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The authors have declared that no competing interests exist.

Conceived and designed the experiments: FX. Performed the experiments: FX. Analyzed the data: FX. Contributed reagents/materials/analysis tools: CIA. Wrote the paper: FX. Supervised study design: CIA. Developed methodology: CIA. Helped write the paper: CIA. Participated in the design of the algorithm: JM. Provided technical support: JM. Provided critical comments and suggestions: JM.

Genetic imprinting is the most well-known cause for parent-of-origin effect (POE) whereby a gene is differentially expressed depending on the parental origin of the same alleles. Genetic imprinting is related to several human disorders, including diabetes, breast cancer, alcoholism, and obesity. This phenomenon has been shown to be important for normal embryonic development in mammals. Traditional association approaches ignore this important genetic phenomenon. In this study, we generalize the natural and orthogonal interactions (NOIA) framework to allow for estimation of both main allelic effects and POEs. We develop a statistical (Stat-POE) model that has the orthogonal estimates of parameters including the POEs. We conducted simulation studies for both quantitative and qualitative traits to evaluate the performance of the statistical and functional models with different levels of POEs. Our results showed that the newly proposed Stat-POE model, which ensures orthogonality of variance components if Hardy-Weinberg Equilibrium (HWE) or equal minor and major allele frequencies is satisfied, had greater power for detecting the main allelic additive effect than a Func-POE model, which codes according to allelic substitutions, for both quantitative and qualitative traits. The power for detecting the POE was the same for the Stat-POE and Func-POE models under HWE for quantitative traits.

Genetic imprinting frequently affects genes during embryogenesis and is the most well-known parent-of-origin effect (POE). Imprinting causes the differential expression of genes based on the parental origin of the chromosome

Several statistical approaches have been developed for modeling POEs and imprinting effects. Shete et al. implemented a variance-components method for testing genetic linkage by incorporating an imprinting parameter

Most traditional association approaches assume that the two alleles from the parents contribute equally to the trait, thereby ignoring the potentially important genetic phenomenon, POEs. These approaches estimate the main allelic effect, which could also be considered as the overall genetic effect, without considering POEs. Thus, it is important to develop new methods applicable to genome-wide scans that model the differential contribution of paternal and maternal alleles. It is desired that a method that allows for POE also maintain the power to detect the main allelic effect after adding one or more parameters to the model. Therefore, the proper and orthogonal decomposition of genetic variance renders this framework meaningful and useful to estimate main allelic effects along with the POE.

The natural and orthogonal interactions (NOIA) model was originally developed as a framework for estimating genetic effects for a quantitative trait and gene-gene (GxG) interactions

Genome-wide association studies (GWASs) have achieved great success in identifying genetic susceptibility loci associated with human disorders and traits in the past decade, such as cancer, diabetes, hypertension and heart diseases

In this study, we generalize the NOIA framework to incorporate POEs. We show that more disease-associated genes could be detected by incorporating POEs with orthogonal models than by using traditional models, and that the NOIA POE model would fulfill the requirement of maintaining the power to detect the main allelic effect for complex diseases when multiple loci contribute to disease risk. The orthogonality of the statistical formulation of NOIA framework is important, especially when multiple loci are contributing to the outcome. Using Kronecker product rule, our one-locus NOIA POE formulation can be easily extended to the general case of multiple loci (and environmental factors) to model general GxG/GxE interactions in the presence of imprinting effect, making NOIA a unified framework for detecting GxG/GxE interactions along with imprinting effect. Here we focus on one-locus association analysis for quantitative trait, implementing NOIA into a POE integrated framework by re-parameterization.

From the NOIA statistical model without POE (Stat-Usual) and the traditional functional model without POE (Func-Usual), we derived the formulas of several different quantitative trait association models, including a statistical POE (Stat-POE) model and a functional POE (Func-POE) model. Then, we evaluated the performance of the Stat-POE and Func-POE models. We also compared the performance of the POE models (Stat-POE and Func-POE) with that of the models without POE incorporated (Stat-Usual and Func-Usual). These studies were all performed for both a simulated quantitative trait dataset and a qualitative trait dataset. We found that the incorporation of POE into the statistical model did not affect the estimation of the main allelic effect. Moreover, the power of the statistical models with POE incorporated was higher in the presence of imprinting than that of the usual models without POE for detecting the main allelic effect for both the quantitative trait and qualitative trait. Although our methods were developed and evaluated for single locus association study, we show that they can be straightforwardly extended to gene-gene interaction or gene-environment interaction models.

We first briefly review the NOIA model without POE detecting. Using the usual approach for genotype-phenotype mapping of a quantitative trait locus (QTL), if the trait is influenced by a single diallelic locus, with alleles

Several methods have been proposed for mapping a quantitative trait controlled by one locus with two alleles. The vector of genotypic values

The vector of genetic effects (

One of the traditional regression models, which we call a functional model, is given by:

The inverse is

Here, the reference point

A second approach to modeling, the “statistical model” which we call the Stat-Usual model, was proposed by Alvarez-Castro and Carlborg for estimating genetic effects for a quantitative trait and gene-gene (GxG) interactions

The genetic effects,

We notice the Stat-Usual and Func-Usual models have same estimators for the dominant effect and different estimators for the additive effect as

In this section, we extend the models described above by incorporating the POE and evaluate the performance of these extended models in detecting both the overall genetic effect and POE. For a gene with POE, the vector of genotypic values

Similar to

We have two different ways to construct a functional and a statistical model with POE. First, we can do so by decomposing the additive effects into two paternal and maternal additive effects, resulting in Model 1 for the functional model and Model 2 for the statistical model (see

Let

Let

Therefore, according to

The POE functional model (Func-POE) and statistical model (Stat-POE) are related by

We have previously showed that the Stat-Usual model was orthogonal in the sense that the estimates of the four parameters were uncorrelated

Note that

Also,

To show that the additive variance,

And

The two additive variance components

We performed simulation studies for both a quantitative trait and a qualitative trait (case-control) using an approach similar to that used in

To simulate samples of independent individuals with a quantitative trait controlled by a diallelic locus, we assumed that the gene is under HWE. The case that a gene is not under HWE was not considered in our study, and will be investigated in our future work. For a given value of the minor allelic frequency (

In the simulation study of a quantitative trait, three scenarios were simulated with different levels of POE (

Scenario 1 | 90.0 | 3.0 | −3.0 | 1.2 |

Scenario 2 | 90.0 | 3.0 | −2.0 | 1.2 |

Scenario 3 | 90.0 | 3.0 | −1.0 | 1.2 |

Scenario 1 | 100.0 | 2.0 | −2.0 | 0.5 |

Scenario 2 | 100.0 | 2.0 | −0.6 | 0.5 |

Ma et al.

Briefly, we used the logistic model and Bayes’ theorem to set the genotype of each individual according to the prespecified genetic effect terms,

For each replicate, 1000 cases and 1000 controls were generated, and a total of 1000 replicates were simulated. The MAF

To determine whether the setting of the MAF value influence the performance of the models, we also simulated two additional scenarios with different MAF values (0.03 and 0.48) for both quantitative traits and qualitative traits.

First we performed a simulation study for a quantitative trait in three scenarios with strong, moderate, or weak imprinting effect while the main allelic additive effect remained the same (

The pre-specified minor allele frequency was 0.28. The values of the four parameters were

To evaluate the performance of these models in detecting main allelic additive effect and POE, we calculated the statistical power of four models under different critical values of P values obtained using a Wald test (

(a) Power for detecting the main allelic additive effect in scenario 1 when strong POE exists. Power for detecting POE of the Stat-POE and Func-POE models was compared for scenario 1 (b), scenario 2 (c), and scenario 3 (d).

To evaluate whether the MAF influences the estimation of the genetic effects by these models, we also performed analyses for quantitative traits when the MAF was 0.03 and 0.48, respectively (

Similarly, we also performed analyses for simulated case-control data. The simulating values for each of the two scenarios are shown in

The pre-specified minor allele frequency was 0.28; the true values of the four parameters were

The minor allele frequency was 0.28.

Another simulation was performed with a moderate POE for case-control data (

The minor allele frequency was 0.28.

Simulations were also performed when MAF was set as 0.03 and 0.48 for case-control design, respectively (

We summarized the detailed power comparison of the Stat-POE and Func-POE models in

MAF = 0.03 | MAF = 0.28 | MAF = 0.48 | ||||||

Strong POE | Weak POE | Strong POE | Weak POE | Strong POE | Weak POE | |||

0.01 | 0.01 | 0.36 | 0.35 | 0.75 | 0.75 | |||

0.4 | 0.1 | 0.61 | 0.1 | 0.77 | 0.03 | |||

0.4 | 0.1 | 0.61 | 0.1 | 0.77 | 0.02 | |||

0.005 | 0.007 | 0.07 | 0.06 | 0.12 | 0.15 | |||

0.005 | 0.007 | 0.07 | 0.06 | 0.12 | 0.15 | |||

1 | 1 | 1 | 1 | |||||

0.001 | 0.001 | 1 | 1 | 1 | 1 | |||

0.8 | 0.33 | 1 | 0.33 | 1 | 0.61 | |||

1 | 1 | |||||||

0.04 | 0.05 | 0.29 | 0.35 | 0.8 | 0.9 | |||

0.04 | 0.05 | 0.29 | 0.35 | 0.8 | 0.9 |

Type I error was evaluated for both the quantitative trait and the qualitative trait by simulating a null scenario where there was no main genetic effect or POE. We estimated the type I error for the main additive effect, POE and dominant effect for both quantitative traits and case-control traits when the MAF was set as 0.03, 0.28 or 0.48 (

MAF = 0.03 | MAF = 0.28 | MAF = 0.48 | |||||||

Models/MAF | Add | POE | Dom | Add | POE | Dom | Add | POE | Dom |

0.055 | 0.036 | 0.059 | 0.056 | 0.037 | 0.048 | 0.052 | 0.042 | 0.043 | |

0.048 | 0.06 | 0.056 | 0.048 | 0.053 | 0.044 | ||||

0.048 | 0.06 | 0.056 | 0.048 | 0.053 | 0.044 | ||||

0.01 | 0.063 | 0.017 | 0.049 | 0.047 | 0.046 | 0.049 | 0.048 | 0.039 | |

0.045 | 0.017 | 0.047 | 0.047 | 0.047 | 0.038 | ||||

0.045 | 0.017 | 0.047 | 0.047 | 0.047 | 0.038 |

In this study, we extended the NOIA framework, which was initially developed for epistatic analysis of quantitative traits, by incorporating POE for genetic association analysis. Herein, we propose a unified framework for one-locus association study that allows for main allelic additive effect, the dominant effect and POE estimation via linear regression or logistic regression. Using simulation, we illustrated the statistical properties of this extended framework on one-locus association study. Although the Func-POE model sometimes presented slightly greater power than the Stat-POE model for estimation of POE for qualitative trait. The Stat-POE model are always preferred than the Func-POE model in detecting overall additive effect for quantitative traits and qualitative traits, because of its much greater power.

We conducted genetic variance decomposition to show that the Stat-POE model was orthogonal when either HWE or equal minor and major allele frequencies is satisfied for quantitative traits (

Using simulation, we demonstrated that the statistical models, including the Stat-POE and Stat-Usual models, had better performance for detecting the main allelic additive effect than the functional models, Func-POE model and Func-Usual for both quantitative traits and qualitative traits. These two POE models on detecting the POE had the same power when

However, the performance of these two models for detecting the main allelic effect and dominance effect had a different pattern in qualitative traits (

We also illustrate why the proposed model can detect more disease-associated genes than the traditional models in model setting as follows. First, the orthogonal (Stat-Usual) model proposed by Alvarez-Castro et al. orthogonalizes the estimation of the additive and dominant effects but the usual model (Func-Usual) does not. We constructed the test statistics of the Stat-Usual and Func-Usual models for quantitative traits with and without dominance components effects (

Several recent studies have incorporated POEs in association analyses for quantitative traits. Genome-wide rapid association using mixed model and regression (GRAMMAR) and its extension are a recently developed approach that is based on a measured genotype approach and has been shown to have greater power than the transmission disequilibrium test (TDT)-based tests

Although our extension of the NOIA is expected to provide new insights into disease gene mapping, pedigree data are needed for our framework to be used to estimate transmitting information of each heterozygotes or homozygotes locus. Obtaining the transmitting status of one locus is possible for non-informative pedigrees determined by nearby linked loci or haplotype phasing. Or we could use weighted analysis in which the probabilities of each genotype are used. A future direction of our next step will be to extend our formulations to incorporate non-deterministic genotypes due to insufficient parental information or missing data. Alvarez-Castro and Carlborg handled internal mapping by implementing the Haley-Knott Regression with NOIA

The motivation of our implemented framework was based on the orthogonality property of NOIA which allows for easy model selection and variance component analysis. A next step is to extend the formulation proposed here to multi-locus and/or environmental factor cases, including gene-gene interaction and gene-environment interaction analyses when POE is incorporated. Conceptually, this generalization should be fairly straightforward using the Kronecker product rule as in

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