Volume 4 Supplement 1
Genetic Analysis Workshop 13: Analysis of Longitudinal Family Data for Complex Diseases and Related Risk Factors
Locating disease genes using Bayesian variable selection with the Haseman-Elston method
- Cheongeun Oh^{1},
- Kenny Q Ye^{2}Email author,
- Qimei He^{3} and
- Nancy R Mendell^{2}
DOI: 10.1186/1471-2156-4-S1-S69
© Oh et al; licensee BioMed Central Ltd 2003
Published: 31 December 2003
Abstract
Background
We applied stochastic search variable selection (SSVS), a Bayesian model selection method, to the simulated data of Genetic Analysis Workshop 13. We used SSVS with the revisited Haseman-Elston method to find the markers linked to the loci determining change in cholesterol over time. To study gene-gene interaction (epistasis) and gene-environment interaction, we adopted prior structures, which incorporate the relationship among the predictors. This allows SSVS to search in the model space more efficiently and avoid the less likely models.
Results
In applying SSVS, instead of looking at the posterior distribution of each of the candidate models, which is sensitive to the setting of the prior, we ranked the candidate variables (markers) according to their marginal posterior probability, which was shown to be more robust to the prior. Compared with traditional methods that consider one marker at a time, our method considers all markers simultaneously and obtains more favorable results.
Conclusions
We showed that SSVS is a powerful method for identifying linked markers using the Haseman-Elston method, even for weak effects. SSVS is very effective because it does a smart search over the entire model space.
Background
In this work, we analyzed the slope of the cholesterol increase with age in the simulated data (Problem 2). Our objective was to identify the markers that are linked to the disease genes related to a high rate of increase in cholesterol. Genetic Analysis Workshop 13 provided information that the disease genes are located on chromosomes 7(s7), 15(s8), and 21(s9), respectively, and that the gene on chromosome 21(s9) only affects cholesterol rate in the females, i.e., it interacts with gender. The Haseman-Elston [1] method allowed one to apply linear regression methods for linkage analysis. For each sibling pair, it used the number of alleles identical by descent (IBD) at each marker as the explanatory variables and a statistic measuring similarity of values of the quantitative traits in the sibling pair as the response variable. The original Haseman-Elston method [1] used the squared difference between the traits of the siblings. In a recent publication, Elston et al. [1] proposed the cross-product of the two trait values in a sib pair as the response, which was used in this paper. Suh et al. [2] applied Stochastic Search Variable Selection (SSVS), a Bayesian variable selection method proposed by George and McCulloch [3] for the linear regression model, to the Haseman-Elston method. Although the scope of Suh et al was very preliminary, with only the IBD values at the linked markers plus 10 unlinked markers used as candidate explanatory variables in the variable selection, it showed the Bayesian variable selection approach to be very promising. The study presented here extended these methods in two respects. First, we took advantage of SSVS by including all 399 markers as candidate explanatory variables. It is computationally impossible to consider all subsets of 399 markers using a traditional frequentist approach. Secondly, a hierarchical prior probability structure as discussed by Chipman [4] was imposed on the model space to study the interaction effects (epistasis). The results were reported and compared with those obtained with the more traditional forward and backward step-wise regression.
Methods
Haseman-Elston method
We chose to analyze the rate of change in cholesterol over time in the simulated data. First, for each individual, we obtained the least square (LS) estimate for the slope of cholesterol over the time. For the i^{th} sibling pair, using the LS estimate of slope as the trait (Y_{1i}, Y_{2i}), we computed their cross-product CP_{ i }= (Y_{1i}-m)(Y_{2i}- m) as our response values, where m is the mean of the slopes over all siblings in the same family. Elston et al. [1] introduced the cross-product CP, as the replacement of the squared difference . In our regression analysis, we adopted CP as the response, and also used squared-difference for comparison. For simplicity, we assumed the errors to be independent but a correlation structure could be implemented into our method in a straightforward way.
There are about 1500 full sib pairs and a few half sib pairs in each replicate. In the replicate we considered there are 1522 full sib pairs. The number of alleles shared in each pair was obtained for each sib pair at each marker using the SIBPAL program of the SAGE software [5]. There were a total of 399 markers. We had
where the ε ~ N(0, σ^{2}) were assumed to be independent and X values were IBD scores.
To study the effect of gender, we also included the genders of the siblings as an explanatory variable. It was in fact coded as two dummy explanatory variables as follows: (male, male) = (0, 0), (male, female) = (0,1), and (female, female) = (1,1).
SSVS
George and McCulloch [3] proposed a Bayesian model selection method for variable selection based on the Gibbs sampler. The criterion of interest was taken to be the posterior probability of a model conditional on the data that could be obtained using the stochastic search variable-selection. For the simplest case of linear regression with normal errors:
Y = X' β + ε, ε ~ N(0, σ^{2}I),
where β may contain main effects or interactions effects. They set the prior distribution of β as mixtures of two normal distributions by introducing the latent variable γ:
β_{ k }|γ_{ k }~ (1 - γ_{ k }) N (0, τ^{2}) + γ_{ k }N (0, c^{2} τ^{2}),
where much larger variance (c > 1) allowed for γ_{ k }= 1 to have a large influence. A recommended choice for these parameter values is given by George and McCulloch [3]. The value of c was set equal to 10 in our analysis. A model was represented by a vector γ = (γ_{1}, γ_{2},..., γ_{ p }), where γ_{ k }= 0 or 1. If γ_{ k }= 0, then the marker X_{ j }was considered to be excluded from the model and if γ_{ k }= 1, it was considered to be included in the model. Note that β_{0} was taken to be always included, thus we could set β_{0} N((0, c^{2}τ^{2}). With appropriate prior on γ = (γ_{1}, γ_{2},..., γ_{ p }) and σ^{2}, we obtained a posterior distribution of γ using Gibbs sampling. Therefore, by examining the posterior probability of γ, we identified the optimal model with the largest posterior probability and rank the markers using the marginal distribution of each γ_{ k }. A prior for γ corresponds to a prior on the model. The commonly used independence prior implies that the importance of any variable is independent of any other variable. In other words, under this prior, each X_{ i }enters the model independently of the other coefficients, with probability p(γ_{ i }= 1) = 1 - p(γ_{ i }= 0) = p_{ i }. A smaller p_{ i }can be used to downweight X_{ i }values that are costly or of less interest. For our case, a useful reduction was to set p_{ i }= p, in which p is the a priori expected proportion of X_{ i }values in the model. When only main effects but no interaction were considered, the importance of any variable was independent of the importance of any other variable. Thus the independence prior implied that the prior of γ was simply set as prob(γ) = p^{n}, where n is the number of ones in γ. Increased weight on parsimonious models could instead be obtained by setting p small. So in our case, p was set to be small, 0.02 first, and next to see how our method is robust to this choice of p, we chose a new value of p = 0.002 for comparison. The details on the MCMC algorithms can be found in George and McCulloch [3].
We applied SSVS to select markers linked to cholesterol rate from all 399 markers under consideration. Since it is impractical to track the complete posterior of γ, only the marginal posterior of each marker is obtained. Although both posterior probability of the models and marginal probabilities of each marker are sensitive to the prior settings, especially c and p, we showed that the ranking of the marginal posterior of the markers are not.
Figure 2 illustrates the robustness through plots of the ranking of the markers obtained using two different priors p = 0.02 and 0.002. Other prior settings showed similar high correlations in the rankings of the markers.
Hierarchical prior structure
When interaction effects (epistasis) are considered in the model selection, the model space becomes enormous and the common independence prior for γ is not appropriate anymore. With interactions, the prior for gamma can capture the dependence relation between the importance of a higher order term and those lower order terms from which it was formed. Chipman [4] proposed a hierarchical prior structure for this model space. The importance of the interactions such as X_{ i }X_{ j }will depend only on whether the main effects X_{ i }and X_{ j }are included in the model. This belief can be expressed by a prior for of the form
The probability that the term X_{ i }X_{ j }is active may take on four different values, depending on the values of the pair .
In our analysis, we set (p_{00}, p_{01}, p_{10}, p_{11}) = (0, 0, 0, p). This corresponded to the prior belief that if the interaction effect between two factors exists in a model, the main effects of the two factors must be included in the same model.
Our study was conducted in two stages. At the first stage, all 399 candidate markers and gender were the candidate variables in SSVS, but interactions were not considered. At the second stage, SSVS was applied to the same sib-pair responses with the top 30 candidate variables selected from the first stage and their interactions as the candidate variables. Among the third were the gender and 29 markers. This brought the total number of candidate variables in SSVS to 465. We chose only the top 30 variables from the first stage for two reasons. First, it is reasonable to assume that only a few linked loci exist and they should be contained in the top 30. Second, this is the maximum size that the current SSVS algorithm handles comfortably in the second stage.
Step-wise regression
In order to compare the traditional method to our method, we used a step-wise method based on Akaike information criterion (AIC) [6] to select a formula-based model, which was implemented under R, the "GNU S". The details of this method can be found in the R manual [7].
Results
Only the first of the 100 simulated data sets was used. Figure 1 displays the marginal posterior of each marker obtained from SSVS with all 399 markers but no interactions. The marginal posterior was computed from the relative frequency of each markers in the MCMC sample of γ. It clearly showed that the high posterior values are concentrated on chromosomes 7, 15, and 21. Table 1 shows the top 30 markers, a marker from chromosome 7 is rated as most significant, and there are seven, four, and two markers from chromosomes 7, 15, and 21, respectively. The variable gender was ranked as 15^{th}. Table 2 shows the most significant 20 markers obtained from the univariate LS regression and from the step-wise regression. These markers were very much evenly distributed in all chromosomes.
Top rank markers based on their marginal posterior probability.
Posterior Ranking | Markers |
---|---|
1 | M160 (chr7) ^{A} |
2 | M254 |
3 | M315(chr15) |
4 | M173(chr7) |
tied at 5 | M336, M325, M387(chr21) |
tied at 8 | M260, M257, M201, M169(chr7) |
tied at 12 | M385, M351, M306(chr15) |
tied at 15 | 400(gender), M367, M281, M279, M182 |
tied at 19 | M175(chr7), M120 |
tied at 22 | M391(chr21), M178(chr7) |
tied at 24 | M309(chr15), M179(chr7) |
tied at 26 | M311(chr15) |
tied at 27 | M164(chr7) |
Top 20 markers selected in step-wise regression and univariate regression.
Chromosome | Step-wise | Univariate |
---|---|---|
1 | M13^{A} | M3 |
2 | M57^{A} | M43 |
3 | M64^{A}, M69^{A}, M70^{A}, M81^{A} | |
4 | ||
5 | M114^{A} | M114^{A}, M128^{A} |
6 | M142^{A} | M142, M158^{A} |
7 | M162^{A} | M169^{A} |
8 | M182^{A}, M193^{A} | M182^{A} |
9 | M211^{A}, M216^{B} | M211^{A}, M216^{B} |
10 | M227^{A} | |
11 | M244^{A}, M253^{A} | M257^{A} |
12 | ||
13 | M278^{C} | M278^{B} |
14 | ||
15 | M313^{A}, M318^{A} | M318^{A} |
16 | M341 | |
17 | M347^{A} | M347^{B} |
18 | M359^{A} | M355^{A} |
19 | M372^{C} | |
20 | ||
21 | M389^{B} | |
22 | M397^{A} |
The same analysis was also carried out on the third simulated data set and similar results were obtained.
Conclusion
We showed that SSVS is a powerful method in identifying linked markers using the Haseman-Elston method, even for weak effects. SSVS is very effective because it does a smart search over the entire model space, while the frequentist best subset model selection procedures are constrained by computing power required to examine all candidate models. The former can work on problems with many more candidate variables, which is essential when interaction effects are studied. By using the prior structures that reflect the relation among the candidate variables, SSVS can accommodate a good number of candidate markers as well as their interactions. The two-stage strategy used in this study worked well. It identified the chromosomes of the linked markers in the first stage and the interaction effects were located in the second stage. Both univariate regression and the step-wise regression failed to identify the chromosomes of the linked markers.
Discussion
Authors’ Affiliations
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