic cell score were farmers and collected every month by Milk Record-
positive: one unit change in PTA corresponded to an ing technicians until March 1, 1997. Reliability and
increase of 36 in mastitis incidence. Rogers et al. completeness of disease recording was assessed by
1996 found similar results for US bulls used in an additional survey conducted by the Milk Record-
Denmark and Sweden. In two Swedish breeds, ing agents. Herds 36 for which clinical cases
Philipsson et al. 1995 reported a linear relationship recording was considered to be incomplete by the
between Relative Breeding Values RBV for clinical technicians were excluded from the study. Herds
mastitis and SCC, with a 0.35 increase in RBV for without any clinical mastitis event recorded were
mastitis per unit of RBV in SCC. All these studies also discarded. The information collected was the
suggested that SCC could and should be decreased to date of the monthly test following the event. If
the lowest possible value. mastitis occurred around calving 28 to 1 8 days,
Other authors investigated the relationship be- this was additionally stated. Consequently, the confi-
tween SCC at a given time, e.g., the onset of dence interval of the exact date of clinical mastitis
lactation, and occurrence of intramammary infection was 35 days during the lactation average interval
or clinical mastitis later in lactation Coffey et al., between two consecutive test day or 16 days around
1986b; Schukken et al., 1994; Beaudeau et al., calving. Accordingly, for processing this informa-
1998. These studies, however, were carried out with tion, a mastitis during lactation was arbitrarily sup-
a limited amount of data or did not account for the posed to occur 16 days before test day and a mastitis
time period up to mastitis. around calving was supposed to occur the day of
Alternatively, in addition to the presence or ab- calving. The whole data set included 25,833 cows,
sence of mastitis events, survival analysis can also out of them 5156 20 had at least one clinical
account for the length of the time period up to the mastitis in first lactation. To avoid the possible effect
¨ event Grohn et al., 1997, i.e., the number of days
of a previous clinical mastitis event on the first SCC, up to first mastitis. Survival analysis is based on the
1940 cows with a clinical mastitis recorded before 35 concept of hazard rate, defined as the probability of
days after first calving were discarded. Finally, to occurrence of some event at time t, given that it did
study the relationship between early SCC and clini- not happen just before t. This methodology provides
cal mastitis in a range of low to moderate cell estimates of relative risks of an event for groups of
counts, only cows with a first SCC lower than cows defined according to given characteristics. The
400,000 cells ml were considered. Accordingly, objective of this study was to determine if low SCC
2779 cows were discarded. These edited cows were cows are at greater risk to first clinical mastitis than
found to have high mastitis frequency, as 476 17 cows with somewhat higher SCS. Relationship be-
of them had at least one clinical mastitis in first tween SCC at initial test in first lactation and time to
lactation. Finally, cows culled before 35 days and first mastitis later in first and second lactation was
herds with less than three selected cows 692 were assessed by survival analysis.
discarded. After edits, the final data set consisted of 20,422 cows in 2611 herds, with 13 of these cows
having at least one clinical mastitis. Distribution of
2. Material and methods clinical events is shown in Fig. 1.
The variable analyzed was the interval to mastitis 2.1. Data
event. Because of editing rules, it was defined as the number of days from 35 days after first calving to
The data consisted of Holstein cows from Mor- first mastitis occurring in first or second lactation. If
` bihan and Finistere regions in Western France, with
no mastitis occurred during the first lactation and if first calving between September 1, 1995 and August
the cow did not calve a second time before March 1, 31, 1996. Cow calving dates, parity, test day milk
1997, her record was censored the day of her last test yield and SCC were extracted from the national data
day. If a cow was still present in the herd and still base. The first test day record was required to have
unaffected at the end of the period under study,
172 R
. Rupp, D. Boichard Livestock Production Science 62 2000 169 –180
Fig. 1. Distribution of time to first clinical mastitis CM j and relationship between total number of clinical events and time to first clinical mastitis s.
whether she had begun a new lactation or not, her estimate constant relative risks RR associated with
record was also censored at March 1, 1997. a factor w. The risk of effect w1 relative to w2 is
computed as the ratio of the hazard function of the corresponding groups of animals:
2.2. Model RR 5
lt; w1 lt; w2
The survival model used was the proportional 5
l t expw19u l t expw29u
hazard model Cox, 1972, which is based on the
5 exp[w1 2 w29 u ] 5 constant
concept of a hazard function lt, where lt was the
limiting probability for a cow to have her first The baseline function
l t was assumed to follow a mastitis at time t expressed in days, given that she
Weibull hazard distribution: was still unaffected just prior to t. The hazard rate
r 21
was defined as the product of a baseline hazard l t 5 lrlt
function l t, which acts as an average hazard
with parameters l and r. This distribution is flexible
function, and of a function w of explanatory vari-
and has been shown to usually adequately fit bio- ables, including the initial SCC:
logical data Ducrocq et al., 1988a. In the Weibull regression model, the hazard function can be sim-
lt; w 5 l t expw9u plified as:
where u is a vector representing the effects of the
r 21
lt; w 5 rt exp
r logl 1 w9u explanatory covariables.
Use of proportional model makes it possible to Therefore only the parameter
r describes the
R . Rupp, D. Boichard Livestock Production Science 62 2000 169 –180
173
baseline hazard function and r logl appears as an
these levels included 8 to 25 of cows. The intercept on the logarithmic scale. The effects in-
variable IMY was also categorized into six groups: cluded in the model were the random herd-year
1 less than 21.5 kg; 2 21.5–23.5 kg; 3 23.5– effect and the fixed effects of lactation stage, month
25.5 kg; 4 25.5–27.5 kg; 5 27.5–29.5 kg; and 6 of first calving, initial SCC, and initial milk yield.
more than 29.5 kg. Each of the six IMY classes The herd-year effect, with 5222 levels, was as-
included approximately the same number of records. sumed to be time-dependent, with changes at Sep-
In a preliminary analysis, the effect of days in milk tember 1, 1996, to allow for a modification in
at first test day was found not to be significant and mastitis hazard in a herd from one year to the next,
was removed from the final model. accounting for differences in udder health manage-
ment, bacteriological pressure, and other environ- 2.3. Method
mental changes. It was assumed to be random and to follow a log–gamma distribution. The latter assump-
Effect estimates were obtained by a maximum tion allowed to algebraically integrate the herd-year
likelihood technique, using the ‘Survival Kit’, a set effect out of the joint posterior density, decreasing
of
FORTRAN
programs written by Ducrocq and dramatically the number of parameters to estimate
¨ Solkner 1994. The random herd-year effect was
Ducrocq et al., 1988b. assumed to follow a log–gamma distribution with
Stage of lactation was a fixed time-dependent parameter
g Ducrocq et al., 1988a. The marginal effect with five classes starting at day 35, 91, 181
posterior distribution was obtained, after algebrai- after first calving, 10 days before and 30 days after
cally integrating the herd-year effect out of the joint second calving, respectively. Such a definition al-
posterior distribution. Finally, fixed possibly time- lowed for possible hazard changes during the life of
dependent effects, the herd-year parameter g and
the cow, and particularly for a decrease in hazard the Weibull parameters
r and r logl were esti- during lactation and an increase around calving. A
mated by maximization of the resulting logarithm of slight drawback for the use of time-dependent
the marginal posterior density Ducrocq et al., covariates is that comparison of hazard of cows at
1988a; Ducrocq, 1993. Standard errors of estimates different points in time must be done by combining
r, r logl, and fixed effects were computed as the the estimates of all time-dependent covariates with
square root of the diagonal elements of the inverse of the values of the baseline hazard function at these
the Hessian matrix. The moments of the distribution time points. Similarly, estimates of time-dependent
of the herd-year effect were: effects must be interpreted jointly with the values of
1
ˆ ˆ
¨ the baseline hazard function Grohn et al., 1997.
ˆ ˆ
ˆ E 5
Cg 2 logg and Var 5C g ,
Month of first calving was treated as a fixed
1
ˆ ˆ
discrete variable with 12 levels, from September where
Cg and C g are the digamma and the
ˆ 1995 to August 1996.
trigamma function evaluated at g, respectively
Because somatic cell count and milk yield at first Kalbfleisch and Prentice, 1980. Likelihood ratio
test day ISCC and IMY, respectively were highly tests of explanatory variables were obtained by
dependent on days in milk, they were pre-adjusted comparing the full model with reduced models
for days in milk by regression with linear and excluding one variable at a time.
quadratic terms. Regression coefficients were previ- The analysis was carried out on the complete data
ously estimated from the data set. Then ISCC and set as well as on different herd subgroups. First, two
IMY were treated as fixed categorical variables to subgroups were defined according to herd mastitis
account for any nonlinear association with mastitis frequency less or more than 20 affected lacta-
occurrence. Six levels were defined for ISCC: 1 tions, which was calculated from data of all parities
less than 35,000 cells ml; 2 35,000 to 50,000 and all cows in each herd, even those not selected for
cells ml; 3 50,000 to 75,000 cells ml; 4 75,000 the survival analysis. Secondly, two subgroups were
to 150,000 cells ml; 5 150,000 to 215,000 cells also defined according to herd somatic cell score
ml; and 6 215,000 to 400,000 cells ml. Each of mean below or above 3.4. Test day somatic cell
174 R
. Rupp, D. Boichard Livestock Production Science 62 2000 169 –180
scores SCS were defined in a classical way through computed using Kaplan–Meier’s formula 8. This
a logarithmic-transformation of test day SCC: SCS5 plot displayed a straight line, which validated the
log SCC 100,00013 and a lactation mean of all assumption of Weibull proportional hazard model.
2
test day SCS was computed for each cow. As for Likelihood ratio tests are shown in Table 1 for the
herd mastitis frequency, herd SCS mean was calcu- complete data set. All factors were found to be
lated from data of all parities and all cows in each highly significant.
herd. Separate analyses allowed to account for possible differences in the baseline hazard function
3.1. Estimates of the Weibull parameters and of and changes in effects of covariates.
the explanatory covariates Table 2 presents the estimates of Weibull parame-
3. Results ters
