Noncentrality Parameter Research Articles

Characterization of Slow and Fast Fading in Off-Body Communication at 2.45 GHz with Space Diversity Scheme in an Indoor Environment

The characterization of slow and fast fading in wireless body area networks with space diversity scheme has been presented. The analysis, based on the measurements at 2.45 GHz in an indoor environment, has shown that for all investigated configurations of receiving wearable antennas, the correlation coefficient values of the received signals’ parameters are below the assumed value of 0.5, being close to zero for the vast majority of cases. It has been shown that the slow fading component may be modelled by a lognormal distribution with zero average and the standard deviation from the range of [1.43, 1.98] dB. The fast fading component is the best modelled by a Rice distribution with the noncentrality parameter and the scale parameter being in the range [0.8125, 0.9624] and [0.5269, 0.6954], respectively.

Estimation of a covariance matrix in multivariate skew-normal distribution

This article addresses the problem of estimating a covariance matrix in a multivariate skew-normal distribution relative to two different losses. The estimation problem can be reduced to that of a scale matrix of a noncentral Wishart distribution. The noncentrality parameter matrix, which is a nuisance parameter, brings about non optimality of the best triangular invariant estimators which are minimax under normality. Some improving techniques under normality are proven to remain robust under the multivariate skew-normal distribution.

On Wishart and noncentral Wishart distributions on symmetric cones

Necessary conditions for the existence of noncentral Wishart distributions are given. Our method relies on positivity properties of spherical polynomials on Euclidean Jordan algebras and advances an approach by Peddada and Richards [Ann. Probab. 19 (1991), pp. 868–874] where only a special case (positive semidefinite matrices, rank $1$ noncentrality parameter) is treated. Not only do the shape parameters need to be in the Wallach set—as is the case for Riesz measures—but also the rank of the noncentrality parameter is constrained by the size of the shape parameter. This rank condition has recently been proved with different methods for the special case of symmetric, positive semidefinite matrices (Letac and Massam; Graczyk, Malecki, and Mayerhofer).

SU35 - EVALUATING THE IMPACT OF PHENOTYPIC DEFINITIONS ON STATISTICAL POWER IN MAJOR DEPRESSION GENETICS

Background Obtaining sufficient statistical power has been a key issue in the study of genetics in depression; only recently have samples become large enough to identify any genome-wide significant hits for this phenotype. Researchers have used a range of different approaches to tackle the issue of balancing sample size with phenotypic depth, and here we evaluate the statistical power achieved with these different methods. Methods We collated details of published genome-wide association studies exploring major depression or depressive symptoms. For each study we assessed sampling methods, phenotyping criteria and sample size, and use GPC (Genetics Power Calculator, Purcell et al. 2003) and extension methods (Yang et al. 2010, Traylor et al. 2015) to calculate statistical power, and compare approaches within the literature. We compared the statistical power (as captured by the non-centrality parameter) of each cohort across a range of allele thresholds and genetic effect sizes, contrasting different methodologies of cohort ascertainment within the depression literature. We evaluate how statistical power is affected by the selection of potentially more homogenous but less prevalent depression phenotypes, the potential impact of misclassification in large cohorts which have employed brief phenotypic screening questions, and the value of dimensional measures of depression symptoms in population-based cohorts. Results We confirm the findings of Traylor et al. apply in the case of depression; that in the analysis of specific subtypes of depression performed in datasets including PGC2 (2013, secondary analysis of recurrent, early-onset recurrent and typical-like depression, observed in approximately 75%, 50% and 40% of cases respectively) and CONVERGE (2015, secondary analysis of melancholia observed in approximately 85% of cases), very modest increases in genotypic relative risk (GRR) are required to achieve equivalent power with the whole sample analyses, particularly for less common risk allele frequencies and lower index GRR. We also compare the statistical power of the PGC2 case-control sample (discovery sample n=18,759) and the study from the CHARGE Consortium (discovery sample n= 34,549, Hek et al. 2013) looking at depression symptoms, observing that with a sample size which is 54% of that in CHARGE, the PGC2 dataset has less than half the statistical power to detect genetic effects. Discussion Given range of approaches, here we give evaluation of expected statistical power across a range of approaches specific to the field of major depression genetics. We compare the statistical power achieved in the current literature and evaluate how different phenotypic approaches affect this. If the ultimate aim of our research is to improve outcomes for patients with depression, our phenotypic definitions must be informed by clinical relevance. But nevertheless, an understanding of how statistical power is impacted by these definitions is key to optimal study design for the identification of the genetic variants associated with depressive disorders.

Coexistence of MIMO Radar and FD MIMO Cellular Systems With QoS Considerations

In this work, the feasibility of spectrum sharing between a multiple-input multiple-output (MIMO) radar system (RS) and a MIMO cellular system (CS), comprising of a full duplex (FD) base station (BS) serving multiple downlink and uplink users at the same time and frequency is investigated. While a joint transceiver design technique at the CS's BS and users is proposed to maximise the probability of detection (PoD) of the MIMO RS, subject to constraints of quality of service (QoS) of users and transmit power at the CS, null-space based waveform projection is used to mitigate the interference from RS towards CS. In particular, the proposed technique optimises the performance of PoD of RS by maximising its lower bound, which is obtained by exploiting the monotonically increasing relationship of PoD and its non-centrality parameter. Numerical results show the utility of the proposed spectrum sharing framework, but with certain trade-offs in performance corresponding to RS's transmit power, RS's PoD, CS's residual self interference power at the FD BS and QoS of users.

Heritability informed power optimization (HIPO) leads to enhanced detection of genetic associations across multiple traits.

Genome-wide association studies have shown that pleiotropy is a common phenomenon that can potentially be exploited for enhanced detection of susceptibility loci. We propose heritability informed power optimization (HIPO) for conducting powerful pleiotropic analysis using summary-level association statistics. We find optimal linear combinations of association coefficients across traits that are expected to maximize non-centrality parameter for the underlying test statistics, taking into account estimates of heritability, sample size variations and overlaps across the traits. Simulation studies show that the proposed method has correct type I error, robust to population stratification and leads to desired genome-wide enrichment of association signals. Application of the proposed method to publicly available data for three groups of genetically related traits, lipids (N = 188,577), psychiatric diseases (Ncase = 33,332, Ncontrol = 27,888) and social science traits (N ranging between 161,460 to 298,420 across individual traits) increased the number of genome-wide significant loci by 12%, 200% and 50%, respectively, compared to those found by analysis of individual traits. Evidence of replication is present for many of these loci in subsequent larger studies for individual traits. HIPO can potentially be extended to high-dimensional phenotypes as a way of dimension reduction to maximize power for subsequent genetic association testing.

Asymptotically optimal one-bit quantizer design for weak-signal detection in generalized Gaussian noise and lossy binary communication channel

Abstract In this paper, quantizer design for weak-signal detection under arbitrary binary channel in generalized Gaussian noise is studied. Since the performances of the generalized likelihood ratio test (GLRT) and Rao test are asymptotically characterized by the noncentral chi-squared probability density function (PDF), the threshold design problem can be formulated as a noncentrality parameter maximization problem. The theoretical property of the noncentrality parameter with respect to the threshold is investigated, and the optimal threshold is shown to be found in polynomial time with appropriate numerical algorithm and proper initializations. In certain cases, the optimal threshold is proved to be zero. Finally, numerical experiments are conducted to substantiate the theoretical analysis.

CFAR subspace detectors with multiple observations in system-dependent clutter background

Abstract In this paper, the target detection problem is studied in system-dependent clutter (SDC) background where the clutter and signal of interest (SOI) propagate in the same channels. The received data model applied here, named as multiple observations data model, consists of multiple observations (primary data) and a set of secondary data (free of SOI). Three detectors with the constant false alarm rate feature are designed by using the generalized likelihood ratio, Rao and Wald tests. Interesting findings are: (1) Theoretical derivation proves that these three detectors are equivalent. (2) The primary data size D and the secondary data size K have nearly the same contribution to computational complexity and the probability of detection (PD) is more sensitive to D. (3) For desired probability of false alarm and PD, proper D and K can be selected based on the approximately estimated noncentrality parameters involved in probability density functions. A selection procedure is provided. (4) Based on those estimated noncentrality parameters, it is demonstrated that the relationship between PD and clutter-to-noise ratio is determined by the system response matrix with respect to clutter and SOI. Simulation results validate that the designed new detectors outperform other classic detectors in SDC background.

Calculating power for the general linear multivariate model with one or more Gaussian covariates

ABSTRACTWe describe a noncentral power approximation for hypotheses about fixed predictors in general linear multivariate models with one or more Gaussian covariates. The results apply to both single and multiple parameter hypotheses. The approach extends power approximations for models with only fixed predictors, and for models with a single Gaussian covariate. The new method approximates the noncentrality parameter under the alternative hypothesis using a Taylor series expansion for the matrix-variate beta distribution of type I. We used a Monte Carlo simulation to evaluate the accuracy of both the novel power approximation, and published power approximations. The simulation study accounted for randomness in both the predictors and the errors. We varied the number of outcomes, the number of parameters in the hypothesis, the per-treatment sample size, and the correlations between the random predictors and the outcomes. We demonstrate that our approximation is more accurate than published methods, both in small and large samples. We show that the run time for a single power calculation with the new method is on the order of milliseconds, compared to an average empirical simulation time of roughly three minutes. Approximate and simulated power can be calculated using the free, open-source rPowerlib package (http://github.com/SampleSizeShop/rPowerlib).

A characterization of Wishart processes and Wishart distributions

A characterization of the existence of non-central Wishart distributions (with shape and non-centrality parameter) as well as the existence of solutions to Wishart stochastic differential equations (with initial data and drift parameter) in terms of their exact parameter domains is given. These two families are the natural extensions of the non-central chi-square distributions and the squared Bessel processes to the positive semidefinite matrices.

Angle-of-Arrival-Dependent Interference Modeling in Rician Massive MIMO

In this paper, we study the uplink in a single-cell massive multiple-input–multiple-output system. The base station (BS) is equipped with three antenna arrays, each covering one third of the cell area. Each antenna array comprises a large yet finite number of antennas. The single-antenna users are randomly and uniformly distributed in the cell, transmitting to the BS utilizing full channel inversion power control. All users experience Rician fading. Receiver maximum-ratio-combining is performed at the BS. Under such a setting, we focus on one cell sector and analyze the intrasector interference in a realistic situation where the number of BS antennas is not extremely large compared with the user number. In particular, we show that, due to the line-of-sight (LoS) component of the channel, the interference is partially determined by the angles of arrival of the signals. We approximate the LoS component interference by a Beta mixture. The interference in Rician fading is then modeled as a noncentral chi-square distribution with a random noncentrality parameter, corresponding to the LoS component. The approximate interference distribution can be used to compute signal-to-interference-ratio-dependent metrics such as outage probability and average throughput.

Robust MIMO Beamforming for Cellular and Radar Coexistence

In this letter, we consider the coexistence and spectrum sharing between downlink multi-user multiple-input-multiple-output (MU-MIMO) communication and an MIMO radar. For a given performance requirement of the downlink communication system, we design the transmit beamforming such that the detection probability of the radar is maximized. While the original optimization problem is non-convex, we exploit the monotonically increasing relationship of the detection probability with the non-centrality parameter of the resulting probability distribution to obtain a convex lower-bound optimization. The proposed beamformer is designed to be robust to imperfect channel state information (CSI). Simulation results verify that the proposed approach facilitates the coexistence between radar and communication links, and illustrates a scalable tradeoff between the two systems' performance.

Use of Automated Scoring Features to Generate Hypotheses Regarding Language-Based DIF

This study uses the feature sets employed by two automated scoring engines to determine if a “linguistic profile” could be formulated that would help identify items that are likely to exhibit differential item functioning (DIF) based on linguistic features. Sixteen items were administered to 1200 students where demographic information was collected on gender and socioeconomic status (SES). Textual features were extracted and analyzed using Differential Item Functioning Analysis System (DIFAS) and the Mantel-Haenszel chi-square, the Liu-Agresti cumulative common log-odds ratio, and Cox's noncentrality parameter estimator to assess the probability that the focal groups (e.g., females) differed significantly from the reference groups (e.g., males). Two of the 14 items were flagged for possible DIF on gender and four were flagged for possible DIF on SES. The responses from the focal and reference groups were then analyzed using nine machine learning algorithms to determine if there were significant differences on specific linguistic features. For gender, the procedure did not reliably identify items manifesting DIF. The results for SES were more encouraging with a reliable prediction of the presence of DIF, but not its direction.

A unified approach to estimation of noncentrality parameters, the multiple correlation coefficient, and mixture models

We consider a class of mixture models for positive continuous data and the estimation of an underlying parameter θ of the mixing distribution. With a unified approach, we obtain classes of dominating estimators under squared error loss of an unbiased estimator, which include smooth estimators. Applications include estimating noncentrality parameters of chi-square and F-distributions, as well as ρ 2/(1 − ρ 2), where ρ is amultivariate correlation coefficient in a multivariate normal set-up. Finally, the findings are extended to situations, where there exists a lower bound constraint on θ.

Statistical power of likelihood ratio and Wald tests in latent class models with covariates

This paper discusses power and sample-size computation for likelihood ratio and Wald testing of the significance of covariate effects in latent class models. For both tests, asymptotic distributions can be used; that is, the test statistic can be assumed to follow a central Chi-square under the null hypothesis and a non-central Chi-square under the alternative hypothesis. Power or sample-size computation using these asymptotic distributions requires specification of the non-centrality parameter, which in practice is rarely known. We show how to calculate this non-centrality parameter using a large simulated data set from the model under the alternative hypothesis. A simulation study is conducted evaluating the adequacy of the proposed power analysis methods, determining the key study design factor affecting the power level, and comparing the performance of the likelihood ratio and Wald test. The proposed power analysis methods turn out to perform very well for a broad range of conditions. Moreover, apart from effect size and sample size, an important factor affecting the power is the class separation, implying that when class separation is low, rather large sample sizes are needed to achieve a reasonable power level.

Power comparison of Cochran-Armitage trend test against allelic and genotypic tests in large-scale case-control genetic association studies.

The Cochran-Armitage trend test (CA) has become a standard procedure for association testing in large-scale genome-wide association studies (GWAS). However, when the disease model is unknown, there is no consensus on the most powerful test to be used between CA, allelic, and genotypic tests. In this article, we tackle the question of whether CA is best suited to single-locus scanning in GWAS and propose a power comparison of CA against allelic and genotypic tests. Our approach relies on the evaluation of the Taylor decompositions of non-centrality parameters, thus allowing an analytical comparison of the power functions of the tests. Compared to simulation-based comparison, our approach offers the advantage of simultaneously accounting for the multidimensionality of the set of features involved in power functions. Although power for CA depends on the sample size, the case-to-control ratio and the minor allelic frequency (MAF), our results first show that it is largely influenced by the mode of inheritance and a deviation from Hardy-Weinberg Equilibrium (HWE). Furthermore, when compared to other tests, CA is shown to be the most powerful test under a multiplicative disease model or when the single-nucleotide polymorphism largely deviates from HWE. In all other situations, CA lacks in power and differences can be substantial, especially for the recessive mode of inheritance. Finally, our results are illustrated by the comparison of the performances of the statistics in two genome scans.

Computing the noncentral-F distribution and the power of the F-test with guaranteed accuracy

The computations involving the noncentral-F distribution are notoriously difficult to implement properly in floating-point arithmetic: Catastrophic loss of precision, floating-point underflow and overflow, drastically increasing computation time and program hang-ups, and instability due to numerical cancellation have all been reported. It is therefore recommended that existing statistical packages are cross-checked, and the present paper proposes a numerical algorithm precisely for this purpose. To the best of our knowledge, the proposed method is the first method that can compute the noncentrality parameter of the noncentral-F distribution with guaranteed accuracy over a wide parameter range that spans the range relevant for practical applications. Although the proposed method is limited to cases where the the degree of freedom of the denominator of the F test statistic is even, it does not affect its usefulness significantly: All of those algorithmic failures and inaccuracies that we can still reproduce today could have been prevented by simply cross-checking against the proposed method. Two numerical examples are presented where the intermediate computations went wrong silently, but the final result of the computations seemed nevertheless plausible, and eventually erroneous results were published. Cross-checking against the proposed method would have caught the numerical errors in both cases. The source code of the algorithm is available on GitHub, together with self-contained command-line executables. These executables can read the data to be cross-checked from plain text files, making it easy to cross-check any statistical software in an automated fashion.

Indicators of Evidence for Bioequivalence

Some equivalence tests are based on two one-sided tests, where in many applications the test statistics are approximately normal. We define and find evidence for equivalence in Z-tests and then one- and two-sample binomial tests as well as for t-tests. Multivariate equivalence tests are typically based on statistics with non-central chi-squared or non-central F distributions in which the non-centrality parameter λ is a measure of heterogeneity of several groups. Classical tests of the null λ ≥ λ 0 versus the equivalence alternative λ < λ 0 are available, but simple formulae for power functions are not. In these tests, the equivalence limit λ 0 is typically chosen by context. We provide extensions of classical variance stabilizing transformations for the non-central chi-squared and F distributions that are easy to implement and which lead to indicators of evidence for equivalence. Approximate power functions are also obtained via simple expressions for the expected evidence in these equivalence tests.

Bootstrapping Confidence Intervals for Fit Indexes in Structural Equation Modeling

Bootstrapping approximate fit indexes in structural equation modeling (SEM) is of great importance because most fit indexes do not have tractable analytic distributions. Model-based bootstrap, which has been proposed to obtain the distribution of the model chi-square statistic under the null hypothesis (Bollen & Stine, 1992), is not theoretically appropriate for obtaining confidence intervals (CIs) for fit indexes because it assumes the null is exactly true. On the other hand, naive bootstrap is not expected to work well for those fit indexes that are based on the chi-square statistic, such as the root mean square error of approximation (RMSEA) and the comparative fit index (CFI), because sample noncentrality is a biased estimate of the population noncentrality. In this article we argue that a recently proposed bootstrap approach due to Yuan, Hayashi, and Yanagihara (YHY; 2007) is ideal for bootstrapping fit indexes that are based on the chi-square. This method transforms the data so that the “parent” population has the population noncentrality parameter equal to the estimated noncentrality in the original sample. We conducted a simulation study to evaluate the performance of the YHY bootstrap and the naive bootstrap for 4 indexes: RMSEA, CFI, goodness-of-fit index (GFI), and standardized root mean square residual (SRMR). We found that for RMSEA and CFI, the CIs under the YHY bootstrap had relatively good coverage rates for all conditions, whereas the CIs under the naive bootstrap had very low coverage rates when the fitted model had large degrees of freedom. However, for GFI and SRMR, the CIs under both bootstrap methods had poor coverage rates in most conditions.

Accelerating Monte Carlo power studies through parametric power estimation.

Estimating the power for a non-linear mixed-effects model-based analysis is challenging due to the lack of a closed form analytic expression. Often, computationally intensive Monte Carlo studies need to be employed to evaluate the power of a planned experiment. This is especially time consuming if full power versus sample size curves are to be obtained. A novel parametric power estimation (PPE) algorithm utilizing the theoretical distribution of the alternative hypothesis is presented in this work. The PPE algorithm estimates the unknown non-centrality parameter in the theoretical distribution from a limited number of Monte Carlo simulation and estimations. The estimated parameter linearly scales with study size allowing a quick generation of the full power versus study size curve. A comparison of the PPE with the classical, purely Monte Carlo-based power estimation (MCPE) algorithm for five diverse pharmacometric models showed an excellent agreement between both algorithms, with a low bias of less than 1.2 % and higher precision for the PPE. The power extrapolated from a specific study size was in a very good agreement with power curves obtained with the MCPE algorithm. PPE represents a promising approach to accelerate the power calculation for non-linear mixed effect models.