RESTRICTED M A X I M U M LIKELIHOOD TO E S T I M A T E GENETIC P A R A M E T E R S - IN PRACTICE

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1 RESTRICTED M A X I M U M LIKELIHOOD TO E S T I M A T E GENETIC P A R A M E T E R S - IN PRACTICE K. M e y e r Institute of Animal Genetics, Edinburgh University, W e s t M a i n s Road, Edinburgh EH9 3JN, Scotland, U.K. S U M M A R Y An efficient computing strategy for estimating variance and covariance components between and within random effects by Restricted Maximum Likelihood is described. It pertains to data with one random factor and equal design matrices for all traits or traits recorded on different sets of animals. INTRODUCTION Until recently, the estimation of variance (and covariance c o m p o n e n t s f r o m animal breeding d a t a has relied a l m o s t exclusively on Henderson's (1953) m e t h o d s for unbalanced m i xed models. M e t h o d III especially has found w i d e s p r e a d use, greatly aided by t h e availability of a 'general purpose' c o m p u t e r p r o g r a m tailored t o w a r d s the estimation of genetic p a r a m e t e r s (Harvey, 1960 and 1977). Interest has g r o w n since in m a x i m u m likelihood (ML) and related procedures w h i c h yield estimators w i t h m u c h m o r e desirable properties (Harville, 1977). The so-called Restricted M a x i m u m Likelihood (REML), d e veloped by P a t t e r s o n and T h o m p s o n (1971), has b e c o m e accepted as t h e prefered m e t h o d to e s t i m a t e (co)variance c o m p o n e n t s in animal breeding. T h o m p s o n (1982) discussed R E M L for estimating genetic p a r a m e t e r s in different cases. In spite of an extensive theoretical treatment, R E M L has b e e n put to comparatively little u s e in practice. Algorithms are complicated a n d c o m putational r e q u i r e m e n t s are extensive, especially for multivariate analyses or models w i t h m o r e than one r a n d o m factor. Although R E M L or analogous procedures (MINQUE, MIVQUE) are n o w available in s o m e general statistical c o m p u t e r packages (e.g. GENSTAT, Robinson et a/., 1982, or SAS), t h ese are often n o t suitable for the analysis of large d a t a sets f r o m livestock i m p r o v e m e n t schemes. This paper describes a c o m putational a p p r o a c h to R E M L estimation of (co)variance c o m p o n e n t s b e t w e e n and within genetic groups, considering a simple m o d e l but possibly large a m o u n t s of data, as appropriate, for instance, to the analysis of dairy cattle data. M O D E L O F ANALYSIS The computing strategy described allows for a linear m o del w i t h : One "major" fixe d e ffe c t. This c a n be simply an overall m e a n or, on the o t her extreme, a s y s t e m a t i c environmental factor w i t h a very large n u m b e r of levels, as, for instance, h e r d - y e a r - s e a s o n effects for dairy cattle data. All levels of this effect are absorbed and e s t i m a t e s are not obtained. Several additional ("minor") fixe d e ffe c ts (optional). These factors are a s s u m e d to h a v e relatively f e w levels and estimates of their effects are determined. For dairy records this could be, for example, sire country of origin or m o n t h of calving within season. Several covariables (optional). A linear or higher order regression on each covariable can be fitted. E s t i m a t e s of regression coefficients are obtained. For t h e dairy e x a m p l e this could be age at calving fitted as a linear and quadratic covariable. 454

2 One random e ffe c t. This is c o m m o n l y s o m e genetic grouping, e.g. sires or families. Optionally, s o m e of the r a n d o m effect levels can be t r e a t e d as fixed, i.e. records pertaining to t h ese levels contribute to t h e estimation of the within but n o t b e t w e e n g r o u p (co)variance c o m ponents. This option is of particular relevance in t h e analysis of dairy cattle d a t a w h e r e young, unselected sires are used c o n t e m p o r a n e o u s l y to proven, selected sires. T o avoid bias due to selection only the variance b e t w e e n y o u n g sires should be utilised to e s t i m a t e the additive genetic variance. H o w e v e r, for such d a t a the m o d e l of analysis often includes h e r d - y e a r - s e a s o n s as fixed effects. Considering y o u n g sire records only then results in v e r y small subclass sizes while including records for proven sires, t r e a t e d as fixed, increases subclass sizes a n d improves the c o n n e c t e d n e s s in the d a t a and thus enhances the accuracy of estimation (Meyer, 1983; V a n Vleck, 1985). For each trait, b o t h r a n d o m effects and residual errors are a s s u m e d to be identically distributed w i t h m e a n zero. All covariances a m o n g errors a n d b e t w e e n errors a n d r a n d o m effects are a s s u m e d to be zero. A specific covariance structure a m o n g the r a n d o m effect levels due to relationships b e t w e e n animals can optionally be t a k e n into a c c o u n t by including the n u m e r a t o r relationship matrix. C O M P U T I N G STRATEGY Largely the s a m e c o mputational steps are required for univariate analyses a n d multivariate analyses w h e r e the w i t h i n g r oup ("error") covariances b e t w e e n traits are zero. The latter e n c o m p a s s e s t w o special cases -. Firstly, different traits are m e a s u r e d on distinct sets of animals. Secondly, the design matrices for all traits are equal. T h e n a transformation to canonical scale is feasible w h i c h leaves all traits uncorrelated, b o t h genetically and phenotypically (Thompson, 197 6). Figure 1 s u m m a r i z e s the sequence of steps of analysis. Absorption I T h e first step is to set u p the m i x e d m o d e l equations (MME) (Henderson, 1973) for the r a n d o m effect a n d a n y additional fixed effects a n d covariables, absorbing the major fixed effect. Ordering d a t a according to levels of this effect and r a n d o m levels w i t h i n subclasses, this can be d o n e very efficiently for one level at a time. A suitable p r o g r a m m i n g strategy is outlined by Schaeffer (1975). This yields the coefficient matrix, a right hand side (RHS) for each trait a n d the s u m s of squares and crossproducts (SS/CP) within subclasses. Absorption II If additional fixed effects or covariables are fitted, these are a b s o r b e d simultaneously into the r a n d o m effect levels, after all major fixed effect subclasses h a v e been processed. This requires the direct inverse of a matrix of order equal to the n u m b e r of additional fixed effects levels and regression coefficients. The inverse t o g e t h e r w i t h o t h e r parts of the M M E required to backsolve for the a b s o r b e d effects are saved. Residual SS/ CP are adjusted for the variation removed. Absorption III If a n y levels of the r a n d o m effect are to be t r e a t e d as fixed these are a b s o r b e d now. Again an inverse of order equal to t h e n u m b e r of levels to be a b s o r b e d is required, information to obtain backsolutions is saved an d the residual SS/CP are adjusted. 455

3 Relationship M a t r i x In m i x e d m o d e l methodology, the relationships b e t w e e n animals are usually t a k e n into a c c o u n t by adding the inverse of the n u m e r a t o r relationship matrix, w e i g h t e d according to a function of the within and b e t w e e n (co)variances, to the coefficient matrix for t h e r a n d o m effects. This is feasible for a large n u m b e r of r a n d o m levels as the inverse can be set up directly f r o m a list of pedigree information (Henderson, 1976). For an iterative estimation procedure, however, use of a n equivalent model, as suggested by Quaas (1984), is m o r e efficient. L e t u d e n o t e the vector of r a n d o m effects for a trait, Z the design matrix relating it to the d a t a vector ^ a n d A the relationship matrix. The variance matri^c of u is2 t h e n A o 0 a n d its contribution to the variance of y is ZAZ'c>b w i t h ob the variance b e t w e e n r a n d o m effects. D e c o m p o s i n g A into LL' jjjives the latter as ZLL'Z'o 2. Hence defining a n e w design m a t r i x Z =ZL yields an equivalent 8 m o d e l w h i c h incorporates the relationships b e t w e e n animals but has V(u)=Io 2. L is a l o w e r triangular matrix w h i c h also has s o m e interpretation in describing the gene flow b e t w e e n generations. In computational t e r m s this requires to pre- and postmultiply the coefficient m a t r i x for r a n d o m effects w i t h L' a n d L, respectively, a n d to premultiply each RHS by L'. L c a n be set up f r o m a list of pedigree information one column at a t i m e as described by Quaas (197 6). The necessary multiplications can be carried out likewise, considering only the n o n - z e r o elements of each c o l u m n of L. Due to the triangular f o r m of L, m o s t of the n e w M M E can be stored in the original arrays (Meyer, 1985a). Tridiagonalisation T h e major c o m p u t a t i o n a l r e q u i r e m e n t in estimating (co)variance c o m p o n e n t s by R E M L is the inversion a matrix of order equal to a multiple to the n u m b e r of r a n d o m effect levels for each r o und of iteration. E v e n for a multiple of one (uni- or multivariate 'canonical' analysis) resources required can be considerable. Fortunately, in this case, a t r a n s f o r m a t i o n can eliminate the n e e d for a direct inverse. Q u a a s and S m ith (1981, cit. Taylor et a/., 1985) s u g g e s t e d to apply a series of Householder transformations to t h e M M E for r a n d o m effects w h i c h leaves the resulting coefficient matrix tridiagonal. R a n d o m effects solutions can t h e n be obtained in linear time and, moreover, the trace of the inverse of t h e coefficient matrix can be a c c u m u l a t e d indirectly, utilising t e r m s w h i c h arise in obtaining t h e solutions. A detailed outline of the procedure can be found in S m i t h a n d Graser (1985) and Taylor et a/. (1985). Transforming the M M E, coefficient matrix and all RHS, requires just s o m e w h a t m o r e time t h a n to invert a matrix of the s a m e size once. For N r a n d o m levels, N-2 Householder matrices are required, the i-th m a t r i x characterised by a vector of length N-i. T h ese vectors are s a ved to allow a backtransformation. Estimation In the univariate analysis o n e RHS is considered at a time. Variances b e t w e e n a n d within r a n d o m effects are e s t i m a t e d using an E M - a l g o r i t h m (Dempster et a/., 1977) on the tridiagonal scale t o g e t h e r w i t h a reparameterisation to speed up convergence, as suggested by T h o m p s o n a n d M e y e r (1985), iterating until the c h a n g e in e s timates b e t w e e n rounds reaches a specified minimum. At convergence, r a n d o m effect solutions are t r a n s f o r m e d to the original scale. If applicable, backsolutions for the fixed levels of the r a n d o m effect and subsequently a n y additional fixed effects a n d regression coefficients are obtained. 456

4 W i t h a M e t h o d of Scoring algorithm, large sample s t a n d a r d errors of the e s t i m a t e s w o u l d arise as a b y - p r o d u c t f r o m t h e inverse of the information matrix. To determine the information m a t r i x requires the trace of the square of the inverse of the coefficient matrix. W i t h the EM-algorithm, a good approximation of this quantity, a n d hence the sampling errors, can be obtained by numerical differentiation (Thompson, pers. comm.), requiring computational effort equivalent to less t h a n half a r o und of iteration. Using a t r a n s f o r m a t i o n to canonical scale, the multivariate analysis of q traits w i t h equal design matrices reduces to q corresponding univariate analyses (Meyer, 1985b). For an E M - a l g o r i t h m e s t i m a t e s can t h e n again be obtained on the tridiagonal scale w i t h o u t inverting the coefficient matrix; see Q u a a s and S m i t h (1981, cit. Taylor e t at., 1985) for a detailed description. A reparameterisation to increase the speed of c o n v e r g e n c e analogous to the univariate case w a s disregarded here to exploit the property of the E M - a l g o r i t h m t h a t it forces estimates to be w i t h i n the p a r a m e t e r space (Harville, 1977). A t convergence, backsolutions on the canonical scale are obtained as a b o v e for one trait at a time and t r a n s f o r m e d to the original scale subsequently. Again numerical differentiation (Thompson, pers. comm.) is utilised to d e t e r m i n e additional traces required in the information matrix. This requires t i m e equivalent to about q /2 multivariate (or q2/2 univariate) rounds of iteration. A p p r o x i m a t e sampling variances of the b e t w e e n and within c o m p o n e n t s o n the canonical scale are d e t e r m i n e d using formulae for the e l e m e n t s of (twice) the information m a t r i x given by M e y e r (1985b). T r a n s forming these then gives a half stored s y m m e t r i c m a t r i x of order q(q+1), describing the (co)variance structure of e s timates on the original scale. A p p r o x i m a t e sampling errors of genetic parameters, derived f r o m the e s timated (co)variance c o mponents, are t h e n calculated using formulae obtained by statistical differentiation. If different traits are m e a s u r e d on differen t sets of animals, d a t a h a v e to be split according to traits a n d t h e first 4 steps (if applicable) be carried o u t for each seperately. The resulting M M E are t h e n combined, yielding a coefficient matrix of order n u m b e r of traits * n u m b e r of r a n d o m levels to be inverted for each r o u n d of iteration. Variances and covariances b e t w e e n and variances within r a n d o m effects are e s t i m a t e d using a M e t h o d of Scoring t y p e algorithm (Schaeffer et a/., 1978). At c o n v e r g e n c e backsolutions for a b s o r b e d effects are obtained as a b ove for one trait at a time. C O M M E N T S The c o m p u t i n g strategy presented has b e e n u s e d to analyse dairy cattle d a t a w i t h up to 22 traits and m o r e t h a n 600 sires. T h o u g h requiring s o m e care in manipulating input a n d o u t p u t files f r o m the various steps, it has proven to be e a s y - t o - u s e a n d efficient. All information specifying the model of analysis and r u n - t i m e options (convergence criteria, trait(s) to be analysed a n d optionally starting values) are specified interactively in free format. T h e univariate analysis a p p e a r e d to be robust against starting values quite different f r o m the eventual estimates. It has b e e n found g o o d practice to use univariate variance estimates, together w i t h g u e s s e d or zero covariances, as starting values for a multivariate analysis, gradually increasing the n u m b e r of traits considered simultaneously. P r o g r a m s are w r i t t e n in F O R T R A N 77 and self-contained; listings can be m a d e available on request. 457

5 ACKNOW LEDGEMENTS This study w a s supported by the Agricultural and Food Research Council. REFERENCES D E MPSTER, A.P., LAIRD,N.M. and RUBIN, D.B M a x i m u m likelihood f r o m incomplete d a t a w i t h t h e E M algorithm. J. Roy. Stat. Soc. B 39,1-22. HARVEY, W.R L e a s t squares analysis of data w i t h unequal subclass numbers. USDA, ARS HARVEY, W.R Users guide for LSML76. M i x e d m o d e l least squares and m a x i m u m likelihood c o m p u t e r program. USDA, ARS. HARVILLE, D.A M a x i m u m likelihood approaches to variance c o m p o n e n t estimation and to related problems. J. Amer. Stat. Ass. 72, HENDERSON, C.R Estimation of variance and covariance components. Biometrics 9, HENDERSON, C.R Sire evaluation and genetic trends. Proc. Anim. Breed. Genet. Symp. in Honour o f Dr.J.L. Lush, Amer. Soc. Anim. Sci., pp HENDERSON, C.R A simple m e t h o d for c o m p u t i n g t h e inverse of a n u m e r a t o r relationship matrix used in prediction of breeding values. Biometrics 32, MEYER, K M a x i m u m likelihood procedures for estimating genetic p a r a m e t e r s for later lactations in dairy cattle. J. Dairy Sci. 66, MEYER, K. 1985a. Restricted M a x i m u m likelihood estimation of variance c o m p o n e n t s accounting for relationships b e t w e e n animals. Unpubl. man. MEYER, K. 1985b. M a x i m u m likelihood estimation of variance c o m p o n e n t s for a multivariate m i x e d m o d e l w i t h equal design matrices. Biometrics 41, PATTERSON, H.D. and T H O M P S O N, R Recovery of inter-block information w h e n block sizes are unequal. Biometrika 58, QUAAS, R.L C o m p u t i n g the diagonal elements and inverse of a large n u m e r a t o r relationship matrix. Biometrics 32, QUAAS, R.L Linear prediction. In BLUP School Handbook, AGBU, Univ. N e w England, Armidale, pp ROBINSON, D.L., T H O M P S O N, R. and DIGBY, P.G.N R E M L - a p r o g r a m for the analysis of n o n - o r t h o g o n a l d a t a by Restricted M a x i m u m Likelihood. In Proc. COMPSTAT Symp. : Short communications, A u g u s t SCHAEFFER, L.R Dairy sire evaluation fo r milk and fa t production. Dept. Anim. Poultry Sci., Uni. Guelph. SCHAEFFER, L.R., WILTON, J.W. and THOMPSON, R Simultaneous estimation of variance a n d covariance c o m p o n e n t s f r o m multitrait m i x e d m o del equations. Biom etrics 34, SMITH, S.P. and GRASER, H.U Estimating variance c o m p o n e n t s in a class of models by Restricted M a x i m u m Likelihood. J. Dairy Sci. in press. TAYLOR, J.F., BEAN, B., M A R S H A L L, C.E. and SULLIVAN, J.J Genetic and environmental c o m p o n e n t s of s e m e n production traits of artificial insemination Holstein bulls. J. Dairy Sci. 68, THOMPSON, R Estimation of quantitative genetic parameters. In Proc. Int. Conf. Quantit. Genet., Ames, Iowa,pp THOMPSON,R M e t h o d s of estimation of genetic parameters. In Proc. 2nd World Congr. Genet. Appl. Livest. Prod., Madrid, V, THOMPSON, R. and MEYER, K Estimation of variance c o m p o n e n t s : W h a t is missing in the E M algorithm? S u b m i t t e d to J. S tatist. Comput. V A N VLECK, L.D Including records of daughters of selected bulls in estimation of sire c o m p o n e n t s of variance. J. Dairy Sci. 68,

6 Figure 1 : Schematic outline of computational steps for the analysis of d a t a w i t h one r a n d o m factor and equal design matrices for all traits or traits m e a s u r e d on different animals. DATA I ABSORPTION I SET UP MME ABSORBING THE MAJOR FIXED EFFECT ABSORPTION II MME ABSORBING ALL FIXED EFFECTS ABSORPTION III MME FOR RANDOM LEVELS ONLY UNIVARIATE ANALYSIS A * f MULTIVARIATE ANALYSIS FOR EQUAL DESIGN MATRICES 459