1. 2003 Paul M. VanRaden Animal Improvement Programs Laboratory Agricultural Research Service, USDA, Beltsville, MD paul@aipl.arsusda.gov Comparison of Longevity and Fertility Trait Definitions in Theory and Simulation

2. Interbull Workshop March 2-3, 2003 (2) P.M. VanRaden 2003 Goals  Compare methods to evaluate non- normally distributed traits Linear analysis of length of time Binomial analysis of culling rate Survival analysis for log(culling rate)  Try to understand parameters  Evaluate sires from simulated data

3. Interbull Workshop March 2-3, 2003 (3) P.M. VanRaden 2003 Current Longevity Evaluations  Linear animal or sire models Binomial culling rate – Single EBV = c (SWE, Cornell) – Multiple EBV = c t (CAN, AUS, IRL) Productive life – Single EBV = 1/c (USA, GBR, ISR) – Multiple EBV = 1/c t (NZL)  Non-linear sire-MGS models Survival analysis – Single EBV = log(c * env t ) (DEU, FRA, NLD, ITA, BEL, CHE)

4. Interbull Workshop March 2-3, 2003 (4) P.M. VanRaden 2003 Animal vs Sire-MGS Models  Value of maternal relatives Average of 5 maternal brothers for bulls proven in USA Extra info is worth 10-20 daughters when heritability is low (.10 to.03) Animal vs sire models differ more than linear vs non-linear (Boettcher et al 1999 JDS 82:1034)  All relatives or non-linear math?

5. Interbull Workshop March 2-3, 2003 (5) P.M. VanRaden 2003 Distribution of Longevity

6. Interbull Workshop March 2-3, 2003 (6) P.M. VanRaden 2003 ≤ 50 ≥ 250 Distribution of Days Open Holstein Calvings 1990 - 2001 Cows culled for reproductive reasons

7. Interbull Workshop March 2-3, 2003 (7) P.M. VanRaden 2003 Toss Coin Until Head Observed

8. Interbull Workshop March 2-3, 2003 (8) P.M. VanRaden 2003 Maximum Likelihood  Define fraction of cows culled in each lactation = c and number of lactations survived by cow i = y i Prob(y i | c) = c (1 – c) (Yi – 1) ML estimate of c = n / Σ y i Linear estimate of (1/c) = Σ y i / n  Culling rate is just reciprocal of number of lactations (no info lost)

9. Interbull Workshop March 2-3, 2003 (9) P.M. VanRaden 2003 Prior Distributions  Sire effects normal for: Number of lactations (linear model) Culling rate (binomial model) Log of culling rate (survival model)  Differences are small if heritability is low  Prob (y | c) can be the same, but Prob (c) differs

10. Interbull Workshop March 2-3, 2003 (10) P.M. VanRaden 2003 Comparison of Priors

11. Interbull Workshop March 2-3, 2003 (11) P.M. VanRaden 2003 Simulated Data  500 sires with 10-1000 daughters, 10 replicates  Sire effects normal for log (c /.33) BV for log(relative risk) transformed to culling rate and longevity by non-linear formulas True BV had product-moment (linear) correlations of.98 to.995.  What scale should MACE work on?

12. Interbull Workshop March 2-3, 2003 (12) P.M. VanRaden 2003 Simulation Results  Heritability estimates.106 for number of lactations.038 for culling rate (per lactation).091 for log(number of lactations)  True accuracy, or corr (EBV, BV).72 for number of lactations.72 for culling rate.70 for log(number of lactations)

13. Interbull Workshop March 2-3, 2003 (13) P.M. VanRaden 2003 Conclusions  Including all relatives is more important when heritability is low  EBV from repeated binomial models are reciprocals of PL  True reliabilities are similar, but reported reliabilities differ slightly  Linear functions of data seem OK