Appendix S1. Statistical models of analysis
A. Repeatability mixed linear model for the analysis of immune traits, and weekly health and
lactation traits corresponding to the week of the immune measurement:
Yijklmno    Gi  D j  YM k  LAl  Wm  An  eijklmno
Yijklmno = Animal record
µ = Overall mean
Gi = Fixed effect of ith genetic line (i=1, 2)
Dj = Fixed effect of jth diet group (j=1, 2)
YMk = Fixed effect of kth year by month of calving interaction (k=1, 25)
LAl = Fixed effect of lth lactation number (l=1, 3) by age at calving interaction
Wm = Fixed effect of mth week of lactation when the record was taken (m=1, 44)
An = Random effect of nth individual cow (mean=0, variance=Va)
eijklmno = Random residual effect (mean=0, variance=Ve)
B. Fixed linear model for the analysis of number of health episodes and reproductive traits:
Yijklmo    Gi  D j  YM k  LAl  Wm  eijklmo
Yijklmo = Animal record
µ = Overall mean
Gi = Fixed effect of ith genetic line (i=1, 2)
Dj = Fixed effect of jth diet group (j=1, 2)
YMk = Fixed effect of kth year by month of calving interaction (k=1, 25)
LAl = Fixed effect of lth lactation number (l=1, 3) by age at calving interaction
Wm = Fixed effect of mth week of lactation when the record was taken (m=1, 44)
eijklmo = Random residual effect (mean=0, variance=Ve)
C. Random regression model for repeated measures analysis of lactation traits across the
entire lactation:
2
2
p 0
p 0
Yijkqlmnpo    Gi  D j  YM k  YMR q  LAl   b p PpWm   Apn PpWm  eijkqlmnpo
Yijkqlmnpo = Animal record
µ = Overall mean
Gi = Fixed effect of ith genetic line (i=1, 2)
Dj = Fixed effect of jth diet group (j=1, 2)
YMk = Fixed effect of kth year by month of calving interaction (k=1, 25)
YMRq = Fixed effect of qth year by month of record interaction (q=1, 28)
LAl = Fixed effect of lth lactation number (l=1, 3) by age at calving interaction
Wm = Fixed effect of mth week of lactation when the record was taken (m=1, 44)
bp = Fixed regression coefficient on week of lactation
Pp = pth orthogonal polynomial of week of lactation (p=order of polynomial),
Apn = Random regression coefficient on week of lactation associated with the random
effect of the nth individual cow
eijqklmnpo = Random residual effect (mean=0, variance=Ve)