Appendix S1. Statistical Models of Analysis

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 = Animal record

µ = Overall mean

th Gi = Fixed effect of i genetic line (i=1, 2)

th Dj = Fixed effect of j diet group (j=1, 2)

th YMk = Fixed effect of k year by month of calving interaction (k=1, 25)

th LAl = Fixed effect of l lactation number (l=1, 3) by age at calving interaction

th Wm = Fixed effect of m week of lactation when the record was taken (m=1, 44)

th An = Random effect of n 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 = Animal record

µ = Overall mean

th Wm = Fixed effect of m 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:

Yijkqlmnpo = Animal record

µ = Overall mean

YMRq = Fixed effect of qth year by month of record interaction (q=1, 28)

th Wm = Fixed effect of m week of lactation when the record was taken (m=1, 44) bp = Fixed regression coefficient on week of lactation

th Pp = p 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)