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Genetics. Jun 2007; 176(2): 1169–1185.
PMCID: PMC1894582

Bayesian Shrinkage Analysis of Quantitative Trait Loci for Dynamic Traits

Abstract

Many quantitative traits are measured repeatedly during the life of an organism. Such traits are called dynamic traits. The pattern of the changes of a dynamic trait is called the growth trajectory. Studying the growth trajectory may enhance our understanding of the genetic architecture of the growth trajectory. Recently, we developed an interval-mapping procedure to map QTL for dynamic traits under the maximum-likelihood framework. We fit the growth trajectory by Legendre polynomials. The method intended to map one QTL at a time and the entire QTL analysis involved scanning the entire genome by fitting multiple single-QTL models. In this study, we propose a Bayesian shrinkage analysis for estimating and mapping multiple QTL in a single model. The method is a combination between the shrinkage mapping for individual quantitative traits and the Legendre polynomial analysis for dynamic traits. The multiple-QTL model is implemented in two ways: (1) a fixed-interval approach where a QTL is placed in each marker interval and (2) a moving-interval approach where the position of a QTL can be searched in a range that covers many marker intervals. Simulation study shows that the Bayesian shrinkage method generates much better signals for QTL than the interval-mapping approach. We propose several alternative methods to present the results of the Bayesian shrinkage analysis. In particular, we found that the Wald test-statistic profile can serve as a mechanism to test the significance of a putative QTL.

SOME quantitative traits can be measured repeatedly during the development of life. Such traits are called longitudinal traits in humans, but more often are called dynamic traits in animals and plants. Some genes control the phenotypic values of the dynamic traits at fixed time points and others may alter the transitions of the phenotypes between consecutive time points. The growth pattern of a dynamic trait is called the growth trajectory. Studying the growth trajectory may detect both sets of genes and thus may enhance our understanding of the genetic architecture of the growth trajectory. Dynamic traits are often collected in large animals and plants, such as milk production in dairy cattle, growth rate in pigs, egg production in chickens, and growth rate in forest trees.

A growth trajectory is usually described by a logistic growth function. Wu and colleagues (Ma et al. 2002; Wu et al. 2002, 2003, 2004) developed many logistic regression models for mapping QTL for dynamic traits. The basic model is the logistic regression but different covariance structures of the residual errors are fit to different models. The logistic regression fits only growth trajectories that are sigmoid. Many growth trajectories, however, may show patterns that are not sigmoid. This stimulated Yang et al. (2004, 2005) to adopt a more general model called Legendre polynomial analysis. Legendre polynomial has been extensively used by animal geneticists and breeders to fit milk production and other dynamic traits (Kirkpatrick and Heckman 1989; Kirkpatrick et al. 1990; Schaeffer 2004). Milk production varies across days both within the same season and over different seasons. The curves are definitely not logistic, which explains why animal breeders do not use logistic regression to fit milk production curves. In addition to the flexibility to fit biological curves with arbitrary shapes, the Legendre polynomial is a linear model, as such theory and methodology extensively developed in linear models apply directly to Legendre polynomial analysis.

Macgregor et al. (2005) recently applied Legendre polynomial to QTL mapping for longitudinal traits in pedigrees. They adopted the traditional random regression model (RRM) in which the vector of polynomial regression coefficients (genetic effects) for each animal is treated as a random vector sampled from a multivariate normal distribution. The identity-by-descent-based variance component method (Goldgar 1990; Schork 1993; Amos 1994; Fulker and Cardon 1994; Xu and Atchley 1995; Almasy and Blangero 1998; Yi and Xu 2000a,b) is applied to estimate and test the variance components of a putative location of the genome. Yang et al. (2006) used Legendre polynomial to map QTL for dynamic traits in line crosses. The method of Yang et al. (2006), however, may not be called RRM because the polynomial regression coefficients for QTL effects are treated as fixed effects. Both the Macgregor et al. (2005) and the Yang et al. (2006) methods are implemented via the interval-mapping approach where the model includes the effects of a single QTL and a complete QTL analysis requires scanning the entire genome. Macgregor et al. (2005) included a polygenic effect in their RRM. Yang et al. (2006) included an individual specific residual effect in their fixed regression model. Both the polygenic effect in Macgregor et al. (2005) and the individual specific residual effect in Yang et al. (2006) can absorb, to some extent, the effects of QTL in regions other than the current one being tested. The models, in terms of QTL mapping, are still single-QTL-effect models. As a consequence, results are hard to summarize because QTL effects for different genome locations are estimated from different models.

In this study, we employed the multiple-QTL model developed for single-trait QTL analysis (Xu 2003; Wang et al. 2005) to map multiple QTL for dynamic traits. Although the maximum-likelihood method has been developed for multiple-QTL mapping (e.g., Jansen 1993; Jansen and Stam 1994; Kao et al. 1999), the Bayesian method has become more and more popular because of its ability to handle more complicated models (Satagopan et al. 1996; Heath 1997; Uimari and Hoeschele 1997; Sillanpää and Arjas 1998, 1999; Daw et al. 1999; Hurme et al. 2000; Xu and Yi 2000; Yi and Xu 2000a,b, 2002; Yuan et al. 2000; Uimari and Sillanpää 2001; Bink et al. 2002; Corander and Sillanpää 2002; Yi et al. 2003a,b, 2005; Yi 2005). The most cumbersome issue in multiple-QTL mapping is how to determine the optimal number of QTL. Variable selection via stepwise regression is commonly used in maximum-likelihood (ML) mapping (Kao et al. 1999). Reversible-jump Markov chain Monte Carlo (RJ–MCMC) is the corresponding variable selection procedure used in Bayesian analysis (Green 1995, 2003). However, recent studies show that RJ–MCMC is subject to poor mixing and slow convergence to the stationary distribution (Godsill 2001; Green 2003). Bayesian shrinkage analysis (Xu 2003; Wang et al. 2005) and stochastic search variable selection (SSVS) (Yi et al. 2003a,b) are more efficient methods than RJ–MCMC. In these methods, no variable selection is conducted in an explicit manner; rather, a treatment similar to variable selection is made implicitly by shrinking the effects of excessive QTL to zero. QTL with estimated effects shrunk to zero in the shrinkage analysis are equivalent to being excluded from the model in a variable selection approach. The advantages of the shrinkage analysis over an explicit variable selection are twofold: (1) improving the convergence of MCMC and (2) reducing the chance of missing QTL.

THEORY AND METHODS

Genetic model:

We use a backcross (BC) mating design as an example to describe the genetic model for dynamic traits. Let equation M1 be the phenotypic value of individual i (equation M2) measured at time t (equation M3), where n is the sample size and t is a standardized time point between −1 and 1. Let T be the time point in the original scale. The time point in the standardized scale is obtained using equation M4, where equation M5 and equation M6 are the starting and ending time points (see Kirkpatrick et al. 1990). The single-QTL model used by Yang et al. (2006) is modified and adopted here to describe equation M7,

equation M8
(1)

where equation M9 is the population mean at time t, equation M10 is an individual specific random environmental effect with a equation M11 distribution, and equation M12 is a random environmental error (independent of time) distributed as equation M13. This is a multiple-QTL model with the effect of the jth QTL at time point t denoted by equation M14 for equation M15, where q is the number of QTL included in the model. The single-QTL model of Yang et al. (2006) is a special case when equation M16. Variable equation M17 is a genotype indicator variable for individual i at locus j and defined as equation M18 for one genotype and equation M19 for the other genotype. For example, if the genotypes of the two parents for a QTL are AA and aa, the genotype of the F1 hybrid is Aa. There are two types of BC design. Assuming that the BC family is generated by crossing F1 with the AA parent, the two genotypes of the BC family are AA and Aa, respectively. Therefore, if a BC progeny has a genotype of AA, equation M20; otherwise, equation M21.

Following Yang et al. (2006), we use a Legendre polynomial of order d to express each time-dependent variable in model (1) as a linear function of a time-independent vector of parameters, such as equation M22, equation M23, and equation M24, where each one of equation M25, equation M26, and equation M27 is a equation M28 column vector of time-independent parameters and equation M29 is a equation M30 row vector of constants (Yang et al. 2006). Model (1) is then rewritten as

equation M31
(2)

Since equation M32 is a vector of random effects, we assume equation M33, where equation M34 is a equation M35 unstructured variance–covariance matrix (a full symmetric positive definite matrix).

To estimate the parameters, we need to collect molecular marker data as well as phenotypes of repeated measurements of the dynamic trait. Let equation M36 (equation M37) be the kth time point at which the phenotype equation M38 is measured. Let equation M39 be an equation M40 column vector for the repeated measurements of the dynamic traits. In matrix notation, model (2) becomes

equation M41
(3)

where equation M42 is a equation M43 matrix of constants (see Yang et al. 2006) and equation M44 is now an equation M45 vector for the residual errors with an assumed equation M46 distribution. Note that the equation M47 in Equations 1 and 2 is scalar because it represents the residual error for the dynamic trait at a single time point whereas the equation M48 in Equation 3 is a vector because it contains an array of the residual errors for equation M49 time points. The assumption of equation M50 means that the residual errors are i.i.d. normal across all time points. The heterogeneous distribution of the environmental errors over time has been taken into account by the time-dependent environmental effect, equation M51.

For simplicity, we assume that each individual has observations at all time points and the time points are common for all individuals. If this assumption is violated, equation M52 will have a different dimension across different i and matrix equation M53 will also vary in both dimension and content over different individuals. The technical details for variable time points across individuals have been discussed by Yang et al. (2006).

Data, parameters, and missing values:

The phenotypic values of the dynamic trait of interest are a source of data. Marker genotypes and the relative positions of the markers along the genome (or called the marker map) are another source of data. Let us denote the array of phenotypic values by Y and the marker information by M. The data now are denoted by equation M54. In QTL mapping, the most important parameter is the number of QTL (denoted by q in this study). In the Bayesian shrinkage analysis, however, we treat q as a constant (see Wang et al. 2005 for justification). The parameters of direct interest in the Bayesian shrinkage mapping include the position (equation M55) and the effect (equation M56) of QTL (equation M57). Other parameters include equation M58, equation M59, and equation M60. The genotype indicator variable x and the individual specific effect equation M61 also appear in model (3), but they are called missing values rather than parameters. These missing values are not missing observations; rather, they may be better called the latent variables or nuisance parameters to avoid confusion. In Bayesian analysis, however, missing values and parameters are treated equally and both are called unobservables, denoted by equation M62. Each unobservable is a random variable following a certain distribution. The distribution of a missing value is usually described as a function of the existing parameters. The distribution of each parameter is called the prior distribution, which also has its own parameters called the hyperparameters. The hyperparameters are constants chosen by the investigator or estimated from the data if they are described by a higher level prior distribution. If the prior of a hyperparameter is used to estimate the hyperparameter, the model is called the hierarchical model (Gelman 2005). Let equation M63 be the vector of hyperparameters. The prior density is denoted by equation M64. The probability density of the data given the parameters is denoted by equation M65, which is also called the likelihood. The posterior distribution of the parameter vector is

equation M66
(4)

Given the likelihood function and the prior distribution, the specific form of the posterior distribution can be found or inferred through MCMC sampling.

The likelihood can be written as equation M67, where equation M68 is multivariate normal and equation M69 is modeled by a Markov distribution (Wang et al. 2005). The position of each QTL is assigned a uniform prior within the interval bracketed by two markers. Each QTL effect is assigned a multivariate normal prior with mean zero and an unknown variance matrix. This variance matrix is further assigned a hyperprior so that the variance matrix can be estimated from the data. An inverse Wishart prior is chosen for each variance matrix. The inverse Wishart distribution is a multivariate version of the scaled inverse chi-square distribution and is often used to model a variance–covariance matrix (Meuwissen and Goddard 2004). The residual variance is assigned a scaled inverse chi-square prior distribution.

Gibbs sampler:

Let equation M70 be the initial values used for all the unknown variables. Let equation M71 be a component of vector equation M72 and equation M73 be the remaining components of equation M74, excluding equation M75. The conditional posterior probability of equation M76 given equation M77 is denoted by equation M78, which is used to simulate equation M79. Once equation M80 is simulated, it is moved to the list of known elements and the conditional posterior probability of the next component of equation M81 is calculated and a value is drawn from this distribution. Once all elements of equation M82 are drawn, we complete the first iteration and the value of equation M83 is denoted by equation M84. The sampling process may continue for N iterations to form a Markov chain, denoted by equation M85, where N is a very large number. The sample mean for the kth element, equation M86, is the empirical posterior mean of equation M87, which may be considered as a Bayesian point estimate.

Here we focus on each variable whose equation M88 has an explicit form so that samples can be directly drawn from that distribution. This process of directly generating samples from the posterior is called the Gibbs sampler (see Gelman et al. 1995). Variables that can be generated with the Gibbs sampler are described below.

The prior distribution for the population mean is uniform across the real (uninformative prior), which leads to a normal posterior distribution. Therefore, the population mean equation M89 is drawn from a multivariate normal distribution with mean

equation M90
(5)

and variance

equation M91
(6)

where equation M92 is the variance–covariance matrix of vector equation M93. Note that we use equation M94 and equation M95 to denote the conditional posterior expectation and variance, respectively. Similar notation applies to other parameters.

Let equation M96 be the multivariate normal prior for the effect of the jth QTL with mean 0 and variance equation M97, where equation M98 is a equation M99 positive definite matrix. It is a multivariate version of the shrinkage parameter (Wang et al. 2005). The conditional posterior distribution for equation M100 is multivariable normal with mean

equation M101
(7)

and variance

equation M102
(8)

The Bayesian shrinkage method differs from the usually Bayesian regression analysis in that here we treat equation M103 (a hyperparameter) as an unknown variable rather than as a fixed constant. Because of this, equation M104 is described by another prior equation M105, called the inverse Wishart distribution with a prior belief equation M106 and a equation M107 positive definite scale matrix equation M108. This special treatment is called hierarchical modeling (e.g., Gelman et al. 1995; Gelman 2005). The conditional posterior of equation M109 remains inverse Wishart; i.e.,

equation M110
(9)

This posterior distribution is used to draw equation M111. Because equation M112 is sampled, it varies from one iteration to another, reflecting the stochastic nature. More importantly, equation M113 varies across j (different loci), reflecting the selective nature of the shrinkage. When equation M114 is large, the posterior mean and variance of equation M115 will resemble the “least-squares” estimates (no shrinkage). When equation M116 is small, both the posterior mean and the posterior variance of equation M117 will approach zero, leading to the “shrinkage estimate” of equation M118. It is easy to understand where the shrinkage occurs under the univariate model of QTL mapping (Wang et al. 2005), but hard to understand the nature of shrinkage under the multivariate version of the QTL model. We have provided an explanation for this in the beginning of the discussion.

The individual-specific equation M119 is sampled from a multivariate normal distribution with mean

equation M120
(10)

and variance

equation M121
(11)

Assume that the prior distribution of equation M122 is equation M123. Given equation M124, the posterior distribution of equation M125 is

equation M126
(12)

Finally, the residual error variance is sampled from a scaled inverse chi-square distribution,

equation M127
(13)

where equation M128 and equation M129 are the hyperparameters in the inverse chi-square prior distribution, and

equation M130
(14)

is the residual error.

Metropolis–Hastings sampler:

We are able to use the Gibbs sampler to generate variables so far because the conditional posterior for each of the aforementioned variables has an explicit form of distribution. The conditional posterior distribution of the position of a QTL, however, has no explicit form. Therefore, the general Metropolis–Hastings (Metropolis et al. 1953; Hastings 1970) algorithm is required to sample equation M131. Since the genotype of QTL (equation M132) depends on the QTL position equation M133, we decide to sample equation M134 jointly as a block but proceed with the sampling with one locus at a time. The Gibbs sampler turns out to be a special case of the general Metropolis–Hastings (MH) algorithm (Gelman et al. 1995). Following Wang et al. (2005), we assume that there is one QTL in each marker interval. The position of the QTL varies within the marker interval that contains the QTL. The prior distribution of equation M135 is

equation M136
(15)

where equation M137 and equation M138 are the positions of the left and the right markers that define the interval. Let equation M139 be the current position of the locus of interest and equation M140 be the genotype array of all individuals at the locus. We first sample a new position for the QTL, called the proposed position and denoted by equation M141, where equation M142 is sampled from equation M143 and s is a small positive number (tuning parameter) such as 1 cM. For the new position, we simulate the genotypes for all individuals, denoted by equation M144. We then use the MH rule to decide whether equation M145 should be accepted or not. If equation M146 is accepted, we update both the position and the genotype using equation M147 and equation M148. Otherwise, the old values of equation M149 and equation M150 are carried over so that equation M151 and equation M152. A detailed formula of the MH acceptance rule can be found in Wang et al. (2005) and Zhang and Xu (2005). The one-QTL-per-interval approach to sampling QTL positions is called the fixed-interval approach (Wang et al. 2005). It works well when markers are more or less evenly distributed and the marker density is intermediate. For highly dense marker maps, a slight modification is required.

Figure 1 shows the problem of dealing with a highly dense marker map and a proposed solution. This modified method is called the moving- (or variable-) interval approach as opposed to the fixed-interval approach described above and by Wang et al. (2005). The moving-interval approach was also introduced by Wang et al. (2005), but only very briefly. Here, we use an example to illustrate the moving-interval process. In this example, there are 18 dense markers (17 intervals) covering a chromosome. In the modified method, we place five QTL evenly on the chromosome. The upper limit for the number of QTL is now equation M153. Each QTL interval may expand to >2 markers. We assume that each QTL interval carries one QTL. Before the MCMC starts, we simply place the five QTL evenly on the chromosome. We then update one QTL at a time given the positions of the remaining QTL. The prior of each QTL position is uniform between the positions of the left QTL and the right QTL; i.e., equation M154. Because the prior for a QTL position depends on the current positions of other QTL, this approach is called the adaptive Bayesian approach (Gilks et al. 1995). The sampling process remains the same as that described earlier for the fixed-interval approach except that the length of the prior interval for each QTL varies and moves from one iteration to another. A complete cycle of updates for QTL positions is illustrated in Figure 2.

Figure 1.
Illustration of the problem with dense marker map and the sketch of a proposed solution. The line at the top shows 7 markers (equation M292) with intermediate density and 6 QTL (equation M293), one in each marker interval. The Bayesian shrinkage method works well without modification. ...
Figure 2.
Illustration of the moving-interval search algorithm for QTL on a chromosome with saturated markers. The first line shows the marker map and the initial positions of five QTL. The second line shows that the position of equation M298 can move between equation M299 and equation M300. Once the ...

Block updating of QTL positions and genotypes is still needed in the moving-interval approach. The moving-interval approach creates high correlation between parameters because the range of one QTL position depends on positions of other QTL. This may potentially disturb the mixing of the sampler. However, our previous simulation study did not show any notable disturbance of the mixing of the sampler (Wang et al. 2005).

Post-MCMC analysis:

Unlike maximum-likelihood analysis, the product of MCMC sampling is a realized sample of all unknown variables drawn from the joint posterior distribution and the results should be interpreted in a different way. The most important parameters of interest are the locations of the QTL and their effects. The variance matrix equation M155 for the jth QTL is not of interest by itself but serves as a mechanism to control the estimate of the QTL effect (i.e., shrinkage). In conventional Bayesian mapping (e.g., Sillanpää and Arjas 1998, 1999; Yi and Xu 2000a,b; Wang et al. 2005), the marginal posterior distribution of QTL position can be depicted via plotting the number of hits by QTL in a short segment (say a 1-cM segment), called a bin, against the genome position where the bin is located. The curve is called the QTL intensity profile, denoted by equation M156, where equation M157 represents a location of the genome. Here we have a situation where each marker interval is assumed to have a QTL and thus all marker intervals are hit by the same number of times. If a marker interval contains a true QTL, we expect the QTL intensity profile within the interval to show a peak. Otherwise, the profile should be flat.

In addition to the QTL intensity profile, there are several alternative profiles to present the result of MCMC. One profile is the quadratic QTL effect profile, denoted by equation M158, where equation M159 is the average genetic effect (a vector) over the posterior samples for the QTL whose positions fall in the bin located at equation M160. If the bin at equation M161 is not hit by any QTL, equation M162 is set to 0. Normally, the number of hits of a bin by QTL is proportional to the size of the QTL for that bin. However, it is possible that a bin may be hit by one or a few large QTL, which makes equation M163 an uninformative representation of QTL position. Therefore, a more informative profile may be the profile of the quadratic effect weighted by the frequency, denoted by equation M164. This profile is a better indicator for the position of the genome repeatedly hit by large QTL and thus is more informative than either equation M165 or equation M166. Another alternative profile may be the Wald test-statistic profile denoted by equation M167, where equation M168 is the sample variance matrix for the QTL effect at position equation M169. A potential problem is that for a position not frequently hit by QTL, equation M170 is not a good representative of the variance matrix of the sampled equation M171. Therefore, we may use the average theoretical conditional posterior variance matrix of equation M172 given in Equation 8 in place of equation M173 to draw the Wald statistic profile. The Wald test statistic may be used as an objective way to test the significance of QTL at position equation M174. Under the null hypothesis, i.e., there is no QTL at position equation M175, equation M176 may follow a chi-square distribution with equation M177 d.f. Therefore, a critical value may be used to declare statistical significance at position equation M178. This critical value is equation M179, the equation M180 percentile of a chi-square distribution with equation M181 d.f., where equation M182 is the type I error (not the vector of QTL effect). Let equation M183 be the kth element of vector equation M184 and equation M185 be the kth diagonal element of matrix equation M186 for equation M187. We may propose to use equation M188 as the univariate test statistical profile for the kth element, where equation M189 is the chi-square test, a univariate version of the Wald test statistic. The critical value used to declare statistical significance is equation M190 for equation M191 and equation M192 for equation M193. Note that

equation M194
(16)

is the conditional expectation of equation M195 given equation M196 (the remaining elements of vector equation M197) and

equation M198
(17)

is the conditional variance of equation M199 given equation M200. A matrix or a vector with a subscript equation M201 means a collection of the remaining elements of the corresponding matrix or vector that excludes element k.

APPLICATIONS

Simulated data analysis:

Fixed-interval approach:

We simulated a dynamic trait measured at equation M202 time points from equation M203 BC individuals. The original eight time points denoted by a vector T and the standardized time points denoted by a vector t are given below:

equation M204

A genome consisting of a single large chromosome of 600 cM was simulated, which was covered by 61 evenly placed markers (10 cM per marker interval). The growth pattern of the dynamic trait was assumed to be controlled by 10 QTL with positions and effects listed in Table 1. The order of the polynomial that generated the growth trajectory was equation M205. The population mean was equation M206, the residual variance was equation M207, and the individual-specific environmental error variance matrix was

equation M208
TABLE 1
QTL parameters used in the simulation experiment

Phenotypic values of the dynamic trait were simulated using model (3). A single sample was simulated and analyzed with the Bayesian shrinkage method. Marker genotypes and the phenotypic values of the 150 simulated individuals are available from the authors on request.

The trajectories for the two genotypes of each QTL are shown in Figure 3, where equation M209 is the trajectory for one genotype and equation M210 is the trajectory for the alternative genotype. Figure 3 does not show the plot for QTL 4 because (a) it is close to QTL 3 and (b) with this additional plot Figure 3 would not be as neat as it is.

Figure 3.
The trajectories of genotypic effects for 9 of the 10 simulated QTL (the fourth QTL effects are not included in the plots). The genotypic value for one genotype is equation M305 and the value for the alternative genotype is equation M306, where t is the standardized time point ...

Although we simulated 10 QTL, the working model included equation M211 QTL. Here are the actual values for the hyperparameters used in the data analysis: equation M212, equation M213, equation M214, and equation M215. The initial values of all variables were sampled from their prior distributions. The MCMC was run for 6000 cycles as a burn-in period (deleted) and then for an additional 30,000 cycles after the burn-in. The chain was then thinned to reduce serial correlation by saving one observation in every 60 cycles. The posterior sample contained 5000 (equation M216) observations for the post-MCMC analysis. The simulation experiment was replicated five times. The result showed very little variation among the replicates. Therefore, we report the result of one replicate only.

Figure 4 shows the four proposed profiles of the QTL parameters. The QTL density profile (Figure 4a) shows nine peaks at the locations of nine simulated QTL. The third and fourth QTL are too close to each other (5 cM distance) and thus could not be separated by the method. This profile shows the expected behavior of the shrinkage analysis and the characteristic of the profile is similar to that observed for the single-trait shrinkage mapping (Wang et al. 2005). The quadratic effect profile is shown in Figure 4b and the signals are very clear for positions occupied by the simulated QTL. The shrinkage effect is obvious: effects of intervals without QTL have been shrunk to zero and thus no variable selection is needed. Peaks of the quadratic effect profile are not sharp. However, the weighted quadratic effect profile (Figure 4c) shows sharp peaks at the positions where the simulated QTL occur. Finally, the Wald test-statistic profile (Figure 4d) also shows clear signals for the simulated QTL. Although Bayesian analysis is not designed for significance tests, the Wald test statistic can be used for a significance test. All nine peaks passed the critical value (equation M217) and no peaks elsewhere on the genome are higher than the critical value. In real data analysis, one may simply choose the weighted quadratic effect profile equation M218 or the Wald test-statistic profile equation M219 for presentation.

Figure 4.
Profiles of QTL parameters drawn from the fixed-interval analysis: (a) QTL intensity profile equation M307, (b) quadratic effect profile equation M308, (c) weighted quadratic effect profile equation M309, and (d) Wald test statistic profile equation M310. The horizontal reference line on the W plot (plot ...

We fit the polynomial model with order equation M220 and thus each QTL effect equation M221 is a equation M222 vector, equation M223. Each component can be tested using the Z-test statistic. The equation M224 profiles are shown in Figure 5. Clearly, some components of the simulated QTL did not pass the critical value. This means that not all QTL have the same order. Some are linear and others are quadratic and so on. Again, no variable selection is required because the coefficients corresponding to the nonsignificant one are shrunk to zero.

Figure 5.
The Z-test-statistic profile for each of the four components of the QTL effect equation M312 obtained from the fixed-interval analysis, where equation M313 is the component for order k of the polynomial. The reference line on each plot is the critical value for the Z-test (equation M314).

The estimated QTL locations and effects (posterior means) are summarized in Table 2, showing that Bayesian analysis provides quite accurate estimates of the locations and effects of QTL (see Table 1). The estimated population mean and covariance matrix for the individual specific environmental effects (posterior means) are

equation M225

and

equation M226

respectively, where the numerical values in parentheses are the posterior standard deviations. The estimated residual variance is equation M227.

TABLE 2
QTL parameters for the simulated data estimated from the fixed-interval approach

Figure 6 shows the estimated trajectories for the nine QTL detected. They are close to the true trajectories shown in Figure 3.

Figure 6.
Estimated growth trajectories for the effects of the nine detected QTL with the fixed-interval approach, where equation M315 and equation M316 are the trajectories of the two alternative genotypes for each QTL.

Finally, as a comparison, we ran the interval-mapping program for dynamic traits (Yang et al. 2006) on the simulated data. The likelihood-ratio profile is shown in Figure 7. Interval-mapping analysis only detected eight simulated QTL using a permutation generated critical value of 25.01 (from 500 permuted samples). The peaks for QTL 2 and QTL 9 are not sufficiently clear. For the two pairs of closely linked QTL, the Bayesian shrinkage method gained an extra power to separate one pair of QTL whereas the likelihood analysis separated neither pair.

Figure 7.
The likelihood-ratio (LR) test-statistic profile for the simulated data generated by the dynamic trait interval-mapping program. The horizontal line indicates the empirical critical value at equation M317 obtained from 500 permuted samples. The true location of each ...

Moving-interval approach:

For the same design of experiment as described in the fixed-interval experiment, we simulated data with unevenly distributed markers. This time, we placed 121 markers on the same genome with one marker at each end of the genome and the remaining 119 markers were distributed randomly. On average, there was 1 marker in every 5 cM. The positions of the 10 QTL remain the same as those in the fixed-interval experiment. With the fixed-interval approach, we would have to deal with a model with 120 QTL effects. Multicolinearity would be a problem because some markers were too closely linked. With the moving-interval approach, however, we included only 15 QTL in the model. In the beginning of the MCMC process, each QTL interval expands 40 cM of the genome (8 markers on average). Note that the length of the simulated chromosome is equation M228. With equation M229 evenly placed QTL on the chromosome, each QTL covers a range of equation M230. As the MCMC proceeded, the intervals moved from one cycle to another.

Figure 8 shows the QTL profiles for the moving-interval analysis. This time, the QTL intensity profile (Figure 8a) is much more informative than that shown in the fixed-interval approach (Figure 4a). Because of the high density of markers, all 10 simulated QTL have been identified. The Wald test-statistic profile (Figure 8d) shows that all 10 QTL are significant and nowhere else (other than the 10 QTL identified) in the genome shows peaks higher than the critical value. The Z-test profiles for each of the components of QTL effects are depicted in Figure 9. Compared with the results of the fixed-interval approach, there were less significant components, but for each simulated QTL there was at least one significant component. The estimated effects for the 10 QTL are given in Table 3. The moving-interval approach is highly recommended in real data analysis. It takes less computing time and produces better results than the fixed-interval approach. For the simulated data analysis, the fixed-interval approach took ~8 hr on a Pentium IV PC with a 3.60-GHz processor and 3.00 GB RAM while the moving-interval approach took ~3 hr on the same computer.

Figure 8.
Profiles of QTL parameters drawn from the moving-interval analysis: (a) QTL intensity profile equation M318, (b) quadratic effect profile equation M319, (c) weighted quadratic effect profile equation M320, and (d) Wald test-statistic profile equation M321. The horizontal reference line on the W plot (plot ...
Figure 9.
The Z-test-statistic profile for each of the four components of the QTL effect equation M323 from the moving-interval analysis, where equation M324 is the component for order k of the polynomial. The reference line on each plot is the critical value for the Z-test (equation M325).
TABLE 3
QTL parameters for the simulated data estimated from the moving-interval approach

A real-life example:

A doubled-haploid (DH) rice population consisting of 110 lines was established by anther culture from a cross between an indica rice variety Gui-630 and a japonica rice variety Taiwanjing. Ten plants from two inner rows in each plot were chosen for trait evaluation. The number of developed leaves and the length of the developing leaf on the main stem of each plant were recorded every 5–7 days, beginning from the 30th day (May 5) after the seeds were planted until the end of leaf development. The trait subject to QTL mapping was called “leaf age,” which was defined as

equation M231

(Zhou et al. 2001). The trait was measured eight times after May 5. The time points of measurement counted by the numbers of days after May 5 were

equation M232

On the basis of the DH population, a linkage map comprising 175 RFLP markers and covering a total length of equation M233 cM of the rice genome was constructed (Weng et al. 2000). This map consists of the 12 largest linkage groups for each parental map. These 12 linkage groups represent roughly the 12 chromosomes of the rice genome.

We analyzed the data with both the Bayesian shrinkage method and the ML interval-mapping procedure. In the Bayesian analysis, considering the unequal marker intervals, we adopted the variable-interval approach. We placed a total of equation M234 QTL in the whole genome. These QTL were distributed among the 12 chromosomes with the number of QTL in each chromosome proportional to the size of the chromosome. With this q-value, the average QTL interval was 17.50 cM, i.e., one QTL in every 17.5 cM. With this QTL density, we can be sure that no major QTL would be missed. The MCMC experiment was set up in the same way as that in the simulation study. For comparison, we also analyzed the data with the ML-based interval mapping (Yang et al. 2006). The empirical critical value for the significance test was obtained from 500 permutations. In both methods, the order of the Legendre polynomial was set at equation M235, the same order that we chose for the simulation study.

The Wald test-statistic (W) profile for the Bayesian method and the likelihood-ratio (LR) test-statistic profile for the ML interval-mapping method are depicted in Figure 10. Similar to the results of the simulation study, the Bayesian shrinkage method generated clear contrasts between the signals and the noises. Overall, there were eight peaks (QTL) with the W-values >equation M236. The ML-based interval mapping, however, detected only five QTL, all of which were detected by the Bayesian shrinkage method (compare Figure 10a with 10b). The fact that the Bayesian shrinkage method detected three more QTL implies that Bayesian shrinkage analysis has higher power than the ML-based interval mapping.

Figure 10.
The test-statistic profiles for QTL mapping from the rice data analysis: (a) The Wald test-statistic (W) profile generated by the Bayesian shrinkage method; (b) the likelihood-ratio (LR) test-statistic profile drawn from the maximum-likelihood interval-mapping ...

The estimated equation M237 and equation M238 obtained from the means of the posterior sample are

equation M239

and equation M240, respectively, where the values in parentheses are the posterior standard deviations. The estimated positions and effects for the eight detected QTL for the Bayesian method are listed in Table 4. Overall, the real data analysis shows the same behavior as the simulation study. The Bayesian method has successfully shrunk spurious QTL to zero, generating clear contrasts between the signals and the noises.

TABLE 4
Estimated QTL positions and effects obtained from the Bayesian shrinkage method for the leaf-age trait in the rice-crossing experiment

DISCUSSION

The Bayesian shrinkage mapping strategy was originally developed for a single quantitative trait. The study showed that a multivariate version of the shrinkage analysis works equally well for dynamic traits. The selective shrinkage property for the univariate Bayesian QTL mapping of Wang et al. (2005) is obvious, but it is not apparent for the multivariate version of the Bayesian QTL mapping. After some algebraic manipulation, Equations 7 and 8 can be rewritten as

equation M241
(18)

and

equation M242
(19)

respectively. The selective shrinkage property for the multivariate model is now obvious. The inverse of matrix equation M243 can be expressed as equation M244, where equation M245 is a matrix with the same dimension as that of equation M246 and equation M247 is the determinant of equation M248 (a scalar). When matrix equation M249 is “large” (large deviation from 0), equation M250 will be large so that equation M251 becomes “small” (close to 0). In this case, the posterior mean of equation M252 will be equivalent to the least-squares estimate (no shrinkage). When matrix equation M253 is small (close to 0), equation M254 will be small so that equation M255 becomes extremely “large.” A large equation M256 will dominate over equation M257 so that equation M258 is effectively very large, leading to a very small equation M259. A small equation M260 in turn will lead to a small equation M261 because of the relationship given in Equation 18. As a result, the value of equation M262 will be shrunk toward 0.

Shrinkage mapping differs from mapping with variable selection in that the model dimension is fixed. This means that we must predetermine the number of QTL (q) before the MCMC starts. Wang et al. (2005) defined q as the “maximum number of QTL.” This maximum number depends on the knowledge of the investigator on the traits of interest. It also depends on the marker density and the sample size. A denser marker map and larger sample size allow a larger q to be specified. Wang et al. (2005) and Zhang and Xu (2005) found that the q-value should be selected in such a way that, on average, a QTL interval covers ~45 cM. This optimal value was obtained empirically via simulation studies. For dynamic traits, the average QTL interval should be shorter because we expect the true number of QTL to be larger for a dynamic trait than for a single quantitative trait. Assuming that 45 cM is the optimal average interval for a single trait, it is reasonable to take equation M263 cM as the optimal interval for a dynamic trait, because a dynamic trait fit with a polynomial of order d is equivalent to d independent single traits. In the simulation studies, we chose the average interval of 40 cM because we knew that equation M264 is sufficient for these data, where 600 is the length of the simulated chromosome measured in centimorgans. We also tried shorter intervals, e.g., 25 cM, and the results were not much different from that when 45 cM was used (data not shown). In real data analysis, one may choose a few alternative q-values and present the results with the “optimal” q. The optimal q-value may be found by trial and error using a cross-validation technique, where we use part of the sample to estimate the parameters and test the estimated parameters using the remaining sample. We then choose the q that has the minimum prediction error.

Is the optimal q very important in the shrinkage analysis? The answer is no. It may be important for other methods, e.g., variable selection, but certainly not in shrinkage analysis. Therefore, the rule of thumb is to choose a q as large as the marker density and the sample size allow. How do we deal with the excessive QTL? By definition, the shrinkage analysis will shrink them to zero. The positions of these excessive QTL will be distributed uniformly across the regions where these QTL are started. For example, in the moving-interval analysis, several excessive QTL were trapped between QTL 2 and QTL 3, but their positions were almost uniformly distributed between the two QTL.

The order of the polynomial will determine the shape of the QTL effect trajectories. There is no reason to believe that the shape of equation M265 is the same as that of equation M266. Similarly, the shape of equation M267 may be different across different QTL. Therefore, the ideal treatment would be to use different orders for different effects. The Z-test statistic profile provides such information, which can be used to select different orders for different effects. However, it is tedious to implement the Z-test approach interactively for order selection during the MCMC process. Fortunately, the shrinkage method provides an automatic mechanism to shrink higher-order coefficients to zero. For simplicity, we may simply choose the highest possible order and use it for all QTL and let the shrinkage mechanism take care of the excessive coefficients. In addition to the order of the polynomial, the effects of the polynomial regression coefficients also determine the shape of a function. It is very likely that two polynomial functions of the same order have quite different shapes because the regression coefficients are different. If a higher-order polynomial is fit to a curve that can be adequately described by a lower-order polynomial, the regression coefficients for the orders higher than necessary will be shrunk to zero. The shape of the curve fit with the higher-order polynomial that contains extra zero regression coefficients will remain the same as that fit with the lower-order polynomial.

The SSVS approach to QTL mapping proposed by Yi et al. (2003a,b) is a method similar to the shrinkage analysis, both of which employed models with fixed dimensions. SSVS, however, used a latent indicator variable to indicate the status of a QTL: inclusion or exclusion. Wang et al. (2005) compared SSVS with the shrinkage analysis and found that the two approaches performed equally well (slightly in favor of the shrinkage method but difference barely notable). A multivariate version of SSVS has been developed by Meuwissen and Goddard (2004), which can be adopted here to map dynamic traits. Let equation M268 be the indicator variable for QTL j. Let equation M269 be the effect for the jth QTL, where equation M270 is a equation M271 vector. The prior distribution for equation M272 in SSVS is

equation M273

where equation M274 and equation M275. This is a multivariate version of the following single-variable mixture prior,

equation M276

where equation M277 here is a single variable. Although equation M278 has been suggested (George and McCulloch 1993; Yi et al. 2003a,b), a more informative prior for equation M279 is equation M280, where equation M281 is further described by a Beta prior so that its value can be inferred from the data. The multivariate SSVS discussed here is a simple extension of the univariate SSVS. The method of Meuwissen and Goddard (2004), however, goes one step further by assigning a scaled inverse chi-square prior to equation M282. As a result, equation M283 can be sampled from its posterior distribution. Furthermore, the authors set equation M284 so that sampling for equation M285 has been avoided. Since the multivariate SSVS method works for multiple-QTL mapping (Meuwissen and Goddard 2004), there is no reason to believe that it does not work for dynamic-trait QTL mapping.

An even more general curve-fitting approach, called the B-splines (de Boor 2001), may be used in place of the Legendre polynomials. The statistical method is identical to that reported here except that equation M286 is defined differently for the B-splines. If a biological curve has a complexity that must be described by a polynomial with order 5 or higher, B-splines are recommended. Especially, when the curve is irregular and there is no global trend to fit the data, B-splines will be more useful than the polynomials. With the B-splines, one can divide the entire time course by a few segments. The dividing points are called the knots. Within each segment, a polynomial curve with a particular order is fit and the orders of polynomials across different segments are the same. This is called local polynomial fitting. The number of knots and the order of the polynomial within each segment determine the shape of the entire time-course function. The overall degree of freedom is the sum of the polynomial order and the number of knots (rather than the product of the two). This allows a low degree of freedom of the B-splines to fit a more complicated curve than a global polynomial fitting. One example may be the lifetime milk production curve for dairy cattle. Daily milk production varies within seasons and across seasons. The lifetime curve usually shows many cycles, making B-spline an attractive alternative to the Legendre polynomial. Most biological curves, however, may be described by polynomials of order ≤5. Biological curves with higher order are perhaps very rare and hard to find in reality. Therefore, the Legendre polynomial analysis proposed in this study is sufficient for most dynamic traits.

The Bayesian shrinkage mapping procedure is developed on the basis of an “additive” model in the sense that epistatic effects of QTL pairs have been ignored. Yang et al. (2006) demonstrated that the Legendre polynomial can be easily extended to include epistatic effects. Their model (Equation 42 of Yang et al. 2006) can be directly adopted here. The only difference between the epistatic effect model and the additive model is that the former may have substantially more model effects than the latter. Therefore, the moving- or variable-interval approach is recommended for the epistatic effect model.

Finally, Legendre and other orthogonal polynomial fittings for a covariance function (Kirkpatrick and Heckman 1989) have been criticized by Pletcher and Geyer (1999). They gave five reasons why a polynomial analysis is not suitable for fitting a covariance function: (1) positive semidefinite property of the covariance function is not guaranteed; (2) there is no theoretical justification to choose Legendre as opposed to choosing any other orthogonal polynomials; (3) polynomials with a high degree are wiggly and do not have asymptotes; (4) polynomial analysis is nonparametric, which is inferior to the parametric analysis; and (5) parameters of polynomials have no biological interpretation. While these criticisms may be true for fitting a covariance function for the genetic effects, our polynomial fitting deals mainly with QTL effects, not covariance functions, and the QTL effects are fixed effects rather than random effects. Therefore, the criticisms do not apply to the QTL effects. Some of the criticisms do apply to the covariance function for the residuals, equation M287, which is a highly structured covariance function. However, this covariance matrix is always positive definite as long as equation M288. The biological interpretation of equation M289 is not important here in the QTL mapping problem because we are more interested in the QTL effects. In addition, we are dealing with a multiple-QTL model. When all the QTL effects have been included in the model, the time-dependent covariance may be negligible, and equation M290 should be close to zero. Finally, fitting this covariance structure is more convenient than fitting a parametric structure, e.g., autoregressive model with order 1 [AR(1)], as demonstrated in the MCMC iterations described earlier. Yang et al. (2006) compared this nonparametric fitting with the AR(1) structure for the covariance function and did not find any improvement in the parametric approach. In fact, the covariance structure described by equation M291 is more flexible than the parametric structure because we can actually choose a different degree of the polynomial to fit a covariance structure with a different degree of complexity. If the polynomial order is as high as the number of time points, the covariance function is fully unstructured (maximum degree of complexity). If the polynomial order is one or zero, the covariance function is highly structured, e.g., a diagonal covariance function for a zero degree of polynomial. The same MCMC sampling algorithm applies to all degrees of polynomials and thus to a covariance function with arbitrary complexity. This flexibility is actually a desirable property rather than a property for criticism.

Acknowledgments

We are grateful to two anonymous reviewers for their helpful comments on an earlier version of the manuscript. This research was supported by the National Institutes of Health grant R01-GM55321 and the National Science Foundation grant DBI-0345205 to S.X. and by the Chinese National Natural Science Foundation grant 30471236 to R.Y.

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