Genetic model

Genetic cross between two parental inbred lines P₁ and P₂ is performed to produce an F₁ population, consist of all heterozygotes genotypes, which are used to produce the segregating progeny B₁ = F₁× parent (backcross) or an F₂ = F₁×F₁ (intercross). The genetic model for intercross population is as follows (1)

Here G₂, G₁ and G₀ are the genotypic values of genotypes QQ, Qq and qq. D is known as the genetic design matrix and E = [a, d]^T is the vector of genetic parameters. The first and second columns of D, denoted by D₁ and D₂, represent the status of the additive and dominance effects.

The genetic model of backcross population can be written as (2)

All notations are similar as described in the upper of this section.

CIM statistical model for QTL mapping

Phenotypic traits might be controlled by several QTLs. Estimators of simple interval mapping approach might be biased for confounding effects of multiple background QTLs [18]. Usually QTLs are highly linkage disequilibrium with corresponding flanking markers, therefore highly significant markers choosing through stepwise procedure could be a good representative set of background QTLs. Composite interval mapping approach modified the simple interval-mapping approach by including several significant markers as cofactors. Some others confounding factors (e.g. sex, age, diet etc.) also might have influence on traits, can be used as cofactor in the model to adjust their effects.

Suppose, we want to test for a QTL on a marker interval for F₂ population, then the statistical model for composite interval mapping can be written as (3) where y_j is the phenotypic value of the j^th individual; a is the additive effect of the testing QTL position; d is the dominance effect of testing QTL position; X_j is the matrix, may contain some chosen markers and other explanatory variables; γ is the vector of partial regression coefficients including the general mean effect μ; ε_j is a random error. The value of (, ) is (1, -0.5) if QTL genotype is QQ, (0, 0.5) if genotype is Qq, and (-1, -0.5) if genotype is qq.

Statistical genetic model for backcross population can be written as (4)

Notations of this equation are similar as described in upper of this section. The value of is 1 if QTL genotype is QQ, and 0 if genotype is Qq.

Robustification of CIM approach using beta-likelihood estimators

Composite interval mapping approach use the estimators derived from classical likelihood function to estimate genetic parameters, which might produce false positive or reduce detecting power of true loci in presence of phenotypic outliers. Instead of classical estimators of genetic parameters, β-Composite Interval Mapping uses the robust estimators derived from beta-likelihood function for robustly estimating the QTL positions in presence and absence of phenotypic outliers.

In the model (3), observation y_j’s are influenced by three QTL genotypes QQ, Qq, and qq; therefore, each phenotypic observation (y_j) is assumed to follow a mixture of three possible gaussian densities with different means and mixing proportion. The distribution function of each phenotypic observation (y_j’s) can be defined as where θ = (p,a,d,γ,σ²), ϕ() is a standard normal probability density function, , , and . The mixing proportions p_ji’s, which are functions of the QTL position parameter p, are the conditional probabilities of QTL genotypes given marker genotypes. For n individuals, the objective function for estimating θ is defined as (5) where l_β(θ|X) = [∫{f(y|θ,X}^β+1dy]^β/(β+1)

It was induced from the beta-divergence [19] for estimation of the parameters. It reduces to the log likelihood function for β →0. That is

Therefore, the objective function (5) called as beta-likelihood function. Maximization of this type of beta-likelihood function is equivalent to the minimization of beta-divergence [19] for estimating model parameter θ. The β-LOD score for the evidence of a QTL in a marker interval from the robustness point of view is defined by (6) where θ₀ and θ are the restricted and unrestricted parameter spaces. For β →0, the LOD_β reduces to the classical LOD criterion. The threshold value to reject the null hypothesis can be computed by permutation test [19].

Generalized form of the formulas for QTL mapping

The formulas for robust estimators of genetic parameters were elaborately described in related publications [12, 13]. In this section, we described the generalized forms of the robust estimators for QTL mapping. If k marker intervals (k putative QTLs) are considered jointly in mapping, the dimension of the genetic design matrix D augment to 2^k × k for a backcross population and to 3^k × 2k for an F₂ population when epistasis is ignored. Let us consider a backcross population as an example to see how to use these general formulas for other genetic models and populations. If we want to consider three marker intervals (three putative QTLs) simultaneously and use an additive model, the genetic model can be defined as (7) where G₁₁₁, G₁₁₀, G₁₀₁, G₁₀₀, G₀₁₁, G₀₁₀, G₀₀₁ and G₀₀₀ represent genotypic values of QTL genotypes AABBCC, AABBCc, AABbCC, AABbCC, AABbCc, AaBBCC, AaBBCc, AaBbCC and AaBbCc respectively. The notations a₁, a₂ and a₃ represent the effects of QTLs A, B, and C respectively. The genetic design matrix D with dimension 8 × 3 specifies that the corresponding likelihood is a mixture of 8 normal densities and has 3 genetic parameters (excluding μ) to be estimate. Accordingly, the matrix of β weighted posterior probabilities Π_β, defined in the related publication [12], is an n × 8 matrix. In deriving the β-estimators, in the E-step Π_β of the eight QTL genotypes are updated, and in the M-step the following equations (8) (9) (10) are applied to maximize Q(θ | θ^(t)). Here,

δ is an indicator variable. For details about Q(θ | θ^(t)) and Π_β, see the related publication [12]. To infer the joint conditional probability matrix Q for the three putative QTLs, we use the property that if there is no interference in crossing over, the conditional distributions of the individual putative QTL genotypes given the flanking marker genotypes are independent, irrespective of whether the QTLs are linked or not. This independence property simplifies the inference of Q matrix. If pair-wise epistasis of QTLs A × B, A × C, and B × C are also analyzed, the dimensions of genetic design matrix D in Eq (7) augment to 8 × 6. Columns 4, 5, and 6, which are the products of columns 1 and 2, 1 and 3, and 2 and 3, of the genetic design matrix represents the status of the epistatic parameters of different genotypes. The Q matrix is the same as that for the additive model. If higher order of QTL epistasis is considered, the corresponding column vector of D can be extended by the same procedure. For more details discussion about general formulas for CIM algorithm, please see the paper of Kao and Zeng [20].

Robustness

Let G be the distribution function of g, then we can view the β-estimator, which is a function of G defined by (11) where (12) is the objective function. Therefore, the robustness of the β-estimator can be investigated by the influence function. The influence function (IF) for the β-estimator at y under the distribution function G is defined as (13) where Δ_y is the probability measure that puts mass 1 at the point y. An estimator is said to be B-robust if its influence function is a bounded function of y [21]. Since the β-estimator satisfies the properties of M-estimator, the influence function for the robust estimator also can be written as (14) where ψ_β(y;θ) = ∂Ω_β(y;θ)/∂θ is the estimating function for the β-estimator and (15) is a matrix which does not depend on y; thus, the B-robustness is equivalent to the boundness of the estimating function for the M-estimator as well as the β-estimator [14, 22]. To prove the boundedness of the estimating function ψ_β(y;θ) for the β-estimator, let us consider the general form of estimating function as defined by (16)

Obviously, the boundedness of the estimating function depends only on the second term of the right-hand side of (16), since C_β(θ) is independent on observations. In the second term of the right-hand side of (16), we have (17) (18) (19) (20)

Thus, we can conclude that if β > 0, then all components of estimating function are bounded with respect to y and X. This is because all the terms corresponding to the Eq (17–20) are of the form with f(z) being polynomial in z, which is bounded in . In case of β = 0 or equivalently the maximum likelihood estimator, all the terms corresponding to the Eq (17–20) become unbounded. Typically, for example, . Thus, we may conclude that the estimating function ψ_β(z;θ) for the β-estimator is bounded for β > 0. Therefore, the β-estimators are B-robust against outliers.

Selection of the tuning parameter β

The value of the tuning parameter β plays a key role in the performance of the BetaCIM method. It controls the trade-off between robustness and efficiency of estimators. This method shows good performance for a wide range of β. A large β decreases the efficiency and increases the robustness of an estimator, and vice-versa for the smaller β. However, an optimum value for β depends on the initialization of model parameters, data contamination rates, type of data contamination, type of datasets and so on. So heuristic selection of the tuning parameter β, may produces misleading results in some satiations. To find an optimum β for minimum β-divergence method, Mollah et al. [16, 17] used β-divergence with a fixed value β₀ of β as a measure for evaluation of the minimum β-divergence estimators. In this paper, we also use the same measure for β selection using cross validation. To define the measure for β selection using K-fold cross validation, the entire dataset into K subsets , , …, and let . Then the measure for β selection by K-fold cross validation can be defined by (21) where are estimated using dataset and (22) with (23) where n_k is the number of observation in the subset . Under null hypothesis the distribution function is , where . We select an appropriate β by the minimizer of for β. To compute β-LOD score for testing the evidence of a QTL in a specific position in a chromosome, we need to select β by cross validation under null hypothesis and alternative hypothesis separately. So, it takes a lot of time to compute the genome-wide β-LOD scores. However, we can use the same β by cross validation under null hypothesis, for each of alternative cases also to save the computational time, since the estimators show good performance for a wide range of β > 0 also. If cross validation results find the value of β significantly larger than 0, it indicates that outliers contaminate the dataset. If cross validation results find the value of β very close to 0, it indicates that the dataset is not contaminated by outliers and the robustified CIM algorithm reduce to the traditional CIM.

Simulation study

Simulations studies were conducted for Backcross (BC) and Intercross (F₂) populations based on the assumptions of multiple linked and unlinked QTLs, and in presence and absence of phenotypic outliers. LOD scores were calculated for classical interval mapping and composite interval mapping approaches and β-LOD scores were calculated for BetaCIM approach that is equivalent to classical LOD scores for β→0. Trait specific optimum β was selected by using k-fold cross-validation procedure, implemented in our BetaCIM R-package. We simulated genotype data by using the implemented functions (sim.map and sim.cross) of popular R/qtl package. Then by setting effects corresponding to some specific markers the phenotypic data sets were generated. Mean sum of squares of the parameters were calculated by using the following formula-

Detection power of the QTL was calculated as the numbers of times calculated LOD scores of a QTL position exceed the significant threshold level divided by the number of simulations. Threshold value of the test statistic was calculated by using permutation test.

Multiple unlinked QTLs

We simulated data for backcross population by assuming the QTLs located far enough to be linkage equilibrium to each other’s. Four QTLs situated in different chromosomes were considered for simulation study. Four chromosomes each with fifteen markers separated in 10cM intervals were simulated. Phenotypic data sets were generated by assuming some of the specific marker positions (Chromosome-Marker: C₁M₃, C₂M₆, C₃M₄ and C₄M₄) as QTLs. Variations of phenotypic data was contributed 50% by QTLs and 50% by random error. We generated 100 simulated data and analyzed by using IM, CIM, and BetaCIM approaches, and then plotted the average LOD scores (Fig 1). Sample size for each simulation was 300. Moreover, simulation was conducted with 5% contaminated phenotype data to investigate the robustness property of classical approaches and BetaCIM approaches. Same approach was used to generate contaminated data described in previous publications [12, 15], randomly selected 5% of the observations and then added random numbers.

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Fig 1. Simulations results without and with phenotype outliers in the case of multiple unlinked QTL.

(A) Analysis results in absence of outliers; (B) analysis results in presence of 5% outliers. Threshold for each method were calculated using permutation test with 1000 replicates.

https://doi.org/10.1371/journal.pone.0208234.g001

Simulation results showed that all approaches could identify the true loci in absence of phenotypic outliers, and provided good estimates of genetic parameters with smaller standard error (Table 1).

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Table 1. Detection power of different approach for multiple unlinked QTLs.

https://doi.org/10.1371/journal.pone.0208234.t001

Although, detection powers of 3 QTL positions were similar for IM, CIM and BetaCIM approaches, for QTL C₄M₄ detection power of IM approach was low (power = 64%) as compared to others two methods. Therefore, in the case of multiple unlinked QTL, detection power of the QTLs is better for CIM and BetaCIM approaches as compared to IM. Moreover, in the case of 5% phenotypic outlying observations, IM and CIM approaches failed to detect true QTL positions (Fig 1B). Detection powers of these approaches were very low (8~27% for IM, and 6~31% for CIM), as well as standard errors of the parameters were very high, implying the estimated genetic effects highly varied across different simulated data sets (Table 1). Interestingly, the BetaCIM method provides consistent results, just as there is no outlying observations, identified all QTL positions with high power and provided good estimates of genetic parameters with smaller standard errors (Table 1). Therefore, BetaCIM approach significantly improved the performance over classical CIM approach in the case of multiple unlinked QTLs and presence of phenotypic outliers.

Multiple linked QTLs

One of the crucial properties of CIM is the ability of identifying multiple linked QTLs. In this scenario, we conducted simulation with similar parameter setting as previous publication for CIM [6]. We simulated data from four chromosomes, each with 16 markers and separated in 15 10cM intervals. The traits were controlled by 10 QTLs with positions and effects given in Table 2. Among the 10 QTLs, 9 were in 1^st three chromosomes and 1 was in chromosome 4.

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Table 2. Parameters and point estimates of effects with and without phenotypic outliers.

https://doi.org/10.1371/journal.pone.0208234.t002

Together, the QTLs account for 50% and 70% of the phenotypic variance for two different sets of simulated data. Sample size was 300. In Fig 2, we plotted the average LOD scores for 300 simulations under two different scenarios, with and without phenotypic outliers, and for two different heritability settings. Results showed that BetaCIM can provide similar LOD score profile as CIM approach in absence of phenotypic outliers (Fig 2A and 2C).

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Fig 2. Simulation results in presence and absence of phenotypic outliers in the case of multiple linked QTLs.

Total genetic heritability was set as 50% and 70% for two different sets of simulated data. 5%-contaminated data was added to phenotypes to investigate the robustness property of the approaches. (A) Analysis results in the case of 50% genetic heritability and no outlying observation; (B) analysis results in the case of 50% genetic heritability and 5% outlying observations; (C) Analysis results in the case of 70% genetic heritability and no outlying observation; (D) analysis results in the case of 70% genetic heritability and 5% outlying observations. Threshold for each method were calculated using permutation test with 1000 replicates.

https://doi.org/10.1371/journal.pone.0208234.g002

With 50% genetic heritability both approaches identified 6 QTLs out of 10 QTLs at 5% level of significance (Fig 2A). These approaches provided lower picks at others five true QTL positions. Although, with 70% genetic heritability both approaches detected one additional QTL positions, referring that increasing of genetic heritability of the phenotypic traits can increase the detection power of the approaches. IM approach detected 4 true QTL positions with 50% and 70% genetic heritability respectively. This approach provided wider picks, failed to separate multiple linked QTLs and detected several wrong QTL positions (Fig 2A and 2C). Therefore, in the case of multiple linked QTLs, CIM and BetaCIM provided better result than IM approach.

Simulation with single replicate showed that IM approach could provide larger mean sum square error (MSE = 0.124) compared to CIM and BetaCIM approaches, referring that parameter estimation of IM approach could be biased in the case of multiple linked QTL. CIM and BetaCIM approaches provided similar estimates of the genetic parameters and equal mean sum square error (MSE = 0.007), indicating the CIM and BetaCIM approaches could provide better results compared to the IM approach in the case of multiple linked QTL.

With 5% contaminated observations, CIM and IM failed to detect true QTL positions (Fig 2B and 2D). However, BetaCIM provided similar results as without outliers, detected the true QTL positions (Fig 2B and 2D). Therefore, BetaCIM is robust against phenotypic contamination, can detect true QTL positions in presence and absence of outliers. Outlying observations had large impact on parameter estimation of IM and CIM approaches. Estimation was biased upward or downward for these approaches (Table 2), and provided larger MSE. BetaCIM approach significantly improved the performance in effects estimation and identifying true QTLs in the case of outlying observations, provided smaller MSE. Simulation with F₂ population provided similar results for the approaches (Text A, Figs A and B, and Table A in S1 File).

Beta selection for QTL analysis

In simulation study, we observed that BetaCIM approach significantly improved the analysis results in presence and absence of phenotypic outliers. Tuning parameter β plays key role for controlling effects of contaminated data. Value of β depends on proportion of contamination in phenotypic data; its value can increase with respect to increasing outlying observations. We calculated the average value of β with different proportion of phenotypic outliers (Fig 3).

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Fig 3. Optimum value of the tuning parameter beta with different proportion of phenotypic outliers.

Triangles are locating at the median optimum value of beta. (A-E) plots for β selection by cross-validation in presence of 0%, 2%, 5%, 10% and 20 outliers, respectively.

https://doi.org/10.1371/journal.pone.0208234.g003

In 100 simulations without phenotypic outliers, the median of optimum values of the tuning parameter β was 0.001, referring that without phenotypic outlier optimum value of β is very small. We observed that optimum value of the tuning parameter β increase with respect to increasing outlying observations. For example, in the case of 2~20% of outlying observations median of optimum values of β were varies to 0.041~0.291. Therefore, selecting trait specific tuning parameter β by using cross validation is crucial for the BetaCIM approach. Selecting the tuning parameter could be useful for real data analysis that may help in efficient estimation of QTL positions and effects. Implemented function of “BetaCIM” R-package can select trait specific optimum beta for analysis.

Real data analysis

Data from an experiment on multiple traits in the mouse was downloaded from mouse phenome database (https://phenome.jax.org/projects/Feng1). The data was for an intercross between 129S1/SvlmJ and A/J inbred mouse strains. There were several phenotypic traits scored in the cross. We analyzed left and right kidney weight of mouse to identify the QTLs underlying these traits. There were total 336 intercross individuals, aged 8 weeks, and typed at 91 markers. For more details about the data see the related publication [23]. IM, CIM and BetaCIM approaches were used for analyses (Fig 4).

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Fig 4. Analysis of kidney weight data of mouse intercross population (129 S1/SvlmJ× A/J).

(A, D) Plots of phenotypic distributions for right and left kidney weights, respectively; (B, E) Plots for β selection by cross-validation from left and right kidney weights, respectively. (C, F) LOD score profiles for left and right kidney weights, respectively. Threshold for each method were calculated using permutation test with 1000 replicates.

https://doi.org/10.1371/journal.pone.0208234.g004

From Fig 4A, we observed that the right kidney weight (RKW) data symmetrically distributed and there had no extreme observations. None of the phenotypic observation was larger than Q₃+3*IQR or smaller than Q₁-3*IQR, where Q₁, Q₃ and IQR are first quartile, 3^rd quartile, and inter quartile range of RKW data. In this case, optimum value of beta was 0.001 that also indicated there had no contaminated observations in the data set. Analysis with CIM and BetaCIM approaches provided similar LOD score profile and identified 3 QTL positions at 5% level of significance (Fig 4C). Highest picks of the identified QTL positions by using CIM and BetaCIM approaches were in 42.14 cM of chromosome 9 (LOD = 5.18, LOD_β = 5.17) with nearby marker rs3676158 (position = 40.876 cM, LOD = 5.16, LOD_β = 5.15); in 67.941 cM of chromosome 10 at the marker rs3674646 (LOD = 3.90, LOD_β = 3.89); and in 45.055 cM of chromosome 13 at the marker rs3716022 (LOD = 3.53, LOD_β = 3.42). IM approach also identified three QTL positions; the highest significant picks were 45.132 cM of chromosome 9 (LOD = 3.79) with nearby marker rs3676158 (LOD = 3.67), in 67.941 cM of chromosome 10 at the marker rs3674646 (LOD = 3.46); and 35.299 cM of chromosome 13 within the marker interval rs3676930 (position 24.6737 cM) and rs3716022 (position 45.055 cM). The candidate markers rs3676158 and rs3674646 are the intron variants of genes Unc13c, and Grip1 respectively, however gene information of another variant rs3716022 is unknown. Unc13c is responsible for an additional step of molecular and/or positional "superpriming" that substantially increases the efficacy of Ca(2+)-triggered release [24]. Its play a role in vesicle maturation during exocytosis as a target of the diacylglycerol second messenger pathway, and may be involved in the regulation of synaptic transmission at parallel fiber (http://www.uniprot.org/uniprot/Q8K0T7). Grip1 play a role as a localized scaffold for the assembly of a multiprotein signaling complex and as mediator of the trafficking of its binding partners at specific subcellular location in neurons (http://www.uniprot.org/uniprot/Q925T6).

We analyzed the left kidney weight (LKW) data of the same population. In this case, there was evidence of one outlying phenotypic observation (phenotypic value of one individual observation was larger than Q₃+3*IQR, where Q₃ is the third quartile and IQR is interquartile range of LKW data). Optimum value of the tuning parameter β was larger than before (opt β = 0.041), indicated there had some contaminated observations in the data set. For LKW, IM and CIM approaches identified none of the QTL positions at 5% level of significance (Fig 4F). However, BetaCIM approach identified 2 QTL positions, which are 12.43~21.43 cM of chromosome 5 and 20.30~29.30 cM of chromosome 13. LOD scores for CIM and IM approaches at the QTL positions were high but not significant at 5% level of significance (Fig 4F). These two QTLs hold the markers rs3023765 (position = 16.550 cM, LOD_β = 3.78) and rs3676930 (position = 24.6737 cM, LOD_β = 4.54), which are the variant of the genes Otof and A330033J07Rik. Mutations in human orthologous of mice Otof gene are a cause of neurosensory nonsyndromic recessive deafness, hearing loss [25]. Individual with moderate chronic kidney disease (CKD) have a higher prevalence of hearing loss than those of the same age without CKD (https://www.kidney.org/news/ekidney/november10/HearingLoss_November10). Mice lacking Otof display hearing loss. It expressed in the cochlear IHC, vestibular type I sensory hair cells, eye, heart, skeletal muscle, liver, kidney, lung and testis (http://www.uniprot.org/uniprot/Q9ESF1). Another novel gene A330033J07Rik also expressed in kidney (https://www.ncbi.nlm.nih.gov/geoprofiles/7902881). Therefore, the candidate genes may have relevant function for kidney weight.