Animal Models 1 Introduction An animal model is one in which there are one or more observations per animal, and all factors affecting those observations are described including an animal additive genetic effect. The animal additive genetic effects are random variables with an expected value of zero, and a covariance matrix that is equal to A, the additive genetic relationship matrix. Assumptions are that the trait of interest is influenced by an infinite number of loci each with a small, relatively equal effect, and that the population is randomly mating. Animal models were first used in 1989, but the theory about these models was known since 1969. Animal models were not used before 1989 because computer power was not sufficient to handle so many equations. As computers became faster and had more memory, then the statistical models became more realistic, but also more complex. Example Situation Sheep are scanned at maturity by ultrasound(us) to determine the amount of fat surrounding the muscle. A model (equation) might be where Year-month of birth is fixed, USFat = YearMonth + FMG + b(age at US) +Animal + Residual FMG is a flock-year-management group effect (random), Age at ultrasound is a covariate, Animal additive genetic effects, and Residual effects. Fat thickness is in millimeters. Relationships among animals will be used. The purpose of the analysis is to estimate the variances, and afterwards to estimate the breeding values of the animals. Animals are assumed to have only one US Fat measurement each, and that they have not been pre-selected on the basis of any other trait. The sex of the animal is assumed 1
to not have any effect on the measurements. Within a FMG, all sheep are assumed to be treated and fed in the same manner. 3 Estimation of Variances There are two methods of estimating variances that are used in animal breeding today. One is called Restricted Maximum Likelihood (or REML). REML has several different ways of being calculated. One is called Derivative Free REML (DFREML), and another is called Average Information REML (AIREML, ASREML). Other computational methods are too cumbersome or slow. Software is available for DFREML and ASREML from various sources (Denmark, Australia). To employ REML one needs to assume that the observations follow a normal distribution. Then the likelihood function can be written for the particular model. Both DFREML and ASREML try to maximize the log of the likelihood function, but in different ways. If both methods operate correctly, then both methods should give the same final answers. This does not always happen. The details of the methodology are too complex for this course. 3.1 Bayesian Methods Bayesian statisticians differ from traditional statisticians (known as Frequentists) because Bayesians assume that everything in a model is random. That means everything in the model comes from a population with a certain mean and variance. However, the Bayesians do not necessarily assume a normal distribution for everything. Even the variances that are to be estimated are assumed to be a random variable, and variances tend to have Chisquared distributions. Fixed effects are assumed to have uniform distributions. Animal genetic effects and residual effects are usually assumed to have normal distributions. The distribution of every factor in the model equation, including the variances should be described. These distributions are combined into an overall likelihood as the product of the likelihoods of all the factors in the model equation. This is called the Joint Probability Function. Maximizing the likelihood of the Joint Probability Function provides the estimators within the Bayesian Methods. Unfortunately, the Joint Probability function is too complex to take derivates to find the maximum. To get around this problem Bayesians use a Gibbs sampling process. The marginal probability function of each factor assuming the parameters of all the other variables are known is derived. Then a value is computed for an unknown parameter (based on its marginal probability function), and then a random amount is added or subtracted from that value depending on its expected variance. Each unknown parameter is treated this way, one at a time. One pass through all of the unknown parameters is one Gibbs sample. Bayesians perform hundreds of thousands of samples. After thousands (50,000?) samples, then
the sample values of the unknown parameters begin to approximate samples from the joint probability function. The early samples are known as the burn-in period and are discarded. The averages of the sample values after the burn-in period give an estimate of that parameter. The standard deviation of the sample values give the standard error of the estimates. The Bayesian method is less limiting than REML because distributions other than normal can be utilized. The sampling process can take a long time, but software is easy to write for the Bayesian method. A good random number generator is needed for Gibbs sampling. 4 Comments 4.1 Examples To illustrate either method of estimating variances is nearly impossible using small examples. Small examples tend not to give good results. If large examples are used, then too many pages of details need to be given. Thus, a good example is difficult to present. 4. Amount of Data The estimation of variances requires data on at least a few thousand animals (000 or more). The more animals that are included then the sharper will be the peak at the maximum of the likelihood function or joint probability function. With too few observations the peaks are less pronounced and find the maximum becomes more difficult. Success also depends on the model and the number of unknown parameters in the model. 4.3 Changes in Variances Variance parameters tend to not change very much over time. This means that variances do not need to be re-estimated very often. Usually parameters need to be re-estimated every time the model is changed (adding or deleting factors to the model). Using estimates of variances that match the model is preferred. Of course, this will depend on the changes that were made. 3
4.4 Breed or Country Differences Variance parameters may be specific to a breed. For example, the Holstein breed in dairy cattle generally has larger variances for milk production because Holsteins produce more milk than the other breeds. Charolais beef cattle grow more rapidly than Hereford or Angus. Variances may also be specific to a breed within a particular country. Holsteins in Canada have larger variances than Holsteins in South Africa or New Zealand. 4.5 Genetic Evaluation and Rankings of Animals If heritability is estimated to be 0.30, then genetic evaluations that are calculated using either 0.0 or 0.40 would not greatly re-rank animals. By using 0.0 instead of 0.30, the estimated breeding values will have a smaller range in values, and using 0.40 the estimated breeding values will have a bigger range than those calculated using 0.30. Using the correct variance is important for measuring genetic trends, but not for ranking animals for selection. 5 Repeated Records on Animals There are many situations where animals are observed more than once for a trait, such as Fleece weight of sheep in different years; Calving records of a beef cow over time; Test day records within a lactation for a dairy cow; Litter size of sows over time; Antler size of deer in different seasons; and Racing results of horses from several races. Usually the trait is considered to be perfectly genetically correlated over the ages of the animal. Besides an animal s additive genetic value for a trait, there is a common permanent environmental (PE) effect which is a non-genetic effect assumed to be common to all observations on the same animal (Lush, 1945). Permanent implies stability and a constant presence. The proposition in this paper is that new permanent environmental effects can appear over time as the animal ages, or gains experience in the trait events that are being recorded, and are, therefore, cumulative, as illustrated in Figure 1. 4
Figure 1. Diagram of genetic(g), permanent(pe) and temporary environmental(te) factors affecting repeated records of an animal. Note that PE1 affects records 1,, and 3 of the animal while PE affects only records and 3, and PE3 affects only record 3. Thus, the environmental effects are permanent, but only affect records after they occur. Also, PE1 could be opposite in effect to PE or PE3. The variance of PE1 effects could differ from the variances of PE and PE3. The purpose of this sectiom is to present a model to accommodate cumulative PE effects in a genetic evaluation model. 6 Usual Repeatability Model A repeated records animal model is usually written as where y = Xb + ( 0 Z ) ( a 0 a r b = vector of fixed effects, ) + Zp + e, 5
( ) a0 a r = ( animals without records animals with records p = vector of PE effects of length equal to a r, and e = vector of residual effects. The matrices X and Z are design matrices that associate observations to particular levels of fixed effects and to additive genetic and PE effects, respectively. In a repeated records model, Z is not equal to an identity matrix. Also, ), a A, σa N(0, Aσa) p I, σp N(0, Iσp) e N(0, Iσe) ( Aσ G = a 0 0 Iσp ). Repeatability is a measure of the average similarity of multiple records on animals across the population (part genetic and part environmental), and is defined as a ratio of variances as r = σ a + σp, σa + σp + σe which is always going to be greater than or equal to heritability, because h = σ a. σa + σp + σe Henderson s (1984) mixed model equations would be written as X X 0 X Z X Z 0 A 00 k a A 0r k a 0 Z X A r0 k a Z Z + A rr k a Z Z Z X 0 Z Z Z Z + Ik p ˆb â 0 â r ˆp = X y 0 Z y Z y, where A 1 = ( A 00 A 0r A r0 A rr is partitioned according to animals with and without records. Notice that if the fourth equation of the MME is subtracted from the third equation, then the result is A r0 k a â 0 + A rr k a â r Ik pˆp = 0. Thus, the solutions for PE effects are a function of the solutions for animal additive genetic effects and the elements in A 1. Consequently, the tendency (although not the rule) is for animals to rank similarly for their genetic and PE effects. 6 )
A small example is given in Table 1. Assume that k a =, and k p = 3. The model contains the fixed factor of year, and random factors for animal additive genetic and animal permanent environmental effects. Table 1. Repeated records example. Year 1 Year Year 3 Animal Sire Dam y 1jk y jk y 3jk 1 0 0 0 0 69 53 3 0 0 4 0 0 37 47 5 0 0 6 0 0 55 7 1 39 51 6 8 3 4 48 7 9 5 6 71 77 96 10 1 4 38 56 47 11 3 6 51 71 86 1 5 64 46 is Animals are not inbred, so that the inverse of the additive genetic relationship matrix A 1 = 1 4 1 0 1 0 0 0 0 0 0 1 4 0 0 1 0 0 0 0 0 0 0 4 1 0 1 0 0 0 0 1 0 1 4 0 0 0 0 0 0 0 1 0 0 4 1 0 0 0 0 0 0 1 0 1 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4. The solutions to the MME are given in Table. Each animal has one additive genetic solution and those with records have one PE solution. The estimate of the residual variance from this analysis was 107.69. Table. Solutions to usual repeated records animal model analysis. 7
Year Effects Yr 1 = 5.44 Yr = 59.64 Yr 3 = 7.38 Animal Additive Genetic PE solution 1-4.78 0.01 1.98 3.48 4-5.99-3. 5 3.6 6 4.66-0.53 7-4.69-3.07 8-0.44 1.76 9 8.4 5.71 10-7.87-3.31 11 4.74 1.15 1 1.16-0.88 7 Cumulative PE Repeated Records Model The cumulative PE repeated records model can be written as where y = Xb + ( 0 Z a ) ( a 0 a r ) + Z p p + e, ( a0 a r b = vector of fixed effects, ) ( ) animals without records =, animals with records p = vector of PE effects of length equal to the number of records, and e = vector of residual effects. The matrices X and Z a are design matrices that associate observations to particular levels of fixed effects and to additive genetic effects, respectively. In a cumulative PE repeated records model, Z p is not a typical design matrix. Rows of Z p may have more than a single 1. If an animal has two records, then in the row for the second record there will be two 1 s. If an animal has four records, then in the row for the first record there will be one 1, for the second record two 1 s, for the third record three 1 s, and for the fourth record four 1 s. Using the example data in Table 1, the Z p matrix for animals, 4, 6, and 7 would 8
appear as Z p p = 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 Thus, p 1 contributes to records 1 and of animal, while p contributes to record and any subsequent records that animal may have. Note that there are as many PE effects to be estimated as there are records. Also, Z pz p = 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 3 1 0 0 0 0 0 1 0 0 0 0 0 Because of the special form of Z p, the PE effects can be estimated separately from temporary environmental effects, and from the additive genetic effects for animals, which rely on the additive genetic relationship matrix. Also, p 1 p p 41 p 4 p 61 p 71 p 7 p 73.. a A, σa N(0, Aσa) p I, σp N(0, Pσp) e N(0, Iσe) ( Aσ G = a 0 0 Pσp ). The matrix P is assumed to be a diagonal matrix with 1 s on the diagonal for the first records made by every animal. For second records, one might suspect that the variance of permanent environmental effects might be less than that for first records, and so the diagonals for second records may be reduced. Similarly, the variance of PE effects for third and later records may also be reduced further. The point is that an allowance must be made for the variances of PE effects to be different depending on the record number for that animal. The residual variance is assumed to be the same for all records, but this too could vary. The genetic variance is assumed to be constant and the genetic correlation between records is still assumed to be unity. 9
V ar The variance of three records, for example, on one non-inbred animal would be y 1 y y 3 = σ a + σp1 σp1 σp1 σp1 (σp1 + σp) (σp1 + σp) σp1 (σp1 + σp) (σp1 + σp + σp3) + 1 0 0 0 1 0 0 0 1 σ e. This implies that the variance of repeated records is getting larger over time. traditional repeated records model, the same variances would be V ar y 1 y y 3 = σ a + σ p + 1 0 0 0 1 0 0 0 1 σ e. In the The example data from Table 1 was analyzed with the cumulative PE repeated records model. Let the diagonals of P be 1 for first records,.9 for second records, and.8 for third records, and let k a = and k p = 3. The mixed model equations are of order 36 (i.e. 3 years, 1 animal additive genetic, and 1 PE effects). The solutions to the mixed model equations are given in Table 3. Table 3. Solutions to the cumulative PE repeated records animal model analysis. Year Effects Yr 1 = 5.44 Yr = 59.58 Yr 3 = 71.69 Animal Additive Genetic PE1 PE PE3 1-4.8 0.69.15 -.17 3 1.83 4-5.7-3.14-0.86 5 3.47 6 4.0-0.36 7-4.0 -.96-0.73-0.4 8-0.90 1.40.75 9 7.53 5.05.75 1.89 10-7.06 -.74-1.08 -.91 11 3.70 1.03.78 1.43 1 1.76-0.4-3.44 The residual variance from this analysis was 93.80 which was smaller than that for the usual repeated records model. Thus, more residual variation was accounted for by the cumulative PE effects, but that could be anticipated because the cumulative PE model has more parameters to estimate. There were only 1 animals, but the correlation between additive genetic estimates from the two analyses was.9957 with only slight differences in magnitude. The PE1 estimates were also highly correlated with the PE effects from the first model. 10
Most probable producing abilities (MPPA) are predictions of how animals might perform if they made another record, and are based on the genetic and PE estimates (Lush, 1933). In the usual repeated records model this is not a problem because the PE effects are assumed to be constant for all records. However, in the cumulative PE repeated records model, to predict a fourth record for animal 7, for example, the prediction would be MPPA = â 7 + ˆp 71 + ˆp 7 + ˆp 73 + ˆp future, where the prediction of the future PE effect would be 0, the mean of the distribution from which a future PE effect would be sampled. The variance of prediction error of MPPA would be increased by the variance of the future PE effects. Similarly, repeatability would vary according to record number. Assuming the new PE effects are not correlated with previous PE effects, then repeatability would increase with record number because the PE variances would be additive. If PE effects for second records, for example, depends on the PE effect for the first records, then those covariances (positive or negative) may decrease repeatability with record number. Without an analysis of real data and estimates of the PE variances, these comments are only speculative, and the results could vary depending on species and traits. A Bayesian approach using Gibbs sampling could be used to estimate the PE variances, one for each record number, just as one estimates different residual variances for different years or herds. 8 Discussion There are other alternative models. Let each observation on an animal be considered a different trait that is not perfectly genetically correlated, then a multiple trait animal model could be used, such that permanent environmental effects are confounded with temporary environmental effects, and therefore, estimated jointly. If the observations are truly perfectly genetically correlated, then the multiple trait approach may have difficulty in estimating a genetic correlation of 1 (Van Vleck and Gregory, 199). Another possibility is to consider a random regression model where the observations are functions of age of the animal, and follow a trajectory, such as a lactation curve. Test day models include random regression coefficients for permanent environmental effects such that their variability differs over the course of a lactation. However, if animals do not make very many records, then there could be estimation problems. The random regression model suggests that permanent environmental effects close together in time are more similar in magnitude and effect than PE effects farther apart in time. Thus, an autoregressive correlation model might be assumed. The model for 11
autocorrelated PE effects can be written as where y = Xb + ( 0 Z a ) ( a 0 a r ) + Z p p + e, ( a0 a r b = vector of fixed effects, ) ( ) animals without records =, animals with records p = vector of PE effects of length equal to the number of records, and e = vector of residual effects. In this model, Z p is an identity matrix within animals, so that the PE effects for each record are separated, but 1 ρ ρ ρ 1 ρ P = ρ ρ 1 σ p....... The example data were re-analyzed with the autoregressive model where ρ = 0.6 was used and k a = and k p = 3. The results are given in Table 4. Table 4. Solutions to an autoregressive PE repeated records animal model analysis. Year Effects Yr 1 = 5.44 Yr = 59.71 Yr 3 = 7.6 Animal Additive Genetic PE1 PE PE3 1-5.8-0.04 3.10 0.36 3.64 4-6.36 -.84 -.5 5 4.00 6 5.04-0.6 7-5.19 -.54 -.0-1.98 8-0.44 0.53.50 9 9.8 3.76 4.30 4.67 10-8.57-1.71-1.37-3.61 11 5.06-0.36 1.59.31 1 1.3 0.68 -.9 Ranking of the EBV of animals was the same, but with greater differences among animals. PE effects were estimated for each record comparable to the results in Table 3, but PE effects were more similar across records within animals than in Table 3. For example, animal had PE estimates in the cumulative PE model of.15 and -.17, but 1
in the autoregressive model the PE estimates were 3.10 and 0.36. The autoregressive correlation is forcing the estimates to be more similar. The estimate of the residual variance from the autoregressive model was 104.7, which is less than 107.69 for the usual repeated records model, but more than that of the cumulative PE effects model (93.80). Thus, the first step would be to use the cumulative PE model, estimate the PE effects and determine correlations among PE estimates within animals. If these correlations are high, then perhaps an autoregressive model may be more appropriate. If the correlations are close to zero, then the cumulative PE model would be preferred. If the permanent environmental effects are cumulative with time, then there is no reason to assume that PE effects for records 1 and are more similar than PE effects for records 1 and 4. The model presented in this paper allows the magnitude and sign of the PE effects to vary with time, such that the correlation between PE effects for each record are zero. Philosophically, every event encountered by an animal has some bearing or influence on the reaction of the animal to future events. Hence PE effects may be correlated to some degree as well as cumulative. 9 Conclusions The traditional repeated records animal model is not very realistic with the concept of a constant PE effect contributing to every record an animal makes. With time, every living creature experiences new environmental conditions that change their attitude, their habits, or their physical being, as they age. Thus, PE effects are accumulating over time, and once they occur they affect every record thereafter. The cumulative PE repeated records model should better fit repeated records on animals. The traditional repeated records animal model should be extended to the cumulative PE repeated records animal model wherever the traditional model is currently used. Data analyses should determine if the variances of PE effects increase, stay the same, or decrease with record number. Also, the effect on prediction error variances of animal additive genetic effects should be quantified, and whether the estimated breeding values are more accurate with the new model. Perhaps some computing simplifications may be possible given the special structure of Z p. 10 References Henderson, C. R. 1984. Applications of Linear Models in Animal Breeding. University of Guelph, Guelph, Ontario, Canada. Lush, J. L. 1945. Animal Breeding Plans. Iowa State University Press. Eleventh printing. 13
Van Vleck, L. D., K. E. Gregory. 199. Differences in heritability estimates from multiple-trait and repeated records models. J. Anim. Sci. 70:994-998. 14