Journal of Heredity

Journal of Heredity Advance Access published online on July 10, 2007
Journal of Heredity, doi:10.1093/jhered/esm041

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Mapping Binary Trait Loci in the F_2:3 Design

Chengsong Zhu, Ju Huang, and Yuan-Ming Zhang

From the Section on Statistical Genomics, State Key Laboratory of Crop Genetics and Germplasm Enhancement/National Center for Soybean Improvement, Nanjing Agricultural University, Nanjing 210095, China

Address correspondence to Dr Y.-M. Zhang to the address above, or e-mail: soyzhang{at}njau.edu.cn.

In the inheritance analysis of quantitative trait with low heritability, the precision is relatively low. In this situation, an F_2:3 design, which is genotyped in F₂ plants and phenotyped in the F_2:3 progeny, is applied to increase the precision in the detection of quantitative trait loci (QTL). This is because that residual variance on the basis of family-mean–based observations has been significantly decreased by increasing the number of F_2:3 progeny. Our previous results showed that the mixture distribution for the F_2:3 family of heterozygous F₂ plant can significantly increase the power of QTL detection relative to the classical F₂ design. In this article, we extended our previous method from continuous traits to binary traits in the F_2:3 design. The method here also takes full advantage of the mixture distribution. However, the method presented here differs from our previous method in 2 aspects. One is that the penetrance model is integrated with the liability model for mapping binary trait loci (BTL), and another is that the phenotypic data used in the analysis are the sum of phenotypic values of F_2:3 progeny derived from each F₂ plant rather than the average of F_2:3 progeny due to the fact that the distribution of the sum follows binomial distribution. In addition, the threshold in the liability model could also be estimated. Therefore, a new framework of mapping BTL on the basis of a single BTL model was set up and implemented via the Expectation–Maximization algorithm. Results of simulated studies showed that the proposed method provides accurate estimates for both the effects and the locations of BTL, with high statistical power even under the low heritability. With the new method, we are ready to map BTL, as we can do for quantitative traits under the F_2:3 design. The computer program performing the analysis of the simulated data is available to users for real data analysis.

---

In line–crossing experiments, a segregating population derived from the crosses of some carefully chosen inbred lines, that is, F₂ or backcross populations, is widely used to map quantitative trait loci (QTL). If the heritability of a trait is low, one can take family mean as a unit of phenotypic measurement (Mather and Jinks 1982; Lynch and Walsh 1998; Zhang et al. 2003; Zhang and Xu 2004). This is an F_2:3 design in plant genetics (Austin and Lee 1996; Cockerham and Zeng 1996; Fisch et al. 1996; Jiang and Zeng 1997; Chapman et al. 2003; Zhang and Xu 2004; Kao 2006) and daughter or granddaughter design in animal genetics (Weller et al. 1990). These designs are frequently used in mapping QTL in both plant and animal kingdom. This is because that the family-mean–based heritability can be significantly increased by increasing the number of progeny, and the progeny are only phenotyped after the individuals at F₂ are genotyped, so leading to substantial cost saving. Mapping QTL for such a trait requires a special technique to handle the distribution for the statistic of phenotypic values of F_2:3 progeny. Therefore, it is necessary to pay more attention on the F_2:3 design.

Statistical methods for F₂ and recombinant inbred line populations have been well developed (Lander and Botstein 1989; Zeng 1994; Doerge 2001). In the real data analysis, the current method for the F_2:3 design is adopted by simply replacing the F₂ phenotype by the average value of the F_2:3 progeny (Zhang et al. 2003). Obviously, it neglects the mixture distribution for the F_2:3 family of heterozygous F₂ plant. Zhang and Xu (2004) show that the mixture distribution in the F_2:3 design can be used to significantly increase the power of QTL detection relative to the classic F₂ design, even if only a single F_2:3 progeny is collected from each F_2:3 family. In this article, the mixture distribution would be further incorporated into mapping binary trait loci (BTL) in the F_2:3 design.

Most intensively studied characters in plants, animals, and human are complex traits with discontinuously phenotypic variations. Many diseases show dichotomous phenotype (0 and 1) but do not follow a simple Mendelian pattern of inheritance. Binary traits, that is, susceptibility/resistance, sterility/fertility, and mortality/survival, while not quantitative in nature, are equally important in the detection of QTL. Too much complication in binary traits comes from seemingly simple descriptions meanwhile in fact the underlying biological model may be complicated. One could take a QTL mapping approach by treating binary phenotype as a quantitative trait (Visscher et al. 1996). From a theoretical point of view, standard QTL mapping can not be applied to discrete trait locus mapping. Analyses with dichotomous traits may be theoretically more challenging than those with continuous traits because the former requires modeling the link between the observable phenotype and the corresponding latent variable. Therefore, Yi and Xu (1999) developed a threshold model with heterogeneous error variance for mapping BTL in multiple full-sib families based on Fisher's scoring algorithm, Kadarmideen et al. (2000) proposed a method to detect multiple BTL using generalized and regression interval mapping in multiple-family half-sib designs, Yi and Xu (2000) suggested a Bayesian approach for mapping BTL using reversible jump Markov Chain Monte Carlo algorithm, and McIntyre et al. (2001) put forward a probability model particularly suitable for binary disease locus mapping and treated the probabilities of disease (penetrance) as parameters. Recently, Xu et al. (2005) presented a new Expectation–Maximization (EM) algorithm by treating both the unobserved genotypes and the disease liability as missing values, and Coffman et al. (2005) modeled selection in BTL mapping. However, the above investigations have been seldom addressed in mapping BTL using the so-called F_2:3 design.

To extend our previous study to BTL mapping, we firstly investigated the mixture distribution for the sum of binary trait phenotypes of F_2:3 progeny derived from each F₂ plant in the F_2:3 design and then integrated the penetrance model with the liability model to map BTL by incorporating the mixture distribution mentioned above in the F_2:3 design. Finally, simulation studies were used to test the proposed method in this article.

	Theory and Methods

Top Theory and Methods Discussion Funding Appendix: Solving for the... References

 
Genetic Model of F_2:3
We consider cosegregation of a gene at an autonomic locus that affects a dichotomous trait. The phenotype of a trait is assumed to be distributed as a binary variable. The phenotype of individual j within the ith family is modeled by

(1)

where t is threshold for the underlying liability of an imaginary trait z_ij, which is formulated as

(2)

In which µ is the overall population mean, g_ij is genotypic value of individual at a BTL, and _ij is a normally distributed residual variable with mean zero and standard deviation 1.0. Three genotypes at the BTL, AA, Aa, and aa, are assumed to have genotypic values a – d/2, d/2, and –a – d/2, respectively, with a and d indicating additive and dominant effects. The residual variance can be arbitrarily defined without affecting the conclusion due to the fact that the liability is a hypothetical variable. Thus, the genetic effects of the BTL are measured by units of the residual deviations of the liability.

The liability of each genotype will follow a truncated distribution with a cumulative probability indicated by

(3)

where = (µ,a,d,t,) with being the position of considered BTL and G_ij = k and f_k indicate the kth genotype and its penetrance of the BTL, respectively.

Maximum-Likelihood Analyses
Let the phenotypic data in the analyses be the sum of phenotypic values of the n F_2:3 plants but not the observations of the F₂ plants. The variable y_i = y_ij has a binomial distribution: in which p_i is the probability of y_ij = 1.

The log-likelihood function is

(4)

where m is sample size of F₂ population and h_ik is the prior probability of the kth BTL genotype for the ith F₂ individual inferred from the known marker information.

Exact EM Algorithm
Let n_i1, n_i2, and n_i3 be the number of the 3 possible genotypes in the ith F_2:3 family, respectively, where n = n_ik. This leads to

(5)

where f₁ f₃ are same as those in equation 3. If an F₂ plant is AA, then the F_2:3 progeny will be AA, so leading to n_i1 = n and n_i2 = n_i3 = 0, and thus p_i = f₁; if an F₂ plant is aa, then the F_2:3 progeny will be aa, so leading to n_i3 = n and n_i1 = n_i2 = 0, and thus p_i = f₃. However, if an F₂ plant is Aa, (n_i1, n_i2, n_i3) follows a multinomial distribution with a probability

(6)

Thus the log-likelihood function can be expressed as

(7)

Based on the fact that the variable y_i follows a binomial distribution, so

(8)

where defines the domain of all possible values of the n's subject to the restriction of n = n_ik.

There are several ways to find the maximum likelihood estimates (MLE) of the model parameters; we adopt an EM algorithm (Dempster et al. 1977) and treat the numbers of the 3 possible genotypes for each F_2:3 family as missing data. The unknown parameters involve =(µ,a,d,t,). We regard as constant for the moment, now the parameter set will be =(µ,a,d,t). The E step and M step of the EM algorithm are described as follows.

E step: The parameters (µ, a, d, t) are initialized with zero. The liability of each genotype (f_k) can be obtained from solving equation 3 by numerical algorithm (Press et al. 2001). The posterior probabilities of the 3 BTL genotypes are calculated as

(9)

In addition, we need to calculate the posterior probabilities for F_2:3 progeny of heterozygous F₂ plant, as shown below:

(10)

M step: The MLEs of f₁, f₂, and f₃ can be obtained from solving the following equations:

(11.1)

(11.2)

(11.3)

The details for the derivation of these equations and implementation of the numerical algorithm for solving the equations are given in Appendix.

Once the MLEs of f₁, f₂, and f₃ are obtained, the genetic parameters (µ, a, d) can be estimated from the equation (3) using the numerical algorithm (Press et al. 2001).

Let b₀ be the proportion of the unaffected individuals in the sample. Thompson (1972) suggested the use of b₀ to calculate the maximum-likelihood estimate of the threshold t. In the present context, the following equation

(12)

is numerically searched for the MLE of t on the basis of the above estimates of the other model parameters (Thompson 1972; Luo and Wu 2001), where h_•k = h_ik/m.

The E and M steps are iterated until convergence.

We now can test the null hypothesis that there is no BTL for the particular location . The null hypothesis is formulated as H₀:a = d = 0.0, which can be tested using the likelihood-ratio (LR) test statistic

(13)

The MLE for the position of the BTL can be obtained by examining the LR profile along the genome as commonly done in interval mapping (Lander and Botstein 1989).

The exact EM algorithm is recommended for small n, say n 5. Note that the number of possible partitions of n is (n + 1)(n + 2)/2, which can be very large as n increases.

Approximate EM Algorithm
When n is large, derived from the heterozygous F₂ may be approximated by their corresponding expectations. In this approximation, we simply replace thus the log-likelihood will be formulated as

(14)

Now the EM algorithm can be simplified. The posterior probabilities of 3 BTL genotypes are calculated as

(15)

The MLEs of the f₁, f₂, and f₃ can be explicitly obtained as follows:

(16.1)

(16.2)

(16.3)

Simulation Studies
The purpose of simulation studies is 1) to test the proposed method in this paper; 2) to confirm that the correct F_2:3 analysis using the new EM algorithm is more efficient than the currently adopted F₂ method; 3) to demonstrate that the approximate EM algorithm is as efficient as the exact EM algorithm in practice; and 4) to compare the power for mapping genetic loci between quantitative traits and binary traits.

Eleven equally spaced markers were simulated on a single-chromosome segment of length 100 cM. A single BTL was located on position 25 cM. The statistical power is determined by counting the number of replicates that have a test statistic (LR score) greater than the empirical critical values obtained from analyses of 1000 additional samples simulated under the null model (zero heritability). In all simulations, the overall means were set at µ = 0 and the residual variance on individual plant was set at ² = 1. The conditional probabilities of BTL genotypes given marker information were calculated on the basis of the multipoint method (Jiang and Zeng 1997). The broad heritability involved in the simulation experiments is all expressed on the individual F₂ plant basis. One complication from the multiple-generation problem is that different generations usually have different genetic variances and different definitions of heritability. Here we simply defined the genetic variance and the residual variance on the single-plant basis. This will eliminate such complication.

To demonstrate the first objective of the simulation experiments, we simulated 200 F_2:3 families each with 1, 3, 5, and 10 plants, respectively. A single BTL was simulated with the heritability of 0.05 and 0.10, respectively. The corresponding additive and dominant effects under each level of heritability are listed in Table 1. By maximizing the likelihood equation 7 with the EM algorithm, the parameters are estimated, and the results, including mean, standard deviation and empirical power, are listed in Table 1. The results show that the estimates are close to their corresponding true values except for that the position, and the dominant effect of the BTL are biased while only a single F_2:3 progeny is collected from each F_2:3 family. As we expected, the results show the general behavior of BTL mapping, that is, the standard deviation of estimates decreases and the power of BTL detection increases as the number of plants in each family and the heritability of BTL increase. Moreover, we also investigated the effect of sample size (number of F₂ plants) on mapping BTL. Here the number of F_2:3 progeny was set at 5. Table 2 shows the average values of the MLEs of the parameters more than 100 replicates of simulations and their corresponding standard deviations along with the empirical power of the LR test under the alternative model. Again, the observed trend is consistent with what was expected, that is, the method works well as the sample size or the heritability increases.

View this table: [in this window] [in a new window]   Table 1. Effects of the number of F2:3 progeny derived from each F2 plant on mapping BTL in the F2:3 design

 

View this table: [in this window] [in a new window]   Table 2. Effects of sample size of F2 population on mapping BTL in the F2:3 design

 
The second objective of the simulation experiments is to compare the efficiencies between the new "exact EM" algorithm and the "adopted F₂" method. We simulated 200 F_2:3 families each with 5 plants. The number of resisted plants (y_i) for each F_2:3 families was recorded and used in the analysis of the exact EM method proposed in this study, where the mixture distribution of F_2:3 progeny derived from heterozygous F₂ parents was fully considered. Meanwhile, the average of phenotypic data from F_2:3 family replaced the measurement of F₂ individual and was regarded as categorical data with the analysis of the adopted F₂ method for F₂ population, where the mixture distribution of the F_2:3 sum derived from heterozygous F₂ parents was completely ignored (the adopted F₂ mapping procedure, currently available). The former is proposed for F_2:3 design in this article, and the latter is the method of Xu et al. (2005) for F₂ population. The 2 methods were compared under 3 levels of the BTL size (also called the BTL heritability): h²= 0.01, 0.05, and 0.10. The corresponding additive and dominant effects under each level of h² are given in Table 3. Note that as for the adopted F₂ method, the 5 threshold values were not listed in order to simplify the table. The results showed that the 2 methods differ in the following aspects: 1) the estimates of dominant effect are severely biased for the adopted F₂ method, whereas the biases are largely corrected by using the correct model, the exact EM method and 2) using the correct model can significantly increase the statistical power of BTL detection compared with the old method. The 2 differences clearly demonstrate the superiority of the new method over the old one.

View this table: [in this window] [in a new window]   Table 3. Comparisons of the "adopted F2" (old) method and the "exact EM" (new) method for BTL mapping in the F2:3 design

 
To demonstrate the third objective of the simulation experiments, we compared the efficiencies of the 2 methods developed in the study: the exact EM and the approximate EM method. The number of F₂ plants was set at 200 and each F₂ parent had 5 F_2:3 progeny. The 2 methods were compared under 3 levels of heritability: 0.05, 0.10, and 0.15. The results are listed in Table 4. No significant differences were found for the 2 methods except the statistical power. Therefore, the approximate EM algorithm is concluded as efficient as the exact EM algorithm.

View this table: [in this window] [in a new window]   Table 4. Comparisons of the exact EM method with the approximate EM method for BTL mapping in the F2:3 design

 
The last objective of the simulated studies is to compare mapping BTL for binary traits with mapping QTL for quantitative traits. We previously developed the method of mapping QTL for quantitative traits in the F_2:3 design (Zhang and Xu 2004). In this article, we proposed a new method of mapping BTL for binary traits in the F_2:3 design. Thus, it is possible to compare the 2 methods described above under 3 levels of heritability: 0.01, 0.05, and 0.10, respectively. The results were summarized in Table 5. It shows that the method of mapping QTL for a continuous phenotype is usually more powerful than that for a binary phenotype. The table also shows that the power for QTL detection is about 50% even the heritability is set at 0.01 among 100 replicates, whereas the power for BTL identification is up to 22%.

View this table: [in this window] [in a new window]   Table 5. Comparisons of QTL mapping method with BTL mapping method in an F2:3 design

 

	Discussion

Top Theory and Methods Discussion Funding Appendix: Solving for the... References

 
For mapping QTL in cotton (Zhang et al. 2003), maize (Yan et al. 2006), and cucumber, the F_2:3 design is widely used to improve the power and the precision in the detection of QTL. Following the concept of the mixture distribution of F_2:3 progeny derived from heterozygous F₂ plants, we extended statistical method from mapping QTL for continuous traits (Zhang and Xu 2004) to mapping BTL for binary traits in this paper. In fact, the analyses of binary traits may be theoretically more challenging than that of continuous traits because the former requires modeling the link between the observable phenotype and the corresponding latent variable. As compared with Zhang and Xu (2004), the proposed method in this study also makes use of the characteristics of the mixture distribution but there are some differences. On one hand, for the former, the average value of F_2:3 progeny is analyzed, whereas for the latter, the sum of phenotypic values of F_2:3 progeny (i.e., only the number of the resistant plants is known) is used. On the other hand, for the former, the distribution for F_2:3 progeny derived from heterozygous F₂ plants is the mixture of normal distributions, but for the latter, the corresponding distribution is multinomial distribution. In fact, the main difference is between continual variable in Zhang and Xu (2004) and discrete variable here. Although the method presented here concerns the F_2:3 design, its principal may be applicable to any type of F_x:y design and some similar design (Kao 2006), that is granddaughter design (Weller et al. 1990; Bovenhuis and Weller 1994; Mackinnon and Weller 1995; Ron et al. 2001), NCIII design (Cockerham and Zeng 1996), with minor modifications (Zhang and Xu 2004). For example, the new method is extended to an F_x:y design. In this case, the heterozygous individual in the heterozygous F_x:y family accounts for (1/2)^y–x(y > x), and the prior and posterior probabilities (h_ik and w_ik) of the kth genotype at the BTL for individual i in F_x generation will change. In other words, h_ik and equations 6 and 9 will be changed.

Many complex traits show a binary or categorical phenotypic distribution. Mapping loci of such traits requires method that takes into account these specific phenotypic distributions. McIntyre et al. (2001) developed a probability model treating the probabilities of disease (penetrance) as parameters of interest. However, the genetic parameters of interest for most plant or animal breeders were not estimated. In addition, the penetrance model cannot be used to estimate threshold value, which links the observable phenotype to the corresponding latent variable. Under the threshold model of binary disease, Xu et al. (2003) developed an EM-implemented maximum-likelihood method by treating both the unobserved genotype and the disease liability as missing values in a 4-way-cross mouse family. However, the F_2:3 design differs from the 4-way-cross design in that the F_2:3 progeny is not genotyped. Again, the threshold was not estimated in Xu et al. (2003). Herein, under the single BTL model, we integrated the penetrance model with the liability model to simultaneously estimate the penetrance of each genotype and the genetic parameters of interest as well as the above-mentioned threshold. The method used to estimate the threshold is the same as that in Thompson (1972) and Luo and Wu (2001) although the latter is for natural population. The results from simulation studies here show that the method works well. Therefore, the method may be considered in mapping BTL. In actual BTL-mapping experiments, the number of BTL is most likely greater than one and is usually unknown. In that case, the single locus model may be used to scan for multiple loci, which are implied by multiple peaks on the test statistic profiles, like the original interval mapping procedure of Lander and Botstein (1989). In the future project, it is possible to develop multiple BTL model in the F_2:3 design using the Bayesian statistics.

In practice, the average of phenotype data from F_2:3 family is usually used to replace the measurement of F₂ individual. This is the currently adopted F₂ method for quantitative traits (Zhang et al. 2003). For binary trait, however, it does not work well. The reason is that it is difficult to explain the biological meanings of the estimates for both BTL effects and residual variance although the power and the position for BTL detection can be obtained. In this case, it is necessary to seek another approach. Under the assumption of 5 plants per F_2:3 family, there are 6 numbers for the average 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. Obviously, it is suitable to regard these 6 numbers as categorical data. If so, the method of Xu et al. (2005) for F₂ population may be used to analyze the data and be viewed as the currently adopted F₂ method for binary traits in the second simulation experiment.

The source codes for a C++ program, with which the above calculations can be performed, are available on request for scientific purposes from C.Z. (cszhu@ sina.com) or Y.-M.Z. (soyzhang{at}njau.edu.cn).

	Funding

Top Theory and Methods Discussion Funding Appendix: Solving for the... References

 
The 973 program (2006CB101708), the National Natural Science Foundation of China (No. 30470998), the Specialized Research Fund for the Doctoral Program of Higher Education (20060307008), the 863 program (2006AA10Z1E5), the NCET (NCET-05-0489), and the Talent Foundation of Nanjing Agricultural University (No. 804079) to Y.-M.Z.; the China and Jiangsu Postdoctoral Science Foundation to C.Z. (No. 2005038246); and the Program for Changjiang Scholars and Innovative Research Team in University, the Ministry of Education (IRT0432).

	Appendix: Solving for the MLEs of the Model Parameters

Top Theory and Methods Discussion Funding Appendix: Solving for the... References

 
The expectation of the complete data log-likelihood function as to the equation 7 in the text has a form as

where = (f₁, f₂, f₃), w_i1 w_i3 can be calculated from the equation 9, and _mix(y_i; n,p_i) is same as equation 8. Hence, we have

where C is a constant. The first derivative of L() with respect to can be

As to the second derivatives, because there is a symmetric matrix, only their diagonal and lower diagonal elements are listed as follows:

With these derivatives, The MLEs of the model parameters cab be obtained using the numerical algorithm, for example, in Press et al. (2001), which solve the equations 11 in the text.

	Acknowledgments

 
We are grateful to the Associate Editor Reid G. Palmer and 3 anonymous reviewers for their constructive comments and suggestions that significantly improved the presentation of the manuscript.

	Footnotes

 
Corresponding Editor: Reid Palmer

Received December 1, 2006
Accepted March 8, 2007

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