Functional Data Analysis Techniques Research Articles

Application and interpretation of functional data analysis techniques to differential scanning calorimetry data from lupus patients.

BackgroundDSC is used to determine thermally-induced conformational changes of biomolecules within a blood plasma sample. Recent research has indicated that DSC curves (or thermograms) may have different characteristics based on disease status and, thus, may be useful as a monitoring and diagnostic tool for some diseases. Since thermograms are curves measured over a range of temperature values, they are considered functional data. In this paper we apply functional data analysis techniques to analyze differential scanning calorimetry (DSC) data from individuals from the Lupus Family Registry and Repository (LFRR). The aim was to assess the effect of lupus disease status as well as additional covariates on the thermogram profiles, and use FD analysis methods to create models for classifying lupus vs. control patients on the basis of the thermogram curves.MethodsThermograms were collected for 300 lupus patients and 300 controls without lupus who were matched with diseased individuals based on sex, race, and age. First, functional regression with a functional response (DSC) and categorical predictor (disease status) was used to determine how thermogram curve structure varied according to disease status and other covariates including sex, race, and year of birth. Next, functional logistic regression with disease status as the response and functional principal component analysis (FPCA) scores as the predictors was used to model the effect of thermogram structure on disease status prediction. The prediction accuracy for patients with Osteoarthritis and Rheumatoid Arthritis but without Lupus was also calculated to determine the ability of the classifier to differentiate between Lupus and other diseases. Data were divided 1000 times into separate 2/3 training and 1/3 test data for evaluation of predictions. Finally, derivatives of thermogram curves were included in the models to determine whether they aided in prediction of disease status.ResultsFunctional regression with thermogram as a functional response and disease status as predictor showed a clear separation in thermogram curve structure between cases and controls. The logistic regression model with FPCA scores as the predictors gave the most accurate results with a mean 79.22% correct classification rate with a mean sensitivity = 79.70%, and specificity = 81.48%. The model correctly classified OA and RA patients without Lupus as controls at a rate of 75.92% on average with a mean sensitivity = 79.70% and specificity = 77.6%. Regression models including FPCA scores for derivative curves did not perform as well, nor did regression models including covariates.ConclusionChanges in thermograms observed in the disease state likely reflect covalent modifications of plasma proteins or changes in large protein-protein interacting networks resulting in the stabilization of plasma proteins towards thermal denaturation. By relating functional principal components from thermograms to disease status, our Functional Principal Component Analysis model provides results that are more easily interpretable compared to prior studies. Further, the model could also potentially be coupled with other biomarkers to improve diagnostic classification for lupus.

Bi-level heterogeneity modeling of functional performance degradation for the aging population

ABSTRACTAs the majority of service recipients of the U.S. long-term care system, the elderly with disabilities experience heterogeneous functional limitation and performance degradation. Successful modeling of heterogeneous degradation becomes of great importance for long-term care decision making. Existing heterogeneity modeling approaches are mainly restricted to single-level modeling of functional performance degradation for older adults. This article proposes a novel bi-level heterogeneity modeling and quantification framework to systematically investigate heterogeneous performance degradation at both sub-population and individual levels. Specifically, a Bayesian non-parametric modeling and inference framework is first proposed to characterize sub-population-level degradation heterogeneity. It relaxes the conventional assumption of pre-specifying a fixed number of sub-populations and allows joint model estimation and selection. Functional data analysis techniques are further considered to quantify individual heterogeneity nested within sub-populations. Based on the extracted temporal variation signatures of degradation profiles, post-analysis is performed to identify relevant risk/protective factors in explaining individual heterogeneity. To illustrate the proposed work and demonstrate its validity, the heterogeneous cognitive degradation quantification of a sample of community-dwelling older adults from the Health and Retirement Study is provided.

Inflorescence characteristics as function‐valued traits: Analysis of heritability and selection on architectural effects

AbstractProduction of multiple flowers in inflorescences allows the reproductive phenotypes of individual plants to include systematic among‐flower variation, which could be adaptive. Systematic trait variation within inflorescences could arise from resource competition among flowers, or be a developmentally determined feature of flower position, regardless of resource dynamics. The latter, architectural effect typically manifests as continuous floral variation within inflorescences. For architectural effects to be adaptive, floral trait variation among individuals must covary with reproductive performance and be heritable. However, heritability and phenotypic selection on gradients of variation cannot be estimated readily with traditional statistical approaches. Instead, we advocate and illustrate the application of two functional data analysis techniques with observations of Delphinium glaucum (Ranunculaceae). To demonstrate the parameters‐as‐data approach we quantify heritability of variation in anthesis rate, as represented by the regression coefficient relating daily anthesis rate to inflorescence age. SNP‐based estimates detected significant heritability (h2 = 0.245) for declining anthesis rate within inflorescences. Functional regression was used to assess phenotypic selection on anthesis rate and a floral trait (lower sepal length). The approach used spline curves that characterize within‐inflorescence variation as functional predictors of a plant's fruit set. Selection on anthesis rate varied with inflorescence age and the duration of an individual's anthesis period. Lower sepal length experienced positive selection for basal and distal flowers, but negative selection for central flowers. These results illustrate the utility and power of functional‐data analyses for studying architectural effects and specifically demonstrate that these effects are subject to natural selection and hence adaptive.

Statistical classification of early and late wood through the growth rings using thermogravimetric analysis

The aim of this study is to statistically identify and distinguish wood samples corresponding to different areas of annual rings in trees of temperate regions, using the corresponding thermogravimetric (TG) and their first TG derivative (DTG) curves, and specifically to verify whether late and early wood chestnut samples are different with statistical significance, taking into account their TG and DTG curves. These significant differences are sought by applying statistical procedures based on functional data analysis (FDA), such as the functional ANOVA and the FDA classification methods. Each TG curve is firstly smoothed using the local polynomial regression estimator, and its first derivative is estimated. Then, functional ANOVA based on random projections (RP) is used to identify significant differences between TG or DTG curves of early and late wood samples. In order to know the extent of the differences between early and late wood samples, they are discriminated (and the correct classification proportion obtained) by employing a kernel nonparametric functional data analysis technique, based on the Bayes' rule, as well as functional generalized linear models and functional generalized additive models, allowing to classify materials using more than one type of thermal curves simultaneously. The results are compared with those obtained using some classical multivariate supervised classification methods: linear discriminant analysis, naive Bayes (NBC) and quadratic classification (QDA). The partial least squares (PLS) dimension reduction procedure was previously applied to the TG curves in order to employ these multivariate methods. The application of RP ANOVA shows significant differences between late and early wood regarding mass loss and mass loss rate under combustion. The use of PLS multivariate methods or FDA classification approaches applied to the TG and DTG curves allows to distinguish very accurately between late and early wood. The proposed method could be applied to other species to identify thermooxidative differences, combined with other experimental methods to find their chemical and physical causes.

Global Sensitivity Analysis of Transient Code Output Applied to a Reflood Experiment Model Using the TRACE Code

Appropriate model assessment requires proper understanding of the impact of variations of model parameters and their interactions on results. However, in many thermal-hydraulic (TH) simulations, the code often predicts time-dependent output where the impact of model parameter variations on the overall transient is not easily discernible from quantities of interest (QoIs) based on the maximum, minimum, or average over space and/or time. This work proposes a methodology to analyze transient code output in a more comprehensive manner and coupled with established global sensitivity analysis (SA) methods.This paper first describes how the particularities of transient output variations can be characterized through application of functional data analysis (FDA) techniques. The convoluted amplitude and phase variations in the output are separated first by using a registration technique. The high-dimensional transient output is then projected in reduced dimension by using principal component analysis. Based on this, innovative QoIs that capture the variation over the whole course of the transient are finally derived.The subsequent global SA is a two-step process. First, the noninfluential parameters are identified using the efficient Morris screening method. The exclusion of these parameters reduces the size of the problem and decreases the cost of the downstream analysis. Second, the sensitivity of the influential model parameters with respect to a selected QoI is computed using the Sobol’ method. The method gives a set of sensitivity indices quantifying the contributions of input variations to the QoI variation considering possible interactions among the parameters.The applicability of the proposed methodology is demonstrated by analyzing a 26-parameter reflood experiment simulation model with the TRAC/RELAP Computational Engine (TRACE) TH system code. The FDA-derived QoI on the time-dependent cladding temperature was able to give a deeper insight on particular modes of variations exhibited by the model over the whole course of the reflood transient. That is, the variations in the cladding temperature transient could be simply divided into two modes: the reversal phase (from heatup until shortly after the temperature turnaround) and the descent phase (onward until quenching). The applied SA method was then able to attribute the variation of these two modes to the variation of the model parameters. It was found that the model was additive during the temperature reversal phase with dispersed flow film boiling—related parameters contributing to most of the model output variation, consistent with the adopted phenomenological model. Yet, the variation during the temperature descent could only be explained through parameter interactions indicating a more complex input/output relationship.Through study of this particular case, it is shown that if different QoIs based on the same time-dependent output variable are used—thus inspecting functional aspects of the output—the model might behave differently, being additive with respect to one QoI and interacting with respect to another, thus revealing more information on the model behavior, which might be overlooked in a SA based on extreme or time-averaged QoIs.

Setting a generalized functional linear model (GFLM) for the classification of different types of cancer

This work aims to classify the DNA sequences of healthy and malignant cancer respectively. For this, supervised and unsupervised classification methods from a functional context are used; i.e. each strand of DNA is an observation. The observations are discretized, for that reason different ways to represent these observations with functions are evaluated. In addition, an exploratory study is done: estimating the mean and variance of each functional type of cancer. For the unsupervised classification method, hierarchical clustering with different measures of functional distance is used. On the other hand, for the supervised classification method, a functional generalized linear model is used. For this model the first and second derivatives are used which are included as discriminating variables. It has been verified that one of the advantages of working in the functional context is to obtain a model to correctly classify cancers by 100%. For the implementation of the methods it has been used the fda.usc R package that includes all the techniques of functional data analysis used in this work. In addition, some that have been developed in recent decades. For more details of these techniques can be consulted Ramsay, J. O. and Silverman (2005) and Ferraty et al. (2006).

Dynamical correlation: A new method for quantifying synchrony with multivariate intensive longitudinal data.

In this article, we introduce dynamical correlation, a new method for quantifying synchrony between 2 variables with intensive longitudinal data. Dynamical correlation is a functional data analysis technique developed to measure the similarity of 2 curves. It has advantages over existing methods for studying synchrony, such as multilevel modeling. In particular, it is a nonparametric approach that does not require a prespecified functional form, and it places no assumption on homogeneity of the sample. Dynamical correlation can be easily estimated with irregularly spaced observations and tested to draw population-level inferences. We illustrate this flexible statistical technique with a simulation example and empirical data from an experiment examining interpersonal physiological synchrony between romantic partners. We discuss the advantages and limitations of the method, and how it can be extended and applied in psychological research. We also provide a set of R code for other researchers to estimate and test for dynamical correlation. (PsycINFO Database Record

Dependence Evolution in the Spanish Disabled Population: A Functional data Analysis Approach

Summary In a health context, dependence is defined as a lack of autonomy in performing the basic activities of daily living and requiring care giving or significant help from another person. However, this contingency, if present, changes over one's lifetime. Empirical evidence shows that, once this situation occurs, it is almost impossible to return to the previous state and in most cases increases in intensity. In the paper, the evolution of the intensity of this situation is studied for the Spanish population affected by this contingency. Evolution in dependence can be seen as sparsely observed functional data, where we obtain a curve for each individual that is observed at only those points where changes in his or her condition of dependence occur. We use functional data analysis techniques, such as curve registration, functional data depth and distance-based clustering, to analyse this type of data. This approach proves to be useful in this context because it considers the dynamics of the dependence process and provides more meaningful conclusions than simple pointwise or multivariate analysis. We use the sample statistics obtained to predict the future evolution of dependence. The database analysed originates from the ‘Survey on disability, personal autonomy and dependence situations’ in Spain in 2008. The survey is the largest and most complete survey to be made available in Europe for the study of disability. In addition, the Spanish legislation is one of the most recent in Europe and provides a detailed quantitative scale to assess dependence. In the paper, the scale value according to this legislation has been calculated for each individual included in the survey. Differences between sex, age and the time of first appearance were considered, and a prediction of the future evolution of dependence is obtained.

Functional measure of ozone exposure to model short‐term health effects

This article explores the issue of measuring daily ozone exposure for epidemiological studies of pollutants and their effects on health. Generally, hourly monitored ozone has been reduced to daily summary measures, which are then assumed to represent daily individual exposure to the pollutant. We instead propose the use of a functional measure of ozone exposure, which means, all hourly ozone measurements from a single day, to model short‐term health effects. Functional data analysis techniques are used to convert hourly monitored ozone from a single day into a function accounting for the daily profile of ozone. The role of this functional exposure measure is to better represent daily individual exposure and to identify the periods of daily ozone exposure that are most harmful to health. This is demonstrated by analyzing the relationship between daily hospital admissions and ozone exposure in the Italian city of Milan using functional regression models. A statistically significant association between hospital admissions and this functional measure of ozone exposure was found in the afternoon hours of the day. We propose that the functional measure of ozone exposure has superior predictive performance compared with that obtained by measuring daily exposure through summary measures. Copyright © 2016 John Wiley & Sons, Ltd.

Facing Network Management Challenges with Functional Data Analysis: Techniques & Opportunities

Current fixed and mobile networks’ behavior is rapidly changing, which calls for flexible monitoring approaches to avoid loosing track with such a fast evolutionary pace. Due to the many challenges that this scenario is posing to network managers, we propose the exploration of Functional Data Analysis (FDA) techniques as a mean to easily deal with network management and analysis issues. Specifically, we describe and evaluate several FDA methods with applications to network measurement preprocessing and clustering, bandwidth allocation, and anomaly and outlier detection. Our work focuses on how these FDA-based tools serve to improve the outcomes of traffic data mining and analysis, providing easy-to-understand and comprehensive outputs for network managers. We present the results that we have obtained from real case studies in the Spanish Academic network using throughput time series, comparing them with other alternatives of the state of the art. With this com- parative, we have qualitatively and quantitatively evaluated the advantages of FDA-methods in the networking area.

Genome-wide gene-gene interaction analysis for next-generation sequencing.

The critical barrier in interaction analysis for next-generation sequencing (NGS) data is that the traditional pairwise interaction analysis that is suitable for common variants is difficult to apply to rare variants because of their prohibitive computational time, large number of tests and low power. The great challenges for successful detection of interactions with NGS data are (1) the demands in the paradigm of changes in interaction analysis; (2) severe multiple testing; and (3) heavy computations. To meet these challenges, we shift the paradigm of interaction analysis between two SNPs to interaction analysis between two genomic regions. In other words, we take a gene as a unit of analysis and use functional data analysis techniques as dimensional reduction tools to develop a novel statistic to collectively test interaction between all possible pairs of SNPs within two genome regions. By intensive simulations, we demonstrate that the functional logistic regression for interaction analysis has the correct type 1 error rates and higher power to detect interaction than the currently used methods. The proposed method was applied to a coronary artery disease dataset from the Wellcome Trust Case Control Consortium (WTCCC) study and the Framingham Heart Study (FHS) dataset, and the early-onset myocardial infarction (EOMI) exome sequence datasets with European origin from the NHLBI's Exome Sequencing Project. We discovered that 6 of 27 pairs of significantly interacted genes in the FHS were replicated in the independent WTCCC study and 24 pairs of significantly interacted genes after applying Bonferroni correction in the EOMI study.

Smoothing Wind and Rainfall Data through Functional Data Analysis Technique

The pattern of wind and rainfall throughout Peninsular Malaysia are varied from one region to another, because of strong influences from the monsoons. In order to capture the wind and rainfall variations, a functional data analysis is introduced. The purpose of this study is to convert the wind and rainfall data into a smooth curve by using functional data analysis method. Fourier basis is used in this study since the wind and rainfall data indicated periodic pattern. In order to avoid such overfitting data, roughness penalty is added to the least square when constructing functional data object from the observed data. Result indicated that if we use a small number of bases functions, the difference is very small between with and without roughness penalty, showing that it is safer to smooth only when required. However, when a large basis function is employed, the roughness penalty should be added in order to obtain optimal fit data. Based on the contour plot of correlation and cross-correlation functions of wind and rainfall data, the relationship between both climate functions could be determined.

Valuation of large variable annuity portfolios under nested simulation: A functional data approach

A variable annuity (VA) is equity-linked annuity product that has rapidly grown in popularity around the world in recent years. Research up to date on VA largely focuses on the valuation of guarantees embedded in a single VA contract. However, methods developed for individual VA contracts based on option pricing theory cannot be extended to large VA portfolios. Insurance companies currently use nested simulation to valuate guarantees for VA portfolios but efficient valuation under nested simulation for a large VA portfolio has been a real challenge. The computation in nested simulation is highly intensive and often prohibitive. In this paper, we propose a novel approach that combines a clustering technique with a functional data analysis technique to address the issue. We create a highly non-homogeneous synthetic VA portfolio of 100,000 contracts and use it to estimate the dollar Delta of the portfolio at each time step of outer loop scenarios under the nested simulation framework over a period of 25 years. Our test results show that the proposed approach performs well in terms of accuracy and efficiency.

Forecasting Age-Specific Brain Cancer Mortality Rates Using Functional Data Analysis Models

Incidence and mortality rates are considered as a guideline for planning public health strategies and allocating resources. We apply functional data analysis techniques to model age-specific brain cancer mortality trend and forecast entire age-specific functions using exponential smoothing state-space models. The age-specific mortality curves are decomposed using principal component analysis and fit functional time series model with basis functions. Nonparametric smoothing methods are used to mitigate the existing randomness in the observed data. We use functional time series model on age-specific brain cancer mortality rates and forecast mortality curves with prediction intervals using exponential smoothing state-space model. We also present a disparity of brain cancer mortality rates among the age groups together with the rate of change of mortality rates. The data were obtained from the Surveillance, Epidemiology and End Results (SEER) program of the United States. The brain cancer mortality rates, classified under International Classification Disease code ICD-O-3, were extracted from SEER*Stat software.

A functional regression model for predicting optical depth and estimating attenuation coefficients in sea-ice covers near Resolute Passage, Canada

AbstractThe spectral dependence of natural light transmittance on ice algae concentration and snow depth in Arctic sea ice provides the potential to study the changing bottom-ice ecosystem using optical relationships. In this paper, we consider the use of functional data analysis techniques to describe such relationships. Specifically, we created a functional regression model describing spectral optical depth as a function of chlorophyll a concentration, snow depth and ice thickness. Measurements of the aforementioned covariates and surface and transmitted spectral irradiance were collected on landfast first-year sea ice in the High Arctic near Resolute Passage, Canada, during the spring of 2011 and used as model input. The derived model explains 75–84.5% of the variation in the observed spectral optical depth curves. No prior assumptions of snow/sea-ice optical properties are required in the application of this technique, as the model estimates the attenuation coefficients of each covariate using only the measurements mentioned above. The quality and simplicity of the model highlight the potential of functional data analysis to study the Arctic marine ecosystem.

Load carriage analysis for military using functional data analysis technique : registration and permutation test

This paper focused on the analysis of ground reaction force (GRF) of load carriage for military through functional data analysis (FDA) technique. The main objective of the study is to investigate the effect of 10% bodyweight (BW) load increment for normal walking through FDA method. The experiment on GRF was carried out by ten healthy military soldiers aged 31 ± 6.2 years old; weighing 71.6 ± 10.4 kg with the height of 166.3 ± 5.9 cm. Data was captured using Vicon 1.4 motion analysis system, Kistler force plates and thirty nine body markers attached to the subjects. FDA method mainly B-spline basis formation, time normalization, smoothing, curves registration and permutation tests were applied to analyze the data. The results confirmed that FDA did provide a better technique of smoothing to produce quality data in an easy manner. From this paper we conclude that registration did significantly align all the curves according to curves features providing a better way of distinguishing the amplitude of load increment. Finally, permutation tests confirmed that 10% BW load increased from 0% BW to 40% BW, have significant effect on soldiers regarding their load carriage and GRF.

Classification of wood using differential thermogravimetric analysis

The aim of this study is to propose an alternative methodology to classify wood species using the first (DTG), second (2DTG), and third (3DTG) derivatives of the thermogravimetric curves (TG). Accordingly, the main contribution of this new procedure consists on classifying materials (wood) taking into account the mass loss rate and acceleration with respect to temperature. In our research, each TG curve is firstly smoothed using the local polynomial regression estimator, and the first, second, and third derivatives are estimated. The application of the local polynomial regression estimator provides a reliable way to obtain the TG derivatives, overcoming the noise problem in the TG derivative estimation. Then, using these estimated curves, the different wood classes are discriminated employing a nonparametric functional data analysis (NPFDA) technique, based on the Bayes rule and the Nadaraya-Watson regression estimator, and also novel functional generalized additive models (GAM). The latter allows to classify materials using simultaneously more than one type of thermal curves. The results are compared with those obtained using classical and machine learning multivariate supervised classification methods, such as Linear discriminant analysis, Quadratic classification, Naive Bayes, Logistic regression, \(k\) Nearest neighbors, Neural networks, and Support vector machines. A regression model consisting of the mixture of the first derivatives of four generalized logistic components, one per principal wood constituent (water, hemicellulose, cellulose, and lignin), is applied to fit the DTG curves. The resulting 16 parameters from this fit characterize each curve and are used as datasets to apply the multivariate supervised classification methods. The use of the TG derivatives jointly with the TG curves has proved to be an optimal discriminating feature, when the new functional GAM techniques are employed.

Automated J wave detection from digital 12-lead electrocardiogram.

In this report we provide a method for automated detection of J wave, defined as a notch or slur in the descending slope of the terminal positive wave of the QRS complex, using signal processing and functional data analysis techniques. Two different sets of ECG tracings were selected from the EPICARE ECG core laboratory, Wake Forest School of Medicine, Winston Salem, NC. The first set was a training set comprised of 100 ECGs of which 50 ECGs had J-wave and the other 50 did not. The second set was a test set (n=116 ECGs) in which the J-wave status (present/absent) was only known by the ECG Center staff. All ECGs were recorded using GE MAC 1200 (GE Marquette, Milwaukee, Wisconsin) at 10mm/mV calibration, speed of 25mm/s and 500HZ sampling rate. All ECGs were initially inspected visually for technical errors and inadequate quality, and then automatically processed with the GE Marquette 12-SL program 2001 version (GE Marquette, Milwaukee, WI). We excluded ECG tracings with major abnormalities or rhythm disorder. Confirmation of the presence or absence of a J wave was done visually by the ECG Center staff and verified once again by three of the coauthors. There was no disagreement in the identification of the J wave state. The signal processing and functional data analysis techniques applied to the ECGs were conducted at Duke University and the University of Toronto. In the training set, the automated detection had sensitivity of 100% and specificity of 94%. For the test set, sensitivity was 89% and specificity was 86%. In conclusion, test results of the automated method we developed show a good J wave detection accuracy, suggesting possible utility of this approach for defining and detection of other complex ECG waveforms.

Forecasting Online Auctions via Self‐Exciting Point Processes

ABSTRACTModeling online auction prices is a popular research topic among statisticians and marketing analysts. Recent research mainly focuses on two directions: one is the functional data analysis (FDA) approach, in which the price–time relationship is modeled by a smooth curve, and the other is the point process approach, which directly models the arrival process of bidders and bids. In this paper, a novel model for the bid arrival process using a self‐exciting point process (SEPP) is proposed and applied to forecast auction prices. The FDA and point process approaches are linked together by using functional data analysis technique to describe the intensity of the bid arrival point process. Using the SEPP to model the bid arrival process, many stylized facts in online auction data can be captured. We also develop a simulation‐based forecasting procedure using the estimated SEPP intensity and historical bidding increment. In particular, prediction interval for the terminal price of merchandise can be constructed. Applications to eBay auction data of Harry Potter books and Microsoft Xbox show that the SEPP model provides more accurate and more informative forecasting results than traditional methods. Copyright © 2014 John Wiley & Sons, Ltd.

Generalized functional linear models for gene-based case-control association studies.

By using functional data analysis techniques, we developed generalized functional linear models for testing association between a dichotomous trait and multiple genetic variants in a genetic region while adjusting for covariates. Both fixed and mixed effect models are developed and compared. Extensive simulations show that Rao's efficient score tests of the fixed effect models are very conservative since they generate lower type I errors than nominal levels, and global tests of the mixed effect models generate accurate type I errors. Furthermore, we found that the Rao's efficient score test statistics of the fixed effect models have higher power than the sequence kernel association test (SKAT) and its optimal unified version (SKAT-O) in most cases when the causal variants are both rare and common. When the causal variants are all rare (i.e., minor allele frequencies less than 0.03), the Rao's efficient score test statistics and the global tests have similar or slightly lower power than SKAT and SKAT-O. In practice, it is not known whether rare variants or common variants in a gene region are disease related. All we can assume is that a combination of rare and common variants influences disease susceptibility. Thus, the improved performance of our models when the causal variants are both rare and common shows that the proposed models can be very useful in dissecting complex traits. We compare the performance of our methods with SKAT and SKAT-O on real neural tube defects and Hirschsprung's disease datasets. The Rao's efficient score test statistics and the global tests are more sensitive than SKAT and SKAT-O in the real data analysis. Our methods can be used in either gene-disease genome-wide/exome-wide association studies or candidate gene analyses.