Random regression models: a longitudinal perspective

Analyses of test-day records of dairy cattle and goats in Guelph began in the early 1990s by Ptak and Schaeffer (Livest. Prod. Sci., 34, 23–34). The model had elements for the fixed curves, but only a single genetic value common to all test day records of a cow. This implied that the shape of the lactation curve was the same for all cows, but that they could differ in height. In 1992, Jack Dekkers (now at Iowa State University) suggested using random regressions for the animal additive genetic effects. Random regression models (RRM) were mentioned in a paper by Henderson Jr (1982, Biometrics, 38, 623–640) and in Henderson’s 1984 book, but there were no details about modelling test-day records in those publications. The first RRM models had only random regressions for animal additive genetic effects. The covariates were an intercept, days in milk (DIM), DIM squared, log of DIM and log squared of DIM as per Ali and Schaeffer (1987, Can. J. Anim. Sci., 67, 637–644). The problem with these covariates was that they had high correlations amongst themselves, which made iteration of solutions to mixed model equations converge slowly. The results were promising and led to our 1994 presentation of RRM at the 5th WCGALP in Guelph (18, 443). Kirkpatrick et al. (Genetics, 124, 979–993) described covariance functions, and Meyer (Genet. Sel. Evol., 30, 221–240) showed that the covariance matrices could be obtained by using Legendre polynomials as the covariates in a RRM. Because of the orthogonality of Legendre polynomials, the covariances among the coefficients were smaller than for other kinds of covariates. From this point onwards, Legendre polynomials were utilized in many RRM studies. Karin Meyer and Ignacy Misztal have modified their existing REML and MCMC programs to handle RRM, and have shared them with many researchers. The RRM were applied to growth traits in swine, beef cattle, sheep, and rainbow trout. Growth traits require maternal genetic and maternal permanent environmental effects to be modelled as random regressions in addition to direct genetic and animal permanent environmental effects. Recent work by Sanchez et al. (J. Anim. Sci., 86, 267–277) have incorporated a non-zero covariance matrix between direct and maternal genetic covariance matrices in a RRM. Early technical problems with RRM were the estimation of covariance matrices, the number of covariates to include for each random factor of the model, and computational approaches. Model validation and comparisons to usual linear animal models were also studied. There have also been studies of binary/threshold RRM, use of non-linear functions, and non-Gaussian assumptions about residual effects in connection with RRM applications. The application of RRM to test day yields in dairy cattle gave better removal of environmental effects, and therefore more accurate genetic evaluations of cows and bulls. Not as many test day records were needed to achieve accuracies similar to that of animal models applied to 305-day yields. This finding allowed milk recording programmes to offer less stringent recording options to producers for having their milk tested. The intervals between tests do not need to be close to 30 days, and fat and protein may only be obtained for every other test. The Canadian Test Day Model (CTDM) became official in February 1999 (J. Dairy Sci., 83, 1135–1144). CTDM is a multiple trait RRM model for four traits (milk, fat and protein yields, and somatic cell score) in the first three lactations. A fourth order Legendre polynomial was used for each trait, giving a total of 60 covariates for each animal’s additive genetic effects, and another 60 covariates for each animal’s permanent environmental effects. The model continues to be improved with additions for number of days pregnant and an extension of DIM from 305 to 365. The use of results from CTDM for management purposes in milk recording has not materialized. In the CTDM, every animal receives 60 genetic regression coefficients. These solutions are converted to 12 EBVs (estimated breeding values): for 305-day yields of milk, fat, protein and somatic cell score in the first three lactations. Finally, the EBVs for each trait are weighted across lactations to give one EBV per trait per animal. Thus, the information about the shapes of lactation curves and the changes in level of production from lactation to lactation are obscured in the final results. An EBV is calculated for persistency, but its usage and importance are unknown. Could more useful information be extracted from the very sophisticated and time consuming RRM computations? If not, then maybe a RRM is not necessary. An annoying feature of RRM using Legendre polynomials has been an artefact, where using the estimated covariance matrices to calculate genetic variances over DIM (in dairy cattle, for example), the genetic variances at the beginning and at the end of lactation were usually much greater than those through the middle of the lactation. This results in very high heritabilities at the beginning and end. These numeric values were greater than estimates of heritability obtained from a multiple trait animal model where the yields were classified into 30 time intervals of 10 days each. Probably too much effort has been put into reducing or eliminating the artefact. The individual regression coefficients are the components that are heritable in a RRM. In recent years, spline functions have been proposed to replace Legendre polynomials. In some ways splines are easier to use, but they have been slightly better than Legendre polynomials for growth traits, but not for test-day models. The best method for removing the artefact would be to have a genetic effect for each day in milk. The RRM have been used for analysis of survival data and for genotype by environment interactions using reaction norms. However, their best use is for repeated observations on animals over time. New applications of RRM are reported every year. Future research on RRM will continue to see where it can be used effectively. The comparisons between Legendre polynomials and spline functions will likely continue. The estimation of covariance matrices will remain a challenge for all applications. Multiple trait models that combine longitudinal traits with traits measured only once, for example, test-day milk yield with age at first insemination, or growth with calving difficulties, will begin to appear more frequently. Methodology for these studies is not new, but the applications are new and applicable to many species. Lastly, with the availability of single nucleotide polymorphism (SNP) genotypes, attempts will be made to estimate effects for the genetic regression coefficients in search of quantitative trait loci that control yields and trajectories of various traits. Because there will be many thousands of SNP genotypes per animal, the computing aspects of these problems will be very important. As predicted by Hill (6th WCGALP, 23, 32–39) in Armidale, RRM have nearly replaced all analyses of growth traits and dairy production traits since 1998, but there are still many applications that are possible.

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