Maximum likelihood estimates of two-locus recombination fractions under some natural inequality restrictions
- Ying Zhou^{1},
- Ning-Zhong Shi^{1},
- Wing-Kam Fung^{2} and
- Jianhua Guo^{1}Email author
https://doi.org/10.1186/1471-2156-9-1
© Zhou et al; licensee BioMed Central Ltd. 2008
Received: 11 September 2007
Accepted: 04 January 2008
Published: 04 January 2008
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. Three-locus linkage analysis is an important case of multi-locus 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 two-locus recombination fractions are calculated.
Results
In this paper, we present a study of estimating the two-locus recombination fractions for the phase-unknown triple backcross with two offspring in each family in the framework of some natural and necessary parameter restrictions. A restricted expectation-maximization (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 [1–4]. 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 [5–7], 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, three-locus analysis yields more information than does two-locus analysis [8–11]. Three-locus linkage analysis is also an important case of multi-locus 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 two-locus recombination fractions for phase-unknown triple backcross families with two offspring in each family. The author gave the presentations of the estimates of the two-locus 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 two-locus 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 [9–11].
This estimation problem of two-locus recombination fractions in three-locus linkage analysis belongs to the constrained parameter problems which are not only important but also appear in many areas. The reader is referred to [14–17]. 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 two-locus 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 two-locus 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 A-B-C. 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.
Conditional haplotype probabilities given phase produced by a triply heterozygous parent
Phase | |||||
---|---|---|---|---|---|
i | Haplotype | I | II | III | IV |
1 | ABC | g_{00}/2 | g_{01}/2 | g_{11}/2 | g_{10}/2 |
2 | ABc | g_{01}/2 | g_{00}/2 | g_{10}/2 | g_{11}/2 |
3 | AbC | g_{11}/2 | g_{10}/2 | g_{00}/2 | g_{01}/2 |
4 | Abc | g_{10}/2 | g_{11}/2 | g_{01}/2 | g_{00}/2 |
5 | aBC | g_{10}/2 | g_{11}/2 | g_{01}/2 | g_{00}/2 |
6 | aBc | g_{11}/2 | g_{10}/2 | g_{00}/2 | g_{01}/2 |
7 | abC | g_{01}/2 | g_{00}/2 | g_{10}/2 | g_{11}/2 |
8 | abc | g_{00}/2 | g_{01}/2 | g_{11}/2 | g_{10}/2 |
Total | 1 | 1 | 1 | 1 |
Let θ_{ AB }, θ_{ BC }and θ_{ AC }, respectively denote two-locus 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 non-recombination, 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:
θ_{ AB }= g_{11} + g_{10}, θ_{ BC }= g_{11} + g_{01}, θ_{ AC }= g_{10} + g_{01}. (1)
Phenotype classes for phase-unknown triple backcross families with two offspring
k | (i, j)^{ a } | p _{ k } |
---|---|---|
1 | (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) (7,7), (8,8), (4,5), (3,6), (2,7), (1,8) | ${g}_{11}^{2}+{g}_{10}^{2}+{g}_{01}^{2}+{g}_{00}^{2}$ |
2 | (1,2), (3,4), (3,5), (1,7), (4,6), (2,8), (5,6), (7,8) | 2(g_{11}g_{10} + g_{01}g_{00}) |
3 | (2,3), (1,4), (1,5), (2,6), (3,7), (4,8), (6,7), (5,8) | 2(g_{11}g_{01} + g_{10}g_{00}) |
4 | (1,3), (2,4), (2,5), (1,6), (4,7), (3,8), (5,7), (6,8) | 2(g_{11}g_{00} + g_{10}g_{01}) |
Total | 1 |
Natural inequality restrictions on parameters
These restrictions are natural and necessary.
Proposed algorithm
where θ = (θ_{ AB }, θ_{ BC }, θ_{ AC }). Our goal is to find ${\widehat{\theta}}^{R}=\left({\widehat{\theta}}_{AB}^{R},{\widehat{\theta}}_{BC}^{R},{\widehat{\theta}}_{AC}^{R}\right)$, such that $l({\widehat{\theta}}^{R}|\left\{{n}_{k}\right\})=\underset{\theta}{\mathrm{max}\phantom{\rule{0.1em}{0ex}}}l(\theta \left|\right\{{n}_{k}\})$ under restriction (2), where ${\widehat{\theta}}^{R}$ 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 ${n}_{k}={\displaystyle \sum _{l=1}^{4}{n}_{kl}}$, 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 ${p}_{11}={g}_{00}^{2},{p}_{12}={g}_{01}^{2},{p}_{13}={g}_{10}^{2},{p}_{14}={g}_{11}^{2}$; 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.
Then the restricted estimating problem may be written as
Max Q(g|g^{(s)}, {n_{ k }}), subject to g satisfies restriction (3).
The Hessian matrix of Q(g|g^{(s)}, {n_{ k }}) for g_{10}, g_{01} and g_{11} is negative definite, so Q(g|g^{(s)}, {n_{ k }}) is strictly concave for g_{10}, g_{01} and g_{11}. This implies that there exists one unique point ${\tilde{g}}^{(s+1)}=({\tilde{g}}_{10}^{(s+1)},{\tilde{g}}_{01}^{(s+1)},{\tilde{g}}_{11}^{(s+1)})$ satisfying $Q({\tilde{g}}^{(s+1)}|{g}^{(s)},\left\{{n}_{k}\right\})=\underset{g}{\mathrm{max}\phantom{\rule{0.1em}{0ex}}}Q(g|{g}^{(s)},\{{n}_{k}\})$. Following some calculation, it is easy to obtain that ${\tilde{g}}_{10}^{(s+1)}={a}_{3}^{(s+1)}/n,{\tilde{g}}_{01}^{(s+1)}={a}_{2}^{(s+1)}/n$ and ${\tilde{g}}_{11}^{(s+1)}={a}_{4}^{(s+1)}/n$. If ${\tilde{g}}^{(s+1)}$ satisfies restriction (3), then ${g}^{(s+1)}={\tilde{g}}^{(s+1)}$ in the (s + 1)th iteration for EM algorithm, otherwise, we use the Kuhn-Tucker conditions [19, 20] to deal with problem (5). Thus, we can still find a unique point $\u030c{g}^{(s+1)}=(\u030c{g}_{10}^{(s+1)},\u030c{g}_{01}^{(s+1)},\u030c{g}_{11}^{(s+1)})$, such that $Q(\u030c{g}^{(s+1)}|{g}^{(s)},\left\{{n}_{k}\right\})=\underset{g}{\mathrm{max}\phantom{\rule{0.1em}{0ex}}}Q(g|{g}^{(s)},\{{n}_{k}\})$ under restriction (3), because Q(g|g^{(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 Kuhn-Tucker conditions and the solving process of $\u030c{g}^{(s+1)}$.
We give the complete REM algorithm as follows:
Let ${g}^{(0)}=({g}_{10}^{(0)},{g}_{01}^{(0)},{g}_{11}^{(0)})$ be the starting point (the starting value of g^{(0)} may be taken as ${\widehat{g}}^{U}=({\widehat{g}}_{10}^{U},{\widehat{g}}_{01}^{U},{\widehat{g}}_{11}^{U})$ which can make the REM converge faster, where $({\widehat{g}}_{10}^{U},{\widehat{g}}_{01}^{U},{\widehat{g}}_{11}^{U})$ can be obtained from $({\widehat{\theta}}_{AB}^{U},{\widehat{\theta}}_{BC}^{U},{\widehat{\theta}}_{AC}^{U})$ by equations (1));
E-step: At step s, compute the expected number of recombination events ${a}^{(s+1)}=({a}_{1}^{(s+1)},{a}_{2}^{(s+1)},{a}_{3}^{(s+1)},{a}_{4}^{(s+1)})$ from g^{(s)};
M-step: Compute g^{(s+1)}using a^{(s+1)}. Firstly, compute ${\tilde{g}}^{(s+1)}=({\tilde{g}}_{10}^{(s+1)},{\tilde{g}}_{01}^{(s+1)},{\tilde{g}}_{11}^{(s+1)})=({a}_{3}^{(s+1)}/n,{a}_{2}^{(s+1)}/n,{a}_{4}^{(s+1)}/n)$. If ${\tilde{g}}^{(s+1)}$ satisfies restriction (3), then g^{(s+1)}= ${\tilde{g}}^{(s+1)}$; otherwise, then g^{(s+1)}must belong to one of the following cases (i.e. only one case holds):
The above procedure is iteratively carried out until convergence. Then the restricted MLE ${\widehat{\theta}}^{R}$ of θ in terms of the restricted MLE ${\widehat{g}}^{R}$ can be obtained correspondingly by equations (1).
Compared to the general EM algorithm, the M-step 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 closed-form solution, so it will largely improve the computational efficiency of the parameters. The restricted EM algorithm is convergent, and the restricted MLE ${\widehat{\theta}}^{R}$ 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.
where ${b}_{i}^{(s+1)}$'s have similar expressions with ${a}_{i}^{(s+1)}$'s given previously. Then the other steps of the REM are the same as those for the case of two offspring, except replacing ${a}_{i}^{(s+1)}$'s by ${b}_{i}^{(s+1)}$'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
Phenotype classification when each linkage phase occur with probability h_{i}
k | (i, j)^{ a } | p _{ k } |
---|---|---|
1 | (1,1), (8,8), (1,8) | ${h}_{1}{g}_{00}^{2}+{h}_{2}{g}_{01}^{2}+{h}_{3}{g}_{11}^{2}+{h}_{4}{g}_{10}^{2}$ |
2 | (2,2), (7,7), (2,7) | ${h}_{1}{g}_{01}^{2}+{h}_{2}{g}_{00}^{2}+{h}_{3}{g}_{10}^{2}+{h}_{4}{g}_{11}^{2}$ |
3 | (3,3), (6,6), (3,6) | ${h}_{1}{g}_{11}^{2}+{h}_{2}{g}_{10}^{2}+{h}_{3}{g}_{00}^{2}+{h}_{4}{g}_{01}^{2}$ |
4 | (4,4), (5,5), (4,5) | ${h}_{1}{g}_{10}^{2}+{h}_{2}{g}_{11}^{2}+{h}_{3}{g}_{01}^{2}+{h}_{4}{g}_{00}^{2}$ |
5 | (1,2), (1,7), (2,8), (7,8) | 2((h_{1} + h_{2})g_{00}g_{01} + (h_{3} + h_{4})g_{10}g_{11}) |
6 | (3,4), (3,5), (4,6), (5,6) | 2((h_{1} + h_{2})g_{10}g_{11} + (h_{3} + h_{4})g_{00}g_{01}) |
7 | (2,3), (2,6), (3,7), (6,7) | 2((h_{1} + h_{4})g_{01}g_{11} + (h_{2} + h_{3})g_{00}g_{10}) |
8 | (1,4), (1,5), (4,8), (5,8) | 2((h_{1} + h_{4})g_{00}g_{10} + (h_{2} + h_{3})g_{01}g_{11}) |
9 | (1,3), (1,6), (3,8), (6,8) | 2((h_{1} + h_{3})g_{00}g_{11} + (h_{2} + h_{4})g_{01}g_{10}) |
10 | (2,4), (2,5), (4,7), (5,7) | 2((h_{1} + h_{3})g_{01}g_{10} + (h_{2} + h_{4})g_{00}g_{11}) |
Total | 1 |
Simulation methods
We conduct two simulation studies to evaluate the performance and robustness of the proposed REM. In the simulations, we simulate two-offspring 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.
The averages of estimates over 1000 replicates for 300 two-offspring families by unrestricted method and the REM
Parameters | REM | Unrestricted Method | |||||||
---|---|---|---|---|---|---|---|---|---|
Scenario^{ a } | θ _{ AB } | θ _{ BC } | θ _{ AC } | ${\widehat{\theta}}_{AB}^{R}$ | ${\widehat{\theta}}_{BC}^{R}$ | ${\widehat{\theta}}_{AC}^{R}$ | ${\widehat{\theta}}_{AB}^{U}$ | ${\widehat{\theta}}_{BC}^{U}$ | ${\widehat{\theta}}_{AC}^{U}$ |
CC | 0.05 | 0.05 | 0.06 | 0.0495 | 0.0497 | 0.0604 | 0.0499 | 0.0501 | 0.0597 |
0.075 | 0.0498 | 0.0500 | 0.0751 | 0.0499 | 0.0500 | 0.0752 | |||
0.09 | 0.0500 | 0.0500 | 0.0903 | 0.0496 | 0.0499 | 0.0903 | |||
CM | 0.05 | 0.15 | 0.16 | 0.0502 | 0.1486 | 0.1607 | 0.0502 | 0.1494 | 0.1602 |
0.175 | 0.0502 | 0.1495 | 0.1742 | 0.0502 | 0.1495 | 0.1745 | |||
0.19 | 0.0497 | 0.1508 | 0.1898 | 0.0497 | 0.1509 | 0.1906 | |||
CL | 0.05 | 0.35 | 0.36 | 0.0496 | 0.3531 | 0.3685 | 0.0496 | 0.3300 | 0.3711 |
0.375 | 0.0502 | 0.3532 | 0.3777 | 0.0502 | 0.3287 | 0.3837 | |||
0.39 | 0.0498 | 0.3534 | 0.3939 | 0.0499 | 0.3344 | 0.3990 | |||
MM | 0.15 | 0.15 | 0.16 | 0.1487 | 0.1489 | 0.1643 | 0.1504 | 0.1507 | 0.1611 |
0.225 | 0.1503 | 0.1500 | 0.2248 | 0.1503 | 0.1501 | 0.2252 | |||
0.29 | 0.1497 | 0.1508 | 0.2887 | 0.1498 | 0.1509 | 0.2923 | |||
ML | 0.15 | 0.35 | 0.36 | 0.1505 | 0.3494 | 0.3745 | 0.1505 | 0.3247 | 0.3737 |
0.425 | 0.1505 | 0.3533 | 0.4274 | 0.1507 | 0.3254 | 0.4310 | |||
0.49 | 0.1498 | 0.3481 | 0.4499 | 0.1503 | 0.3331 | 0.4535 | |||
LL | 0.35 | 0.35 | 0.36 | 0.3443 | 0.3470 | 0.3601 | 0.3573 | 0.3312 | 0.3675 |
0.425 | 0.3525 | 0.3517 | 0.4305 | 0.3582 | 0.3315 | 0.4272 | |||
0.49 | 0.3531 | 0.3524 | 0.4554 | 0.3582 | 0.3255 | 0.4505 |
To better show the performance of the REM, we mainly use the following three measures of accuracy to compare ${\widehat{\theta}}^{U}$ and ${\widehat{\theta}}^{R}$:
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 ${\widehat{\theta}}_{i}^{R}$; the ratio of SDs of two kinds of estimates being $\text{rSD}=\text{SD}({\widehat{\theta}}_{i}^{U})/\text{SD}({\widehat{\theta}}_{i}^{R})$, i = AB, BC, AC.
3. The mean absolute error (MAE) of the estimate ${\widehat{\theta}}^{R}$, where $\text{MAE}={\displaystyle \sum _{l=1}^{1000}(|{\widehat{\theta}}_{ABl}^{R}-{\theta}_{AB}|+|{\widehat{\theta}}_{BCl}^{R}-{\theta}_{BC}|+|{\widehat{\theta}}_{ACl}^{R}-{\theta}_{AC}|)/3000}$; the ratio of MAEs being rMAE = MAE(${\widehat{\theta}}^{U}$)/MAE(${\widehat{\theta}}^{R}$).
Comparison of estimation of two-locus recombination fraction for 300 two-offspring families by the unrestricted method and the REM
Parameters | SD | rSD^{ b } | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Scenario^{ a } | θ _{ AB } | θ _{ BC } | θ _{ AC } | ${\widehat{\theta}}_{AB}^{R}$ | ${\widehat{\theta}}_{BC}^{R}$ | ${\widehat{\theta}}_{AC}^{R}$ | ${\widehat{\theta}}_{AB}^{U}$ | ${\widehat{\theta}}_{BC}^{U}$ | ${\widehat{\theta}}_{AC}^{U}$ | MAE^{ c } | rMAE^{ d } | KK ^{ e } |
CC | 0.05 | 0.05 | 0.06 | 0.0089 | 0.0088 | 0.0095 | 1.0606 | 1.0790 | 1.1170 | 0.0072 | 1.0434 | 220 |
0.075 | 0.0090 | 0.0092 | 0.0114 | 1.0029 | 1.0033 | 1.0187 | 0.0078 | 1.0043 | 6 | |||
0.09 | 0.0091 | 0.0093 | 0.0127 | 1.0020 | 1.0022 | 1.0274 | 0.0083 | 1.0062 | 84 | |||
CM | 0.05 | 0.15 | 0.16 | 0.0093 | 0.0180 | 0.0177 | 1.0007 | 1.0603 | 1.0560 | 0.0119 | 1.0223 | 183 |
0.175 | 0.0091 | 0.0182 | 0.0195 | 1.0007 | 1.0168 | 1.0562 | 0.0124 | 1.0140 | 34 | |||
0.19 | 0.0094 | 0.0183 | 0.0209 | 1.0012 | 1.0122 | 1.1352 | 0.0128 | 1.0299 | 197 | |||
CL | 0.05 | 0.35 | 0.36 | 0.0095 | 0.0463 | 0.0481 | 1.0008 | 4.2411 | 1.4711 | 0.0272 | 1.2941 | 502 |
0.375 | 0.0090 | 0.0464 | 0.0482 | 1.0009 | 4.2875 | 1.6143 | 0.0272 | 1.3343 | 487 | |||
0.39 | 0.0093 | 0.0445 | 0.0467 | 1.0006 | 3.8670 | 1.7417 | 0.0267 | 1.3462 | 518 | |||
MM | 0.15 | 0.15 | 0.16 | 0.0156 | 0.0168 | 0.0168 | 1.2658 | 1.1893 | 1.2115 | 0.0131 | 1.0956 | 451 |
0.225 | 0.0181 | 0.0176 | 0.0239 | 1.0078 | 1.0080 | 1.0863 | 0.0159 | 1.0174 | 1 | |||
0.29 | 0.0174 | 0.0187 | 0.0261 | 1.0117 | 1.0098 | 1.6503 | 0.0166 | 1.1165 | 343 | |||
ML | 0.15 | 0.35 | 0.36 | 0.0177 | 0.0452 | 0.0514 | 1.0024 | 5.1260 | 1.4805 | 0.0298 | 1.3264 | 419 |
0.425 | 0.0179 | 0.0459 | 0.0504 | 1.0071 | 5.0395 | 1.5690 | 0.0311 | 1.3790 | 297 | |||
0.49 | 0.0179 | 0.0410 | 0.0584 | 1.0167 | 4.9795 | 1.3348 | 0.0303 | 1.2595 | 304 | |||
LL | 0.35 | 0.35 | 0.36 | 0.0390 | 0.0373 | 0.0454 | 2.0082 | 6.4504 | 1.5975 | 0.0319 | 1.4403 | 604 |
0.425 | 0.0454 | 0.0436 | 0.0498 | 1.4563 | 4.3277 | 1.5612 | 0.0378 | 1.2931 | 278 | |||
0.49 | 0.0460 | 0.0465 | 0.0577 | 1.3683 | 4.5456 | 1.4778 | 0.0375 | 1.3018 | 216 |
It is clear to see that our REM outperforms the unrestricted method for estimating two-locus 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 ${\widehat{\theta}}^{U}$, ${\widehat{\theta}}^{R}$ 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
Evaluation of the effect of interference to estimates of recombination fractions
Scenario | θ _{ AB } | θ _{ BC } | θ _{ AC } | I ^{ a } | SD(${\widehat{\theta}}_{AC}^{R}$) | rSD ^{ b } | MAE^{ c } | rMAE^{ d } | KK ^{ e } |
---|---|---|---|---|---|---|---|---|---|
1 | 0.031 | 0.060 | 0.09 | 0.7312 | 0.0124 | 1.0484 | 0.0078 | 1.0128 | 475 |
0.035 | 0.056 | 0.09 | 0.7449 | 0.0124 | 1.0323 | 0.0080 | 1.0125 | 487 | |
0.039 | 0.052 | 0.09 | 0.7535 | 0.0120 | 1.0250 | 0.0078 | 1.0128 | 472 | |
2 | 0.081 | 0.1301 | 0.19 | 0 | 0.0205 | 1.0537 | 0.0131 | 1.0229 | 64 |
0.085 | 0.1265 | 0.19 | 0 | 0.0196 | 1.0765 | 0.0127 | 1.0315 | 72 | |
0.089 | 0.1229 | 0.19 | 0 | 0.0198 | 1.0505 | 0.0127 | 1.0236 | 48 | |
3 | 0.151 | 0.359 | 0.39 | -0.1068 | 0.0553 | 1.1971 | 0.0327 | 1.3945 | 354 |
0.155 | 0.355 | 0.39 | -0.0904 | 0.0549 | 1.2095 | 0.0321 | 1.2928 | 323 | |
0.159 | 0.351 | 0.39 | -0.0751 | 0.0552 | 1.2156 | 0.0324 | 1.3025 | 300 |
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 two-offspring 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 ${\widehat{\theta}}_{AB}^{R}$ = 0.3166, ${\widehat{\theta}}_{BC}^{R}$ = 0.3738 and ${\widehat{\theta}}_{AC}^{R}$ = 0.3738; and ${\widehat{\theta}}_{AB}^{U}$ = 0.3167, ${\widehat{\theta}}_{BC}^{U}$ = 0.3942 and ${\widehat{\theta}}_{AC}^{U}$ = 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 two-locus 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 M-step having the closed-form 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 two-locus 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 two-locus 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 [23–26]. 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 three-point 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 Kuhn-Tucker Theorem [19, 20]
Suppose that θ* is a solution of
Max f(θ) subject to f_{1}(θ) ≥ 0,⋯, f_{ m }(θ) ≥ 0,
where f, f_{1},⋯, f_{ m }: R^{ N }→ R are C^{1} functions. Then the following conditions hold:
(1) $\frac{\partial}{\partial {\theta}_{i}}f({\theta}^{\ast})+{\displaystyle \sum _{j=1}^{m}{\lambda}_{j}\frac{\partial}{{\theta}_{i}}}{f}_{j}({\theta}^{\ast})=0$, i = 1,⋯, N;
(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 Kuhn-Tucker conditions. Specially, if f(θ) is strictly concave and the set {θ: f_{1}(θ) ≥ 0,⋯, f_{ m }(θ) ≥ 0} is convex, the Kuhn-Tucker conditions are also sufficient, and the solution θ* is unique.
Solving equation (5) when ${\tilde{g}}^{(s+1)}$ does not satisfy restriction (3)
Because Q(g|g^{(s)}, {n_{ k }}) is a strictly concave function and the restriction region (3) is a convex set, there must be a unique solution $\u030c{g}^{(s+1)}$ to equation (5) by the Kuhn-Tucker Theorem. The Lagrangian is
L(g, λ) = Q(g|g^{(s)}, {n_{ k }}) + λ_{1}(g_{01} - g_{11}) + λ_{2}(g_{10} - g_{11}) + λ_{3}g_{11} + λ_{4}(1/2 - g_{01} - g_{10}),
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.
Declarations
Acknowledgements
The authors would like to thank Dr. Wen-Sheng 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), NCET-04-0310, the Jilin Distinguished Young Scholars Program (Grant Number 20030113) and the Program Innovative Research Team (PCSIRT) in University (#IRT0519).
Authors’ Affiliations
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