Skewed X chromosome inactivation (XCI), which is a non-random process, is frequently observed in both healthy and affected females. Furthermore, skewed XCI has been reported to be related to many X-linked diseases. However, no statistical method is available in the literature to measure the degree of the skewness of XCI for case-control design. Therefore, it is necessary to develop methods for such a task. In this article, we first proposed a statistical measure for the degree of XCI skewing by using a case-control design, which is a ratio of two logistic regression coefficients after a simple reparameterization. Based on the point estimate of the ratio, we further developed three types of confidence intervals (the likelihood ratio, Fieller’s and delta methods) to evaluate its variation. Simulation results demonstrated that the likelihood ratio method and the Fieller’s method have more accurate coverage probability and more balanced tail errors than the delta method. We also applied these proposed methods to analyze the Graves’ disease data for their practical use and found that rs3827440 probably undergoes a skewed XCI pattern with 68.7% of cells in heterozygous females having the risk allele T active, while the other 31.3% of cells keeping the normal allele C active. For practical application, we suggest using the Fieller’s method in large samples due to the non-iterative computation procedure and using the LR method otherwise for its robustness despite its slightly heavy computational burden.

The methods commonly used to test the associations between ordinal phenotypes and genotypes often treat either the ordinal phenotype or the genotype as continuous variables. To address limitations of these approaches, we propose a model where both the ordinal phenotype and the genotype are viewed as manifestations of an underlying multivariate normal random variable. The proposed method allows modeling the ordinal phenotype, the genotype and covariates jointly. We employ the generalized estimating equation technique and M-estimation theory to estimate the model parameters and deduce the corresponding asymptotic distribution. Numerical simulations and real data applications are also conducted to compare the performance of the proposed method with those of methods based on the logit and probit models. Even though there may be potential limitations in Type I error rate control for our method, the gains in power can prove its practical value in case of exactly ordinal phenotypes.

In the past decade, genome-wide association studies have identified thousands of susceptible variants associated with complex human diseases and traits. Conducting follow-up genetic association studies has become a standard approach to validate the findings of genome-wide association studies. One problem of high interest in genetic association studies is to accurately estimate the strength of the association, which is often quantified by odds ratios in case-control studies. However, estimating the association directly by follow-up studies is inefficient since this approach ignores information from the genome-wide association studies. In this article, an estimator called GFcom, which integrates information from genome-wide association studies and follow-up studies, is proposed. The estimator includes both the point estimate and corresponding confidence interval. GFcom is more efficient than competing estimators regarding MSE and the length of confidence intervals. The superiority of GFcom is particularly evident when the genome-wide association study suffers from severe selection bias. Comprehensive simulation studies and applications to three real follow-up studies demonstrate the performance of the proposed estimator. An R package, “GFcom”, implementing our method is publicly available at https://github.com/JiyuanHu/GFcom .