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Bayesian inference about parameters of a longitudinal trajectory when selection operates on a correlated trait¹

M. Piles^*,2, D. Gianola, L. Varona and A. Blasco^*

^* Departamento de Ciencia Animal, Universidad Politécnica de Valencia, Valencia, Spain; and Department of Animal Sciences, University of Wisconsin, Madison 53706; and and IRTA-UdL, Alcalde Rovira Roure, 177, Lleida, Spain

	Abstract

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A hierarchical model for inferring the parameters of the joint distribution of a trait measured longitudinally and another assessed cross-sectionally, when selection has been applied to the cross-sectional trait, is presented. Distributions and methods for a Bayesian implementation via Markov Chain Monte Carlo procedures are discussed for the case where information about the selection criterion is available for all the individuals, but longitudinal records are available only in the later generations. Alternative specifications of the residual covariance structure are suggested. The procedure is illustrated with an analysis of correlated responses in growth curve parameters in a population of rabbits selected for increased growth rate. Results agree with those obtained in a previous study using both selected and control populations. The high correlation between samples indicates slow mixing, resulting in small effective sample sizes and high Monte Carlo standard errors.

Key Words: Bayesian Theory • Correlated Responses • Markov Processes • Monte Carlo Method • Selection

	Introduction

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The study of the trajectory of traits evolving over a period of time has been a focal point in much of animal breeding research. Examples are: milk yield, body weight, prolificacy of a female in her productive life and presence or absence of mastitis during lactation. Statistically, the problem can be posed as one of inferring the trajectory of the trait within an individual, and of assessing sources of variation, genetic and nongenetic, among the profiles of individuals. Different approaches have been applied in animal breeding (e.g., random regression models: Jamrozik and Schaeffer, 1997 or, covariance function models: Kirkpatrick et al., 1994; Meyer, 1999). Maximum likelihood (Zucker et al., 1995) or Bayesian procedures for linear or nonlinear hierarchical models (Wakefield et al., 1994; Varona et al., 1997) have been employed for inference. Advantages of the latter procedures are illustrated in applications to cattle (Varona et al., 1998; Rekaya et al., 2000, 2001) poultry (Mignon-Gastreau et al., 2000) and rabbits (Blasco et al., 2003).

Consider a population that has undergone selection for some trait. Inferences about parameters of the "direct" or of a correlated trait can be obtained ignoring selection only if the entire history of the selection process is contained in the data employed for analysis. Otherwise, a distribution under selection must be constructed (Gianola and Fernando, 1986; Sorensen and Gianola, 2002). When information on both the selection criterion and on longitudinal data are available for all individuals, the Bayesian procedure can be applied as in Varona et al. (1998), and Rekaya et al. (2000, 2001). However, longitudinal information may be available only for some individuals in some generations. We present a Bayesian hierarchical probability model that allows extracting finite sample inferences about parameters of interest when longitudinal data are available only for a subset of animals and selection is for some correlated trait.

	Materials and Methods

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Method

Suppose that only a subset of animals has both longitudinal and cross-sectional data; other individuals have records for the trait under selection (cross-sectional), but not for the longitudinal process. For clarity, we shall assume that there are no repeated records for the trait under selection, although this can be relaxed in a straightforward manner.

The model is based on the hierarchical Bayesian scheme of Wakefield et al. (1994). The joint posterior density of an unknown vector of parameters () and hyper parameters (), given the data (y), is expressed as:

where: p(y|) is the likelihood (first-stage of the hierarchical model), p(|) is the prior density of the parameters given the hyper parameters (second-stage) and, p() is the prior density of the hyper parameters. In this model, uncertainty about unknowns at some level is accounted for when inferring unknowns at other levels. The model accounts for all (co)variances between observations, leading to a correct statement of precision of estimates of parameters of the longitudinal trajectory, breeding values and (co)variance components (Varona et al., 1998).

Data Structure. Let the set of N individuals having records on the trait selected directly (D) be represented as B = {B_L,B}. At some point of the selection process longitudinal records (L) are obtained from a random sample, B_L, of M individuals. The complementary subset (B) includes individuals with records on D only. The data vector for the ith individual measured for L(i B_L) is denoted as , where is the phenotypic value of D for individual i, and is a vector of the n_i longitudinal measurements for individual i. For animals belonging to the set B, the only information is . From here onwards, parameters of the model pertaining to the selection criterion will be denoted by the superscript D, and parameters pertaining to the correlated traits will be denoted by the superscript C.

First Stage of The Model: The Longitudinal Trajectory. An observation on individual i at time j is:

where _i (i = 1,2,...,M) is a vector of order p of individual-specific parameters of the mathematical function, f_i(•), describing the expected value of the longitudinal trajectory for the animal i at time j, (t_ij). The residual (_ij) can be interpreted as the inability of the function f_i(•) to reproduce the observed data exactly. The relationship between observed data (e.g., body weights or milk production) and parameters may be linear or nonlinear.

In general, it is sensible to assume that the first-stage residuals are independent between individuals, but serial dependence within trajectories of individuals may exist. Assuming normality of the residuals, the first-stage distribution is:

where is the vector of records on animal i, f_i(_i,t_i) is the vector of expected values, t_i is the vector of order n_i of known times at which longitudinal data were recorded, and R_i() is the first-stage residual variance-covariance matrix of order n_i x n_i, written as a function of an unknown vector of dispersion parameters . The specific form of this matrix depends on the specification of the model, for example:

1. If the first-stage residuals are independently distributed and homoscedastic, R_i() = I_ni, i, where is the first-stage residual variance, so = .
   
2. If the residuals are independently distributed, but heteroscedastic across individuals, then:
   
   

   
   

3. The residuals may be independently distributed, but heteroscedastic across times. Here, there would be a first-stage variance component for each of the times at which the trajectory is evaluated (if this is done at fixed times in the experimental context):
   
   

   
   

4. Residuals may be neither homoscedastic nor independently distributed. For example, there may be a first-order auto-regressive process, with heterogeneous variance across the times at which measurements are taken, such that Cov[_it, _i(t+k)] = ^k(t) where is a first-stage residual correlation and (t) is the residual variance at time t.
   
5. A structural model may be imposed on the variance. For example, the residual variance may follow a trajectory over time with the parameters of the function describing the evolution of the standard deviation, assumed homogeneous across individuals. An even more ambitious model consists of building up a hierarchical model, where the parameters vary within individuals according to systematic factors, including the possibility of accommodating genetic variation in dispersion. A genetic component would be identifiable from the data if an appropriate relationship structure is available. Such a model may be useful when breeding for uniformity or for plasticity.
   

The density of the first-stage distribution is then:

and the assumption of conditional independence between individuals leads to:

where: y^C is the complete vector of longitudinal data, and is the [(M x p) x 1] vector of parameters of all animals.

Second Stage of The Model: Inter-Individual Variation. The second stage of the model describes the sources of variation of the observable (y^D) and nonobservable "traits" () between individuals. The specification is:

where: ß^D and ß^C are environmental effects affecting and _i; are permanent environmental effects; and are additive genetic effects for the selection criterion and the trajectory parameters of individual i, respectively, and denote incidence vectors and matrices, respectively, and and are second-stage residuals.

At this stage, the assumption made about the conditional distribution of the individual attributes is:

where: is the second-stage residual variance of the selection criterion , and is the second-stage residual (co)variance matrix of order (p+1) x (p+1) between the selection criterion and the p parameters of the longitudinal trajectory, with typical element _Di, where i = 1, 2, . . . p.

Thus, for each animal with longitudinal data, it follows that:

with: being a vector of regression coefficients, _Dj being the second stage covariance between and _ij, for all i. Further:

where _C.D is the p x p conditional covariance matrix between parameters of the growth function, given . It is important to note that can be expressed as:

Data gathering is expensive and, often, most individuals with data for the selection criterion will not have longitudinal information. From a computational point of view, this means that residuals for the missing longitudinal records must be generated. However, time to convergence in a Markov Chain Monte Carlo scheme could be too large depending on the percentage of missing data (Meng and Rubin, 1993). The preceding representation of the residual covariance matrix between the selection criterion and the parameters of the trajectory () allows addressing this problem alternatively via a parameterization in terms b, _C.D and instead of .

Given the parameters, the second-stage residuals are assumed to be mutually independent across individuals but not within individuals. Thus, the density of the conditional distribution of all individual "traits" can be expressed as:

where:

Third Stage of The Model: Uncertainty About First And Second Stage Parameters. A Bayesian probability model requires assigning prior distributions to all unknown quantities in the statistical system. Let , and take as joint prior density

where G is the (p+1) x (p+1) unknown variance-covariance matrix between additive genetic effects of the selection criterion and the trajectory parameters () and is the variance of the permanent environmental effects. Specific forms of the prior distributions above may be:

1. all uniform within boundaries, to ensure propriety of the posterior distributions.
   
2. 
   
3. , where A is a known q x q additive genetic relationship matrix between all the individuals in the genealogy, including those without records.
   

The uniform prior distributions above attempt to convey vague prior knowledge about parameter values within the specified regions. However, one of the advantages of the Bayesian approach resides in the possibility of incorporating external information into the analysis. If such information is available, it could be incorporated, obtaining a different specification of the probability model.

Joint Posterior Density. The joint posterior density of the unknowns can be written using Bayes theorem as (e.g., Blasco, 2001):

Residuals of the longitudinal trajectory are assumed to be independent of the inter-individual (second-stage) residuals, given the vector of parameters of the trajectory _i. They are also assumed to be independent between individuals, thus,

Explicitly

Fully Conditional Posterior Distributions And Sampling Method. Monte Carlo methods were used to draw samples from the joint posterior, with the appropriate coordinate of each sample being a draw from the corresponding marginal. From the samples, several features of the posterior distribution were estimated (e.g., the mean, mode, median, and variance). In particular, Markov Chain Monte Carlo Methods (MCMC), such as Gibbs sampling or Metropolis-Hastings, were employed. Those are described in Casella and George (1992), Chib and Greenberg (1995), and Gelman et al. (1995). We discuss an implementation based on a combination of Gibbs sampling and Metropolis-Hastings algorithms. When the fully conditional distribution of a scalar or vector was identifiable, then the sample is drawn directly using the Gibbs sampler. Otherwise, a Metropolis-Hastings step is implemented.

The conditional posterior distribution of the trajectory parameters can be expressed as:

Given the location effects and the matrix , the trajectory parameters of different individuals are conditionally independent. Hence, we can write:

This implies that, given all parameters and y, all vectors are mutually independent, for i = 1, 2,..., M. Let _-i be the vector of the trajectory parameters without the elements corresponding to animal i. Then,

which implies that the parameters can be sampled individual by individual in the MCMC scheme.

When the trajectory is not linear in the parameters, the kernel of the density of the above distribution does not have a known form, so direct drawing is not feasible. On the other hand, if the trajectory is linear in _i:

for some known incidence matrix T_i. Then, the conditional posterior distribution is normal, with parameters:

where µ_i is .

In this case, sampling is straightforward, as the draws involve M independent p-variate normal distributions, one for each individual.

The conditional posterior distributions of the second-stage location effects, l = {ß^C,ß^D,c^D,u^C,u^D}, are obtained in the same way from the joint posterior density:

where l_-i is the vector l without the ith element.

This is the posterior distribution of "fixed" and random effects in a (p+1)-variate Gaussian mixed effects model where the observed "traits" are the selection criterion and the trajectory parameters. Specifically,

The right hand side (RHS_i) and the corresponding coefficients, g_ij, come from the mixed model equations:

and

= incidence matrix linking c with y*; and = incidence matrix linking u with y*.

The form of the conditional posterior distribution of the first stage dispersion parameters depends on the specification of the covariance matrix. Thus:

1. If the first stage residuals are independently distributed and homocedastic, then:
   
   

   
   where in an inverted chi-square distribution with degrees of freedom.
   

2. If the first stage residuals are independently distributed but heterocedastic across individuals, then:
   
   

   
   where is an inverted chi-square distribution with n_i - 2 degrees of freedom.
   

3. If the first stage residuals are independently distributed but heterocedastic across times, then the conditional posterior distribution of each variance is also scaled inverted chi-square, with the scaled parameters replaced by the normed sum of squared deviations of observations from the expected trajectory, across individuals, for each time. For example, if M(t) animals are weighted at time t, then the degrees of freedom would be M(t)-2.
   
4. For the other cases a "black-box" sampling procedure, such as the Metropolis-Hasting algorithm is needed for drawing the coefficient of correlation or the parameters of a structural model imposed on the variance.
   

The conditional posterior density of the regression vector b is:

calling: .

The conditional posterior distribution of _C.D is:

where W^-1(_C.D, S) is a scaled inverse Wishart distribution with parameters _C.D = M - p - 1 and .

The conditional posterior distribution of is:

where is a scaled inverse Chi-square distribution with parameters: N-2 and .

The conditional posterior distribution of is:

where is a scaled inverse Chi-square distribution with parameters: N - 2 and [c'c].

The conditional posterior distribution of the genetic covariance matrix is:

where W^-1(_g,U) is a scaled inverse Wishart distribution with parameters: _g = q - (p + 1) - 1, where q is the number of individuals in the pedigree, and the matrix U is equal to:

Application

Overview. Breeding schemes for meat production in rabbits often involve a specialized sire line selected for increased growth rate. Selection for growth rate can modify the whole pattern of growth, changing the age at which commercial slaughter weight is reached, the slope of the growth curve, or the adult weight. Animal growth can be described using some nonlinear functions. Such functions have few parameters, each with biological meaning. It is reasonable to expect that the parameters change due to selection for growth rate. The correlated response of growth curve parameters, considered as nonobservable traits, can be assessed by inferring growth curves of control and selected lines. When a control is not available, a multivariate analysis should be performed to account for the effects of selection. In a multivariate analysis, genetic values for the growth curve parameters are estimated using the longitudinal data taken on some individuals, plus all the data employed for selection decisions. Correlated response to selection is then estimated by calculating the averages of these genetic values in each generation.

In rabbits, the Gompertz curve seems to describe growth patterns better than other functions (Gómez and Blasco, 1992; Seeland et al., 1996; Fiorello and German, 1997). This curve is defined by three parameters (a, b, k). Parameter a is the asymptotic body weight and can be interpreted as weight at the adult stage. Parameter b is related to weight at birth and parameter k is the exponential rate of decay of the specific growth rate, also called maturation rate.

Blasco et al., (2003) estimated correlated responses of parameters of the Gompertz growth curve in rabbits due to selection for increased growth rate from differences between a selected and a control population. In this section, we implement the methodology described above to infer this correlated response without a control population, but using all data employed in the selection process plus the longitudinal data recorded in some animals.

Animals. Rabbits came from a synthetic line selected for increased ADG during the fattening period. Average daily gain was recorded on 10,151 animals during 11 generations of selection. A random sample of 137 animals was drawn from two different generations of selection and each individual was weighed weekly until 40 wk of age. Further details are given in Blasco et al. (2003).

Model. At the first stage of the hierarchical model, the observation on individual i at time j can be written as:

with being the vector of animal-specific parameters of the Gompertz growth curve for animal i. We assume that, given the parameters, data are normally distributed, that residuals are independent between and within individuals, and that all animals have the same residual variance at the same time j. Then:

where R_i() is a diagonal residual variance-covariance matrix of order n_i x n_i. Due to a scale effect, it was assumed that the residual standard deviation increased with time until adult weight was reached and then remained constant. We also assumed that the residual standard deviation followed a Gompertz law in time, so the elements in the diagonal of R_i() are given by:

where _j is the standard deviation of the residuals at time j, so = {a,b,k}.

Environmental and genetic effects included in the model for the selection criterion (ADG) were: year-season at weaning (each season consisted of 13 wk), litter size in which the animal was born with 8 levels (<6, 6, 7, 8, 9, 10, 11, > 11), parity order with 3 levels (first, second, third and higher), and common litter and direct additive genetic effects. In the model for the parameters of the Gompertz function (second state of the hierarchy), the effects included were sex and direct additive genetic value of the animal. The joint prior distribution of all unknowns was assumed to be as presented in "Third Stage of the Model".

Sampling From Posterior Distributions Using MC. Inferences were obtained from the samples of the marginal posterior distributions. Gibbs sampling was used for parameters with recognizable conditional distributions. In the case of b_i, k_i and of the parameters included in samples were drawn using a Metropolis-Hastings algorithm within Gibbs sampling, with a uniform proposal distribution centered at the current values of and . This led to acceptance rates of approximately 45%. This is a reasonable rate. If the rate of acceptance is very high the algorithm would move very slowly and the effective sample size would be very small. If it is very low the sampler could not effectively visit the parameter space.

Four chains of 555,000, 790,000, 747,500 and 600,000 samples, each starting from different initial values, were used for the analysis. The procedure of Raftery and Lewis (1992) was applied to estimate the number of initial iterations to be discarded as "burn in." This procedure was carried out using the 22,200, 31,600, 29,900 and 24,000 saved samples from each chain respectively (one sample saved each 25 iterations). Gelman and Rubin’s (1992) diagnostic test was used to assess convergence. This test is based on a comparison of the within and between chain variances of two or more chains, each started from different initial values which are over-dispersed with respect to the true posterior distribution. For each unknown, the factor (shrink factor) by which the scale parameter of the marginal posterior distribution might be reduced if the chain were run to infinity is estimated from this comparison. Convergence may be diagnosed if the shrink factor is approximately 1.0. Statistics of marginal posterior distributions were calculated directly from the samples.

	Results and Discussion

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Visual inspection of sample trace plots indicates whether the samples from each chain followed overlapping paths, and provides an idea about the needed length of the burn in period. Figures 1 to 4 show trace plots of the correlated response to selection of the three growth curve parameters and of the direct response to selection. The four chains are overlaid in each figure. Trace plots of the chains do not overlap completely, but they do not remain near to the starting values of the parameters. Due to the high correlation between samples, it may be that chains were not long enough to allow all regions of the parameter space to be "visited" during the simulation process. If convergence of each chain to the stationary distribution can be assumed, mixing samples retained after the burn-in from independent chains with different starting values of the parameters could provide a more precise approximation to the posterior distributions.

View larger version (17K): [in this window] [in a new window]   Figure 1. Trace plot of correlated response to selection, R(a) (in grams), of parameter a, of the Gompertz growth function for each of the four chains.

View larger version (16K): [in this window] [in a new window]   Figure 4. Trace plot of direct response to selection, R(ADG). ADG: average daily gain (in grams/day) for each of the four chains.

Direct and correlated responses to selection for growth rate were estimated by averaging the posterior means of the breeding values of the animals belonging to the last generation of selection. Marginal posterior distributions of the direct and correlated responses to selection, are summarized in Table 1. In spite of the rather small effective sizes of the chains, results for a and b are similar to those obtained by comparing the selected line with a contemporary control (Blasco et al., 2003). Adult weight (parameter a) increased with selection, whereas parameters related to the slope of the curve (b and k) practically did not change. Selection for ADG was effective and response to selection obtained here is similar to values reported by Estany et al. (1992), in the same breed, and Rochambeau et al. (1989). No estimates of response to selection for growth curve parameters were found in the literature.

	Implications

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View larger version (20K): [in this window] [in a new window]   Figure 2. Trace plot of correlated response to selection, R(b), of parameter b, of the Gompertz function for each of the four chains.

View larger version (14K): [in this window] [in a new window]   Figure 3. Trace plot of correlated response to selection, R(k), of parameter k, of the Gompertz function for each of the four chains.

	Footnotes

² Correspondence: IRTA-Unitat de Cunicultura, Torre Marimón s/n, 08140 Caldes de Montbuí, Barcelona, Spain (phone: +34 93 865 1011, fax: +34 93 865 3777, E-mail: miriam.piles{at}irta.es).

Received for publication September 26, 2002. Accepted for publication June 12, 2003.

	Literature Cited

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