Robust alternatives to best linear unbiased prediction of complex traits

Transcription

1 Robust alternatives to best linear unbiased prediction of complex traits

2 WHY BEST LINEAR UNBIASED PREDICTION EASY TO EXPLAIN FLEXIBLE AMENDABLE WELL UNDERSTOOD FEASIBLE UNPRETENTIOUS NORMALITY IS IMPLICIT

3 DRAWBACK: GAUSSIAN RESIDUALS SENSITIVE TO OUTLYING DATA POINTS Hampel et al. (1986) Rousseau and Leroy (1986) Lange et al. (1989) Seber and Lee (2003)

4 ACCOMODATING OUTLIERS DISCARD DATA WITH AD-HOC RULES FIT ROBUST RESIDUAL DISTRIBUTION ANIMAL BREEDERS HAVE DONE IT FOR INFERENCE, NOT PREDICTION! -STRANDEN AND GIANOLA (1998, 1999) -ROSA ET AL. (2003, 2004) -KIZILKAYA ET A. (2003) -CARDOSO ET AL. (2006 BAYESIAN MCMC USED (ADVANTAGES, DRAWBACKS, PITFALLS) -tricky parameters (df in the t-distribution) -Intensive computation -Involved convergence diagnostic -Monte Carlo error swamping statistical error -Not practical for routine industry application (explains why BLUP used, but consider BOLT)

5 OBJECTIVES PRESENT ROBUST ALTERNATIVES TO BLUP MODEL USES t OR LAPLACE (DOUBLE EXPONENTIAL) RESIDUAL DISTRIBUTIONS BAYESIAN NON-MCMC APPROACH EVALUATION WITH -wheat (grain yield) -Arabidopsis (plant diameter, gene expression, flowering time) -Brown Swiss cows (milk yield)

6 BLUP (BAYESIAN INTERPRETATION) Fixed or Flat prior Pedigree, genomic, similarity matrix Spread parameters Bayesian sampling model CONDITIONAL POSTERIOR MODE (GIVEN SPREAD)= CONDITIONAL POSTERIOR MEAN λ controls regularization MODEL COMPLEXITY

7 TMAP (Maximum a posteriori with t-residuals) Scale degrees of freedom Sampling model CONDITIONAL POSTERIOR DENSITY

8 LOCATE SOME MODE BY ITERATING WITH Diagonal matrix with elements scale parameter instead of residual variance here Weight changes iteratively -smaller when larger residuals -smaller at smaller ν and smaller scale -n 1if 1 observation per phenotype

9 LMAP (Maximum a posteriori with Laplace residuals) Sampling model LOG- CONDITIONAL POSTERIOR DENSITY

10 LOCATE SOME MODE BY ITERATING WITH Diagonal matrix with elements 2 Weight changes iteratively -smaller when larger residuals -n 1if 1 observation per phenotype

11 ZERO MEAN-MODEL (y=g+e) BLUP TMAP LMAP

12 PREDICTIVE ALGORITHM (e.g., TMAP)

13 CASE 1: BROWN SWISS TEST- DAY MILK YIELD n=991 cows, pre-corrected daily milk yield p= 37,568 SNP Grid of MINQUE(guesses): (0.05 increments), followed by MINQUE (all cows) GBLUP, LMAP, TMAP (df= 4, 8, 12, 16) LMAP and TMAP iterated 300 times (overkill) Gianola and Schoen (2016) used to calculate LOO predictions indirectly, assuming constant variances Bootstrap (15,000 samples) emulated repeated sampling from joint distribution [predictands, LOO predictions]

14 FLAG OUTLIERS IN TMAP

15 COWS GOOD OR BAD BY GBLUP NOT THAT GOOD OR THAT BAD IN TMAP (4 OR 8)

16 Bootstrap distribution (b=15,000 samples) of predictive mean squared error (PMSE) and predictive correlation (PCOR) for GBLUP, TMAP (df=4) and LMAP at selected genomic heritability values (guesses of 0.05 and 0.50 produced MINQUE estimates of 0.07 and 0.15, respectively): test day milk yield in Brown Swiss cows. LMAP BEST FOLLOWED BY TMAP4 AND THEN BY GBLUP

17 CASE 2: WHEAT YIELD n=599 inbred lines Analyses for 4 different environments p= 1279 allelic markers (DaRT) Training (n=300)-testing (n=299) 200 random repetitions GBLUP and ABLUP [additive models] TMAP (df= 4, 6, 8) and LMAP [additive models] (200 iterations: overkill)

18 Distribution (200 replicates, training testing layout) of predictive mean squared error for BLUP (B), LMAP (L) and TMAP (4, 6, 8 df) for wheat yield in four environments. Genome (red) and pedigree based (blue) distributions

19 Distribution (200 replicates, training testing layout) of predictive correlation for BLUP (B), LMAP (L) and TMAP (4, 6, 8 df) for wheat yield in four environments. Genome and pedigree based distributions in red and blue

20 Frequency with which a given method had the largest predictive correlation over 200 replications: pedigree (A) based models, wheat ( winner in boldface). YIELD TRAIT ABLUP ALMAP ATMAP4 ATMAP6 ATMAP (1+2) (1+3) (1+4) (2+3) (2+4) (3+4) (1+2+3) (1+2+4) (1+3+4) (2+3+4) ( )

21 Frequency with which a given method had the largest predictive correlation over 200 replications: genome (G) based models, wheat ( winner in boldface) YIELD TRAIT GBLUP GLMAP GTMAP4 GTMAP6 GTMAP (1+2) (1+3) (1+4) (2+3) (2+4) (3+4) (1+2+3) (1+2+4) (1+3+4) (2+3+4) ( )

22 CASE 3: ARABIDOPSIS n=199 accessions (Atwell et al. 2010) Flowering time (n=194), plant diameter (n=180), FRIGIDA expression (n=164) p= 215,947 LOO with variances (MINQUE) re-estimated at each training instance GBLUP, TMAP (df=4, 8, 12, 16, 20), LMAP 50,000 bootstrap samples from [y, predictions] PMSE, PCOR, PREDICTIVE REGRESSION (ALPHA, BETA)

23 Bootstrap distribution (b=50,000 samples) of intercept (ALPHA) and slope (BETA) of regressions of predictands on predictors: flowering time, frigida expression and plant diameter in Arabidopsis

24 Bootstrap distribution (b=50,000 samples) of predictive mean squared error (PMSE) and predictive correlation (PCOR): flowering time, frigida expression and plant diameter in Arabidopsis

25 Table 1. Fraction of bootstrap samples (50,000) in which GBLUP) attained a smaller PMSE or a larger predictive PCOR than either LMAP or TMAP GBLUP vs LMAP GBLUP vs TMAP4 GBLUP vs TMAP8 GBLUP vs TMAP12 GBLUP vs TMAP16 GBLUP vs TMAP20 FLOW GBLUP UNIFORMLY WORSE PMSE PCOR FRIG GBLUP MOST OFTEN WORSE PMSE PCOR DIAM GBLUP UNIFORMLY BETTER PMSE PCOR

26 CONCLUDING REMARKS The Bayesian alphabet goes environmental! BLUP WIDELY USED SIMPLE, UNDERSTOOD, FEASIBLE, FLEXIBLE EXTENSIVE SOFTWARE AVAILABLE DRAWBACK: NOT ROBUST TO OUTLIERS SIMPLE (GLIM-TYPE) METHODS PRESENTED FOR t AND LAPLACE RESIDUAL DISTRIBUTIONS EXTENDS EASILY TO ssblup AND RKHS

27 SKEWED RESIDUAL DISTRIBUTIONS

28 MULTIVARIATE OUTLIERS: UNCHARTED WATERS MULTIPLE-TRAIT t VERSION STRAIGHTFORWARD (STRANDÉN, 1996) MULTIVARIATE LAPLACE, NOT MUCH THEORY, BUT (GOMEZ et al. 1998) Power exponential family

29 CHINESE PHILOSOPHY One can have an army with millions of soldiers, but if their weapon is just a fork, a smaller and better equipped rival can be more effective in battle (Sun Tzu and Dan Gian, 6 th century BC)