Sample Size Scenarios Research Articles

A Bayes factor approach with informative prior for rare genetic variant analysis from next generation sequencing data.

The discovery of rare genetic variants through next generation sequencing is a very challenging issue in the field of human genetics. We propose a novel region-based statistical approach based on a Bayes Factor (BF) to assess evidence of association between a set of rare variants (RVs) located on the same genomic region and a disease outcome in the context of case-control design. Marginal likelihoods are computed under the null and alternative hypotheses assuming a binomial distribution for the RV count in the region and a beta or mixture of Dirac and beta prior distribution for the probability of RV. We derive the theoretical null distribution of the BF under our prior setting and show that a Bayesian control of the false Discovery Rate can be obtained for genome-wide inference. Informative priors are introduced using prior evidence of association from a Kolmogorov-Smirnov test statistic. We use our simulation program, sim1000G, to generate RV data similar to the 1000 genomes sequencing project. Our simulation studies showed that the new BF statistic outperforms standard methods (SKAT, SKAT-O, Burden test) in case-control studies with moderate sample sizes and is equivalent to them under large sample size scenarios. Our real data application to a lung cancer case-control study found enrichment for RVs in known and novel cancer genes. It also suggests that using the BF with informative prior improves the overall gene discovery compared to the BF with noninformativeprior.

A smeary central limit theorem for manifolds with application to high-dimensional spheres

The (CLT) central limit theorems for generalized Fréchet means (data descriptors assuming values in manifolds, such as intrinsic means, geodesics, etc.) on manifolds from the literature are only valid if a certain empirical process of Hessians of the Fréchet function converges suitably, as in the proof of the prototypical BP-CLT [Ann. Statist. 33 (2005) 1225–1259]. This is not valid in many realistic scenarios and we provide for a new very general CLT. In particular, this includes scenarios where, in a suitable chart, the sample mean fluctuates asymptotically at a scale $n^{\alpha }$ with exponents $\alpha <1/2$ with a nonnormal distribution. As the BP-CLT yields only fluctuations that are, rescaled with $n^{1/2}$, asymptotically normal, just as the classical CLT for random vectors, these lower rates, somewhat loosely called smeariness, had to date been observed only on the circle. We make the concept of smeariness on manifolds precise, give an example for two-smeariness on spheres of arbitrary dimension, and show that smeariness, although “almost never” occurring, may have serious statistical implications on a continuum of sample scenarios nearby. In fact, this effect increases with dimension, striking in particular in high dimension low sample size scenarios.

Gully Erosion Susceptibility Mapping Using Multivariate Adaptive Regression Splines—Replications and Sample Size Scenarios

Soil erosion is a serious problem affecting numerous countries, especially, gully erosion. In the current research, GIS techniques and MARS (Multivariate Adaptive Regression Splines) algorithm were considered to evaluate gully erosion susceptibility mapping among others. The study was conducted in a specific section of the Gorganroud Watershed in Golestan Province (Northern Iran), covering 2142.64 km2 which is intensely influenced by gully erosion. First, Google Earth images, field surveys, and national reports were used to provide a gully-hedcut evaluation map consisting of 307 gully-hedcut points. Eighteen gully erosion conditioning factors including significant geoenvironmental and morphometric variables were selected as predictors. To model sensitivity of gully erosion, Multivariate Adaptive Regression Splines (MARS) was used while the Area Under the Receiver Operating Characteristic (ROC) Curve (AUC), drawing ROC curves, efficiency percent, Yuden index, and kappa were used to evaluate model efficiency. We used two different scenarios of the combination of the number of replications, and sample size, including 90%/10% and 80%/20% with 10 replications, and 70%/30% with five, 10, and 15 replications for preparing gully erosion susceptibility mapping (GESM). Each one involves a various subset of both positive (presence), and negative (absence) cases. Absences were extracted as randomly distributed individual cells. Therefore, the predictive competency of the gully erosion susceptibility model and the robustness of the procedure were evaluated through these datasets. Results did not show considerable variation in the accuracy of the model, with altering the percentage of calibration to validation samples and number of model replications. Given the accuracy, the MARS algorithm performed excellently in predictive performance. The combination of 80%/20% using all statistical measures including SST (0.88), SPF (0.83), E (0.79), Kappa (0.58), Robustness (0.01), and AUC (0.84) had the highest performance compared to the other combinations. Consequently, it was found that the performance of MARS for modelling gully erosion susceptibility is quite consistent while changes in the testing and validation specimens are executed. The intense acceptable prediction capability of the MARS model verifies the reliability of the method employed for use of this model elsewhere and gully erosion studies since they are qualified to quickly generating precise and exact GESMs (gully erosion sensitivity maps) to make decisions and management edaphic and hydrologic features.

Sample size for determination of the physiological potential of coriander (Coriandrum sativum L.) seeds

The objective of this work was to determine the minimum number of replications and seeds per replication to perform the germination and seed vigor tests with coriander seeds. Two seed lots were compared in terms of water content, vigor and viability. Eight hundred seeds per lot were used. Values of germination first count, germination, germination speed index and mean germination time were analyzed. Sample size scenarios were developed using different combinations of number of replications (from 2 to 10) and the number of seeds per replication (from 20 to 80) by means of a resampling with replacement technique. The reference sample consisted of four replications of 50 seeds, as it is commonly used in researches with this species. Determination of the minimum number of replications and seeds was based on comparing the 95% bootstrapped confidence intervals for the index of variation (CV/n0.5) of each scenario with the confidence interval of the reference sample. It is reasonable to reduce the number of seeds per replication from 50 to 30 in order determine germination first count, germination and germination speed index. Forty seeds per replication are recommended to quantify the mean germination time. However, reductions in the number of replications can affect negatively the accuracy of germination and vigor tests.

A novel scale-space approach for multinormality testing and the k-sample problem in the high dimension low sample size scenario.

Two classical multivariate statistical problems, testing of multivariate normality and the k-sample problem, are explored by a novel analysis on several resolutions simultaneously. The presented methods do not invert any estimated covariance matrix. Thereby, the methods work in the High Dimension Low Sample Size situation, i.e. when n ≤ p. The output, a significance map, is produced by doing a one-dimensional test for all possible resolution/position pairs. The significance map shows for which resolution/position pairs the null hypothesis is rejected. For the testing of multinormality, the Anderson-Darling test is utilized to detect potential departures from multinormality at different combinations of resolutions and positions. In the k-sample case, it is tested whether k data sets can be said to originate from the same unspecified discrete or continuous multivariate distribution. This is done by testing the k vectors corresponding to the same resolution/position pair of the k different data sets through the k-sample Anderson-Darling test. Successful demonstrations of the new methodology on artificial and real data sets are presented, and a feature selection scheme is demonstrated.

Small-sample performance and underlying assumptions of a bootstrap-based inference method for a general analysis of covariance model with possibly heteroskedastic and nonnormal errors.

It is well known that the standard F test is severely affected by heteroskedasticity in unbalanced analysis of covariance models. Currently available potential remedies for such a scenario are based on heteroskedasticity-consistent covariance matrix estimation (HCCME). However, the HCCME approach tends to be liberal in small samples. Therefore, in the present paper, we propose a combination of HCCME and a wild bootstrap technique, with the aim of improving the small-sample performance. We precisely state a set of assumptions for the general analysis of covariance model and discuss their practical interpretation in detail, since this issue may have been somewhat neglected in applied research so far. We prove that these assumptions are sufficient to ensure the asymptotic validity of the combined HCCME-wild bootstrap analysis of covariance. The results of our simulation study indicate that our proposed test remedies the problems of the analysis of covariance F test and its heteroskedasticity-consistent alternatives in small to moderate sample size scenarios. Our test only requires very mild conditions, thus being applicable in a broad range of real-life settings, as illustrated by the detailed discussion of a dataset from preclinical research on spinal cord injury. Our proposed method is ready-to-use and allows for valid hypothesis testing in frequently encountered settings (e.g., comparing group means while adjusting for baseline measurements in a randomized controlled clinical trial).

Multi-state dead recovery mark-recovery model performance for estimating movement and mortality rates

Multi-state mark-recovery models are used to estimate movement and mortality rates of terrestrial and aquatic animals. These models have become especially popular in the last 20 years since technology and statistical techniques have improved to accommodate the extensive data requirements. However, the ability of multi-state mark-recovery models to estimate movement rates has received little evaluation, with few studies exploring the effects of alternative release and recovery designs on the bias and precision of estimates. Our objectives were to evaluate the effects of the spatio-temporal pattern of releases, pattern of recovery efforts, and number of releases on the performance of a multi-state mark-recovery model. We generated mark-recovery data from a spatial model and fitted them using a multi-state dead recovery model that included prior distributions on movement rates, natural mortality, and catchability. We generated data using a spatially variable schedule of releases and effort, and a release size of about 1 million individuals to mimic a mark-recovery study conducted for Atlantic Menhaden Brevoortia tyrannus in the late 1960s. We also ran alternative scenarios of sample size and spatially uniform releases and effort, either by themselves or in combination to determine their effects on the accuracy of the estimates. The model generally produced unbiased estimates of mortality rates with median error < 0.02 yr−1 for all scenarios, but some biases were present for the movement rates. Movement rates and catchability were more accurately and precisely estimated in scenarios that included spatially uniform fishing effort, while spatially uniform releases had little to no effect on bias or precision of estimated movement rates. Increased sample size improved accuracy of all parameter estimates except for the lowest movement rates. Future mark-recovery experiments that use a multi-state dead recovery model may benefit from distributing recovery effort uniformly over time and space.

Number of Source Signal Estimation by the Mean Squared Eigenvalue Error

Detection of the number of source signals (NoSS) in the presence of additive noise is considered. We present a new approach denoted by the mean squared eigenvalue error (MSEE). The MSEE is the mean squared error between the desired noise-free eigenvalues and the available estimated eigenvalues. The approach investigates and analyzes the probabilistic distribution of the available eigenvalue estimates and revisits proper thresholding of these sorted values. The optimum NoSS is provided by minimizing the MSEE. A probabilistic worst-case technique is proposed to estimate the value of the MSEE by using only the available data. It is shown that the proposed method is consistent as the data length increases. It is also shown that the method is consistent as the signal-to-noise ratio (SNR) increases. Simulation results illustrate advantages of the MSEE over competing approaches and confirm effectiveness and robustness of the MSEE even in low-SNR or small sample size scenarios.

An approximate method for generalized linear and nonlinear mixed effects models with a mechanistic nonlinear covariate measurement error model

The literature on measurement error for time-dependent covariates has mostly focused on empirical models, such as linear mixed effects models. Motivated by an AIDS study, we propose a joint modeling method in which a mechanistic nonlinear model is used to address the time-varying covariate measurement error for a longitudinal outcome that can be either discrete such as binary and count or continuous. We implement an inference procedure that uses first-order Taylor approximation to linearize both the covariate model and the response model. We study the asymptotic properties of the joint model based estimator and provide proof of consistency and normality. We then evaluate the performance of estimation in finite sample size scenario through simulation. Finally, we apply the new method to real data in an HIV/AIDS study.

Mapping Burn Severity of Forest Fires in Small Sample Size Scenarios

Mapping burn severity of forest fires can contribute significantly to understanding, quantifying and monitoring of forest fire severity and its impacts on ecosystems. In recent years, several remote sensing-based methods for mapping burn severity have been reported in the literature, of which the implementations are mainly dependent on several field plots. Therefore, it is a challenge to develop alternative method of mapping burn severity using limited number of field plots. In this study, we proposed a support vector regression based method using multi-temporal satellite data to map the burn severity, evaluated its performance by calculating correlations between the predicted and the observed Composite Burn Index, and compared the performance with that of the regression analysis method (based on dNBR). The results show that the performance of support vector regression based mapping method is more accurate (RMSE = 0.46–0.57) than that of regression analysis method (RMSE = 0.53–0.68). Even with fewer training sets, it can map the detailed distribution of burn severity of forest fires and can achieve relatively better generalization, compared to regression analysis burn severity mapping methods. It could be concluded that the proposed support vector regression based mapping method is an alternative to the regression analysis method in small sample size scenarios. This method with excellent generalization performance should be recommended for future studies on burn severity of forest fires.

Comparison among several commonly used sampling methods for a degradation test

AbstractDegradation testing is an effective tool for evaluating the reliability of highly reliable products. There have been many data collection methods proposed in the literature. Some of these assumed that only degradation values are recorded, and some assumed failure times to be available. However, most research has been devoted to proposing parameter estimates or to designing degradation tests for a specific sampling method. The differences between these commonly used methods have rarely been investigated. The lack of comparisons between different sampling methods has made it difficult to select an appropriate means by which to collect data. In addition, it remains unclear whether obtaining extra information (eg, exact failure times) is useful for making statistical inferences. In this paper, we assume that the degradation path of a product follows a Wiener degradation process, and we summarize several data collection methods. Maximum likelihood estimates for parameters and their variance‐covariance matrices are derived for each type of data. Several commonly used optimization criteria for designing a degradation test are used to compare estimation efficiency. Sufficient conditions under which one method could be better than the others are proposed. Upper bounds of estimation efficiency are also investigated. Our results provide useful guidelines by which to choose a sampling method, as well as its design variables, to obtain efficient estimation. A simulated example based on real light‐emitting diodes data is studied to verify our theoretical results under a moderate sample size scenario.

How to improve the standardization and the diagnostic performance of the fecal egg count reduction test?

Although various studies have provided novel insights into how to best design, analyze and interpret a fecal egg count reduction test (FECRT), it is still not straightforward to provide guidance that allows improving both the standardization and the analytical performance of the FECRT across a variety of both animal and nematode species. For example, it has been suggested to recommend a minimum number of eggs to be counted under the microscope (not eggs per gram of feces), but we lack the evidence to recommend any number of eggs that would allow a reliable assessment of drug efficacy. Other aspects that need further research are the methodology of calculating uncertainty intervals (UIs; confidence intervals in case of frequentist methods and credible intervals in case of Bayesian methods) and the criteria of classifying drug efficacy into ‘normal’, ‘suspected’ and ‘reduced’. The aim of this study is to provide complementary insights into the current knowledge, and to ultimately provide guidance in the development of new standardized guidelines for the FECRT. First, data were generated using a simulation in which the ‘true’ drug efficacy (TDE) was evaluated by the FECRT under varying scenarios of sample size, analytic sensitivity of the diagnostic technique, and level of both intensity and aggregation of egg excretion. Second, the obtained data were analyzed with the aim (i) to verify which classification criteria allow for reliable detection of reduced drug efficacy, (ii) to identify the UI methodology that yields the most reliable assessment of drug efficacy (coverage of TDE) and detection of reduced drug efficacy, and (iii) to determine the required sample size and number of eggs counted under the microscope that optimizes the detection of reduced efficacy. Our results confirm that the currently recommended criteria for classifying drug efficacy are the most appropriate. Additionally, the UI methodologies we tested varied in coverage and ability to detect reduced drug efficacy, thus a combination of UI methodologies is recommended to assess the uncertainty across all scenarios of drug efficacy estimates. Finally, based on our model estimates we were able to determine the required number of eggs to count for each sample size, enabling investigators to optimize the probability of correctly classifying a theoretical TDE while minimizing both financial and technical resources.

Confidence intervals for differences between volumes under receiver operating characteristic surfaces (VUS) and generalized Youden indices (GYIs).

This article explores both existing and new methods for the construction of confidence intervals for differences of indices of diagnostic accuracy of competing pairs of biomarkers in three-class classification problems and fills the methodological gaps for both parametric and non-parametric approaches in the receiver operating characteristic surface framework. The most widely used such indices are the volume under the receiver operating characteristic surface and the generalized Youden index. We describe implementation of all methods and offer insight regarding the appropriateness of their use through a large simulation study with different distributional and sample size scenarios. Methods are illustrated using data from the Alzheimer's Disease Neuroimaging Initiative study, where assessment of cognitive function naturally results in a three-class classification setting.

Clustering and classification problems in genetics through U-statistics

ABSTRACTGenetic data are frequently categorical and have complex dependence structures that are not always well understood. For this reason, clustering and classification based on genetic data, while highly relevant, are challenging statistical problems. Here we consider a versatile U-statistics-based approach for non-parametric clustering that allows for an unconventional way of solving these problems. In this paper we propose a statistical test to assess group homogeneity taking into account multiple testing issues and a clustering algorithm based on dissimilarities within and between groups that highly speeds up the homogeneity test. We also propose a test to verify classification significance of a sample in one of two groups. We present Monte Carlo simulations that evaluate size and power of the proposed tests under different scenarios. Finally, the methodology is applied to three different genetic data sets: global human genetic diversity, breast tumour gene expression and Dengue virus serotypes. These applications showcase this statistical framework's ability to answer diverse biological questions in the high dimension low sample size scenario while adapting to the specificities of the different datatypes.

Development of a copula‐based particle filter (CopPF) approach for hydrologic data assimilation under consideration of parameter interdependence

AbstractIn this study, a copula‐based particle filter (CopPF) approach was developed for sequential hydrological data assimilation by considering parameter correlation structures. In CopPF, multivariate copulas are proposed to reflect parameter interdependence before the resampling procedure with new particles then being sampled from the obtained copulas. Such a process can overcome both particle degeneration and sample impoverishment. The applicability of CopPF is illustrated with three case studies using a two‐parameter simplified model and two conceptual hydrologic models. The results for the simplified model indicate that model parameters are highly correlated in the data assimilation process, suggesting a demand for full description of their dependence structure. Synthetic experiments on hydrologic data assimilation indicate that CopPF can rejuvenate particle evolution in large spaces and thus achieve good performances with low sample size scenarios. The applicability of CopPF is further illustrated through two real‐case studies. It is shown that, compared with traditional particle filter (PF) and particle Markov chain Monte Carlo (PMCMC) approaches, the proposed method can provide more accurate results for both deterministic and probabilistic prediction with a sample size of 100. Furthermore, the sample size would not significantly influence the performance of CopPF. Also, the copula resampling approach dominates parameter evolution in CopPF, with more than 50% of particles sampled by copulas in most sample size scenarios.

Superpixel-Based Multitask Learning Framework for Hyperspectral Image Classification

Due to the high spectral dimensionality of hyperspectral images as well as the difficult and time-consuming process of collecting sufficient labeled samples in practice, the small sample size scenario is one crucial problem and a challenging issue for hyperspectral image classification. Fortunately, the structure information of materials, reflecting region of homogeneity in the spatial domain, offers an invaluable complement to the spectral information. Assuming some spatial regularity and locality of surface materials, it is reasonable to segment the image into different homogeneous parts in advance, called superpixel, which can be used to improve the classification performance. In this paper, a superpixel-based multitask learning framework has been proposed for hyperspectral image classification. Specifically, a set of 2-D Gabor filters are first applied to hyperspectral images to extract discriminative features. Meanwhile, a superpixel map is generated from the hyperspectral images. Second, a superpixel-based spatial–spectral Schroedinger eigenmaps (S4E) method is adopted to effectively reduce the dimensions of each extracted Gabor cube. Finally, the classification is carried out by a support vector machine (SVM)-based multitask learning framework. The proposed approach is thus termed Gabor S4E and SVM-based multitask learning (GS4E-MTLSVM). A series of experiments is conducted on three real hyperspectral image data sets to demonstrate the effectiveness of the proposed GS4E-MTLSVM approach. The experimental results show that the performance of the proposed GS4E-MTLSVM is better than those of several state-of-the-art methods, while the computational complexity has been greatly reduced, compared with the pixel-based spatial–spectral Schroedinger eigenmaps method.

Addressing prior-data conflict with empirical meta-analytic-predictive priors in clinical studies with historical information

ABSTRACTA common question in clinical studies is how to use historical data from earlier studies, leveraging relevant information into the design and analysis of a new study. Bayesian approaches are particularly well-suited to this task, with their natural ability to borrow strength across data sources. In this paper, we propose an eMAP approach for incorporating historical data into the analysis of clinical studies, and we discuss an application of this method to the analysis of observational safety studies for a class of products for patients with hemophilia A. The eMAP prior approach is flexible and robust to prior-data conflict. We conducted simulations to compare the frequentist operating characteristics of three approaches under different prior-data conflict assumptions and sample size scenarios.

Detecting Pairwise Interactive Effects of Continuous Random Variables for Biomarker Identification with Small Sample Size.

Aberrant changes to interactions among cellular components have been conjectured to be potential causes of abnormalities in cellular functions. By systematic analysis of high-throughput-omics data, researchers hope to detect potential associations among measured variables for better biomarker identification and phenotype prediction. In this paper, we focus on the methods to measure pairwise interactive effects among continuous random variables, representing molecular expressions, with respect to a given categorical outcome. Together with a comprehensive review on the existing measures, we further propose new measures that better estimate interactive effects, especially in small sample size scenarios. We first evaluate the performance of the existing and new methods for both small and large sample sizes based on simulated datasets that shows our proposed methods outperform previous methods in general. The best performing method for small sample size scenarios suggested by simulation experiments is then implemented to estimate interactive effects among genes with respect to the metastasis outcome in two breast cancer studies based on micro-array gene expression datasets. Our results further demonstrate that integrating detected interactive effects together with individual effects can help in finding more accurate biomarkers for breast cancer metastasis, which are indeed involved in important pathways related to cancer metastasis based on gene set enrichment analysis.

Pooled shrinkage estimator for quadratic discriminant classifier: an analysis for small sample sizes in face recognition

The quadratic discriminant classifier (QDC) is a well-known parametric Bayesian classifier that has been successfully applied to statistical pattern recognition problems. One such application is in automatic face recognition where the number of training images per subject is often found to be much less than the length of the facial features. In such a case, the QDC cannot be used because the class-specific covariance matrix on which it depends is either poorly estimated or singular thereby resulting in unacceptable classifier performance. High dimensional covariance estimation techniques such as shrinkage can alleviate this problem but only to a certain extent. This paper presents a computationally simple yet effective solution for further improving the QDC performance in small sample size scenarios. The proposed technique adopts a strategy of combining the class-specific shrinkage estimates of the covariance matrix to obtain a pooled shrinkage estimate, which is then plugged into the QDC. Experiments indicate that the proposed classifier leads to remarkable improvement in face recognition accuracy as compared to the existing classifiers such as the nearest neighbor, support vector machine and naive Bayes, irrespective of the nature of the database and feature extraction method. Monte Carlo simulations reveal that this improvement is due to the much lower mean squared error of the pooled shrinkage estimator which offers greater stability to the QDC.

Assessing the impact of restricted follow-up and small sample sizes on survival estimations in prostate cancer using registry data.

294 Background: Economic evaluations in oncology aim to assess the value of new therapies in the long term based on clinical trial data that often have restricted follow-up times (&lt; 5 years) and small sample sizes (&lt; 500 patients). This requires the use of extrapolation assumptions on long-term survival that go beyond the observed data. In this analysis, differences in survival extrapolation methods are tested in samples of sizes and follow-up reflecting typical clinical trials against a background of known survival in prostate cancer from a US based cancer registry. Methods: Data from the National Cancer Institute's Surveillance Epidemiology and End Results (SEER) registry on long-term survival in patients with stage IV prostate cancer were employed. The data set comprised those patients diagnosed between 1988 and 2003, with follow-up data available until 2012. Additional survival for those who received surgery (compared to those who did not), was estimated based on extrapolations using standard parametric statistical models (exponential, Weibull, log-logistic, log-normal, Gamma) fitted to the observed data. Survival analyses were run for 5 sample size scenarios (n = 27,670, 1000, 500, 200, 50) and 6 follow-up scenarios (follow-up years = 25, 20, 10, 5, 2, 1) yielding 30 combination scenarios. Performance of the methods was tested relative to the maximum follow-up, maximum sample size scenario (i.e. reference case) from the SEER registry. Results: Log-logistic and log-normal models were associated with flat tails which led to inflated survival estimations. For scenarios with smaller sizes, gamma models often did not converge. Exponential models were the most frequently reported as best model fit (in approximately 50% of scenarios). Also, gains in OS were consistent when exponential models were selected, and closely matched gain in OS from the reference case. Conclusions: Since clinical trials in oncology are often associated with small patient sample sizes and restricted follow-up, selecting an exponential model may lead to the most consistent and stable results based on the experiment constructed here. Further research should confirm these results for other types of cancer.