Abstract
Background
The goal of linkage analysis is to determine the chromosomal location of the gene(s) for a trait of interest such as a common disease. Threelocus linkage analysis is an important case of multilocus problems. Solutions can be found analytically for the case of triple backcross mating. However, in the present study of linkage analysis and gene mapping some natural inequality restrictions on parameters have not been considered sufficiently, when the maximum likelihood estimates (MLEs) of the twolocus recombination fractions are calculated.
Results
In this paper, we present a study of estimating the twolocus recombination fractions for the phaseunknown triple backcross with two offspring in each family in the framework of some natural and necessary parameter restrictions. A restricted expectationmaximization (EM) algorithm, called REM is developed. We also consider some extensions in which the proposed REM can be taken as a unified method.
Conclusion
Our simulation work suggests that the REM performs well in the estimation of recombination fractions and outperforms current method. We apply the proposed method to a published data set of mouse backcross families.
Background
Molecular genetics has made much progress in recent years, among which linkage analysis fulfills an important role. Genetic linkage refers to the ordering of genetic loci on a chromosome and to estimating genetic distances among them, where these distances are determined on the basis of a statistical phenomenon. Statistical machinery has been used to analyze family data and to detect linkage [14]. The degree of linkage can be measured by recombination fraction. The proportion of recombinant haplotypes (or offspring) potentially produced by a doubly heterozygous parent is called recombination fraction, which is also the probability of occurrence of a recombination. Many map functions under different assumptions have been derived [57], from which the genetic distance and the recombination fraction can be mutually transformed. Human gene mapping is now an important field of science. A critical first step in finding gene loci that contribute to a genetic trait is to demonstrate linkage with a gene of known location (marker). So estimating the recombination fractions is important in linkage analysis.
In several respects, threelocus analysis yields more information than does twolocus analysis [811]. Threelocus linkage analysis is also an important case of multilocus problems. Methods for detecting multilocus linkage in humans and estimation of recombination have been proposed by Lathrop et al. [12], and Lathrop [13]. More recently, Ott [3] has considered the estimation of twolocus recombination fractions for phaseunknown triple backcross families with two offspring in each family. The author gave the presentations of the estimates of the twolocus recombination fractions. Wu et al. [9] considered simultaneous estimation of linkage and linkage phases in outcrossing species. However, as mentioned in Ott [3], the estimates suggested by the author may not satisfy some natural restrictions which twolocus recombination fractions should satisfy in fact. One may not obtain a reasonable interpretation on the recombination phenomenon among loci based on the estimates. Furthermore, illegimate estimates of recombination fractions may also reduce the power to detect linkage which can provide irresponsible evidence to the researchers. In addition, the restrictions on recombination fractions given in the context are necessary in linkage analysis. For example, they can be applied to determine the locus order on the chromosome [911].
This estimation problem of twolocus recombination fractions in threelocus linkage analysis belongs to the constrained parameter problems which are not only important but also appear in many areas. The reader is referred to [1417]. However, the methods provided in the literatures cannot be directly applied to the above genetics problem.
Motivated by this unsolved problem that the restrictions on recombination fractions have not been considered sufficiently, in this paper, we consider the estimation of the twolocus recombination fractions under some natural and necessary restrictions. We develop a restricted EM algorithm, called REM, which gives estimating results through taking account of the natural inequality restrictions on the twolocus recombination fractions, and the algorithm has been implemented by computer. Moreover, this algorithm can be easily generalized to other cases, and the REM performs well as a unified approach. Simulation studies show that our new method works well in each scenario and has advantages over current method, in other words, the major advantages of our method is its robustness and efficiency. An example is used to validate the application of our method to linkage analysis.
Methods
Consider three biallele marker loci, where alleles are designed as A, a; B, b; C, c at loci A, B, C, respectively, with the order of loci being ABC. Assume a triply homozygous parent abc/abc, and a triply heterozygous parent (A/a, B/b, C/c). For the latter, there are four possible phases: (I) ABC/abc, (II) ABc/abC, (III) AbC/aBc, (IV) Abc/aBC. As Ott [3] pointed out, under regular conditions (linkage equilibrium), each of these phases occurs with probability 1/4. When it is not the case, we let the prior probability be h_{i }(i = 1, 2, 3, 4) in a later section, and give corresponding feasible approach.
Each offspring only receives haplotype abc from the triply homozygous parent, but receives one of the eight possible kinds of haplotypes from the heterozygous parent, which can be seen at the second column of Table 1. The last four columns of Table 1 give the conditional probabilities with which the offspring phenotypes occur given the parental phase, and the first column presents the code for each haplotype that we will use. For the phaseunknown triple backcross, each haplotype symbol listed in Table 1 just corresponds to one offspring phenotype of the markers.
Table 1. Conditional haplotype probabilities given phase produced by a triply heterozygous parent
Let θ_{AB}, θ_{BC }and θ_{AC}, respectively denote twolocus recombination fractions between loci A and B, between loci B and C, and between loci A and C; g_{00}, g_{01}, g_{10 }and g_{11 }denote joint recombination fractions, where the subscript 1 represents recombination, and 0 represents nonrecombination, e.g., g_{10 }is the probability of single recombinant with a recombination for loci A and B but none for loci B and C. So it is clear that the following equations hold:
Ott [3] groups all possible twooffspring haplotype pairs into four phenotype classes with probability p_{k }(k = 1, 2, 3, 4) according to linkage analysis regulation. These classes are reproduced in Table 2, in which the second column represents twooffspring haplotype pairs, corresponding to two phenotypes. Taking (i, j) = (5, 6) as an example, we say one of the sib pair expresses phenotype aa/Bb/Cc, and the other expresses phenotype aa/Bb/cc. There is no order relationship between i and j. The probabilities of occurrence for all 8 × 9/2 = 36 possible pairs of offspring's phenotypes can be calculated easily, e.g., the joint probability of occurrence of phenotypes aa/Bb/Cc and aa/Bb/cc (diplotypes aBC/abc and aBc/abc) is (g_{11}g_{10 }+ g_{01}g_{00})/4. It then turns out that, among the 36 probabilities, only four different values occur so that phenotypes with the same probabilities may be combined a single class and four classes are obtained.
Table 2. Phenotype classes for phaseunknown triple backcross families with two offspring
Let the total number of families (or sib pairs) observed be n, and the number of families which are grouped into class k be n_{k }(k = 1, 2, 3, 4). Then (n_{1}, n_{2}, n_{3}, n_{4}) is multinomial distributed with probability (p_{1}, p_{2}, p_{3}, p_{4}), and . The MLEs of p_{k}'s are (k = 1, 2, 3, 4). Using the function relationships given in equations (1) and Table 2, as well as the property of MLE, the MLEs of θ_{AB}, θ_{BC }and θ_{AC }can be obtained as Ott [3]. We call this method the unrestricted method that gives unrestricted estimates, and let denote the unrestricted MLE, where
Natural inequality restrictions on parameters
In parameter estimation, not only the data structure but also the restrictions on the parameters should be considered, otherwise the MLEs obtained may be unreasonable. For twolocus recombination fractions, the following inequality restrictions: θ_{AB }≤ θ_{BC }+ θ_{AC}, θ_{BC }≤ θ_{AB }+ θ_{AC}, θ_{AC }≤ θ_{AB }+ θ_{BC}, and 0 ≤ θ_{AB}, θ_{BC}, θ_{AC }≤ 1/2 must be considered. For the given order of loci ABC, additional restrictions: θ_{AB }≤ θ_{AC }and θ_{BC }≤ θ_{AC }are required. Combining all these inequalities, the following equivalent restrictions are obtained:
These restrictions are natural and necessary.
Proposed algorithm
In this section, we propose an approach to calculate MLEs of twolocus recombination fractions under restriction (2), which works well in application. From equations (1) and Table 2, p_{k}'s are functions of independent parameters g_{10}, g_{01 }and g_{11}, and also functions of θ_{AB}, θ_{BC }and θ_{AC}, so the loglikelihood function can be written as the following form
where θ = (θ_{AB}, θ_{BC}, θ_{AC}). Our goal is to find , such that under restriction (2), where denotes the restricted MLE of θ.
We propose our restricted EM algorithm (REM) on the basis of the EM algorithm of Dempster et al. [18] as follows:
Augment the observed data {n_{k}, k = 1, 2, 3, 4} by latent variables {n_{kl}, k, l = 1, 2, 3, 4} to obtain a complete data set, where , and {n_{kl}, k, l = 1, 2, 3, 4} is multinomial distributed with probability {p_{kl}, k, l = 1, 2, 3, 4}. Here, p_{kl }are components of p_{k }in Table 2 with ; p_{21 }= g_{00}g_{01}, p_{22 }= g_{00}g_{01}, p_{23 }= g_{10}g_{11}, p_{24 }= g_{10}g_{11}; p_{31 }= g_{00}g_{10}, p_{32 }= g_{00}g_{10}, p_{33 }= g_{01}g_{11}, p_{34 }= g_{01}g_{11}; p_{41 }= g_{00}g_{11}, p_{42 }= g_{00}g_{11}, p_{43 }= g_{01}g_{10}, p_{44 }= g_{01}g_{10}. n_{kl }have its interpretation, for example, n_{11 }can be interpreted as the number of the families: (phase I → (1,1) or (8,8) or (1,8)), or (phase II → (2,2) or (7,7) or (2,7)), or (phase III → (4,4) or (5,5) or (4,5)), or (phase IV → (3,3) or (6,6) or (3,6)), where (phase I → (1,1)) denotes the event that the families have phase I, and the haplotype pairs of their offspring are (1,1), and other notations are analogous to interpret.
Because parameters θ_{AB}, θ_{BC}, and θ_{AC }are equivalent to independent parameters g_{10}, g_{01 }and g_{11}, we still consider parameters g_{10}, g_{01 }and g_{11 }here, and restriction (2) is equivalent to the following restriction (3):
Thus, finding MLE (the restricted MLE of g = (g_{10}, g_{01}, g_{11}), such that ) under restriction (3) implies finding MLE of θ under (2). The complete data loglikelihood function can be written as
where p_{kl}'s are functions of g as given above. The conditional expectation of l(g{n_{kl}}) when the sth step parameter values g^{(s) }are given is
where
Then the restricted estimating problem may be written as
The Hessian matrix of Q(gg^{(s)}, {n_{k}}) for g_{10}, g_{01 }and g_{11 }is negative definite, so Q(gg^{(s)}, {n_{k}}) is strictly concave for g_{10}, g_{01 }and g_{11}. This implies that there exists one unique point satisfying . Following some calculation, it is easy to obtain that and . If satisfies restriction (3), then in the (s + 1)th iteration for EM algorithm, otherwise, we use the KuhnTucker conditions [19,20] to deal with problem (5). Thus, we can still find a unique point , such that under restriction (3), because Q(gg^{(s)}, {n_{k}}) is a strictly concave function for g_{10}, g_{01 }and g_{11 }and the restriction region is a convex set. See Appendix for the KuhnTucker conditions and the solving process of .
We give the complete REM algorithm as follows:
Let be the starting point (the starting value of g^{(0) }may be taken as which can make the REM converge faster, where can be obtained from by equations (1));
Estep: At step s, compute the expected number of recombination events from g^{(s)};
Mstep: Compute g^{(s+1) }using a^{(s+1)}. Firstly, compute . If satisfies restriction (3), then g^{(s+1) }= ; otherwise, then g^{(s+1) }must belong to one of the following cases (i.e. only one case holds):
case 1. , if the following inequalities hold simultaneously
The above procedure is iteratively carried out until convergence. Then the restricted MLE of θ in terms of the restricted MLE can be obtained correspondingly by equations (1).
Compared to the general EM algorithm, the Mstep of the REM is a little more complex. It needs some necessary discrimination, then g^{(s+1) }can be obtained based on a^{(s+1)}. Note that g^{(s+1) }has the closedform solution, so it will largely improve the computational efficiency of the parameters. The restricted EM algorithm is convergent, and the restricted MLE from the proposed restricted EM algorithm is a consistent estimator of the parameter θ.
Case for more offspring
It is an important fact that more offspring in each family will provide more information in linkage analysis, therefore, and we need to extend the REM algorithm to cases of multiple offspring (sibship) in each family.
We develop a strategy for estimating the twolocus recombination fractions for this case, and the proposed REM algorithm works as a unified method. Taking threeoffspring case as an example, we group the observed families into 5 classes according to linkage analysis regulation, with the observed data {n_{k}, k = 1,⋯, 5}. After data augmentation, we obtain complete data {n_{kl}, k = 1, 2, 3, 4, 5, l = 1, 2, 3, 4}. Furthermore, the conditional expectation of the completedata loglikelihood is
where 's have similar expressions with 's given previously. Then the other steps of the REM are the same as those for the case of two offspring, except replacing 's by 's. More offspring's cases are analogous completely. It is helpful to construct and analyze a linkage map using this kind of family data.
Case for unequal prior probabilities of linkage phases
Affected by many factors (e.g., linkage disequilibrium), each phase of a triply heterozygous parent's genotype may in fact not occur with equal prior probability, but the proposed REM can also be applied to the case of unequal phase probability as a unified method. Let each phase occur with probability h_{i }(i = 1, 2, 3, 4), where h_{i }is any fixed number that satisfying 0 ≤ h_{i }≤ 1, and . In this case, twooffspring family data needs to be grouped into 10 different phenotype classes according to linkage analysis regulation (see Table 3), and we can obtain the observed data {n_{k}, k = 1, 2,⋯, 10}. Then we augment the observed data {n_{k}, k = 1, 2,⋯, 10} by latent variables {n_{kl}, k = 1, 2,⋯, 10, l = 1, 2, 3, 4} with corresponding probabilities {p_{kl}, k = 1, 2,⋯, 10, l = 1, 2, 3, 4}. The major difference from the procedure of the REM for h_{i }= 1/4 (i = 1, 2, 3, 4) lies in the expression of conditional expectation for each n_{kl }(k = 1, 2,⋯, 10, l = 1, 2, 3, 4). Take n_{11 }as an example, , where h_{1 }is the assigned prior probability of phase I. Repeating the similar procedure given in the REM for h_{i }= 1/4 (i = 1, 2, 3, 4), we find that the conditional expectation of the loglikelihood of the complete data still has the form of (4), and only the expressions of the components of a^{(s+1) }are more complex than those given previously. Using the REM algorithm, we can obtain the restricted MLEs of the twolocus recombination fractions easily.
Table 3. Phenotype classification when each linkage phase occur with probability h_{i}
Simulation methods
We conduct two simulation studies to evaluate the performance and robustness of the proposed REM. In the simulations, we simulate twooffspring family data.
Comparing the REM and the unrestricted method
Let θ_{0 }= (θ_{AB}, θ_{BC}, θ_{AC}) denote the true value of the recombination fraction. In genetics, loci A and B are said to be closely linked when 0 ≤ θ_{AB }≤ 0.1, moderately linked when 0.1 ≤ θ_{AB }≤ 0.2, and loosely linked when 0.2 ≤ θ_{AB }≤ 0.5. To show the advantage of the REM algorithm, we consider six scenarios according to the different combinations of linkage states of loci AB and loci BC: CC, CM, CL, MM, ML, and LL, where C, M, and L denotes close, moderate, and loose linkage, respectively. In each scenario, θ_{AB }and θ_{BC }are respectively taken as 0.05, 0.15, and 0.35 for close, moderate, and loose linkage. θ_{AC }is taken as three equally spaced values which all guarantee that (θ_{AB}, θ_{BC}, θ_{AC}) satisfies the natural restriction (2), and the smaller value and the larger one are near the boundary of the region composed by restriction (2), and the moderate one is inside the region. Since the triply homozygous parent only produces haplotype abc in triple backcross family, we can only consider the sampling from the heterozygous parent. For demonstrate purpose, we give the process of generating data for each θ_{0 }in detail:
1. According to equal probability 1/4, We randomly assign a linkage phase of the heterozygous parent in one family.
2. Generate two haplotypes of two offspring from the heterozygous parent in the family according to the conditional probabilities given in Table 1. The haplotype pair (or the family) is easily classified into one of the four classes in Table 2.
3. Repeat step 1 and 2 for n = 300 times, then data {n_{k}} for n simulated families can be obtained.
In each scenario of our simulations, for each θ_{0}, we calculate and by the unrestricted method and the REM, respectively. Repeating the whole process for M = 1000 times, we obtain the averages of and over 1000 replicates by the two methods (see Table 4). As expected, the averages of over 1000 replicates agree better with θ_{0 }than the averages of .
Table 4. The averages of estimates over 1000 replicates for 300 twooffspring families by unrestricted method and the REM
To better show the performance of the REM, we mainly use the following three measures of accuracy to compare and :
1. The number, denoted by KK, for which the unrestricted methods give unreasonable estimates based on 1000 replicates.
2. The standard derivations (SDs) of the estimate ; the ratio of SDs of two kinds of estimates being , i = AB, BC, AC.
3. The mean absolute error (MAE) of the estimate , where ; the ratio of MAEs being rMAE = MAE()/MAE().
The comparisons of estimations of twolocus recombination fraction by the unrestricted method and the REM are listed in Table 5. In each scenario, the unrestricted method gives lots of unreasonable results, i.e., the estimates do not satisfy the natural restriction (2), whereas the estimates obtained by the proposed REM all satisfy the restriction. The number KK of unreasonable estimates is larger when the true value θ_{0 }is near the boundary of the restriction region (2), which corresponds to the larger or smaller true values of θ_{AC}, and KK is somewhat smaller when θ_{0 }is inside the region, which corresponds to the moderate values of θ_{AC}. In the former situation the resulting could be obtained in the whole parameter space but not in the restriction region (2). When θ_{0 }is near the boundary of the restriction region (2), is liable to be near the boundary of the region and hence likely to lie outside the boundary. However the proposed method can guarantee that must be inside the restriction region at any time.
Table 5. Comparison of estimation of twolocus recombination fraction for 300 twooffspring families by the unrestricted method and the REM
It is clear to see that our REM outperforms the unrestricted method for estimating twolocus recombination fractions in each simulated scenario. The estimates obtained by the REM have smaller SDs than the unrestricted method, which is more obvious especially at least one of the intervals of AB and BC is loosely linked. This suggests that the accuracy of estimates by the REM is more higher than by the unrestricted method, and that the natural restriction (2) should be taken into account in estimating, otherwise it would have significant impact on the accuracy on practical inference. Compared to , is closer to the true value θ_{0 }(rMAE > 1 for all groups in Table 5).
It also can be seen that the proposed REM is a robust algorithm. The REM can still give better results than the unrestricted method in each scenario even when KK is small (e.g., 1).
Evaluating the effect of interference to estimates
Interference refers to the phenomenon that crossovers in nearby intervals along a chromosome do not occur independently. Let I denote the value of interference. According the definition of interference in Strickberger [21], we have . To better evaluate the effect of interference to the two kinds of estimations, we consider three scenarios: positive, null and negative interferences. In each scenario, we choose equal θ_{AC }and different θ_{AB }and θ_{BC }corresponding to different interference values (see Table 6). For each scenario, we also simulate 300family data, and the REM and the unrestricted method are applied to the simulated data, respectively. The whole process is repeated for 1000 times to compute the measures of accuracy given previously. The simulation results listed in Table 6 firstly show that the values of KK are very large when there exists positive (or negative) interference, and the values are small when there is no interference, while the REM gives reasonable estimates at any time. That is to say the estimating results by the unrestricted method are much affected by the interference, but the results by our REM is less affected. Secondly, the less fluctuations of SD() in scenario 1 (or 3) also validate that the REM is less affected by interference. Finally, the REM outperforms the unrestricted method in each scenario (rSD > 1, rMAE > 1), especially, when negative interference is present.
Table 6. Evaluation of the effect of interference to estimates of recombination fractions
In addition, we find that the restricted EM estimate is little changed when different starting values are taken. These above results indicate that the use of the REM can yield better performance than the current unrestricted method.
A worked example
We applied our proposed method to a real data set from published literature [22]. The data set comprised of 134 individuals from a backcross of mice. Here we consider the three ordinal marker loci D2Mit365, D2Mit272 and D2Mit456 on the linkage map of chromosome 2, and we still use A, B and C to denote the three loci. According to the genotypes given in the data set, we record a haplotype code of each individual, where the haplotype is from the heterozygous parent. Two individuals are randomly grouped into one family, and we consider they are really from that family, where the treatment will not affect linkage information, because all offspring's genotypes are independent conditional on the genotypes of all parents for the data. Then we obtain n = 67 twooffspring families, and n_{1 }= 21, n_{2 }= 17, n_{3 }= 14 and n_{4 }= 15 by the classification given in Table 2. We used the proposed REM and the unrestricted method to estimate the recombination fractions based on (n_{1}, n_{2}, n_{3}, n_{4}). The MLEs of the recombination fractions are = 0.3166, = 0.3738 and = 0.3738; and = 0.3167, = 0.3942 and = 0.3634, respectively. Obviously, the unrestricted estimates do not satisfy the second one of the natural restriction (2), and thus estimates contradict with the true order of the three markers on the linkage map of chromosome 2 [22]. According to our simulation and practical experience, the accuracy of estimation by the REM will improve by increasing sample size or by using the unrestricted estimates as initial values.
Discussion
We developed a restricted EM algorithm to calculate numerically the MLEs of twolocus recombination fractions that initially studied by Ott [3]. The method in Ott [3] may not always provide the parameter estimates satisfying the natural restriction (2), since the approach does not take the inequality restrictions into account. Our method can deal with this problem, and the real data were handled very well with the proposed method.
The performance of the REM is also illustrated using simulated data. Our simulation shows that the unrestricted method gives some unreasonable estimate results in each scenario, and thus such estimates may not provide correct interpretation of the recombination phenomenon in practice. The major advantage of the REM is its robustness and efficiency. The REM can give better results even when the number for which the unrestricted method gives unreasonable estimate results is small (e.g., KK = 1), and our estimates are more precise than those obtained by the unrestricted method. Moreover, the REM is less affected by interference, and the estimate of parameter g in Mstep having the closedform solution largely improves the computational efficiency of the parameter.
On the other hand, noticing the important fact that more offspring in each family can really provide more information in linkage analysis, we develops a strategy for estimating the twolocus recombination fractions when each observed family has more offspring, and the proposed REM algorithm works as a unified method. In practice, the method developed by Lu et al. [10] can be first adopted to obtain the estimates of probabilities h_{i}'s of linkage phases when considering multiple offspring, then the REM is used to obtain the restricted MLEs of recombination fractions, which may improve the estimation precision. It is helpful to construct and analyze a linkage map using this kind of family data.
Recent research in genetics has shown that statistical inference about the twolocus recombination fraction offers an effective approach for constructing and analyzing a linkage map between the genetic marker and the genetic disorders. Reasonable estimates of the recombination fractions are important in gene mapping, especially in interval mapping [2326]. Only the reasonable estimate result may identify the actual genes responsible for some trait, and it is feasible to embed the REM into interval mapping to improve the efficiency of mapping.
It is noticed that our analysis is focused on three biallelic loci. The above constrained parameter problem may become complicated if the number of loci is more than three, or some markers may have more alleles than others, for example, in outcrossing plant species. When the number of loci is more than three, we suggest that every three adjacent loci are subject to threepoint analysis. We can obtain two different estimates of the recombination fraction for the same marker interval, and a better way to combine these estimates is to take a weighted mean. More alleles for each markers mean more possible linkage phases [10], which bring some difficulty to linkage analysis, however, the idea of considering the natural restriction (2) on recombination fractions should also be emphasized. Further investigation in this area is warranted.
Appendix
The KuhnTucker Theorem [19,20]
Suppose that θ* is a solution of
where f, f_{1},⋯, f_{m}: R^{N }→ R are C^{1 }functions. Then the following conditions hold:
(2) λ_{j}f_{j}(θ*) = 0, j = 1,⋯, m;
(3) f_{j}(θ*) ≥ 0, j = 1,⋯, m;
(4) λ_{j }≥ 0, j = 1,⋯, m,
where (λ_{1},⋯, λ_{m}) are Lagrangian multipliers. The four conditions are called KuhnTucker conditions. Specially, if f(θ) is strictly concave and the set {θ: f_{1}(θ) ≥ 0,⋯, f_{m}(θ) ≥ 0} is convex, the KuhnTucker conditions are also sufficient, and the solution θ* is unique.
Solving equation (5) when does not satisfy restriction (3)
Because Q(gg^{(s)}, {n_{k}}) is a strictly concave function and the restriction region (3) is a convex set, there must be a unique solution to equation (5) by the KuhnTucker Theorem. The Lagrangian is
where λ = (λ_{1}, λ_{2}, λ_{3}, λ_{4}), and λ_{i}'s are Lagrangian multipliers. Then is a unique solution to
To solve the above equations, we need to consider all possible cases for λ_{i }= 0 or λ_{i }> 0, i = 1, 2, 3, 4. There are totally seven possible solutions for the above equations which were just given in the previous REM algorithm.
Authors' contributions
YZ derived the genetic and statistical model and wrote computer programs. NZS and WKF provided insightful comments to the presentation. JG conceived of ideas and algorithm. All authors read and approved the final manuscript.
Acknowledgements
The authors would like to thank Dr. WenSheng Zhu for helpful discussions and comments on a draft of the paper. This research was supported by the National Natural Science Foundation of China (Grant Numbers 10431010 and 10701022), National 973 Key Project of China (2007CB311002), NCET040310, the Jilin Distinguished Young Scholars Program (Grant Number 20030113) and the Program Innovative Research Team (PCSIRT) in University (#IRT0519).
References

Elston RC, Stewart J: A general model for the analysis of pedigree data.
Hum Hered 1971, 21:523542. PubMed Abstract

Risch N: Linkage strategies for genetically complex traits.
Am J Hum Genet 1990, 46:222253. PubMed Abstract  PubMed Central Full Text

Ott J: PhaseUnkown Triple Backcross with Two Offspring. In Analysis of Human Genetic Linkage. 3rd edition. The Johns Hopkins University Press: Baltimore; 1999:122124.

Thompson EA: Statistical Inference from Genetic Data on Pedigree. Institute of Mathematical Statistics Beachwood: Ohio; 2000.

Haldane JBS: The recombination of linkage values and the calculation of distances between the loci of linked factors.

Morgan TH: The Theory of Genes. Yale University Press: New Haven; 1928.

Felsenstein J: A mathematically tractable family of genetic mapping functions with different amounts of interference.
Genetics 1979, 91:769775. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Thompson EA: Information gain in joint linkage analysis.
IMA J Math Appl Med Biol 1984, 1:3149. PubMed Abstract  Publisher Full Text

Wu RL, Ma CX, Painter I, Zeng ZB: Simultaneous maximum likelihood estimation of linkage and linkage phases in outcrossing populations.
Theor Pop Biol 2002, 61:349363. Publisher Full Text

Lu Q, Cui YH, Wu RL: A multilocus likelihood approach to joint modelling of linkage, parnet diplotype and gene order in a fullsib family.
BMC Genet 2004, 5:20. PubMed Abstract  BioMed Central Full Text  PubMed Central Full Text

Wu RL, Ma CX, Casella G: Statistical Genetics of Quantitative Traits: Linkage, Maps, and QTL. Springer: New York; 2007.

Lathrop GM, Lalouel JM, Julier C, Ott J: Strategies for multilocus linkage analysis in humans.
Proc Natl Acad Sci USA 1984, 81:34433446. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Lathrop GM: Multilocus linkage analysis in humans: Detection of linkage and estimation of recombination.
Am J Hum Genet 1985, 37:482498. PubMed Abstract  PubMed Central Full Text

Dykstra RL: An algorithm for restricted least squares regression.
J Am Statist Assoc 1983, 78:837842. Publisher Full Text

Robertson T, Wright FT, Dykstra R: Order Restricted Statistical Inference. Wiley: New York; 1988.

Liu C: Estimation of discrete distribution with a class of simplex constraints.
J Am Stat Assoc 2000, 95:109120. Publisher Full Text

Shi NZ, Zheng SR, Guo JH: The restricted EM algorithm under inequality restrictions on the parameters.
J Multivariate Anal 2005, 92:5376. Publisher Full Text

Dempster AP, Laird NM, Rubin DB: Maximum likelihood from incomplete data via the EM algorithm (with discussion).

Mokhtar SB, Shetty CM: Nonlinear Programming: Theory and Algorithms. John Wiley and Sons: New York; 1979.

Anthony LP, Francis ES, Uhl JJ Jr: The Mathematics of Nonlinear Programming. SpringerVerlag: New York; 1992.

Strickberger MW: Genetics. third edition. MacMillan: New York; 1985.

Clemens KE, Churchill G, Bhatt N, Richardson K, Noonan FP: Genetic control of susceptibility to UVinduced immunosuppression by interacting quantitative trait loci.
Genes and Immunity 2000, 1:251259. Publisher Full Text

Lander ES, Botstein D: Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps.
Genetics 1989, 121:185199. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Jansen RC, Stam P: High resolution of quantitative trait into multiple loci via interval mapping.
Genetics 1994, 136:14471455. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Kao CH, Zeng ZB, Teasdale RD: Multiple interval mapping for quantitative trait loci.
Genetics 1999, 152:12031216. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Chen Z: The full EM algorithm for the MLEs of QTL effects and positions and their estimated variance in multipleinterval mapping.
Biometrics 2005, 61:474480. PubMed Abstract  Publisher Full Text