Abstract
The mutational input of genetic variance per generation (
) is the lower limit of the genetic variability in inbred strains of mice, although greater values could be expected due to the accumulation of new mutations in successive generations. A mixed-model analysis using Bayesian methods was applied to estimate 
and the across-generation accumulated genetic variability on litter size in 46 generations of a C57BL/6J inbred strain. This allowed for a separate inference on 
and on the additive genetic variance in the base population (
). The additive genetic variance in the base generation was 0.151 and quickly decreased to almost null estimates in generation 10. On the other hand, 
was moderate (0.035) and the within-generation mutational variance increased up to generation 14, then oscillating between 0.102 and 0.234 in remaining generations. This pattern suggested the existence of a continuous uploading of genetic variability for litter size (
). Relevant genetic drift was not detected in this population. In conclusion, our approach allowed for separate estimation of 
and 
within the mixed-model framework, and the heritability obtained highlighted the significant and continuous influence of new genetic variability affecting the genetic stability of inbred strains.
THE importance of new mutations on polygenic variability has been suggested by several investigators in the last decades (Hill 1982a,b; Caballero et al. 1991; Keightley 1998). Direct evidence of new mutations with large effects in experimental selection lines was initially reported during the second half of the 20th century (Macarthur 1949; Yoo 1980; Bradford and Famula 1984). The mutational input of genetic variance per generation (
) can be viewed as the ultimate source of polygenic variation and thus as the raw material for the maintenance of genetic variability in populations (Hill 1982a). Estimates of mutational heritabilities [
, 
being the residual variance of the trait] in animals and cereal crops have ranged between 
and 
(Lynch 1988; Houle et al. 1996). Within this context, it is well known that spontaneous mutation continually contributes new alleles to the pool of genetic variation, allowing for response to long-term artificial selection experiments in both animals (Caballero et al. 1991; Keightley 1998) and plants (Hill 2007).
Most estimates of 
come from experiments focused on the rate of divergence between sublines from a highly inbred base population (Festing 1973; Mackay et al. 1994). These studies typically assumed a stringent scenario under mutation–drift equilibrium, which does not necessarily hold in experimental populations (Hill 1982a). Moreover, a long time is needed for generating the strains, and the analysis using the response to selection typically ignores information on covariances between relatives within lines, a proportion of which can be genetic (Keightley and Hill 1992). Alternatively, Wray (1990) developed a straightforward approach to account for mutation effects in mixed models, using the numerator relationship matrix, allowing for estimation of 
in unselected populations. This methodology has not been widely applied, although some 
estimates have been obtained in mice (Keightley and Hill 1992; Keightley 1998). A topic of interest in studies with laboratory mice is the genetic homogeneity of inbred strains across generations (Taft et al. 2006; Stevens et al. 2007) and Wray's (1990) approach seems to be the only available methodology for testing mutation effects, given that selection is avoided in these genetically controlled strains. Nevertheless, there are no published results on 
in highly inbred unselected mice, and the magnitude or impact of 
on the phenotypic variance remains unknown.
Taking the infinitesimal model (Fisher 1918) as the starting point, the additive genetic variance for a given phenotypic trait in a population characterizes the amount of genetic variability and the potential change due to natural or artificial selection, or genetic drift. Interestingly, analyses performed on the “new” genetic variance originated by mutation are commonly focused on the increment of genetic variation per generation (Caballero et al. 1991; Keightley 1998), but they do not estimate the accumulated genetic variability existing between individuals. Even in highly inbred strains, the genetic variance in a given generation of interest could be viewed as the balance of an equilibrium between 
coming from current and previous generations and the loss of genetic variability due to selection, genetic drift, and/or inbreeding (Hill 1982a,b). Analyses of the mutation phenomenon in laboratory species have been mainly focused on 
(Festing 1973; Keightley and Hill 1992; Keightley 1998), whereas the magnitude of the overall within-generation genetic variability due to the accumulation of new mutations remains unclear.
In this study, we report estimates of mutation variance on litter size in C57BL/6J mice reared for 46 generations without selection. Wray's (1990) algorithm was modified to estimate the amount of genetic variance in the inbred base population and the increment of the per generation variance due to mutation. In addition, we applied the Bayesian approach proposed by Sorensen et al. (2001) to estimate the within-generation genetic variability, to examine if new mutation variance compensates for losses in genetic variability.
MATERIALS AND METHODS
Mice data source:
Mouse strain and breeding scheme:
A C57BL/6J inbred strain was kept in our vivarium at the University of California (Davis, CA) for 46 nonoverlapping generations (G1–G46), between October 1988 and May 2005. This strain was founded with two C57BL/6J males and six C57BL/6J females from The Jackson Laboratory (Bar Harbor, ME). Two to 5 generations per year were produced. Each generation involved between 2 and 28 males and between 6 and 49 females, producing an average of 21.6 litters (Table 1). In August 1995, a subline was derived from G21 and was maintained for five nonoverlapping generations (G22b–G26b), with a large number of litters per generation (Table 1). For each generation, males and females were selected at random from the offspring of a few litters of the previous generation, and full-sibs matings were favored. Only single (one male/one female) and group matings (one male/several females) were used to avoid multiple paternities. Each male and female produced an average of 3.5 and 1.8 litters, respectively, ranging from 1 to 16 litters for males and from 1 to 5 litters for females. Note that this strain was maintained to provide stock research mice in our colony and therefore a variable number of litters per generation were generated, depending on mice demand. All mice were fed with Purina 5008 diet (Ralston Purina, St. Louis; 23.5% protein, 6.5% fat, 3.3 kcal/g) and water was offered ad libitum. Mice were housed in polycarbonate cages under controlled conditions of temperature (21° ± 2°), humidity (40–70%), and lighting (14 hr light, 10 hr dark, lights on at 7 am) and managed according to the guidelines of the American Association for Accreditation of Laboratory Animal Care (AAALAC) (http://www.aaalac.org).
Number of males and females mated per generation (in parentheses, contributors to the next generation), number of litters, and average litter size per generation
Data set and trait analyzed:
Reproductive data were recorded accurately in all generations. Sire, dam, date of mating, date of birth, and number of pups at birth (alive and dead) were recorded for each litter, and pups were individually marked by ear notching at weaning (3 weeks after birth). Records were available on 1986 litters providing 15,044 pups. This study focused on litter size (LS), defined as the sum of live and dead pups at birth. Phenotypic records of LS ranged between 1 and 14 pups, with 7.58 pups per litter on average. The pedigree file included 572 males and 1116 females with a complete knowledge of all parental relationships.
Bayesian analysis:
Model:
Litter size in the C57BL/J6 strain was analyzed with the following linear mixed model,
where 
was the vector of phenotypic data and 
was the vector of residuals after accounting for systematic (
), environmental (
and 
), and additive genetic effects (
and 
). Note that 
, 
, 
, and 
are appropriate incidence matrices. More specifically, 
corrected for two systematic effects, parity number of the dam with the two levels proposed by Kirkpatrick et al. (1988; first parity and following parities), and generation number with 50 levels, accounting for environmental variability between generations (see Falconer 1960). Two environmental sources of variation common to all pups were fitted to the model, the effect of the sire (
; Schilling et al. 1968) and the nongenetic effect of the dam (
), with 572 and 1116 levels, respectively. Following in part Wray (1990), the infinitesimal genetic effect (
; Fisher 1918) was partitioned into two terms, 
, the breeding value inherited from the genetic variability in the base generation (
; Henderson 1973) and from the additional genetic variability originated by mutation (m).
Prior distributions:
Following a standard Bayesian development, the joint posterior distribution of the mixed model outlined above was constructed by multiplying the Bayesian likelihood with the prior distribution of all parameters in the model,
where 
, 
, 
, 
, and 
were the appropriate variance components for 
, 
, 
, 
, and 
, respectively, 
was the standard numerator relationship matrix (Wright 1922), and 
was Wray's (1990) numerator relationship matrix adapted to accommodate the occurrence of mutations in the genome. Note that 
was defined as 
, where t is the number of generations, 
is the numerator relationship matrix of additive genetic effects attributed to mutations arising in time unit k, and 
(see the appendix). The elements of 
are the additive genetic relationships if ancestors born in time unit 
are ignored (Wray 1990).
Litter size data were assumed to be normally distributed as follows,
with 
being an identity matrix with dimensions equal to the number of records in 
. Model parameters 
, 
, 
, 
, 
, and 
were assumed mutually independent. A priori distributions for 
and 
were defined as multivariate normal,
where 
and 
were identity matrices with dimensions equal to the number of elements in 
and 
, respectively. Invoking the infinitesimal model (Fisher 1918), 
and 
were assumed to follow the multivariate normal distributions
Note that mutational effects are assumed a priori independent of 
(Wray 1990) and therefore genetic correlation between 
and 
was arbitrarily fixed to 0. Improper uniform prior distributions were assumed for 
, 
, 
, and 
to approximate vague prior knowledge about systematic, environmental, and residual sources of variation.
According to Henderson (1973), Gianola et al. (1989), and Im et al. (1989), 
measures additive genetic variance at linkage equilibrium in the base population (G1; see Table 1). Although 
should be null or very small in an inbred strain, if it exists, 
must originate from short-term mutations arising in previous generations and is highly related to 
. It seems reasonable to expect a similar behavior for 
and 
and therefore the same prior was assumed for both variance components. To evaluate the effects of a priori information on 
and 
, four different scaled inverted 
-prior distributions with hyperparameters 
and 
were assumed and tested independently on our data set. The first prior (PR1) generalized the scaled 
-distribution to an improper uniform distribution by setting 
and 
(Figure 1). This prior ignores previous knowledge on 
and 
, this being a typical assumption for variance components, where the variance is allowed to take any value between 0 and the phenotypic variance. Three more priors (Table 2) were defined on the basis of information from the literature and they varied on a trial and error basis until the desired shape of the distribution was obtained (Figure 1), following in part Blasco et al. (1998). Given the range of mutational heritabilities reviewed by Lynch (1988; 
–
) and the moderate phenotypic variance observed in our data set (4.1 pups2), it seems reasonable to expect a 
between 
and 
, without disallowing for more extreme values. The second 
-prior (PR2) illustrated a strong a priori opinion (sharp-contour distribution) about the probable distribution of the variance components, its mode being placed at the lower bound of Lynch's (1988) range. Prior 3 (PR3) was an attempt to cover the range of most plausible values, although a left-skewed prior (PR4) gave a vague a priori opinion of the distribution (Figure 1) of the variance components, its mode being 
. Note that PR2 and PR4 were proper priors although they did not have a well-defined variance. See Table 2 for a detailed description of hyperparameters for all the scaled inverted 
-priors.
A priori distributions for 
and 
.
χ−2 hyperparameter specifications and deviance information criterion (DIC) estimates
Additionally, a mixed model without 
and 
effects was analyzed (PR0), with its Bayesian likelihood reduced to 
and the prior distributions for 
, 
, 
, 
, 
, and 
were the same as in the full model. It can be also viewed as the general model with 
and 
, thereby allowing for testing of the biological relevance of 
and 
effects in terms of model adequacy.
Markov chain Monte Carlo sampling:
Within a Bayesian context, inferences are made on the joint posterior distribution or, for a given parameter of interest, on the relevant marginal posterior distribution. Given the multidimensional form of these posterior distributions, direct integration cannot be applied. Markov chain Monte Carlo (MCMC) techniques easily bypass this limitation and allow us to obtain draws from the appropriate marginal posterior distribution. For the mixed model described above, samples from the marginal posterior distribution of all unknowns in the model were obtained by Gibbs sampling (Gilks et al. 1996), following the procedures described by Sorensen et al. (1994).
For each prior distribution of 
and 
(PR1–PR4), as well as for PR0, three independent MCMC chains were launched, with 500,000 iterations after discarding the first 100,000 as burn-in. Convergence was confirmed on variance components by visual inspection and by Raftery and Lewis's (1992) approach. To arrive at the most preferable model, the deviance information criterion (DIC) (Spiegelhalter et al. 2002) was calculated.
Genetic drift and within-generation additive genetic variances:
As mentioned above, breeding mice were randomly picked from the previous generation and no selection was applied along the 46 generations. Nevertheless, the small number of mice contributing to the next generation (see Table 1) could produce genetic drift on litter size if 
and/or 
were not null. Within this context, changes on the within-generation average breeding value (a and m) and environmental effects (
and 
) were evaluated with the Bayesian approach described by Sorensen et al. (1994).
Following Sorensen et al. (2001), both additive genetic and mutation variance components were estimated within generations, using data from all individuals in the population. By definition, the additive genetic value 
of an individual randomly picked from generation t is a random variable with variance as defined by Sorensen et al. (2001),
where 
is the mathematical expectation of additive genetic values in generation t, 
is the ith additive genetic value in generation t, and 
is the number of individuals in generation t. Inferences on the within-generation additive genetic variance were made on their marginal posterior distribution, estimated via Markov chain Monte Carlo methods. A Gibbs sampler was applied following Sorensen et al. (2001). The same approach was applied for the within-generation mutation variance.
RESULTS
Litter size in this C57BL/6J strain averaged 7.58 ± 0.05 pups per litter with substantial variability between generations, from 9.42 ± 0.29 pups (G1) to 5.78 ± 0.40 pups (G31). Larger litter sizes were observed in earlier (G1, G3, and G5) and later generations (G42–G45) with smaller values in the intermediate ones, although a relevant phenotypic trend was not observed (Table 1).
Table 2 shows the hyperparameters of the scaled 
-priors (
and 
) tested for 
and 
. The values chosen for these hyperparameters generated a wide range of shapes for this distribution that reflected different a priori knowledge on the expected values of both variance components. Model fit and complexity were evaluated with the DIC (Spiegelhalter et al. 2002), and PR3 was favored (DIC = 3975.50) although the difference was slight with respect to PR1 and PR4 (DIC = 3976.79 and 3976.71, respectively; Table 2). PR2 was penalized with a 2-units greater DIC than PR3, and the model without genetic effects (PR0) was discarded (DIC = 3981.91). Note that differences in DIC > 3 are generally considered as statistically relevant (Burnham and Anderson 1998; Spiegelhalter et al. 2002), whereas lower discrepancies do not provide a strong evidence of a better fit and a lower degree of model complexity for a given comparison. It is important to highlight that three different MCMC chains were launched for each model and DIC showed a very small variance within models (Table 2).
As shown in Table 3, PR1, PR3, and PR4 models provided very similar estimates of variance components and their ratios, whereas PR2 had a lower value of 
. Taking the PR3 model as reference and after correcting for systematic effects (generation and parturition number of the female), the most important source of variation was that due to the uncontrolled factors that accounted for 
, its mode being 3.842 pups2 (Table 3). Nevertheless, genetic variances 
and 
were high for an inbred strain, 0.151 and 0.035, respectively. It is important to note that the highest posterior density region at 95% (HPD95) for both variance components was away from zero, starting at 0.066 and 0.017, respectively (Table 3). Mutational heritability was 0.008, and the overall heritability at generation G1 (
) was 0.045 (HPD95 between 0.010 and 0.062), showing that enough additive genetic variance was available to develop a genetic trend under selection or drift. In a similar way, sire and (nongenetic) dam effects had moderate variance components (0.092 and 0.037, respectively), although with wide HPD95 (Table 3).
Modal estimates (and highest posterior density region at 95%) for the variance components and heritabilities
The C57BL/6J strain did not show notable genetic drift during 46 generations. The within-generation average breeding value (
) ranged between 0 (G1, G2, and G22) and 0.02 pups (G44), with all the HPD95 including the null value (Figure 2). Similarly, within-generation average values for environmental effect (
and 
) did not differ from zero. The between-generations phenotypic variability was mainly accounted for by the generation number effects (results not shown). Model fit was worse when this systematic effect was dropped out (DIC = 4018.89).
Mode (solid diamonds) and highest posterior density region at 95% (whiskers) of the average breeding value per generation.
The within-generation 
quickly decreased (Figure 3) whereas the new genetic variance originated by mutation increased up to ∼0.20 and oscillated around 0.15 thereafter (Figure 4). More specifically, 
started with a modal estimate of 0.159 in generation G1 (HPD95 between 0.056 and 0.245) and fell to values <0.01 in <10 generations. Mutational variance accumulated during the first generations and reached its maximum at generation G14 (0.231), although it showed an oscillating pattern around 0.15 from generation G7 (HPD95 reached values up to 0.400).
Mode (solid diamonds) and highest posterior density region (whiskers) of the within-generation additive genetic variance (
).
Mode (solid diamonds) and highest posterior density region (whiskers) of the within-generation mutational variance (
).
DISCUSSION
Prior distributions and Bayesian analysis:
The mixed-model analysis of litter size in C57BL/6J mice was carried out using Bayesian methods. An important characteristic of the Bayesian analysis is that the final inference is based on the posterior distribution, resulting from combining two different sources of information. One of these sources is the experimental data, which are not influenced by arbitrary choices other than the model used for analysis. The other source of information is the assignment of prior distributions, which are arbitrarily chosen from previous knowledge of the parameters of interest. If previous information is not available, prior distributions become a blind choice and they could have a substantial impact on the posterior inference (Gianola and Fernando 1986; Blasco 2001). Genetic components of litter size in highly inbred mice have not been previously analyzed and we lacked accurate information on the expected values of 
and 
. To assess influences of priors for both variance components, the analyses performed here made use of very different prior distributions for 
and 
, covering the range of mutational heritabilities reviewed by Lynch (1988) and Houle et al. (1996) in other traits and species. Model PR0, the one without genetic components, showed the poorest model fit and DIC substantially decreased when 
and 
were included in the model. This provided statistical evidence of the presence of additive genetic variance in this inbred strain. Models PR1–PR4 showed a similar fit, although the stringent prior for 
and 
in model PR2 was moderately penalized. It is important to note that posterior inferences from models PR1, PR3, and PR4 did not differ substantially (Table 3). This reassuring conclusion indicated that the experimental data had enough information content to override moderate influences of prior information, and the model performed better under a vague assumption for 
and 
over the parameter space.
Note that this analysis could also be performed under a frequentist approach by maximizing the likelihood function through iterative algorithms. These frequentist methods produce inferences based on the data and the previous knowledge of the distribution of estimators in the sampling space, without using prior information. As highlighted by Blasco (2001), the distribution of the estimator is used for inferences instead of the distribution of the parameter, which leads to a rather unnatural form of expressing uncertainly about the results of an experiment. Within the Bayesian context, conceptual simplicity is gained because inferences are made from probabilities associated with values of the parameter of interest.
Genetic variability:
Reported estimates of mutational heritability found in the literature commonly range between 
and 
(Lynch 1988; Houle et al. 1996), although this parameter has never been estimated for litter size in mice. Our estimate fell within this interval (0.008; Table 3) and was very close to the values reported by Keightley and Hill (1992) and Keightley (1998) for body size in mice. Although Wray's (1990) approach assumes that mutations are small and additive, the inclusion of sire and dam environmental effects accounted for deviations from the infinitesimal model, allowing for a more accurate estimation of 
. Within this context, part of the variability that accounted for 
and 
could have originated from large mutations. Thus, 
likely underestimates all the genetic variation originated by mutation. Moreover, environmental variance could also account for nonadditive genetic mutations (Zhang et al. 2004). Although 
and 
modal estimates suggested a greater impact of 
on litter size, HPD95 were completely overlapping. It is important to note that 
and 
accounted for additive genetic effects whereas nonadditive genetic sources of variation such as inbreeding or heterosis could have a substantial impact on mouse litter size (Bhuvanakumar et al. 1985; Hinrichs et al. 2007). Given the difficulty in accommodating inbreeding and heterosis on the new mutational variability originated at each generation, as well as other nonadditive genetic effects (i.e., epistasis or dominance), we restricted the model to pure additive genetic effects and assumed that the remaining nonadditive genetic influences were accounted for by 
and 
or were absorbed by the residual term.
Mutation variance in the mixed model was modeled as the dispersion term associated with random mutation effects arising in each offspring (Wray 1990). This parameterization has been typically used in mutation experiments (Lynch 1988; Houle et al. 1996) and describes the potential effect of mutations in a very short time interval. Nevertheless, new mutations accumulate in successive generations, sometimes fixed or removed due to selection or genetic drift, and 
must be viewed in highly inbred strains as a lower limit of the accumulated (mutation) genetic variance. As shown in Figure 4, the within-generation mutation variance increased with generation, up to G14. This is in agreement with a continuous input of new mutations with effects on litter size. The oscillating pattern around 0.15 after G14 agrees with 
at G1 and suggests an equilibrium between new mutations and the fixation or loss of mutations due to genetic drift and inbred matings. In a similar way, 
lacks additional sources of new variation and its within-generation estimates have a rapid decrease after a few generations, which is related to the small population size and the mating system. Both within-generation 
and 
estimates support the conclusion that this C57BL/6J strain maintained a substantial degree of genetic variance across generations, accounting for an overall heritability around 0.05 (Table 3). This value is clearly lower than the ones reported in outbred mouse populations (0.15–0.33; Falconer 1960; Joakimsen and Baker 1977; Long et al. 1991) although it indicates a substantial and generally unaccounted for degree of genetic variability in an inbred strain.
Absence of environmental and genetic trend across generations:
Mice bred under inbred mating systems for prolonged periods should fix the vast majority (potentially all) of the genetic contribution to variation (Bailey 1982) and typically, individual mice within an inbred strain are considered genetically identical. Nevertheless, unexpected genetic variability has been observed in highly inbred mouse strains (Keightley and Hill 1992), even allowing for genetic drift (Bailey 1977) and genetic trend on phenotypic traits (Festing 1973; Keightley 1998). Indeed, incongruities between genetic homogeneity and phenotypic variability were first recognized >40 years ago (Wolff 1961) and this issue is an area of concern in laboratory species. The C57BL/6J strain was developed in the early 20th century from a very small founder population and it is considered a classical inbred strain with an almost homozygous genome (Wade and Daly 2005). As reported in other inbred mouse strains (Falconer 1960), our strain showed substantial phenotypic variation for average litter size across generations, average litter size being similar to the estimates reported by other authors (Kirkpatrick et al. 1998; Corva et al. 2004). Note that this population was not under selection for litter size.
Besides the continuous generation of additional genetic variability in our C57BL/6J population, genetic drift did not take place (Figure 2) and changes in the (within-generation) average environmental effect were also negligible. As mentioned above, our mixed-model analysis included generation number as a systematic effect and it accounted for the major differences between generations in litter size. Note that model fit was worse when this effect was dropped out (DIC = 4018.89). Theoretically, this effect must be viewed as the generation-specific contribution of multiple environmental sources of variation (e.g., food, housing, and management among others) although some genetic contributions could be involved too. Additive genetic variability is accounted for by a and m, and residual effects can absorb (individual-specific) nonadditive genetic influences. Nevertheless, a small number of breeding individuals contribute to the next generation and commonly they are closely related (full sibs in the majority of cases). Nonadditive genetic effects from a given ancestor can be widely spread in the following generation, reducing between-individuals variability and therefore being partially accounted for by the within-generation overall mean (generation number effect). Within this context, genetic drift cannot be completely discarded although, if present, it would be due to nonadditive mutations.
In conclusion, we present a new approach to Wray's (1990) method for modeling 
and 
within a Bayesian framework, where all parameters in the mixed model can be inferred using the Gibbs sampling algorithm. The analysis of litter size in the C57BL/6J strain indicated a low mutational input of genetic variance per generation (
= 0.035), although the accumulation of new mutations in successive generations led to a substantial amount of additive genetic variability (
). While genetic uniformity of highly inbred strains is a key point in several research areas (Stevens et al. 2007), our estimates do not support this assumption and confirm a continuous and unavoidable flow of new genetic variability. These results contribute to the understanding of mutation–drift equilibrium in experimental populations.
APPENDIX
Following in part Wray (1990), the additive genetic (co)variance matrix including mutation effects can be partitioned as
where t is the number of generations, 
is the full relationship matrix of additive genetic effects attributed to mutations arising in time unit (or generation) k, and 
is the full relationship matrix including all individuals in the pedigree. For the mixed-model equations, both 
and 
are required, computational efficiency becoming a key point. As developed by Quaas (1976), 
can be recursively computed from a list of individual sire and dam identifications ordered by age of individuals. Following in part Wray (1990), 
can also be computed from an age-ordered pedigree with n individuals and three vectors, 
, 
, and 
, all with dimension 
. Computation efficiency is gained with n rounds, with the following calculations in the ith round,
where 
, 
, and 
are the lth elements in vectors 
, 
, and 
, respectively. For 
,
where 
and 
are parents of the jth individual and 
. For 
,
and for 
,
To construct 
, simply add the following:
If both parents of i are known (p and q), add
where is the element in the kth row and lth column of .If only one parent of i is known (q), add
If neither parent of i is known, add
Note that this construction of 
and 
allows for a separate inference of additive genetic effects in the base population and new mutations arising in successive generations, 
and 
being independent from the remaining parameters in the model. If Bayesian mixed models are applied, this parameterization leads to well-known conditional posterior distributions for all parameters, allowing for standard Gibbs sampling. On the contrary, conditional posterior distributions for several parameters under the original Wray's (1990) approach do not lead to known densities, requiring generic sampling processes (i.e., Metropolis–Hastings sampling) and the intensive reconstruction of the inverse of the relationship matrix within each sampling iteration (Wray 1990).
Acknowledgments
We are appreciative of the excellent efforts of Vince De Vera in mouse husbandry and phenotypic data collection. The useful comments of D. Gianola and an anonymous referee are greatly appreciated. This research was supported by the National Research Initiative grant no. 2005-35205-15453 from the U.S. Department of Agriculture Cooperative State Research, Education, and Extension Service and by the California Agricultural Experimental Station. The research contract of J. Casellas was partially financed by Spain's Ministerio de Educación y Ciencia (program Juan de la Cierva).
Footnotes
Communicating editor: R. W. Doerge
- Received February 14, 2008.
- Accepted May 30, 2008.
- Copyright © 2008 by the Genetics Society of America
