<p>Appendix S1. Statistical models of analysis</p><p>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:</p><p>Yijklmno = Animal record</p><p>µ = Overall mean</p><p> th Gi = Fixed effect of i genetic line (i=1, 2)</p><p> th Dj = Fixed effect of j diet group (j=1, 2)</p><p> th YMk = Fixed effect of k year by month of calving interaction (k=1, 25)</p><p> th LAl = Fixed effect of l lactation number (l=1, 3) by age at calving interaction </p><p> th Wm = Fixed effect of m week of lactation when the record was taken (m=1, 44)</p><p> th An = Random effect of n individual cow (mean=0, variance=Va) eijklmno = Random residual effect (mean=0, variance=Ve)</p><p>B. Fixed linear model for the analysis of number of health episodes and reproductive traits:</p><p>Yijklmo = Animal record</p><p>µ = Overall mean</p><p> th Gi = Fixed effect of i genetic line (i=1, 2)</p><p> th Dj = Fixed effect of j diet group (j=1, 2)</p><p> th YMk = Fixed effect of k year by month of calving interaction (k=1, 25)</p><p> th LAl = Fixed effect of l lactation number (l=1, 3) by age at calving interaction </p><p> 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:</p><p>Yijkqlmnpo = Animal record</p><p>µ = Overall mean</p><p> th Gi = Fixed effect of i genetic line (i=1, 2)</p><p> th Dj = Fixed effect of j diet group (j=1, 2)</p><p> th YMk = Fixed effect of k year by month of calving interaction (k=1, 25)</p><p>YMRq = Fixed effect of qth year by month of record interaction (q=1, 28)</p><p> th LAl = Fixed effect of l lactation number (l=1, 3) by age at calving interaction </p><p> 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</p><p> th Pp = p orthogonal polynomial of week of lactation (p=order of polynomial),</p><p>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)</p>