1. Normality and Skewness of Genetic Evaluations

2. Effects of Skewness • Number of extreme bulls • Rankings are ok within countries • Problems when EBVs with differing skewness merged across countries • Linear vs non-linear models • Economic contribution to index • Values linear on underlying or observed scale?

3. Observed vs Underlying EBV

4. Skewness

5. Data and Methods • May 2006 MACE evaluations • All traits (except conf.) on all scales • Only bulls with domestic data • Skewness computed within year • Removed birth year trend by linear regression • Removal had little effect for traits with small genetic trends

6. Protein, SCS, Longevity Skewness (Within year, for countries sampling most bulls) 1 Protein obtained from Denmark-Finland-Sweden joint evaluation

7. Calving Ease Skewness (Within year) Observed scale Pilot run 1 Protein obtained from Denmark-Finland-Sweden joint evaluation

8. Stillbirth and Mastitis Skewness (Within year) Underlying scale Observed scale

9. Fertility Skewness • Heifer conception rate • Range -.23 to .11 • Days to first insemination • Range -.33 to -.09 • Cow conception rate • Range -.20 to .10 • Calving interval or days open • Range -.35 to -.04

10. Extreme Calving Ease Bulls • USA compared to foreign bulls • Slightly poorer for direct and better for maternal calving ease on average • Percent of extreme 100 bulls across scales • 45% (direct) and 32% (maternal) of BEST • 20 % (direct) and 21% (maternal) of WORST • 34 % (direct) and 22% (maternal) of ALL bulls • Unskewed EBVs are an advantage • Economics linear on observed scale?

11. Transformations • Cumulative normal density • Converts underlying to observed scale EBVs in threshold models • Exponential (Manly, 1976) • [exp(a EBV) – 1] / a • Choose a to make skewness = 0

12. Conclusions • Skewness differs by trait • Yield and longevity fairly normal • SCS always moderately skewed • Calving traits, stillbirth most skewed • Underlying vs observed scale • Nonlinear transformations useful • Consistent skewness is desired