Genetic Analysis Workshop Research Articles

Testing for association between ordinal traits and genetic variants in pedigree-structured samples by collapsing and kernel methods.

In genome-wide association studies (GWAS), logistic regression is one of the most popular analytics methods for binary traits. Multinomial regression is an extension of binary logistic regression that allows for multiple categories. However, many GWAS methods have been limited application to binary traits. These methods have improperly often been used to account for ordinal traits, which causes inappropriate type I error rates and poor statistical power. Owing to the lack of analysis methods, GWAS of ordinal traits has been known to be problematic and gaining attention. In this paper, we develop a general framework for identifying ordinal traits associated with genetic variants in pedigree-structured samples by collapsing and kernel methods. We use the local odds ratios GEE technology to account for complicated correlation structures between family members and ordered categorical traits. We use the retrospective idea to treat the genetic markers as random variables for calculating genetic correlations among markers. The proposed genetic association method can accommodate ordinal traits and allow for the covariate adjustment. We conduct simulation studies to compare the proposed tests with the existing models for analyzing the ordered categorical data under various configurations. We illustrate application of the proposed tests by simultaneously analyzing a family study and a cross-sectional study from the Genetic Analysis Workshop 19 (GAW19) data.

Bivariate quantitative Bayesian LASSO for detecting association of rare haplotypes with two correlated continuous phenotypes.

In genetic association studies, the multivariate analysis of correlated phenotypes offers statistical and biological advantages compared to analyzing one phenotype at a time. The joint analysis utilizes additional information contained in the correlation and avoids multiple testing. It also provides an opportunity to investigate and understand shared genetic mechanisms of multiple phenotypes. Bivariate logistic Bayesian LASSO (LBL) was proposed earlier to detect rare haplotypes associated with two binary phenotypes or one binary and one continuous phenotype jointly. There is currently no haplotype association test available that can handle multiple continuous phenotypes. In this study, by employing the framework of bivariate LBL, we propose bivariate quantitative Bayesian LASSO (QBL) to detect rare haplotypes associated with two continuous phenotypes. Bivariate QBL removes unassociated haplotypes by regularizing the regression coefficients and utilizing a latent variable to model correlation between two phenotypes. We carry out extensive simulations to investigate the performance of bivariate QBL and compare it with that of a standard (univariate) haplotype association test, Haplo.score (applied twice to two phenotypes individually). Bivariate QBL performs better than Haplo.score in all simulations with varying degrees of power gain. We analyze Genetic Analysis Workshop 19 exome sequencing data on systolic and diastolic blood pressures and detect several rare haplotypes associated with the two phenotypes.

Control for population stratification in genetic association studies based on GWAS summary statistics.

Over the past years, genome-wide association studies (GWAS) have generated a wealth of new information. Summary data from many GWAS are now publicly available, promoting the development of many statistical methods for association studies based on GWAS summary statistics, which avoids the increasing challenges associated with individual-level genotype and phenotype data sharing. However, for population-based association studies such as GWAS, it has been long recognized that population stratification can seriously confound association results. For large GWAS, it is very likely that there exist population stratification and cryptic relatedness, which will result in inflated Type I error in association testing. Although many methods have been developed to control for population stratification, only two of these approaches can be used to control population stratification without individual-level data: one is based on genomic control (GC) and the other one is based on linkage disequilibrium score regression (LDSC). However, the performance of these two approaches is currently unknown. In this study, we use extensive simulation studies including populations with subpopulations, spatially structured populations, and populations with cryptic relatedness to compare the performance of these two approaches to control for population stratification using only GWAS summary statistics without individual-level data. Data sets from the genetic analysis workshop 19and UK Biobank are also used to evaluate these two approaches. We demonstrate that the intercept of LDSC can be used as a more accurate correction factor than GC. The results from this study will provide very useful information for researchers using GWAS summary statistics while trying to control for population stratification.

Using potential variable to study gene-gene and gene-environment interaction effects with genetic model uncertainty.

One of the critical issues in genetic association studies is to evaluate the risk of a disease associated with gene-gene or gene-environment interactions. The commonly employed procedures are derived by assigning a particular set of scores to genotypes. However, the underlying genetic models of inheritance are rarely known in practice. Misspecifying a genetic model may result in power loss. By using some potential genetic variables to separate the genotype coding and genetic model parameter, we construct a model-embedded score test (MEST). Our test is free of assumption of gene-environment independence and allows for covariates in the model. An effective sequential optimization algorithm is developed. Extensive simulations show the proposed MEST is robust and powerful in most of scenarios. Finally, we apply the proposed method to rheumatoid arthritis data from the Genetic Analysis Workshop 16 to further investigate the potential interaction effects.

An adaptive combination method for Cauchy variable based on optimal threshold

In sequence study, set-based analysis has been developed as a popular tool for analysing the association of a group rare variant with disease. However, most of the methods are sensitive to the genetic architecture. Besides, by directly combining the association signals of multiple markers within a genomic region can inevitably include a large proportion of noises. To address this issue, we extend the aggregated Cauchy association test (ACAT; Liu et al. 2019) and propose an adaptive Cauchy-variable combination method (ACC). Our proposed method combines Cauchy-variables, which are transformed from variant-level P-values in a given genomic region and adaptively truncate noises by choosing an optimal truncation threshold of the variant-level P-value that is determined by the data. Besides, the ACC method can use summary statistics obtained from open access database when the original genotype and phenotype data are unavailable. Extensive simulation studies and Genetic Analysis Workshop 19 real data analysis show that ACC is more powerful than the other comparative methods when only a small proportion of variants are causal, and ACC is robust to the varied genetic architecture.

Family-based gene-environment interaction using sequence kernel association test (FGE-SKAT) for complex quantitative traits

After the genome-wide association studies (GWAS) era, whole-genome sequencing is highly engaged in identifying the association of complex traits with rare variations. A score-based variance-component test has been proposed to identify common and rare genetic variants associated with complex traits while quickly adjusting for covariates. Such kernel score statistic allows for familial dependencies and adjusts for random confounding effects. However, the etiology of complex traits may involve the effects of genetic and environmental factors and the complex interactions between genes and the environment. Therefore, in this research, a novel method is proposed to detect gene and gene-environment interactions in a complex family-based association study with various correlated structures. We also developed an R function for the Fast Gene-Environment Sequence Kernel Association Test (FGE-SKAT), which is freely available as supplementary material for easy GWAS implementation to unveil such family-based joint effects. Simulation studies confirmed the validity of the new strategy and the superior statistical power. The FGE-SKAT was applied to the whole genome sequence data provided by Genetic Analysis Workshop 18 (GAW18) and discovered concordant and discordant regions compared to the methods without considering gene by environment interactions.

A Powerful Adaptive Cauchy-Variable Combination Method for Rare-Variant Association Analysis

Set-based association analysis has emerged as a popular tool for testing the association of rare variants within a genomic region with complex diseases. However, when only a small proportion of variants are causal, combining the association signals of multiple markers within a genomic region may cause noise due to the inclusion of non-causal variants, which usually decreases the power of a test. Besides, the existing set-based methods are sensitive to the genetic architecture. Therefore, we extend the aggregated Cauchy association test (ACAT) and propose an adaptive Cauchy-variable combination method (AAC). The AAC method adaptively combines Cauchy-variables transformed from variant-level P-values by using the optimal number of P-values that is determined by the data; the AAC method can remove variants with larger P-values. Extensive simulation studies and Genetic Analysis Workshop 19 real data analysis show that AAC is more powerful than the other comparative methods when only a small proportion of variants are causal. And AAC is robust to the varied genetic architecture. In addition, the AAC method may use summary statistics, without requiring the original genotypic and phenotypic data.

A fast and powerful aggregated Cauchy association test for joint analysis of multiple phenotypes.

Pleiotropy is a widespread phenomenon in complex human diseases. Jointly analyzing multiple phenotypes can improve power performance of detecting genetic variants and uncover the underlying genetic mechanism. This study aims to detect the association between genetic variants in a genomic region and multiple phenotypes. We develop the aggregated Cauchy association test to detect the association between rare variants in a genomic region and multiple phenotypes (abbreviated as "Multi-ACAT"). Multi-ACAT first detects the association between each rare variant and multiple phenotypes based on reverse regression and obtains variant-level p-values, then takes linear combination of transformed p-values as the test statistic which approximately follows Cauchy distribution under the null hypothesis. Extensive simulation studies show that when the proportion of causal variants in a genomic region is extremely small, Multi-ACAT is more powerful than the other several methods and is robust to bi-directional effects of causal variants. Finally, we illustrate our proposed method by analyzing two phenotypes [systolic blood pressure (SBP) and diastolic blood pressure (DBP)] from Genetic Analysis Workshop 19 (GAW19). The Multi-ACAT computes extremely fast, does not consider complex distributions of multiple correlated phenotypes, and can be applied to the case with noise phenotypes.

Candidate-gene association analysis for a continuous phenotype with a spike at zero using parent-offspring trios

In this paper, we propose the class of generalized additive models for location, scale and shape in a test for the association of genetic markers with non-normally distributed phenotypes comprising a spike at zero. The resulting statistical test is a generalization of the quantitative transmission disequilibrium test with mating type indicator, which was originally designed for normally distributed quantitative traits and parent-offspring data. As a motivational example, we consider coronary artery calcification (CAC), which can accurately be identified by electron beam tomography. In the investigated regions, individuals will have a continuous measure of the extent of calcium found or they will be calcium-free. Hence, the resulting distribution is a mixed discrete-continuous distribution with spike at zero. We carry out parent-offspring simulations motivated by such CAC measurement values in a screening population to study statistical properties of the proposed test for genetic association. Furthermore, we apply the approach to data of the Genetic Analysis Workshop 16 that are based on real genotype and family data of the Framingham Heart Study, and test the association of selected genetic markers with simulated coronary artery calcification.

Learning genetic and environmental graphical models from family data.

Many challenging problems in biomedical research rely on understanding how variables are associated with each other and influenced by genetic and environmental factors. Probabilistic graphical models (PGMs) are widely acknowledged as a very natural and formal language to describe relationships among variables and have been extensively used for studying complex diseases and traits. In this work, we propose methods that leverage observational Gaussian family data for learning a decomposition of undirected and directed acyclic PGMs according to the influence of genetic and environmental factors. Many structure learning algorithms are strongly based on a conditional independence test. For independent measurements of normally distributed variables, conditional independence can be tested through standard tests for zero partial correlation. In family data, the assumption of independent measurements does not hold since related individuals are correlated due to mainly genetic factors. Based on univariate polygenic linear mixed models, we propose tests that account for the familial dependence structure and allow us to assess the significance of the partial correlation due to genetic (between-family) factors and due to other factors, denoted here as environmental (within-family) factors, separately. Then, we extend standard structure learning algorithms, including the IC/PC and the really fast causal inference (RFCI) algorithms, to Gaussian family data. The algorithms learn the most likely PGM and its decomposition into two components, one explained by genetic factors and the other by environmental factors. The proposed methods are evaluated by simulation studies and applied to the Genetic Analysis Workshop 13 simulated dataset, which captures significant features of the Framingham Heart Study.

Genome-wide association study using truncated likelihood with incomplete information for stratum specific missingness

In many case–control genetic association studies, a secondary phenotype that may have common genetic factors with disease status can be identified. However, the secondary phenotype is observed only for case subjects but may be used as a surrogate marker for disease status. When information on the secondary phenotype is available only for a case group, due to cost and different data sources, fitting a linear regression model that ignores supplementary phenotype data may provide limited knowledge regarding the genetic associations. Herein, existing scientific information that the missing secondary phenotype values of control subjects lie in a certain range is incorporated, and a truncated joint likelihood function is established to estimate the effects of genotype and other covariates in the model. The performance of the proposed method is investigated through numerical studies and, for illustration, the procedure is applied to rheumatoid arthritis data provided by the Genetic Analysis Workshop 16.

Testing gene-environment interactions for rare and/or common variants in sequencing association studies.

The risk of many complex diseases is determined by a complex interplay of genetic and environmental factors. Advanced next generation sequencing technology makes identification of gene-environment (GE) interactions for both common and rare variants possible. However, most existing methods focus on testing the main effects of common and/or rare genetic variants. There are limited methods developed to test the effects of GE interactions for rare variants only or rare and common variants simultaneously. In this study, we develop novel approaches to test the effects of GE interactions of rare and/or common risk, and/or protective variants in sequencing association studies. We propose two approaches: 1) testing the effects of an optimally weighted combination of GE interactions for rare variants (TOW-GE); 2) testing the effects of a weighted combination of GE interactions for both rare and common variants (variable weight TOW-GE, VW-TOW-GE). Extensive simulation studies based on the Genetic Analysis Workshop 17 data show that the type I error rates of the proposed methods are well controlled. Compared to the existing interaction sequence kernel association test (ISKAT), TOW-GE is more powerful when there are GE interactions' effects for rare risk and/or protective variants; VW-TOW-GE is more powerful when there are GE interactions' effects for both rare and common risk and protective variants. Both TOW-GE and VW-TOW-GE are robust to the directions of effects of causal GE interactions. We demonstrate the applications of TOW-GE and VW-TOW-GE using an imputed data from the COPDGene Study.

Hierarchical structural component model for pathway analysis of common variants

BackgroundGenome-wide association studies (GWAS) have been widely used to identify phenotype-related genetic variants using many statistical methods, such as logistic and linear regression. However, GWAS-identified SNPs, as identified with stringent statistical significance, explain just a small portion of the overall estimated genetic heritability. To address this ‘missing heritability’ issue, gene- and pathway-based analysis, and biological mechanisms, have been used for many GWAS studies. However, many of these methods often neglect the correlation between genes and between pathways.MethodsWe constructed a hierarchical component model that considers correlations both between genes and between pathways. Based on this model, we propose a novel pathway analysis method for GWAS datasets, Hierarchical structural Component Model for Pathway analysis of Common vAriants (HisCoM-PCA). HisCoM-PCA first summarizes the common variants of each gene, first at the gene-level, and then analyzes all pathways simultaneously by ridge-type penalization of both the gene and pathway effects on the phenotype. Statistical significance of the gene and pathway coefficients can be examined by permutation tests.ResultsUsing the simulation data set of Genetic Analysis Workshop 17 (GAW17), for both binary and continuous phenotypes, we showed that HisCoM-PCA well-controlled type I error, and had a higher empirical power compared to several other methods. In addition, we applied our method to a SNP chip dataset of KARE for four human physiologic traits: (1) type 2 diabetes; (2) hypertension; (3) systolic blood pressure; and (4) diastolic blood pressure. Those results showed that HisCoM-PCA could successfully identify signal pathways with superior statistical and biological significance.ConclusionsOur approach has the advantage of providing an intuitive biological interpretation for associations between common variants and phenotypes, via pathway information, potentially addressing the missing heritability conundrum.

A general statistic to test an optimally weighted combination of common and/or rare variants.

Both genome-wide association study and next-generation sequencing data analyses are widely employed to identify disease susceptible common and/or rare genetic variants. Rare variants generally have large effects though they are hard to detect due to their low frequencies. Currently, many existing statistical methods for rare variants association studies employ a weighted combination scheme, which usually puts subjective weights or suboptimal weights based on some adhoc assumptions (e.g., ignoring dependence between rare variants). In this study, we analytically derived optimal weights for both common and rare variants and proposed a general and novel approach to test association between an optimally weighted combination of variants (G-TOW) in a gene or pathway for a continuous or dichotomous trait while easily adjusting for covariates. Results of the simulation studies show that G-TOW has properly controlled type I error rates and it is the most powerful test among the methods we compared when testing effects of either both rare and common variants or rare variants only. We also illustrate the effectiveness of G-TOW using the Genetic Analysis Workshop 17 (GAW17) data. Additionally, we applied G-TOW and other competitive methods to test disease-associated genes in real data of schizophrenia. The G-TOW has successfully verified genes FYN and VPS39 which are associated with schizophrenia reported in existing publications. Both of these genes are missed by the weighted sum statistic and the sequence kernel association test. Simulation study and real data analysis indicate that G-TOW is a powerful test.

Multivariate association test for rare variants controlling for cryptic and family relatedness

AbstractIn genetic studies of complex diseases, multiple measures of related phenotypes are often collected. Jointly analyzing these phenotypes may improve statistical power to detect sets of rare variants affecting multiple traits. In this work, we consider association testing between a set of rare variants and multiple phenotypes in family‐based designs. We use a mixed linear model to express the correlations among the phenotypes and between related individuals. Given the many sources of correlations in this situation, deriving an appropriate test statistic is not straightforward. We derive a vector of score statistics, whose joint distribution is approximated using a copula. This allows us to have closed‐form expressions for the p‐values of several test statistics. A comprehensive simulation study and an application to Genetic Analysis Workshop 18 data highlight the gains associated with joint testing over univariate approaches, especially in the presence of pleiotropy or highly correlated phenotypes. The Canadian Journal of Statistics 47: 90–107; 2019 © 2018 Statistical Society of Canada

Genetics of Factor VIII Inhibitor Development in Hemophilia Patients: Novel Statistical Approaches in the PATH Study

Here we apply state-of-the-art statistical genetic approaches toward investigating the genetic architecture of factor VIII (FVIII) inhibitor (FEI) development in Hemophilia A (HA). A total of 442 North American HA patients (237 Whites and 205 Blacks; 88% severely affected) enrolled in the PATH Study were: 1) ImmunoChip genotyped at ~167,000 single nucleotide polymorphisms (SNPs) in genes previously implicated in autoimmune disease risk; 2) Evaluated by DNA sequencing and assays for the recurrent intron (I)1 and I22 inversions to identify their causative F8 mutations; and 3) Tested with the Bethesda assay to determine their FEI status. The ImmunoChip genotypes were used to construct a genetic relationship matrix (GRM), denoted by K, following our previously published method,1 and the F8 sequence data along with results from the I1 and I22 inversion assays were used to construct a shared F8-mutation matrix, denoted by F. We analyzed a dichotomous FEI variable under the statistical genetic threshold/liability model (a probit regression in the fixed effects) in conjunction with a variance components model for the FEI liability phenotypic covariance matrix, denoted by P, to model potentially important random effects. For the latter, we specifically assumed independent additive genetic, F8-mutation, and residual environmental random effects. By the independence assumption, the covariance matrix is then decomposable as a sum of the additive genetic (Va), F8-mutation (Vf), and residual environmental (Ve) variances respectively structured by K, F, and the identity matrix I. The variance component model is given as: P = K*Va + F*Vf + I*Ve. Heritability, denoted by h2, is defined as the ratio of Va to the total phenotypic variance (Vp): h2 = Va / Vp. We can further speak of the total heritability given as: h2t = h2r + h2f + h2snp, where the subscripts t, r, f, and snp respectively denote total, residual additive genetic, F8-mutation-specific, and SNP heritabilities. Using eigenstructure methods,2 we can compute power under a simpler model in which Va and Vf are combined as a single variance component. We computed power to detect genetic association as measured by SNP-specific heritability for a set of 403 SNPs in or near 14 candidate immune response genes previously implicated in FEI risk. To account for multiple hypothesis testing, power was computed at the Bonferroni-adjusted significance level of 0.05/403 = 1.2 × 10-4. Under the simplified model, we computed the statistical power to detect causal SNPs for our sample and study design for the sample FEI prevalences, denoted by Kp, for Whites (22.5%) and Blacks (45%), across a range of total heritabilities, h2t = 15%, 35%, and 55%, where the lattermost total heritability was observed for FEI liability in the current study (Figure 1). It should be noted that because the liability heritability is known to be biased upward, we applied the Dempster-Lerner correction to both the total and SNP-specific heritabilities.3 Close inspection of Figure 1 reveals that varying h2t from 15% to 35% to 55% results in slight decreases in power due to the decreasing ratio of the SNP-specific heritability to the total heritability. However, as seen in all three panels, the more important determinant of power is clearly the FEI prevalence in that the power curve for a Kp of 45% is associated with greater power than the power curve for a Kp of 22.5% across the range of total heritabilities examined. As seen in Figure 1, we have adequate power to detect SNP heritabilities as low as 5% and 6%, respectively, for a Kp of 45% and 22.5%. As noted above, we observed a FEI liability total heritability of 55% consisting of a 47% residual additive genetic heritability (p = 0.019) and 8% F8-mutation specific heritability (p = 0.005). This is the first study to use a GRM based on genotype data and a shared causal F8 mutation matrix to model additive genetic and F8-mutation specific effects.Almeida M, Peralta J, Farook V, …, Blangero J. Pedigree-based random effect tests to screen gene pathways. BMC Proc. 2014; 8(Suppl 1 Genetic Analysis Workshop): S100.Blangero J, Diego VP, Dyer T, …, Göring H. A kernel of truth: statistical advances in polygenic variance component models for complex human pedigrees. Adv Genetics. 2013; 81: 1-31.Glahn D, Williams J, McKay D, …, Blangero J. Discovering schizophrenia endophenotypes in randomly ascertained pedigrees. Biol Psychiatry. 2015; 77(1): 75-83. [Display omitted] DisclosuresChitlur:Baxter, Bayer, Biogen Idec, and Pfizer: Honoraria; Novo Nordisk Inc: Consultancy. Dinh:Haplomics Biotechnology Corporation: Employment, Equity Ownership. Howard:Haplomics Biotechnology Corporation: Equity Ownership, Other: Chief Scientific Officer, Patents & Royalties: Patent applications and provisional patent applications ; CSL Behring: Research Funding.

General retrospective mega‐analysis framework for rare variant association tests

Here, we describe a retrospective mega-analysis framework for gene- or region-based multimarker rare variant association tests. Our proposed mega-analysis association tests allow investigators to combine longitudinal and cross-sectional family- and/or population-based studies. This framework can be applied to a continuous, categorical, or survival trait. In addition to autosomal variants, the tests can be applied to conduct mega-analyses on X-chromosome variants. Tests were built on study-specific region- or gene-level quasiscore statistics and, therefore, do not require estimates of effects of individual rare variants. We used the generalized estimating equation approach to account for complex multiple correlation structures between family members, repeated measurements, and genetic markers. While accounting for multilevel correlations and heterogeneity across studies, the test statistics were computationally efficient and feasible for large-scale sequencing studies. The retrospective aspect of association tests helps alleviate bias due to phenotype-related sampling and type I errors due to misspecification of phenotypic distribution. We evaluated our developed mega-analysis methods through comprehensive simulations with varying sample sizes, covariates, population stratification structures, and study designs across multiple studies. To illustrate application of the proposed framework, we conducted a mega-association analysis combining a longitudinal family study and a cross-sectional case-control study from Genetic Analysis Workshop 19.

Identification of epistatic interactions between the human RNA demethylases FTO and ALKBH5 with gene set enrichment analysis informed by differential methylation

The Genetic Analysis Workshop (GAW) presents an opportunity to collaboratively evaluate methodology relevant to current issues in genetic epidemiology. The GAW20 data combine real clinical trial data with fictitious epigenetic drug response endpoints. Considering the evidence suggesting that networks of interactions between many genes underlie complex phenotypes, we utilize differential methylation status to identify a relevant gene set for enrichment analysis and use this to infer potential biological function underlying drug response. We highlight the pertinence of considering the potential for widespread epistatic interactions in the absence of main effects, and present evidence of epistasis between single-nucleotide polymorphisms (SNPs) on the two RNA demethylases FTO and ALKBH5.

Simulation of a medication and methylation effects on triglycerides in the Genetic Analysis Workshop 20

The GAW20 simulation data set is based upon the companion Genetics of Lipid Lowering Drugs and Diet Network (GOLDN) study fenofibrate clinical trial data set that forms the real data example for GAW20. The simulated data problem consists of 200 simulated replications of what might happen if we were to repeat the GOLDN clinical trial 200 independent times, for these exact same subjects, but using a new fictitious drug (called “genomethate”) that has a pharmaco-epigenetic effect on triglyceride response. For each replication, the pre-genomethate values at visits 1 and 2 are constant (ie, pedigree structures, age, sex, all phenotypes, covariates, genome-wide association study (GWAS) genotypes, and visit 2 methylation values), the same as the real GOLDN data across all 200 replications. Only the post-genomethate treatment data (ie, methylation and triglyceride levels for visits 3 and 4) change across the 200 replications. We postulate a growth curve pharmaco-epigenetic response model, in which each patient’s response to genomethate treatment is individualized, and is dependent upon their genotype as well as the methylation state for key genes.

A generalized, multi-stage adjusted, latent class linear mixed model for testing genetic association

ABSTRACTIn this study, a Generalized, Multi-Stage Adjusted, Latent Class Linear Mixed Model is proposed for modeling the heterogeneous distributed phenotype and genetic information across the whole genome in the presence of both serial and familial correlations. Genome data were analyzed by applying the proposed model to Genetic Analysis Workshop (GAW) data, and the model results were compared to the results of standard models. Moreover, the potential of the model is discussed compared to simulated data. As a result of model comparisons, the information criteria and the genomic control parameter were found to be smaller. The results of a power analysis show that the proposed model is more powerful.