MULTIPLE-QTL MODEL

We describe our method primarily for a mapping population with only two segregating genotypes, e.g., a backcross, double-haploid (DH), or recombinant inbred line (RIL). The method can be extended to other types of population. Assume that we observe many markers across the genome. The aim of QTL mapping is to identify which markers are tightly linked to genes with detectable effects and to estimate the magnitudes of the effects. For a continuously distributed trait, the observed phenotypic value y_i of individual i can be described by the linear regression model

(1)

where p is the number of markers, is the overall mean, denotes the genotype of marker j for individual i and is defined as −0.5 and 0.5 for the two genotypes in the mapping population, the coefficient represents the main effect of marker j, is the residual error assumed to follow a distribution, , and .

In practice, some marker genotypes may be missing. There are two commonly used methods to deal with missing marker data. The first approach takes the uncertainty in missing genotypes into account by treating the missing genotypes as unknowns and sampling them in the MCMC update procedure. The second approach is similar to the regression model of Haley and Knott (1992) and replaces all missing genotypes by their expected values conditioning on the observed marker data, using the multipoint method (Jiang and Zeng 1997). Although the second method ignores the uncertainty of the estimated marker genotypes, it has a big computational advantage over the first method. Both approaches can be incorporated into our Bayesian method. For simplicity and computational convenience, this study considers only the second method.

PRIOR DISTRIBUTIONS

Data are usually sufficient to estimate the overall mean and the residual variance . Thus, we can use any reasonable noninformative prior distributions for these parameters. For example, we can use an independent, flat prior for , i.e., , and a noninformative scale-invariant prior 1/ for .

QTL data typically contain a large number of markers, leading to many coefficients in model (1). However, most of the coefficients are expected to have no or only weak effects on the phenotype. Only a few coefficients may have notable effects (see, e.g., Xu 2003). To incorporate this prior knowledge into our analysis, therefore, we set up a prior distribution that gives each coefficient a high probability of being near zero and in the meantime gives each coefficient a chance to take a large effect.

A commonly used prior that has the above properties is the double exponential (also called Laplace) distribution

(2)

(Tibshirani 1996; Park and Casella 2007), where is a hyperparameter. With the double-exponential prior, the posterior mode estimate of the coefficients is the LASSO estimate (Tibshirani 1996; Park and Casella 2007). Another wide-tailed distribution is the well-known Student's t distribution,

(3)

where the hyperparameters , , and are the degrees of freedom, the location, and the scale parameters, respectively (see, e.g., Sorensen and Gianola 2002; Gelman et al. 2003; Varona et al. 2005).

Both the double-exponential distribution and Student's t distribution can be presented as a two-level hierarchical model (see, e.g., Andrews and Mallows 1974; Griffin and Brown 2006). Compared with their original forms, the two-level hierarchical models are easier to deal with both analytically and computationally. In the two-level hierarchical models, the first level assumes that the coefficients follow independent normal distributions with mean zero and unknown variances ,

(4)

where . At the second level, we assume that the variances follow some specified independent prior distributions

(5)

where includes all the hyperparameters. The above two-level priors result in a scale mixture of normal distributions for the coefficients ,

(6)

where is referred to as the mixing distribution. From this, we see that two-level hierarchical models introduce different variance parameters for different coefficients and hence induce different amounts of shrinkage in the estimates of different coefficients. The variance components are not the parameters of interest, but they are useful intermediate quantities to facilitate better inferences for the individual regression coefficients (e.g., Gelman 2005).

The double-exponential distribution (2) can be presented as a mixture of normal distributions (4) with the mixing distribution being an exponential distribution Expon() or equivalently a gamma distribution Gamma(1, ),

(7)

where is the inverse scale that needs to be carefully preset or estimated from the data.

For any values of , the mode of the exponential distribution Expon() = 0, meaning that the “most likely” values of and thus are zero. However, the variance of is , strongly depending on the value of . Instead of presetting a value for , it is appealing to assign a prior distribution to also so that can be estimated along with other parameters. This procedure obviates the choice of and automatically accounts for the uncertainty in its selection that affects the estimates of regression coefficients (Park and Casella 2007). For convenience, we regard , instead of , as the parameter of interest. Following Park and Casella (2007), we assign a conjugate gamma prior Gamma(a, b), a > 0, b > 0, to the parameter . Assigning a prior distribution to provides a way to estimate . We set a and b as small values (e.g., a = 0.1 and b = 0.1) so that the prior for is essentially noninformative.

Student's t distribution (3) can be expressed as a mixture of normal distributions (4) with the mixing distribution being a conjugate scaled inverse- distribution or equivalently an inverse gamma distribution Inv-gamma(, ),

(8)

where the hyperparameters and represent the degrees of freedom and the scale of the distribution, respectively.

The amount of shrinkage in the prior (8) depends on the values of the hyperparameters and . The improper distribution , which is equivalent to the uniform distribution on , has the form of the inverse- density with and and can be taken as a limit of the proper inverse- or inverse-gamma densities. This improper prior in turn induces an improper prior for of the form (Griffin and Brown 2006). In the class of inverse- densities, has the constant degree of shrinkage in the coefficient estimates. Xu (2003) actually used this improper prior in his shrinkage QTL mapping.

We here treat and as unknown parameters, obviating the choice of (, ) and automatically accounting for the uncertainty in its selection that affects the estimates of the regression coefficients. We assign a uniform density on 1/ for the range (0, 1] and a uniform distribution on for the range (0, A] with A being a large number (see Gelman et al. 2003).

The prior (4) on the coefficients is independent of the residual variance and has been commonly used in the literature. Park and Casella (2007) recently proposed an alternative version of the prior on that is dependent on the residual variance ,

(9)

where . At the second level, the prior distributions on take the same form as (7) or (8). This conditional prior may have some advantages; for example, it results in that is unitless and posteriors induced from (9) generally do not have more than one mode (Park and Casella 2007).

We now have introduced two different priors for and two different priors for . The two priors for are and , respectively. The two priors for are and , respectively. This yields four possible combinations and thus four different models. These four models are called model I with prior , model II with prior , model III with prior , and model IV with prior .

POSTERIOR COMPUTATION

We fit the models using the MCMC algorithm, applied to the joint posterior distribution of all the parameters . The joint posterior distribution can be expressed as

(10)

where , , , represents for models I and II and () for models III and IV, is the normal density function , and other terms in the right-hand side are the priors defined in the last section. For notational convenience, we suppress the dependence on X here and afterward.

With the two-level hierarchical models as set up in the last section, the joint posterior distribution can be simulated using the MCMC algorithm, alternately updating from the normal distribution , updating each from the normal distribution , where represents all elements of except , updating from the inverse- distribution , updating each from the inverse Gaussian (for models I and II) or inverse- distribution (for models III and IV), and updating from the gamma distribution (for models I and II), or updating from the gamma distribution and updating from , using a Metropolis step (for models III and IV).

The two-level hierarchical modeling allows us to easily derive the above conditional posterior distributions (see appendix a). For all the four proposed models, however, the above MCMC algorithms can be performed directly using the Bayesian software WinBUGS14 (Spiegelhalter et al. 2002), and thus we actually do not need to derive the conditional posteriors. However, the expressions of the conditional posteriors allow us to develop faster, purpose-built programs. The WinBUGS code implementing these four hierarchical models can be obtained by request from the first author.

The MCMC algorithm proceeds to draw each unknown from its fully conditional posterior distribution, given the current values of all other unknowns and the observed data. Each iteration of the MCMC algorithm cycles through all elements of . This process continues for a large number of iterations to obtain random samples from the joint posterior distribution. We run multiple parallel chains with overdispersed starting points and use the potential scale reduction factor that compares the between- and within-sequence variances for any scalar estimands, to monitor convergence (Gelman et al. 2003). We use the condition of < 1.1 for all scalar estimands of interest as the criteria of convergence. Once our simulations have converged, we continue to draw thousands of simulations and treat them as a sample from the joint posterior distribution. We thin the sequences by keeping every kth simulation draw from each sequence and discarding the rest (e.g., k = 100).

POSTERIOR SUMMARY AND INTERPRETATION

The proposed Bayesian hierarchical models can provide various graphical and tabular summaries that assess the contribution of individual loci while adjusting for effects of all other possible loci. One of such summaries is the estimate of the genetic effects associated with each marker. Furthermore, using the posterior samples of and the observed values , we can also calculate the proportion of the phenotypic variance explained by each effect (i.e., heritability), , where V_y is the phenotypic variance and is the sample variance of . The heritability is independent of the scale of the phenotype, while the effect size depends on it. The fully Bayesian analysis usually uses the posterior median or mean to estimate and and also can provide the posterior interval for the estimate.

The Bayesian hierarchical models shrink the “insignificant” coefficients to zero, and hence the posterior estimates of and can guide variable selection. However, these posterior summaries do not automatically provide a criterion to declare that an effect is “in” the model. Hoti and Sillanpää (2006) suggest setting a threshold value, c, such that the standardized effect is included if . Note that the summary statistic, heritability, is essentially the same as the standardized effects in Hoti and Sillanpää (2006). Yang and Xu (2007) proposed using a Wald test statistic.

We here construct an alternative measurement that is similar to the widely used LOD score in the traditional QTL mapping approach if the phenotype is affected by a single QTL (Lander and Botstein 1989), following the idea of Yandell et al. (2007) (also see the vignette in the package R/qtlbim). We call this statistic the approximate Bayesian LOD score. The approximate Bayesian LOD score can be conveniently compared with the LOD curve obtained from the traditional interval-mapping approach. Our Bayesian LOD score considers the contribution of a given locus to the LOD after adjusting effects for all other possible loci. For marker j, the Bayesian LOD score (BLOD) is calculated as

(11)

where the values of , , and are the simulated draws in a certain iteration, and , the mean of the conditional posterior . Note that although we adjust for the residual error variance, we use the simulated draws under the full model (including ) to calculate the likelihood function for the reduced model (without ).

REAL DATA ANALYSIS

The well-known barley data set from the North American Barley Genome Mapping Project (Tinker et al. 1996) was analyzed using the four proposed Bayesian hierarchical models. This data set has been extensively analyzed using different methods (Tinker et al. 1996; Xu 2003, 2007; Yi et al. 2003, 2005; Zhang et al. 2005; Xu and Jia 2007). The data were collected from a doubled-haploid population that contained n = 145 lines; each was grown in 25 different environments. The phenotype analyzed was the average value of kernel weight across environments. A total of p = 127 mapped markers covering a genome of 1500 cM (seven chromosomes) were used in the analysis. Observed marker genotypes were coded as −0.5 for one genotype and +0.5 for the other genotype. The data contain ∼5% missing marker genotypes. The missing marker genotypes were replaced by their expected values conditioning on the observed marker data using the multipoint method. We simultaneously modeled all 127 marker effects in the linear regression (1).

For all four proposed models, the MCMC simulations were performed in the Bayesian software WinBUGS14 (Spiegelhalter et al. 2002), linked from the R statistical computing software (Sturtz et al. 2005; R Development Core Team 2006), which we can use to conveniently plot the results. For each analysis, we ran three parallel simulation sequences with starting points randomly generated from the prior distributions. The models with the dependent prior (9) (models III and IV) were found to converge slightly quicker than those with the independent prior (4) (models I and II). For all four models, the potential scale reduction factor values fell below 1.1 for all parameters after ∼5000 iterations, indicating that the chains converged adequately (data not shown). For each of the three parallel sequences, therefore, we ran a total of 20,000 iterations and discarded the first halves to allow adequate convergence. The remaining 10,000 posterior samples of each sequence were thinned to 1000 for estimating the posterior distribution of quantities of interest by keeping every 10th simulation draw from each sequence and discarding the rest. On a dual processor 2.4-GHz machine, each analysis took ∼4.5 hr.

To investigate whether or not posterior inferences are affected by the values of the shape a and the inverse scale b in the gamma prior Gamma(a, b), we analyzed models I and II with four different values of (a, b), a = b = 0.01, 0.1, 0.5, 1. Under the four analyses, the posterior estimates of the parameters were identical, but the inverse scale differed slightly.

Figure 1 shows the posterior medians and the 95% posterior intervals for the marker effects and the proportion of the phenotypic variance explained by each effect (i.e., heritability) . Models I and II produced almost identical results, both being able to identify markers with large effects. The analyses showed four chromosomes (7, 1, 3, and 2) with strong evidence of QTL. The QTL on chromosome 3 was found to have an opposite effect on the phenotype to the other detected QTL and was missed in the analysis of Xu (2003), who used the improper prior . The difference may also be caused by the different ways of treating missing marker genotypes for the two methods. As can be seen in Figure 1, the plots of the heritability can provide a clearer picture than those of the effect.

Figure 2 shows the histograms of the inverse scale for the analyses of models I and II with a = b = 0.1. We can see that the hyperparameter can be estimated with quite high precision. For model I, the inverse scale parameter was estimated at 32.0, with a 95% posterior interval of [20.6, 50.5]. However, model II provided a much smaller estimate, with a posterior median of 5.1 and a 95% posterior interval of [2.8, 9.8]. We know that in the exponential distribution a larger inverse scale forces stronger shrinkage. Figure 1 shows that models I and II perform at the same level of shrinkage in the coefficient estimates. This can also be shown by looking at the estimated , which was 0.3, <1.

Figure 3 depicts the posterior medians and the 95% posterior intervals for marker effect and the heritability from the analyses of models III and IV. Models III and IV produced identical results, similar to those from models I and II. However, the Inv- prior distributions on produced slightly larger posterior medians and wider posterior intervals for and . The degrees of freedom and scale parameters can be estimated from the data. Figure 4 shows the histograms of these two hyperparameters for the analyses of models III and IV. The two models produced similar estimates for the degrees of freedom, with the posterior median at 2.1 and the 95% posterior interval of [1.2, 5.3]. However, the estimate of the scale parameter for model IV was three times as large as that for model III.

Figure 5 (top) shows the posterior medians and the 95% posterior intervals for the Bayesian LOD score for model I. The other three models produced similar estimates (data not shown). For comparison, Figure 5 (bottom) also shows the LOD score curve calculated by the traditional interval mapping based on a single-QTL model. If we use the standard threshold value of 3.2, the interval mapping identified only two regions (on chromosomes 7 and 1, respectively), which have larger effects in our analyses. Our Bayesian models produced much higher estimates of LOD score and detected more significant QTL.

In the proposed two-level hierarchical models, the amount of shrinkage in the coefficient estimates is largely determined by the hyperparameters in the priors. It is expected that optimal values of the hyperparameters depend on the data and the number of variables included, which makes it difficult to preset their values. To illustrate this, we analyzed the data with only markers on chromosomes 1, 3, and 7 included in the model. Figure 6 displays the plots of marker effects, showing that we were still able to detect the same QTL as in our previous analyses. As shown in Figure 7, however, the posterior estimates of the hyperparameters were different from those in the previous analyses.

DISCUSSION

We have presented four Bayesian hierarchical models that can be used to simultaneously fit and estimate all possible genetic effects associated with all molecular markers across the entire genome. We found that these four methods perform equally well. It is known a priori that most markers have no or little effect on the phenotype, but if a marker has an effect, it could be large. To incorporate this prior knowledge into our analysis, we set up a continuous form of prior distribution for genetic effects that gives each effect a high probability of being near zero and also enables us to accommodate large effects. We have considered two such priors—double exponential distribution and Student's t distribution—that can be expressed as scale mixtures of normal distributions with mean zero and unknown variances distributed as exponential and scaled inverse- distributions, respectively. The former prior results in a Bayesian LASSO model (Tibshirani 1996; Yuan and Lin 2005; Park and Casella 2007), and the latter includes Jeffrey's noninformative prior as a special case (Gelman 2006).

Both the exponential and the scaled inverse- distributions include hyperparameters that determine the degrees of shrinkage in the estimates of the variances and thus the genetic effects. A common practice is to preset values for the hyperparameters using methods like cross validation (Tibshirani 1996; Meuwissen et al. 2001; Xu 2003; Bao and Mallick 2004; Xu 2007). However, optimal values depend on the data and thus may be difficult to choose. A novel aspect of our approach is to treat all hyperparameters as unknowns and estimate them along with other parameters. This procedure allows us to control the amount of shrinkage by taking advantage of the characteristics of the data. In contrast, Jeffrey's prior includes no hyperparameter but always forces a constant degree of shrinkage. Xu (2003) showed that Jeffrey's prior yields good performance, but we observed that it converges more slowly than our proposed methods.

We fit the models in a fully Bayesian approach, employing the MCMC simulation to generate posterior samples from the joint posterior distribution, which can be used to make various posterior inferences. Although computationally intensive, it is easy to implement and provides not only point estimates but also interval estimates of all parameters. We have proposed graphical tools to summarize the posterior samples, notably the approximate Bayesian LOD score that can be easily compared with the results from traditional interval mapping. However, we realize that these tools still are informal and descriptive, due to the lack of formal threshold value to select markers. A formal choice of threshold value will be a topic of future research.

The fully Bayesian approach enables us to obtain much richer inferences about the models than most non-Bayesian analyses. In practice, however, it is desirable to have a quicker calculation that merely looks for posterior modes rather than fully investigating the posterior distribution. The original LASSO estimates for linear regression coefficients are equivalent to Bayesian posterior mode estimates when the coefficients have independent and identical double-exponential priors (Tibshirani 1996). The two-level hierarchical formulation enables us to find posterior modes using various algorithms, e.g., conditional maximization (Zhang and Xu 2005), the EM algorithm, and its extensions (Figueiredo 2003; Foster et al. 2007; Xu 2007). However, these methods have been developed on the basis of preset hyperparameters. Further investigation is necessary to extend the mode-searching algorithms to the proposed hierarchical models.

The proposed models have been used to fit all markers and estimate their effects. The methods can be easily extended to detect QTL within marker intervals. The idea is to insert loci within each marker interval as possible positions of QTL and put a reasonable number of loci on each chromosome with positions treated as parameters (Wang et al. 2005). An additional Metropolis step is then used to update the positions. Future research also will consider epistatic effects and extensions to complicated experimental crosses. Computationally efficient algorithms are an essential feature for the practical analysis of these more complicated cases. We are in the process of developing purpose-specific computer programs that would significantly reduce the computing time. Our hierarchical models assign marker effects independent priors with a single prior mean. However, it is desirable to develop hierarchical priors to allow shrinkage of coefficients toward multiple prior means with the locations of these means unknown and to accommodate the spatial covariance structure along markers (Gianola et al. 2003).