Likelihood Ratio Test for Excess Homozygosity at Marker Loci on X Chromosome

Xiao-Ping You; Qi-Lei Zou; Jian-Long Li; Ji-Yuan Zhou

doi:10.1371/journal.pone.0145032

Abstract

The assumption of Hardy-Weinberg equilibrium (HWE) is generally required for association analysis using case-control design on autosomes; otherwise, the size may be inflated. There has been an increasing interest of exploring the association between diseases and markers on X chromosome and the effect of the departure from HWE on association analysis on X chromosome. Note that there are two hypotheses of interest regarding the X chromosome: (i) the frequencies of the same allele at a locus in males and females are equal and (ii) the inbreeding coefficient in females is zero (without excess homozygosity). Thus, excess homozygosity and significantly different minor allele frequencies between males and females are used to filter X-linked variants. There are two existing methods to test for (i) and (ii), respectively. However, their size and powers have not been studied yet. Further, there is no existing method to simultaneously detect both hypotheses till now. Therefore, in this article, we propose a novel likelihood ratio test for both (i) and (ii) on X chromosome. To further investigate the underlying reason why the null hypothesis is statistically rejected, we also develop two likelihood ratio tests for detecting (i) and (ii), respectively. Moreover, we explore the effect of population stratification on the proposed tests. From our simulation study, the size of the test for (i) is close to the nominal significance level. However, the size of the excess homozygosity test and the test for both (i) and (ii) is conservative. So, we propose parametric bootstrap techniques to evaluate their validity and performance. Simulation results show that the proposed methods with bootstrap techniques control the size well under the respective null hypothesis. Power comparison demonstrates that the methods with bootstrap techniques are more powerful than those without bootstrap procedure and the existing methods. The application of the proposed methods to a rheumatoid arthritis dataset indicates their utility.

Citation: You X-P, Zou Q-L, Li J-L, Zhou J-Y (2015) Likelihood Ratio Test for Excess Homozygosity at Marker Loci on X Chromosome. PLoS ONE 10(12): e0145032. https://doi.org/10.1371/journal.pone.0145032

Editor: Dmitri Zaykin, NIH - National Institute of Environmental Health Sciences, UNITED STATES

Received: March 11, 2015; Accepted: November 28, 2015; Published: December 15, 2015

Copyright: © 2015 You et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

Data Availability: All relevant data are within the paper and its Supporting Information files. The rheumatoid arthritis data are from North American Rheumatoid Arthritis Consortium, which are available from Genetic Analysis Workshop 15.

Funding: This work was supported by the National Natural Science Foundation of China (81373098), Science and Technology Planning Project of Guangdong Province, China (2013B021800038) to Professor Ji-Yuan Zhou. The Genetic Analysis Workshops are supported by the National Institutes of Health grant R01 GM031575. The RA data were gathered with the support of grants from the National Institutes of Health (NO1-AR-2-2263 and RO1-AR-44422), and the National Arthritis Foundation. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Association analysis is a useful tool to map disease loci by using markers on autosomes based on family data and case-control data [1–9]. There has been an increasing interest of exploring the association between diseases and markers on X chromosome and the effect of the departure from Hardy-Weinberg equilibrium (HWE) on association analysis on X chromosome [10–17]. Note that there are two hypotheses of interest regarding the X chromosome: (i) the frequencies of the same allele at a locus in males and females are equal and (ii) the inbreeding coefficient in females is zero (without excess homozygosity) in X-specific quality control [18, 19]. As such, excess homozygosity in females and significantly different minor allele frequencies between males and females are used to filter X-linked variants [20, 21]. The inbreeding coefficient is generally estimated by functions of excess homozygosity [22, 23], which may be caused by population substructure, consanguineous mating or factors like null alleles [24, 25]. Overall and Nichols developed an approach to distinguish population substructure and consanguinity by using multilocus genotype data [24]. On the other hand, Zheng et al. proposed two test statistics to test for the equality of the frequencies of the same allele in males and females and the zero inbreeding coefficient in females on X chromosome, respectively [14]. However, they only focused on association analysis on X chromosome and the type I error rates and powers of these two test statistics have not been studied yet. Further, there is no existing method to simultaneously detect both of the issues till now.

Therefore, in this article, we first combine two test statistics proposed in zheng et al. [14] and suggest Z₀ to simultaneously test for (i) the equality of the frequencies of the same allele in males and females and (ii) the zero inbreeding coefficient on X chromosome based on the collected sample. For the purpose of improving the test power for both (i) and (ii), a novel likelihood ratio test on X chromosome is proposed. We write out the likelihood functions of the collected sample under the null hypothesis and alternative hypothesis at a single locus on X chromosome, respectively. Next, we obtain the maximum likelihood estimates (MLEs) of the unknown parameters by expectation-maximization (EM) algorithms [26] and construct the corresponding likelihood ratio test (LRT₀) statistic to test for both (i) and (ii). If the null hypothesis is statistically rejected, we further conduct two hypothesis testing issues to find the underlying reasons why the null hypothesis is violated by proposing another two likelihood ratio tests LRT₁ (for the equality of the frequencies of the same allele in males and females) and LRT₂ (for excess homozygosity). Note that the size of LRT₀ and LRT₂ is conservative from our simulation study. As such, we use parametric bootstrap techniques to evaluate the validity and performance of LRT₀ and LRT₂, which are respectively denoted by LRT_0b and LRT_2b. Moreover, we explore the effect of population stratification on the proposed tests. In addition, the root mean squared error (RMSE) and bias are used to assess the accuracy of the MLEs of the unknown parameters. Finally, the application of the proposed methods to a rheumatoid arthritis (RA) dataset indicates their utility.

Materials and Methods

Background and notations

Consider a biallelic marker locus on X chromosome with alleles M₁ and M₂. Let p_m and p_f be the frequencies of M₁ in males and females, respectively. As such, the frequencies of M₂ in males and females are q_m = 1 − p_m and q_f = 1 − p_f, respectively. In females, let ρ be the inbreeding coefficient, which is generally nonnegative [27–29]. Thus, the frequencies of three genotypes M₁M₁, M₁M₂ and M₂M₂ in females can be expressed as follows:

To this end, there is no excess homozygosity in females when ρ = 0; excess homozygosity exists when ρ > 0. Note that p_m ≠ p_f may be true on X chromosome. So, we construct the null hypothesis denoted by H₀: p_m = p_f and ρ = 0 to test for both of the hypotheses (i) and (ii). If the null hypothesis is violated, we need to investigate which one of p_m ≠ p_f and ρ > 0 is true. As such, we have other two hypothesis testing issues with the null hypothesis being H₀₁: p_m = p_f and H₀₂: ρ = 0, respectively. It should be noted that X chromosome has the problem of X chromosome inactivation and dosage compensation [30], but we do not consider them in this section. The corresponding discussion can be found later (see the Discussion section).

Assume that n_1m and n_0m represent the numbers of males with alleles M₁ and M₂ in a collected sample, respectively; n_2f, n_1f and n_0f denote the numbers of females with genotypes M₁M₁, M₁M₂ and M₂M₂, respectively. Then, N_m = n_1m + n_0m and N_f = n_2f + n_1f + n_0f are respectively the numbers of males and females in the sample, and N = N_m + N_f is the sample size.

Existing methods Z₁ and Z₂ for H₀₁ (equality of the frequencies of the same allele in males and females) and H₀₂ (zero inbreeding coefficient), respectively

Zheng et al. [14] proposed the test statistic to test for H₀₁: p_m = p_f, where and are the estimates of p_m and p_f, , and are the estimates of the variances of and under H₀₁, respectively, with . Under H₀₁, Z₁ asymptotically follows the chi-square distribution with one degree of freedom when the sample size is large enough.

Weir and cockerham [31] introduced the disequilibrium coefficient in females . In other words, testing for Δ_f = 0 is equivalent to testing for ρ = 0. Hence, zheng et al. [14] further developed the following test statistic to test for H₀₂: ρ = 0, where , , and . Under H₀₂, Z₂ approximately follows the chi-square distribution with one degree of freedom when N_f is large enough. It should be noted that the test Z₂ has nothing to do with male individuals and thus only needs female individuals.

Z₀ test for both hypotheses (i) and (ii) of interest regarding the X chromosome

Zheng et al. [14] showed that, under H₀: p_m = p_f and ρ = 0, Z₁ and Z₂ are independent. However, they did not propose the corresponding test statistic for H₀. As such, we suggest the test statistic to test for H₀: p_m = p_f and ρ = 0. Under H₀, Z₀ asymptotically follows the chi-square distribution with the degrees of freedom being 2. Moreover, it should be noted that we can use to estimate the inbreeding coefficient ρ.

Likelihood ratio test for both hypotheses (i) and (ii) of interest regarding the X chromosome

To construct a likelihood ratio test (LRT) for H₀: p_m = p_f and ρ = 0, we give the likelihood function of the sample as follows: (1) where θ = (p_m, p_f, ρ). Firstly, we use the following EM algorithm to estimate the unknown parameters p_m, p_f and ρ under the alternative hypothesis (H₁: p_m ≠ p_f or ρ > 0). Suppose that Y = (Y₁, Y₂, Y₃, Y₄, Y₅) = (n_1m, n_0m, n_2f, n_1f, n_0f) denotes the observed data. (Y₁, Y₂, Y₃, Y₄, Y₅) can be augmented by splitting the third cell into two cells W₁ and W₂, which are unobservable random variables such that Y₃ = W₁ + W₂ for female homozygote M₁M₁ and W₁ and W₂ follow the binomial distributions with success probabilities and , respectively, and by splitting the fifth cell into two cells W₃ and W₄, where Y₅ = W₃ + W₄ for female homozygote M₂M₂ and W₃ and W₄ follow the binomial distributions with success probabilities and , respectively. Thus, the likelihood function of complete data (n_1m, n_0m, w₁, w₂, n_1f, w₃, w₄) is: where the normalizing constant is omitted for brevity.

At the E-step, the Q function at iteration (k + 1) is constructed as where θ^(k) is the estimate of θ at iteration k.

At the M-step, the estimated value θ^(k+1) of θ at iteration (k + 1) can be obtained by maximizing the Q function with respect to θ. Therefore, the MLEs of p_m, p_f and ρ at iteration (k + 1) are respectively

Note that the MLE of p_m is the same for all the iterations, which is also the same as zheng et al. [14]. In the above expressions, (2) (3) (4) (5) where . Given the initial value θ⁽⁰⁾ of θ, the above-mentioned two steps continue until the convergence criterion is satisfied. For example, the absolute differences between the estimates of the parameters at two consecutive iterations are all less than 10⁻⁷. The value of θ obtained at the last iteration is taken as the MLE of θ under H₁.

Note that p_m = p_f and ρ = 0 under H₀. Let p = p_m = p_f, the pooled allele frequency of M₁. Then, L(θ) in Eq (1) can be rewritten as

Thus, the MLE of p under H₀ is , the estimated pooled allele frequency of M₁. Let . Then, we can construct the following LRT to test for H₀ (6) which asymptotically follows a chi-square distribution with the degrees of freedom being 2 when the null hypothesis holds.

Likelihood ratio test for equality of frequencies of the same allele in males and females

Once the null hypothesis (H₀: p_m = p_f and ρ = 0) is rejected based on the result of Eq (6), we further need to consider the following two tests H₀₁: p_m = p_f and H₀₂: ρ = 0. Note that under the null hypothesis H₀₁: p_m = p_f = p, ρ may not be zero and we need to estimate it. Let ϕ = (p, ρ) and q = 1 − p. Thus, the corresponding likelihood function of complete data is

We use the following EM algorithm to estimate ϕ under H₀₁. The corresponding formulas at iteration (k + 1) are as follows where and are respectively the MLEs of p and ρ at iteration (k + 1), and . E_ϕ^(k)(w₁|n_2f), E_ϕ^(k)(w₂|n_2f), E_ϕ^(k)(w₃|n_0f) and E_ϕ^(k)(w₄|n_0f) in the above expressions are similar to E_θ^(k)(w₁|n_2f), E_θ^(k)(w₂|n_2f), E_θ^(k)(w₃|n_0f) and E_θ^(k)(w₄|n_0f) in Eqs (2)–(5), just replacing , and in Eqs (2)–(5) by , and , respectively. Let . Then, we propose the following test statistic LRT₁ to test for the null hypothesis H₀₁: p_m = p_f, (7) which approximately follows a chi-square distribution with the degree of freedom being 1 under H₀₁.

Likelihood ratio test for inbreeding coefficient being zero

Note that under the null hypothesis H₀₂ : ρ = 0, p_m and p_f may be different from each other and we need to estimate them separately. Let ψ = (p_m, p_f) and L(θ) in Eq (1) can be rewritten as

Then, the MLEs of p_m and p_f are and , respectively, which are the same as zheng et al. [14]. Let . As such, we develop the following test statistic to test for H₀₂ : ρ = 0 (8) which asymptotically follows a chi-square distribution with the degree of freedom being 1 under H₀₂. Just like the Z₂ test statistic, LRT₂ only uses female individuals in the sample because the terms based on male individuals in the numerator and the denominator of the fraction are the same, which can be reduced.

Likelihood ratio tests via parametric bootstrap for H₀ and H₀₂

It should be noted from our simulation results (see the Results section) that the simulated type I error rates of LRT₀ and LRT₀₂ respectively for H₀ and H₀₂ are too conservative. On the other hand, several studies showed that the likelihood ratio tests may typically not follow a chi-square distribution asymptotically [31, 32], and hence their exact distributions can be obtained by Monte Carlo simulation [33]. Accordingly, we make use of parametric bootstrap techniques to evaluate the size and power of these two methods. For convenience, we denote these methods via parametric bootstrap by LRT_0b and LRT_2b, respectively. We begin by describing the implementation steps for LRT_0b as follows:

For a collected sample of size N with N_m males and N_f females, calculate the value of LRT₀;
Compute the estimated pooled allele frequency based on the sample as follows: ;
Based on , calculate the frequencies of three genotypes M₁M₁, M₁M₂ and M₂M₂ in females under H₀ in the following: , and , respectively, where ;
According to and , regenerate the alleles of N_m males; based on , and , regenerate the genotypes of N_f females;
Calculate the value of LRT₀ based on the new N_m males and N_f females, denoted by ;
Repeat Steps 4 and 5 B times, which results in B test statistics , , …, ;
The P-value of the original LRT₀ can be estimated as

For LRT_2b, we can conduct the steps similar to those mentioned above. Firstly, after obtaining the value of LRT₂, calculate the frequencies of three genotypes M₁M₁, M₁M₂ and M₂M₂ in females under H₀₂ in the following: , and , respectively, with . The alleles of N_m males stay the same as the original sample and only regenerate the genotypes of N_f females according to , and . Then, carry out the similar procedures of Steps 4–7 and we can obtain the the estimated P-value of LRT₂.

Software implementation

We have written the XHWE software with R (http://www.r-project.org), which includes the eight test statistics: LRT₀, LRT_0b, LRT₁, LRT₂, LRT_2b, Z₀, Z₁ and Z₂. The R package named XHWE is available on CRAN (http://cran.r-project.org/web/packages/XHWE/). The initial values of p_m, p_f, p and ρ in the EM algorithms are taken to be n_1m/N_m, (2n_2f + n_1f)/(2N_f), (n_1m + 2n_2f + n_1f)/(N_m + 2N_f) and 0.02, respectively. The convergence criterion is that the absolute differences between the estimates of the parameters at two consecutive iterations are all less than 10⁻⁷ for the LRT-type statistics. The default maximum number of iterations is 1000. The input data file is the standard pedigree data. The XHWE software only uses the founders with genotypes available in it and will analyze marker loci one by one. The software outputs the values of all the test statistics and the corresponding P-values. Also, the XHWE software outputs the estimates of all the parameters under both the null and alternative hypotheses for each test statistic. The parameter estimates under the alternative hypothesis for the LRT-type test statistics are the same. However, under the respective null hypotheses of the LRT-type test statistics, the estimates may be different. It should be noted that the estimates of p_m and p_f under the null hypothesis of H₀₂ in this article and those in zheng et al. [14] are the same, respectively. The output results will be automatically saved in the text file named “results.txt”.

Simulation settings

Simulation study is conducted to assess the performance of the proposed LRT₀, LRT_0b, LRT₁, LRT₂, LRT_2b and Z₀ test statistics and to compare them with the existing Z₁ and Z₂ under various simulation settings which are similar to those in zheng et al. [14]. The allele frequency p_m in males takes two values: 0.3 and 0.5. When p_m is fixed, the value of p_f in females is taken as p_f = p_m + ϵ, where ϵ = 0, ±0.04 and ±0.05. The inbreeding coefficient ρ in females is set at 0 to 0.1 in increment of 0.05. The sample size is taken as 800 and 1200 with the ratio r = N_m : N_f being 2:1, 1.5:1, 1:1, 1:1.5 and 1:2. As mentioned earlier, when p_m = p_f and ρ = 0, the size of all the eight test statistics is simulated; when p_m = p_f and ρ > 0, the size of LRT₁ and Z₁ is gotten; when p_m ≠ p_f and ρ = 0, the size of LRT₂, LRT_2b and Z₂ is obtained. Otherwise, we simulate the corresponding powers. In addition, it should be noted that for the fixed sample size (800 or 1200) simulated above, the powers of all the three test statistics LRT₂, LRT_2b and Z₂ for H₀₂ : ρ = 0 are not so large, from our simulation results below. On the other hand, these three test statistics only use female individuals. As such, we further obtain the sample size N_f required for LRT_2b to gain 80% simulated power and then simulate the size and powers of LRT₂, LRT_2b and Z₂ under this sample size. To investigate how population structure affects the proposed methods, we also consider the following population stratification model with two subpopulations in our simulation study. p_m = 0.3 (0.5), p_f = p_m + ϵ, ϵ = 0, ±0.04 and ±0.05 in the first (second) subpopulation and the ϵ values are respectively denoted by ϵ₁ and ϵ₂. Assume that ρ = 0 in each subpopulation, and the ratio of each subpopulation constructing the population is set to 0.5. The sample size is taken to be 1800, where each individual is a female or a male with equal probability. Note that under population stratification, the null hypothesis H₀: p_m = p_f and ρ = 0 is generally not true. Thus, we use the population stratification model to study the powers of the proposed methods. The significance level is fixed at 5% and 10000 replications are simulated under each simulation setting. For LRT_0b and LRT_2b via parametric bootstrap, B is set to be 1000. Finally, to compare the efficiency of the parameter estimates of the proposed EM algorithms with those in zheng et al. [14] for each simulation setting, we use the RMSEs and biases to assess the accuracy of the parameter estimates, where and , and β is the parameter which needs to estimate.

Results

Simulation results

Table 1 lists the simulated size of LRT₀, LRT_0b, LRT₁, LRT₂, LRT_2b, Z₀, Z₁ and Z₂ under H₀ : p_m = p_f = p and ρ = 0 with N = 800 and 1200 and p = 0.3 and 0.5 for different values of r = N_m : N_f. According to the table, the size of LRT₁, Z₀, Z₁ and Z₂ is close to the nominal 5% level, while the size of LRT₀ and LRT₂ is too conservative. However, after the parametric bootstrap technique, LRT_0b and LRT_2b stay close to the nominal 5% level.

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Table 1. Simulated size (in %) of LRT₀, LRT_0b, LRT₁, LRT₂, LRT_2b, Z₀, Z₁ and Z₂ under H₀ : p_m = p_f = p and ρ = 0 with N = 800 and 1200 for different values of r and p.

https://doi.org/10.1371/journal.pone.0145032.t001

Fig 1 gives the simulated powers of the eight test statistics against r under H₁ : p_m ≠ p_f and ρ > 0 for different values of ρ (0.05 and 0.1) and N (800 and 1200), having p_m = 0.3 and p_f = 0.35. It is shown in the figure that LRT_0b is more powerful than LRT₀ and Z₀, and LRT₀ and Z₀ have the similar performance in power (Fig 1a-1d in the first row), regarded of the inbreeding coefficient ρ, the sample size N and the ratio r. LRT₁ and Z₁ have almost the same performance in power (Fig 1e-1h in the second row). LRT_2b has much more power than LRT₂ and Z₂, and LRT₂ is a little less powerful than Z₂ (Fig 1i-1l in the third row). The powers of LRT₁ and Z₁ are not so affected by the different values of r, while LRT₀, LRT_0b, Z₀, LRT_2b, LRT₂ and Z₂ are more and more powerful with the number of female individuals increasing (r changing from 2:1 to 1:2) when other parameters are fixed. We also find that the powers of LRT₀, LRT_0b, Z₀, LRT₂, LRT_2b and Z₂ appear great reaction to the different values of ρ when N is fixed. Specially, their powers under ρ = 0.1 (subplots in the second and fourth columns, respectively) are much larger than those under ρ = 0.05 (subplots in the first and third columns, respectively). However, the powers of LRT₁ and Z₁ are almost not influenced by ρ. Further, it can be seen in Fig 1 that LRT₀, LRT_0b and Z₀ with two degrees of freedom (subplots in the first row) are much more powerful than LRT₁, Z₁, LRT₂, LRT_2b and Z₂ with one degree of freedom (subplots in the second and third rows). This is because the true model is p_m ≠ p_f and ρ > 0. In addition, when the sample size changes from 800 (subplots in the first and second columns) to 1200 (subplots in the third and fourth columns), all the test statistics are much more powerful.

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Fig 1. Simulated powers of LRT₀, LRT_0b, LRT₁, LRT₂, LRT_2b, Z₀, Z₁ and Z₂ against r = N_m : N_f under H₁ : p_m ≠ p_f and ρ > 0 based on 10000 replicates with p_m = 0.3 and p_f = 0.35.

In the first column: ρ = 0.05 and N = 800; in the second column: ρ = 0.1 and N = 800; in the third column: ρ = 0.05 and N = 1200; in the fourth column: ρ = 0.1 and N = 1200. In the first row, the powers of LRT₀, LRT_0b and Z₀ for H₀ : p_m = p_f and ρ = 0; in the second row, the powers of LRT₁ and Z₁ for H₀₁ : p_m = p_f; in the third row, the powers of LRT₂, LRT_2b and Z₂ for H₀₂ : ρ = 0.

https://doi.org/10.1371/journal.pone.0145032.g001

Fig 2 displays the simulated size/powers of the eight test statistics against r under H₀₂ : ρ = 0 for different values of p_f, having p_m = 0.3 and N = 1200. The results in the third row of the figure are the size of LRT₂, LRT_2b and Z₂, while those in the first and the second rows of the figure are the powers of LRT₀, LRT_0b and Z₀, and those of LRT₁ and Z₁, respectively. It is shown in the figure that the size of LRT_2b and Z₂ maintains close to the nominal 5% level, while LRT₂ is too conservative. As for the tests for H₀₁ : p_m = p_f, LRT₁ and Z₁ almost have the same simulated power just like Fig 1. On the other hand, the powers of LRT₀, LRT_0b, Z₀, LRT₁ and Z₁ are not so affected by the ratio r. However, their powers are greatly influenced by the absolute difference |ϵ| = |p_m − p_f|. Specifically, their powers under p_f = 0.25 and p_f = 0.35 are much larger than those under p_f = 0.26 and p_f = 0.34. In addition, when the simulation setting is fixed, LRT₁ and Z₁ with one degree of freedom are a little more powerful than LRT₀, LRT_0b and Z₀ with two degrees of freedom, because the true model is p_m ≠ p_f and ρ = 0. By comparing Fig 2d (ρ = 0), Fig 1c (ρ = 0.05) and Fig 1d (ρ = 0.1) under N = 1200, p_m = 0.3 and p_f = 0.35, LRT₀, LRT_0b and Z₀ are more and more powerful with ρ increasing.

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Fig 2. Simulated size/powers of LRT₀, LRT_0b, LRT₁, LRT₂, LRT_2b, Z₀, Z₁ and Z₂ against r = N_m : N_f under H₀₂ : ρ = 0 based on 10000 replicates with p_m = 0.3 and N = 1200.

In the first column: p_f = 0.25; in the second column: p_f = 0.26; in the third column: p_f = 0.34; in the fourth column: p_f = 0.35. In the first row, the powers of LRT₀, LRT_0b and Z₀ for H₀ : p_m = p_f and ρ = 0; in the second row, the powers of LRT₁ and Z₁ for H₀₁ : p_m = p_f; in the third row, the size of LRT₂, LRT_2b and Z₂ for H₀₂ : ρ = 0.

https://doi.org/10.1371/journal.pone.0145032.g002

Figs A–G in S1 File show the corresponding results under other simulation settings with p_m ≠ p_f and ρ > 0, which are similar to those in Fig 1. Figs H and I in S1 File plot the corresponding results under p_m = p_f and ρ > 0, and Figs J–L in S1 File give the corresponding results under p_m ≠ p_f and ρ = 0. The more details refer to S1 File.

Table 2 shows the simulated size of LRT₂, LRT_2b and Z₂ for H₀₂ : ρ = 0 for different values of p_f, having N_m = 0 under the sample sizes N_f required for LRT_2b to obtain 80% simulated power. Table 3 lists the simulated powers under these sample sizes for different values of p_f, having ρ = 0.05 and 0.1. From Table 2, we can see that the type I error rates of LRT₂, LRT_2b and Z₂ are close to the nominal significance level of 5%. It is shown in Table 3 that the power of LRT_2b attains to about 80%, and the difference in power between LRT_2b and Z₂ is about 10%.

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Table 2. Simulated size (in %) of LRT₂, LRT_2b and Z₂, having N_m = 0 and ρ = 0.

https://doi.org/10.1371/journal.pone.0145032.t002

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Table 3. Simulated powers (in %) of LRT₂, LRT_2b and Z₂, having N_m = 0.

https://doi.org/10.1371/journal.pone.0145032.t003

Tables A–J in S1 File list the RMSEs and biases of the estimates of p_m, p_f, the pooled allele frequency p and ρ for different values of p_m, p_f, ρ, r and N. It should be noted that the estimate of p_m based on the EM algorithm is the same as zheng et al. [14]. Further, the estimates and of p_f and p based on the EM algorithms have the similar RMSEs and biases as those from zheng et al. [14], respectively. However, when we focus on the estimate of ρ, we find that although the biases of and based on the EM algorithms are larger than in zheng et al. [14] for some cases, the RMSEs of and are smaller than for all the simulation settings.

Table 4 displays the simulated size/powers of LRT₀, LRT_0b, LRT₁, LRT₂, LRT_2b, Z₀, Z₁ and Z₂ under the population stratification model. When ϵ₁ = ϵ₂ = 0, the size of LRT₁ and Z₁ is obtained. Further, note that the ratios of two subpopulations in the whole population are equal. As such, ϵ₁ = −ϵ₂ will also cause the size of LRT₁ and Z₁. Under other simulation settings, we get the powers of the eight test statistics. To investigate whether or not the population stratification model causes excess homozygosity, we save the values of 10000 ρ estimates for each estimation method (, or ). Then, calculate the corresponding mean and standard deviation (SD), which are also listed in Table 4. The results show that the population stratification model indeed leads to the positive inbreeding coefficient (i.e., excess homozygosity), which is consistent with Overall and Nichols [24]. The mean values ( and ) using the EM algorithm are a little larger than proposed in zheng et al. [14], while and have less standard deviation. On the other hand, the size of LRT₁ and Z₁ is close to the nominal significance level of 5%. The power of LRT_0b is larger than LRT₀ and Z₀, and LRT₀ and Z₀ have the similar powers, irrespective of the ϵ₁ and ϵ₂ values. LRT₁ and Z₁ have almost the same powers. LRT_2b is much more powerful than LRT₂ and Z₂, and the power of LRT₂ is a little smaller than Z₂. If ϵ₁ is fixed and ϵ₂ is changed, the ρ estimate increases with ϵ₂ increasing, and hence LRT₂, LRT_2b and Z₂ are more and more powerful; if ϵ₂ is fixed and ϵ₁ is changed, the ρ estimate decreases with ϵ₁ increasing, and hence LRT₂, LRT_2b and Z₂ are less and less powerful. This may be caused by p_m being taken to be 0.3 and 0.5 in the first and second subpopulations, respectively.

Download:

Table 4. Mean and standard deviation (SD) of ρ estimates over 10000 replications, and simulated size/powers (in %) of LRT₀, LRT_0b, LRT₁, LRT₂, LRT_2b, Z₀, Z₁ and Z₂ under population stratification model.

https://doi.org/10.1371/journal.pone.0145032.t004

Application to RA data

We apply the proposed methods to the RA dataset from North American Rheumatoid Arthritis Consortium for studying their practicability, which is available from Genetic Analysis Workshop 15. In this dataset, there are 1217 families. Note that many individuals’ genotypes are missing. On the other hand, to obtain a sample of which all the individuals are independent, we only select the available founders in this dataset, which results in a sample composed of 369 founders (N_m = 112 and N_f = 257) in the analysis. 293 SNP markers on X chromosome for each founder are included in this application. The significance level is fixed at α = 5%. Table 5 gives the corresponding results based on the P-values of LRT_0b, LRT₁, LRT_2b, Z₀, Z₁ and Z₂. From Table 5, LRT_0b identified 6 loci which Z₀ did not identify, and Z₀ identified 4 additional loci. One locus is detected by LRT₁ that is not found by Z₁, and 4 additional loci are detected by Z₁. There are 12 loci identified by LRT_2b, which can not be identified by Z₂, and only 2 additional loci are identified by Z₂. However, there exist multiple testing problems because we simultaneously analyze 293 loci. So, Bonferroni correction is used (α′ = 0.05/293 = 1.71 × 10⁻⁴) and there is no statistically significant result to occur. The more details can be found in Tables K–M in S1 File.

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Table 5. LRT_0b, LRT₁, LRT_2b, Z₀, Z₁ and Z₂ results of application to rheumatoid arthritis data at 5% level.

https://doi.org/10.1371/journal.pone.0145032.t005

To investigate the computational efficiency of the XHWE software, we implement the code with the default arguments for this dataset (1217 families and 293 SNPs), on a HP 2311f personal computer (Microsoft Windows 7 Enterprise (Service Pack 1), 4GB of RAM and 3.40 GHz Intel(R) Core(TM) i7 Duo processor) and record its computational time. This process needs 977 seconds. Therefore, on the average, the running time for a single SNP is about 3.3 seconds. For the genome-wide case, for example, one would analyze 200000 SNP markers on X chromosome for the family sample of the type mentioned above, which would lead to 1600000 tests for the hypotheses with running time being about 7.6 days on the personal computer of this type.

Discussion

The existing Z₁ and Z₂ tests were respectively proposed to test for H₀₁ : p_m = p_f and H₀₂ : ρ = 0. However, we find that there is no simulation study conducted to assess the validity of Z₁ and Z₂ and their performance [14]. Further, there is no existing method to simultaneously test for H₀ : p_m = p_f and ρ = 0. Therefore, in this article, we first combine these two test statistics and suggest Z₀ = Z₁ + Z₂ to test for the equality of the frequencies of the same allele in males and females and the zero inbreeding coefficient on X chromosome based on the collected sample, because Z₁ and Z₂ are independent of each other. What’s more, for the purpose of improving the test power, we propose several LRT-type test statistics. Firstly, we write out the likelihood functions under H₀ : p_m = p_f and ρ = 0 and H₁ : p_m ≠ p_f or ρ > 0 at a single SNP locus on X chromosome, respectively. Then, we obtain the MLEs of the male allele frequency, the female allele frequency and the inbreeding coefficient by the EM algorithms, where we use the RMSE and bias to assess the accuracy of the MLEs of these unknown parameters and construct the corresponding likelihood ratio test (LRT₀) statistic under the null hypothesis H₀. If H₀ is statistically rejected, we further develop two LRT-type test statistics LRT₁ and LRT₂ respectively for H₀₁ : p_m = p_f and H₀₂ : ρ = 0. Note that LRT₀ and LRT₂ are too conservative from the simulated results. So, we use parametric bootstrap techniques and propose the LRT_0b and LRT_2b test statistics. We simulate the data under different parameter settings. Simulation results show that the proposed bootstrap-based methods LRT_0b and LRT_2b, LRT₁, Z₀ and the existing Z₁ and Z₂ control the type I error rates well under the respective null hypothesis. Power comparison demonstrates that LRT_0b is more powerful than both LRT₀ and Z₀. Under ρ > 0, LRT_2b has much more power than LRT₂ and Z₂, and LRT₂ is a little less powerful than Z₂. In addition, LRT₁ and Z₁ almost have the same power under p_m ≠ p_f.

As for the parameter estimates, the estimate of p_m based on the EM algorithm is the same as that in zheng et al. [14]. Further, the estimates and of p_f and the pooled allele frequency p based on the EM algorithms have the RMSE and bias similar to those from zheng et al. [14], respectively. However, although the biases of and based on the EM algorithms are larger than from zheng et al. [14] for some cases, the RMSEs of and are smaller than for all the simulation settings. In addition, the population stratification model indeed causes excess homozygosity, which is consistent with Overall and Nichols [24]. The mean values ( and ) using the EM algorithm are a little larger than proposed in zheng et al. [14], while and have less standard deviation.

Note that ρ = 0 and ρ > 0 in the null and alternative hypotheses of the likelihood ratio test LRT₀ or LRT₂, respectively, which causes the “boundary” problem and that the corresponding likelihood ratio test is not expected to follow a χ² distribution [31, 33]. This may be the reason why the size of LRT₀ and LRT₂ is too conservative. Therefore, we use parametric bootstrap techniques to obtain the exact distributions of LRT₀ and LRT₂ in this article.

Due to the presence of the X chromosome inactivation (XCI) and dosage compensation (DC), association analysis and excess homozygosity tests on X chromosome are more complicated than those on autosomes [34]. In the presence of XCI, only one allele from a pair of alleles in females is expressed [35]. Consequently, if considering a locus with two alleles M₁ and M₂, the effect of the M₁ allele in males should be equivalent to the difference between M₂M₂ and M₁M₁ homozygous females. As such, when we conduct analyses based on allele-counting, we must either count each allele twice in males or equivalently count each allele in females as 0.5, reflecting a “dosage compensation” for X inactivation [34]. It should be noted that LRT₂, LRT_2b and Z₂ for H₀₂ : ρ = 0 are not affected by XCI and DC because they only use female individuals in the collected sample. Similarly, Z₁ for H₀₁ : p_m = p_f is also not influenced by XCI and DC because it estimates the allele frequencies and the corresponding variances in males and females, respectively. Thus, Z₀ = Z₁ + Z₂ is still valid when XCI and DC exist. To investigate the effect of XCI and DC on LRT₀, LRT_0b, LRT₁ and LRT_1b, where LRT_1b is the bootstrap version of LRT₁, we carry out simulation study under several simulation settings in the presence of XCI and DC. The simulation settings and simulation results are listed in Table 6. It is shown in the table that the size of LRT_2b, Z₀, Z₁ and Z₂ stays close to the nominal 5% level and the size of LRT₂ is still conservative. However, LRT₀ and LRT₁ without bootstrap cannot control the size well. Fortunately, the type I error rates of LRT_0b and LRT_1b with bootstrap are very close to 5%. Furthermore, LRT_0b is more powerful than Z₀ almost for all the cases and LRT_1b and Z₁ almost have the same performance in power. Therefore, in the presence of XCI and DC, LRT_0b, Z₁ and LRT_2b are recommended. Finally, LRT_0b and LRT_2b can deal with samples of small size. However, LRT_0b and LRT_2b are based on the parametric bootstrap techniques, which are more computationally intensive.

Download:

Table 6. Simulated size/powers (in %) of LRT₀, LRT_0b, LRT₁, LRT_1b, LRT₂, LRT_2b, Z₀, Z₁ and Z₂ based on 10000 Monte Carlo replications and 1000 bootstrap replications under X chromosome inactivation and dose compensation, having p_m = 0.3 and the ratio N_m : N_f = 1: 1.

https://doi.org/10.1371/journal.pone.0145032.t006

Supporting Information

S1 File. Supporting Information.

Tables A–J, root mean squared errors (RMSE) and biases of estimates of p_m, p_f and ρ based on EM algorithm and zheng et al. [14] under different simulation settings. Tables K–M, LRT₀, LRT_0b, Z₀, LRT₁, Z₁, LRT₂, LRT_2b, and Z₂ results of application to rheumatoid arthritis data, respectively. Figs A–L, simulated size/powers of LRT₀, LRT_0b, LRT₁, LRT₂, LRT_2b, Z₀, Z₁ and Z₂ against r = N_m : N_f based on 10000 replicates under different simulation settings.

https://doi.org/10.1371/journal.pone.0145032.s001

(PDF)

Author Contributions

Conceived and designed the experiments: XPY JYZ. Performed the experiments: XPY JYZ. Analyzed the data: XPY QLZ JLL. Contributed reagents/materials/analysis tools: XPY QLZ JLL JYZ. Wrote the paper: XPY JYZ.

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Figures

Abstract

Introduction

Materials and Methods

Background and notations

Existing methods Z1 and Z2 for H01 (equality of the frequencies of the same allele in males and females) and H02 (zero inbreeding coefficient), respectively

Z0 test for both hypotheses (i) and (ii) of interest regarding the X chromosome

Likelihood ratio test for both hypotheses (i) and (ii) of interest regarding the X chromosome

Likelihood ratio test for equality of frequencies of the same allele in males and females

Likelihood ratio test for inbreeding coefficient being zero

Likelihood ratio tests via parametric bootstrap for H0 and H02

Software implementation

Simulation settings

Results

Simulation results

Application to RA data

Discussion

Supporting Information

S1 File. Supporting Information.

Author Contributions

References

Existing methods Z₁ and Z₂ for H₀₁ (equality of the frequencies of the same allele in males and females) and H₀₂ (zero inbreeding coefficient), respectively

Z₀ test for both hypotheses (i) and (ii) of interest regarding the X chromosome

Likelihood ratio tests via parametric bootstrap for H₀ and H₀₂