1 Introduction
In genomewide association studies (GWAS) and next generation sequencing (NGS) studies, Xchromosome has been often excluded due to its complexity compared to autosomes. Wise et al. (2013) found that for every GWAS paper published from January 2010 to December 2011 and included in the NHGRI GWAS catalog, “only 33% (242 out of 743 papers) reported including the Xchromosome in analyses”. There are many analytical challenges related to Xinclusive association studies. Some are for both autosomes and Xchromosomes, and some are specific to Xchromosomes.
Throughout this paper, we use to denote phenotype or outcome of interest, which could be binary or continuous, and to denote genotype of a single nucleotide polymorphism (SNP). A single SNP has two alleles: and , one of which is the risk allele with allele frequency and the other is reference allele. It needs to be noted that the major allele could be risk allele and is not necessarily the minor allele frequency (MAF) less than 0.5. An autosome SNP has three genotypes, namely , and . Coding of for each genotype could be for additive effect, and for dominant effect. An Xchromosome SNP has five genotypes, , and for females and and for males. We will discuss the coding of in more details below. The main question of phenotypegenotype association analysis is to test is not associated with . In addition, we use to represent sexspecific effect and to represent other environmental effects. When both effects exist, there may also exist : genotypesex interaction, : geneenvironmental interaction and : sexenvironmental interaction. (with corresponding subscripts) denotes the effect sizes of each covariate.
For any statistical approaches focusing on Xchromosome analysis, we summarize 8 major challenges that must be properly addressed. As we discuss below, challenges C1 to C3 are genomewide, and C4 to C8 are specific to Xchromosome.

C1: Quantitative vs. binary traits/phenotypes

C2: Genotype based vs. allele based

C3: Additive vs. genotypic model (with dominant term)

C4: Sex as a covariate must be included or not

C5: Genotypesex interaction should be included or not

C6: Xchromosome inactivation (XCI) vs no inactivation

C7: If XCI, the inactivation is random vs. skewed

C8: Reference allele vs.
C1 and C2.
Classic allele based tests from casecontrol studies require binary phenotype data so that the Pearson chisquared test statistics can be computed by contingency tables. Hardy–Weinberg equilibrium (HWE) assumption must also be met to achieve correct type I errors. For quantitative phenotypes and any departure from HWE, the most commonly used approaches are genotype based tests under regression models. Regression models support various types of phenotype data and HWE assumption is not required.
Sasieni (1997) had a detailed discussion about allele based tests with HWE assumption versus genotype based tests. Another reason in favor of regression model is that additional covariates such as environmental factors can be easily incorporated in the model.C3. The genotypebased tests require a correct assumption of the genetic model, which has been a long standing controversy. For both autosome and Xchromosome SNPs, the genetic model either assumes a specific relationship between the effects of , and , such as recessive, additive, dominant, multiplicative, or assumes no specific relationship between each genotype (genotypic model), where the total genetic effects are decomposed as a combination of additive and dominant effect. Each assumption leads to a different model, and Bagos (2013) had a good review paper of several model selection approaches. When the true genetic model is unknown, the main idea is to combine each test statistic or pvalue under different models. However, the way to combine these tests are quite ad hoc, and it is lack of theoretical justifications that how and why they should be combined.
On the other hand, a common practice for simplicity is to only examine additive models, as the additive model has reasonable power to detect both the additive and dominant effects (Bush and Moore, 2012). In addition, Hill et al. (2008)
have shown that additive variance typically accounts for over half and often close to 100% of total genetic variance, even if there are nonadditive effects at the level of gene action. It needs to be noted that people are usually reluctant to assume the genotypic model. Although it is the most general assumption, the test is believed to be less powerful due to the extra degree of freedoms of the test statistics. However, we find such belief is not necessary correct in the context of GWAS. We derive the upper bound of the power loss by incorporating the other covariate for dominant effect and compare to the potential power gain, and find it may be worth to allow both the additive and dominant effects in the model.
C4 and C5. The other challenges are specific to Xchromosome, due to the fundamental differences between females and males. First, sexspecific effects may exist in biological point of view. Next, sex is a classic confounder associated to both the genotype and the phenotype. If covariate is not included in the model, the type I error for testing the genotype effect can be inflated. Ozbek et al. (2018) has extensitve simulation studies to show the type I error inflation. Furthermore, different effect sizes of the same SNP in females and males are recognized as genotypesex interaction effects. Proper tests allowing for interaction effects need to be developed.
C6 and C7. The next complications relate to the uncertainty of the biological status of Xchromosome SNPs. Xchromosome inactivation is the phenomenon that one of the two alleles in females is selected to be silenced, so that the effects of female genotypes may be reduced. In brief, the additive coding of , and becomes 0, 0.5 and 1 rather than 0, 1 and 2. The challenge is that although we know about 15% of genes on Xchromosome are escaped from XCI at population level (Carrel and Willard, 2005), we are uncertain if XCI occurs or escapes on each SNP. Even though we are certain that XCI occurs on one particular SNP, at individual level it is still unknown which allele is inactivated. Wang et al. (2014)
discussed various studies suggesting a biological plausibility of skewed inactivation so that one allele is more likely to be inactivated than the other, while the additive model in essence assumes two alleles have equal probability of inactivation.
C8. Lastly, when allele frequency difference is significant, females and males may have different minor alleles. For autosome SNPs, people usually choose the minor allele with allele frequency less than 0.5 as the risk allele, because switching the risk allele and reference allele does not change the statistical inference and thus choosing an arbitrary risk allele does not cause a problem. However, for Xchromosome SNPs, switching the reference allele and risk allele may lead to different statistical models and yield different inferences. When minor allele is different for females and males, the risk allele may be unknown and it becomes a challenge to choose the risk allele. It needs to be noted that sexstratified tests may not solve the challenge, because stratification by sex may result in considerable loss of power (Clayton, 2008), especially when allele frequency difference is significant for females and males.
We summarize the genotype codings after considering all Xspecific challenges in Table 1. If the risk allele and XCI status are both unknown, there are ways to code the additive covariates , and 2 ways to code the genotypesex interaction . We will discuss in section 3 that skewed inactivation can be represented by the dominant effect coding .
Xchromosome  Genotype coding  
Effect  Covariate  Risk allele  inactivated  
Yes  0  0.5  1  0  1  
Additive  Yes  1  0.5  0  1  0  
No  0  1  2  0  1  
No  2  1  0  1  0  
Dominant  Either  Either  0  1  0  0  0  
Interaction  Either  0  0  0  0  1  
Either  0  0  0  1  0  
Sex  Either  Either  0  0  0  1  1 
In recent years, quite a few methods have been proposed for Xchromosome association studies. Zheng et al. (2007) proposed a few tests without considering Xchromosome inactivation. In contrast, Clayton (2008, 2009) discussed analytical strategies assuming Xchromosome is always inactivated. Hickey and Bahlo (2011) and Loley et al. (2011) separately performed simulation studies and gave a thorough comparison of Zheng et al. and Clayton’s tests. Based on these simulation studies, Konig et al. (2014) provided a detailed guideline for including Xchromosome in GWAS. The problem is they suggested different tests under different assumptions of genetic model, interaction effects, XCI status and so on, and it is not always possible to check these assumptions in practice. Gao et al. (2015) developed a software toolset for Xchromosome association studies. Recently, Zhongxue et al. (2017) improved existing sexstratified tests by eliminating assumptions of genetic models, but they still needed to assume same risk allele for females and males, and sexgenotype interaction effects could not be measured. Focusing on XCI status, Wang et al. (2014) proposed a maximum likelihood solution to handle the uncertainty of XCI as well as skewed inactivation, and provided an XCI model selection method in their most recent paper (Wang et al., 2017). In addition, Chen et al. (2017)
used Bayesian model averaging (BMA) method to solve XCI uncertainty. However, both approaches only considered the additive model, and it is unclear how to include nonadditive covariates in regression analysis with unknown XCI status. Furthermore, both approaches were only illustrated by simulation studies, and it would be more appealing to derive a theoretical justification.
After reviewing all uptodate methodology developments on Xchromosome association studies, we believe there is currently no approach which can handle all the 8 challenges discussed above simultaneously. The target of this paper is to propose a theoretically justified robust method that can solve all these challenges in most general framework, while the test powers are well maintained and even improved in most practical situations. The proposed tests are based on regression models, which allow for both quantitative and binary phenotypes as the response variables, departure from HWE and incorporating extra covariates. In section
2, we discuss the longlasting controversy between additive models and genotypic models. We revisit autosome SNPs for better illustrating of the benefits of genotypic model, which leads to a robust and sometimes much more powerful test than additive model. In section 3, we propose our main theory to address the challenges specific to Xchromosome. Section LABEL:application provides supporting evidence to our proposed approach from application studies. Finally, we discuss the limitations of our approach and possible future work in section 4.2 Additive vs. genotypic models
2.1 Theory of chisquared distributions
For completeness and a more clear demonstration of the model selection challenge, we first revisit autosome studies. We find that in general, Hill et al. (2008)’s result does not warrant the exclusion of dominant covariate in regression model. Although the additive effect may account for the majority of total genetic effect, excluding does not necessarily increase test power. In order to determine whether needs to be included or not, two questions must be answered. First, when all genetic effect is additive, what is the power loss by introducing the extra covariate ? Second, when there exists some dominant effect, will the power increase or decrease by introducing ?
To answer these two questions, we first define the additive model and genotypic model in generalized linear regression framework. Let
be the link function. The additive model is defined byand the genotypic model is defined by
Although the HWE assumption is not required, we adopt it only for the purpose of simplifying the computation. The three genotype groups, , and have frequencies , and , and and are coded as 0, 1, 2 and 0, 1, 0 correspondingly. Covariates for environmental factors can be freely added to both models. For notation simplicity we denote the additive model by and genotypic model by below.
We want to compare two tests: under the additive model and
under the genotypic model. The test statistics used most often in regression models are Wald, Score and likelihood ratio statistics, which all follow asymptotic noncentral chisquared distribution under the alternative hypothesis. We define two test statistics by
and , where 1 and 2 denote degree of freedoms, and and are corresponding noncentrality parameters. If the true genotype effect is all additive, then . The power difference of and depends on both the noncentrality parameters and type I error . When or , both tests have no power; when noncentrality parameters are sufficiently large or close to 1, both powers are close to 1. To achieve the maximum power loss of , we expect a moderate value of both the noncentrality parameter and . We show the maximum power loss numerically in Web Appendix A, where the maximum power loss is 0.114 when and . It implies the power loss of using the genotypic model is capped by 0.114, regardless of type I error level, sample size and size of additive effects. It needs to be noted that although we assume additive model is correct, the maximum power loss is same for all 1 vs 2 degree of freedom models. For instance, if the dominant model is correct, the power loss is still capped by 0.114 by using the genotypic model.With capped power loss, we want to investigate the power gain by testing when the true genotype effect is not additive. When dominant effect exists, the noncentrality parameters can be written as , where . For fixed value of , when is close to 0, we still expect to be less powerful than . As increases, there is a threshold value of which makes as powerful as . When is greater than the threshold, is more powerful, and the power goes up to 1 for large . Compared to the maximum power loss, the maximum power gain can be technically as large as when and . In Web Appendix A, we choose a few practical values of and and plot the test powers of and . To clearly illustrate the power gain, we assume the worst case (maximum power loss) scenario where . We show when is as large as , the power gain can be much higher than power loss. Therefore, the genotypic model should not be overlooked in association studies with autosome SNPs.
2.2 Noncentrality parameters and corresponding test power computation
The above power computation is based on the theoretical values of noncentrality parameters, which must be computed from sample size and genotype effect size under the additive or genotypic model. When the sample size , we want each test has a limiting chisquared distribution, but the noncentrality parameter under alternative hypothesis would move toward infinity for fixed value of . As in convention, we assume . Instead of specifying
, we fix the value of constant vector
, so that and the noncentrality parameter under alternative hypothesis converges to finite number as . We provide more discussions about the convergence of asymptotic noncentrality parameters in section 4. We then use standard technique in Cox and Hinkley (1974) to compute the asymptotic noncentrality parameters for the test under genotypic model as described below.We write the generalized linear models in matrix form: where is the design matrix. Suppose we want to test , where is a subset of . To compute the noncentrality parameter, we partition ,
according to the null hypothesis, and the expected Fisher information matrix of
is partitioned accordingly: . Then the noncentrality parameter equals to(1) 
For genotypic model, , , and , . Specifically, we derive under linear model and logistic model and compute corresponding . The mathematical details are given in Web Appendix C. Let be the variance of the error term in linear model. We show when , linear and logistic model has equal asymptotic noncentrality parameters for same and .
The computation of noncentrality parameter under the additive model is less straightforward, because the additive model is indeed misspecified when the dominant effect . Although the derivation is difficult under the canonical parametrization of the genotype as defined above, the result from Begg and Lagakos (1992) implies that a reparametrization of genotype coding may considerably simplify this derivation. Detailed steps are provided in Web Appendix D.
Once the noncentrality parameters are computed, we may compare test powers of the additive and genotypic model when . Choosing , we consider a realistic situation where and change from 0.6 to 0.6. We then plot the power of both tests as a function of in Figure 1, which represents both the logistic model and linear model with . Risk allele frequencies are chosen to be 0.2, 0.5 and 0.8.
Figure 1 indicates that the power gain by using the genotypic model can be as much as 0.4 in realistic situations, which is quite significant compared to the maximum power loss of 0.113. In practice, the strength of dominant effect is usually unknown. In such case including the dominant covariate is more like a riskfree solution: without sacrificing much test power, the potential power gain may be significant.
3 Xchromosome Challenges
3.1 Type I error control and choice of risk allele
We now consider association analysis on Xchromosome SNPs where the covariates are defined in Table 1. When testing for the genotype effects, we note that they are usually correlated with sex effects. The correlation has two implications. First, when genotype effect exists, sex becomes a confounding variable. The sex effect is hard to explain separately and it in fact helps explaining the genotype effect. Second and more importantly, when sex effect exists but genotype effect does not exist, the correlation will lead to an inflated type I error for testing the genotype effect if sex is not included in the model. Including sex as the covariate warrants the correct type I error for testing the genotype effect. Ozbek et al. (2018) provided extensive simulation studies to show both the type I error inflation and correct type I error control when sex is included, and we would agree with their conclusion that sex should always been included in regression models.
As shown in Table 1, the coding of depends on the risk allele, and model may yield different test statistics under different risk allele assumptions. At first sight, it may seem unclear that how we choose the correct way to code when risk allele is unknown. However, with sex as the covariate, there turns out to be a connection between different risk alleles. For instance, we observe two models with no XCI and different risk alleles,
and
where we want to test or under each model. We note that , which yields , so it is equivalent to test and under two models, and we can further show that test statistics under two models are exactly equal. It provides us some intuition that the problem of unknown risk allele is solved when sex is included as the covariate because two tests then become indistinguishable. To make our intuition more rigorous, we propose the following theorem:
For vector of length , Let and be two generalized linear models with same link function , and , where , are design matrices and , are vectors of length . Let and , where and have length and and have length . If there exists a transformation matrix such that , where , are and invertible matrix, then the test statistics (Wald, Score or LRT) for testing and are equal under the technical assumptions given in Web Appendix B.
We prove Theorem 3.1
in Web Appendix B. To make the two test statistics equal, an intuitive explanation of the requirements are: two design matrices must be invertible linear transformations of each other, and two submatrices of the covariates which are not being tested must also be invertible linear transformations of each other. It needs to be noted that the covariates being tested are not required to be linear transformation of each other, e.g.,
and are not linear function of each other. Mathematically speaking, the uncertainty problem arises because four different codings of have no linear transformations. When is included in the model, as we have illustrated above, two design matrices of with different risk alleles but same XCI status become invertible linear transformations of each other. Therefore, different risk alleles result in same test statistic by applying theorem 3.1. This provides another reason to include sex as a covariate in the model. When sex is included, unknown risk alleles of both females and males becomes not a problem. In conclusion, we would recommend the following additive model including the sex effect at this moment:
3.2 Sexgenotype interaction and XCI uncertainty
For Xchromosome SNPs, genotypesex interaction effect may exist, so that the unit effect of one copy of or may not be the same for males and females. The interaction is defined by . It is straightforward to check has two different codings depending on the risk allele of males: and as defined in Table 1.
We have explained when is included in the model, two design matrices with different risk alleles become invertible linear transformations. Furthermore, when both and are included, we can easily show all four design matrices of with different risk alleles and XCI status are invertible linear transformations of each other, and for testing the null hypothesis , the design matrix of the covariates which are not being tested, i.e., , remains unchanged between different coding schemes of and . Therefore, we apply theorem 3.1 to show the tests with different risk alleles and XCI status are equal, and choosing correct coding of and becomes not a issue. The relationship of invertible linear transformation between codings are summarized in Figure 2.
Figure 2 implies that in terms of testing, switching risk allele has no effect when sex is included, and the effect of inactivating Xchromosome alleles is indistinguishable to the effect of sexgenotype interaction. With and included in the model, we do not need to know the risk allele and XCI status, and any one group of covariates of and simply yields the same test statistic. Therefore, we now recommend including both and in regression models to override the uncertainty issues about risk allele and XCI:
3.3 Dominant effects and skewed XCI
The dominant effect defined in Table 1 is invariant to risk allele and XCI status. Similar to autosome SNPs, the first reason to include the dominant effect is to capture any departure from the additive effect of the heterozygous genotype . For Xchromosome, another important reason is that the dominant effect may also characterize the skewness of XCI.
Skewed XCI is the effect that one allele is more likely to be inactivated than the other for female SNPs. For homozygous genotypes and , the genotype effects are always reduced to a half because both alleles have same effect and which allele is inactivated makes no difference. For heterozygous genotype , if one allele is more likely to be inactivated, at population level the effect of will move towards to the effect of either or . For example, we denote the effect of by 0 and effect of by 1. Then will have an effect of either 0 or 1 at individual level depending on the inactivated allele. If two alleles are equal likely to be inactivated, at population level we expect half of and half of are inactivated, so that the averaged group effect of is 1/2. If is more likely to be inactivated, we expect more 1’s than 0’s on average, and has an average effect greater than 1/2. With skewed inactivation, the effect of ranges from 0 to 1, so that the skewness is equivalent to a dominant effect making the the effect of different from 1/2. In conclusion, including the covariates not only captures real dominant effect, but also represents any skewness of inactivation.
When XCI status is unknown, it is more likely that the amount of skewness of the inactivated SNP is also unknown. In such case we recommend including the dominant covariate to explain any possible skewness. Because the coding of is invariant to risk allele and XCI status, including in the model does not change the linear transformation relationships specified in Figure 2. When different covariates are chosen, Table 2 summarizes for each model whether it has problem with inflated type I error and unknown risk allele (Challenge C4 and C8), sexgenotype interaction and XCI uncertainty (Challenge C5 and C6), and dominant effects and skewed XCI (Challenge C7). Other covariates representing environmental effects can be freely added to each model. The ultimate model we recommend is
which resolves all Xchromosome specific challenges, as shown in Table 2.
Model  C4/C8  C6/C7  C5  

3.4 Analytic power comparison
It needs to be noted that all models with sex included as the covariate are valid to test the genotype effects because the type I error is correct, even if they are not capable to handle the XCI uncertainty or skewed XCI. The problem is they may have reduced test powers if the XCI status and skewness of XCI are not correctly specified. On the other hand, the full model may also not be most powerful because it has more degree of freedoms. Hence, a systematic power comparison of the most comprehensive model versus simpler models is desired. In short, we want to compare test powers of to in Table 2.
Similar to autosome SNPs, we need to compute the asymptotic noncentrality parameters of each test statistic for power comparison. After allowing for sex, dominant and interaction effect, it becomes not a issue to specify the true risk allele, XCI status and skewness. We assume HWE for female, equal sex frequency, but unequal allele frequencies for females and males ( and ). Then genotype frequencies of are
We use the same technique as described in section 2 to define and fix the value of so that the noncentrality parameter under alternative hypothesis converges to finite number as . Then the noncentrality parameter for the tests under model can be similarly computed following Cox and Hinkley (1974). , and are misspecified models, and we need a reparametrization of the covariates to simplify the computation of noncentrality parameters. The technical details are provided in Web Appendix C and D.
The theoretical test power of all 4 tests are then computed from the noncentrality parameters. By comparing two chisquared distributions of 1 and 3 degree of freedoms, we show in Web Appendix A that the maximum power loss by omitting both and is 0.188, regardless of type I error, sample size and effect sizes. To see potential power gains by including and
, we want to compute test powers with different dominant effects and interaction effects. Because the XCI status and risk allele are unknown, we specify the averaged effect size (linear regression) or averaged log odds ratios (logistic regression) under each genotype group, i.e.,
and, which can be estimated in practice without knowing the XCI status and risk allele. We fix
and , and change and from 0.6 to 0.6. Fixing and is equivalent to fixing the additive effect, and changing is equivalent to changing the dominant effect, where corresponds to no dominant effect. Similarly, different represent different strengths of interaction effect. We show in Appendix A that the maximum power loss 0.188 is reached when , so we choose to represent the worst case scenario. We keep and risk allele frequency and 0.5. Results under other allele frequencies are presented in Web Appendix E. The test statistics from and depend on the coding of and . Without loss of generality, we use and for all 4 tests. One can easily choose the other codings of and and repeat the power computation. Because the asymptotic noncentrality parameter under logistic model is equal to linear model with variance of error , Figure 3 represents test power comparisons under both linear and logistic model.The 1 df model may have significant power loss compared to the full model as shown by all panels. When allele frequency is 0.2, model may lose power dramatically when both the dominant and interaction effects are strong. When allele frequency is 0.5, model may not perform as good as the full model when dominant effects are strong. Therefore, we conclude that testing from the full model maintains overall best performance. Compared to the maximum power loss of 0.188, the power gain can be as much as 0.7 (e.g., when and ) as shown in Figure 3. It implies the full model is not only robust to all the challenges, but also powerful for testing the additive effects along with various dominant and interaction effects. Therefore, it worths to consider including all , and as the covariates in practice when XCI status and/or skewness is unknown and the strength of dominant and interaction effects is not clear. After the power comparison, we still recommend the full model:
4 Discussion
The assumption that under alternative hypothesis may not be quite intuitive, but it seems to be a common assumption when studying the theoretical properties of chisquared tests (Cox and Hinkley, 1974; Begg and Lagakos, 1992, 1993; Neuhaus, 1998). Let , where the null hypothesis is and is the nuisance parameter not being tested. Among the common practices, there is no doubt to adopt a sequence of alternative hypothesis of converging to 0, but it is not quite clear whether should also be assumed to converge to 0. In the context of GWAS, we believe it is most reasonable to assume and converge to 0 at similar rates, because both parameters denote the genotype effect of the same SNP, and there is no reason to believe the additive, dominant and interaction effects are on different scale for any sample size. Therefore, we assume both and converge to 0 as at the rate of .
We have shown in Xchromosome association study, Sex should be included for correct type I error. For autosome study, sex is usually not included, but the result from Xchromosome suggests that when sex is a confounding variable, e.g., female and male allele frequencies unequal, it should also be included as a covariate for autosome analysis. If the risk allele is uncertain, we also recommend including sex to bypass the uncertainty.
When allele frequency difference is significant and females and males have different minor alleles, it may become unclear that if females and males have the same risk allele or each sex has its own risk allele. As the interaction effects being allowed in the models, we are essentially allowing for different risk alleles for females and males. Switching the risk allele for males is equivalent to adding an interaction effect. Following Theorem 3.1, it is also easy to check when is included, switching the risk allele only for males or females will not change the test statistic.
Although the full model on Xchromosome is robust to XCI uncertainty, it is not capable to determine whether XCI occurs or not. It is possible to detect XCI by biological experiments (Carrel and Willard, 2005), but statistical tests for XCI status may also be desired. For quantitative trait, Ma et al. (2015) proposed a variancebased test for detecting XCI. However, we do not find any statistical approach to testing XCI for binary trait up to date. It is a more challenging problem, because the binary outcomes can only yield a point estimate of percentage of cases, and do not have a variance structure similar to quantitative traits under each genotype group. Further studies are required to develop a statistical test under binary logistic model.
Acknowledgements
The authors would like to thank Dr. Lisa J. Strug for providing the cystic fibrosis application data, and Prof. Mike Evans for suggestions that have improved the presentation of the paper. This research is funded by the Natural Sciences and Engineering Research Council of Canada (NSERC) to RVC and LS, and the Canadian Institutes of Health Research (CIHR) to LS.
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