Large Sample Variance Research Articles

Review of sample size determination methods for the intraclass correlation coefficient in the one-way analysis of variance model.

Reliability of measurement instruments providing quantitative outcomes is usually assessed by an intraclass correlation coefficient. When participants are repeatedly measured by a single rater or device, or, are each rated by a different group of raters, the intraclass correlation coefficient is based on a one-way analysis of variance model. When planning a reliability study, it is essential to determine the number of participants and measurements per participant (i.e. number of raters or number of repeated measurements). Three different sample size determination approaches under the one-way analysis of variance model were identified in the literature, all based on a confidence interval for the intraclass correlation coefficient. Although eight different confidence interval methods can be identified, Wald confidence interval with Fisher's large sample variance approximation remains most commonly used despite its well-known poor statistical properties. Therefore, a first objective of this work is comparing the statistical properties of all identified confidence interval methods-including those overlooked in previous studies. A second objective is developing a general procedure to determine the sample size using all approaches since a closed-form formula is not always available. This procedure is implemented in an R Shiny app. Finally, we provide advice for choosing an appropriate sample size determination method when planning a reliability study.

Evolutionary Insights from a Large-Scale Survey of Population-Genomic Variation.

The field of genomics has ushered in new methods for studying molecular-genetic variation in natural populations. However, most population-genomic studies still rely on small sample sizes (typically, <100 individuals) from single time points, leaving considerable uncertainties with respect to the behavior of relatively young (and rare) alleles and, owing to the large sampling variance of measures of variation, to the specific gene targets of unusually strong selection. Genomic sequences of ∼1,700 haplotypes distributed over a 10-year period from a natural population of the microcrustacean Daphnia pulex reveal evolutionary-genomic features at a refined scale, including previously hidden information on the behavior of rare alleles predicted by recent theory. Background selection, resulting from the recurrent introduction of deleterious alleles, appears to strongly influence the dynamics of neutral alleles, inducing indirect negative selection on rare variants and positive selection on common variants. Temporally fluctuating selection increases the persistence of nonsynonymous alleles with intermediate frequencies, while reducing standing levels of variation at linked silent sites. Combined with the results from an equally large metapopulation survey of the study species, classes of genes that are under strong positive selection can now be confidently identified in this key model organism. Most notable among rapidly evolving Daphnia genes are those associated with ribosomes, mitochondrial functions, sensory systems, and lifespan determination.

Sample Variance in Cosmological Observations with a Narrow Field of View

Surveys with a narrow field of view can play an important role in probing cosmology, but inferences from these surveys suffer from large sample variance, arising from random fluctuations around the cosmic mean. The standard method for computing the sample variance is based on two key approximations: treating perturbations linearly and the survey geometry as a box. We demonstrate that it can lead to a significant underestimate of the sample variance in narrow surveys. We present a new method for accurately computing the sample variance and apply our method to the recent observations of the warm–hot intergalactic medium (WHIM) based on spectroscopic measurements of blazars. We find that the sample variances in these surveys are significantly larger than the quoted measurement errors; for example, the cosmic mean baryon density contained in the WHIM could be lower by 54% at 1σ fluctuation than estimated in one observation. Accurately quantifying the sample variance is essential in deriving correct interpretations of the measurements in surveys with a small field of view.

Estimating the Spectral Density at Frequencies Near Zero

Estimating the spectral density function f(w) for some w ∈ [ − π , π ] has been traditionally performed by kernel smoothing the periodogram and related techniques. Kernel smoothing is tantamount to local averaging, that is, approximating f(w) by a constant over a window of small width. Although f(w) is uniformly continuous and periodic with period 2 π , in this article we recognize the fact that w = 0 effectively acts as a boundary point in the underlying kernel smoothing problem, and the same is true for w = ± π . It is well-known that local averaging may be suboptimal in kernel regression at (or near) a boundary point. As an alternative, we propose a local polynomial regression of the periodogram or log-periodogram when w is at (or near) the points 0 or ± π . The case w = 0 is of particular importance since f(0) is the large-sample variance of the sample mean; hence, estimating f(0) is crucial in order to conduct any sort of inference on the mean. Supplementary materials for this article are available online.

Inferential procedures based on the weighted Pearson correlation coefficient test statistic

In this note, we evaluated the type I error control of the commonly used t-test found in most statistical software packages for testing the hypothesis on H 0 : ρ = 0 vs. 0 $ ]]> H 1 : ρ > 0 based on the sample weighted Pearson correlation coefficient. We found the type I error rate is severely inflated in general cases, even under bivariate normality. To address this issue, we derived the large sample variance of the weighted Pearson correlation. Based on this result, we proposed an asymptotic test and a set of studentized permutation tests. A comprehensive set of simulation studies with a range of sample sizes and a variety of underlying distributions were conducted. The studentized permutation test based on Fisher's Z statistic was shown to robustly control the type I error even in the small sample and non-normality settings. The method was demonstrated with an example data of country-level preterm birth rates.

Conditional information and inference in response-adaptive allocation designs.

Response-adaptive allocation designs refer to a class of designs where the probability an observation is assigned to a treatment is changed throughout an experiment based on the accrued responses. Such procedures result in random treatment sample sizes. Most of the current literature considers unconditional inference procedures in the analysis of response-adaptive allocation designs. The focus of this article is inference conditional on the observed treatment sample sizes. The inverse of information is a description of the large sample variance of the parameter estimates. A simple form for the conditional information relative to unconditional information is derived. It is found that conditional information can be greater than unconditional information. A conditional bootstrap procedure is developed that, in the majority of cases examined, resulted in narrower confidence intervals than relevant unconditional procedures.

Integration of Nonprobability and Probability Samples via Survey Weights

Probability sample encounters the problems of increasing cost and nonresponse. The cost has rapidly been increasing in executing a large probability sample survey, and, for some surveys, response rate can be below the 10 percent level. Therefore, statisticians seek some alternative methods. One of them is to use a large nonprobability sample (S_1 ) supplemented by a small probability sample (S_2 ). Both samples are taken from the same population and they include common covariates, and a third sample (S_3 ) is created by combining these two samples; S_1&amp;nbsp; can be biased and S_2&amp;nbsp; may have large sample variance. These two problems are reduced by survey weights and combining the two samples. Although S_2&amp;nbsp; is a small sample, it provides good properties of unbiasedness in estimation and of survey weights. With these known weights, we obtain adjusted sample weights (ASW), and create a sample model from a finite population model. We fit the sample model to obtain its parameters and generate values from the population model. Similarly, we repeat these processes for other two samples, S_1&amp;nbsp; and S_3&amp;nbsp; and for different statistical methods. We show reduced biases of the finite population means and reduced variances.as the combined sample size becomes large. We analyze sample data to show the reduction of these two errors.

Optimal design of cyclic-stress accelerated life tests for lognormal lifetime distribution under type I censoring

In this paper, optimal ALT plans are developed under the assumptions of cyclic-stress loading, Type-I censoring, lognormal lifetime distribution, log-linear relationship between the scale parameter of the lifetime distribution and stress variable, and the cumulative exposure model for the effect of changing stress levels. In particular, the common floor level of, and the proportions of units allocated to two cyclic-stress conditions are determined such that the large-sample variance of the maximum likelihood estimator of the q-th quantile of the lifetime distribution at the use condition is minimized. In addition, a sample size determination method is developed for the practical use.

On the Use of Aggregate Survey Data for Estimating Regional Major Depressive Disorder Prevalence

Major depression is a severe mental disorder that is associated with strongly increased mortality. The quantification of its prevalence on regional levels represents an important indicator for public health reporting. In addition to that, it marks a crucial basis for further explorative studies regarding environmental determinants of the condition. However, assessing the distribution of major depression in the population is challenging. The topic is highly sensitive, and national statistical institutions rarely have administrative records on this matter. Published prevalence figures as well as available auxiliary data are typically derived from survey estimates. These are often subject to high uncertainty due to large sampling variances and do not allow for sound regional analysis. We propose a new area-level Poisson mixed model that accounts for measurement errors in auxiliary data to close this gap. We derive the empirical best predictor under the model and present a parametric bootstrap estimator for the mean squared error. A method of moments algorithm for consistent model parameter estimation is developed. Simulation experiments are conducted to show the effectiveness of the approach. The methodology is applied to estimate the major depression prevalence in Germany on regional levels crossed by sex and age groups.

Estimation of non-additive genetic variance in human complex traits from a large sample of unrelated individuals

Non-additive genetic variance for complex traits is traditionally estimated from data on relatives. It is notoriously difficult to estimate without bias in non-laboratory species, including humans, because of possible confounding with environmental covariance among relatives. In principle, non-additive variance attributable to common DNA variants can be estimated from a random sample of unrelated individuals with genome-wide SNP data. Here, we jointly estimate the proportion of variance explained by additive (hSNP2), dominance (δSNP2) and additive-by-additive (ηSNP2) genetic variance in a single analysis model. We first show by simulations that our model leads to unbiased estimates and provide a new theory to predict standard errors estimated using either least-squares or maximum likelihood. We then apply the model to 70 complex traits using 254,679 unrelated individuals from the UK Biobank and 1.1M genotyped and imputed SNPs. We found strong evidence for additive variance (average across traits h¯SNP2=0.208). In contrast, the average estimate of δ¯SNP2 across traits was 0.001, implying negligible dominance variance at causal variants tagged by common SNPs. The average epistatic variance η¯SNP2 across the traits was 0.055, not significantly different from zero because of the large sampling variance. Our results provide new evidence that genetic variance for complex traits is predominantly additive and that sample sizes of many millions of unrelated individuals are needed to estimate epistatic variance with sufficient precision.

Large-Sample Variance of Fleiss Generalized Kappa.

Cohen's kappa coefficient was originally proposed for two raters only, and it later extended to an arbitrarily large number of raters to become what is known as Fleiss' generalized kappa. Fleiss' generalized kappa and its large-sample variance are still widely used by researchers and were implemented in several software packages, including, among others, SPSS and the R package "rel." The purpose of this article is to show that the large-sample variance of Fleiss' generalized kappa is systematically being misused, is invalid as a precision measure for kappa, and cannot be used for constructing confidence intervals. A general-purpose variance expression is proposed, which can be used in any statistical inference procedure. A Monte-Carlo experiment is presented, showing the validity of the new variance estimation procedure.

Stable sparse subspace embedding for dimensionality reduction

Sparse random projection (RP) is a popular tool for dimensionality reduction that shows promising performance with low computational complexity. However, in the existing sparse RP matrices, the positions of non-zero entries are usually randomly selected. Although they adopt uniform sampling with replacement, due to large sampling variance, the number of non-zeros is uneven among rows of the projection matrix which is generated in one trial, and more data information may be lost after dimension reduction. To break this bottleneck, based on random sampling without replacement in statistics, this paper builds a stable sparse subspace embedded matrix (S-SSE), in which non-zeros are uniformly distributed. It is proved that the S-SSE is stabler than the existing matrix, and it can maintain Euclidean distance between points well after dimension reduction. Our empirical studies corroborate our theoretical findings and demonstrate that our approach can indeed achieve satisfactory performance.

Maximum Likelihood and Bayesian Estimation of Repeatability Index: Application of Estimating Ratio of Variance Components

An index of repeatability is constructed to evaluate the relative magnitude of measurement error. This index is constructed as a ratio of two variance components. Estimation of the index is derived under the one-way random effects model. We compare the well-known maximum likelihood estimator to the Bayesian estimation procedure using non-informative prior. Large sample variance of the of the maximum likelihood estimator are obtained using the inverse of Fisher’s information matrix and the delta method. Inference procedure using the. We also construct a test statistic on the equality of two repeatability indices using the Monte Carlo integration and sampling Importance re-sampling method. We illustrate the methodologies on the estimation of the index of repeatability of Gamma-glutamyl-transferase, an enzyme found in many organs all over the human body, with the highest concentrations found in the liver. This enzyme’s level is raised in the blood in most diseases that cause damage to the liver or bile ducts and is considered an essential serum marker for alcohol-related liver disease.

Accurate computation of single and product moments of order statistics under progressive censoring

ABSTRACTAn accurate procedure is proposed to calculate approximate moments of progressive order statistics in the context of statistical inference for lifetime models. The study analyses the performance of power series expansion to approximate the moments for location and scale distributions with high precision and smaller deviations with respect to the exact values. A comparative analysis between exact and approximate methods is shown using some tables and figures. The different approximations are applied in two situations. First, we consider the problem of computing the large sample variance–covariance matrix of maximum likelihood estimators. We also use the approximations to obtain progressively censored sampling plans for log-normal distributed data. These problems illustrate that the presented procedure is highly useful to compute the moments with precision for numerous censoring patterns and, in many cases, is the only valid method because the exact calculation may not be applicable.

Coronagraphic Observations of Si x λ14301 and Fe xiii λ10747 Linearly Polarized Spectra Using the SOLARC Telescope

Abstract The forbidden Si x emission line at 14301 Å has been identified as a potentially valuable polarized diagnostic for solar coronal magnetic fields; however, the only polarized Si x measurements achieved to date have been during eclipses and at comparatively low spatial and spectral resolution. Here we report spectropolarimetric observations of both the Si x 14301 Å and more well-established Fe xiii 10747 Å coronal lines acquired with the 0.45 m aperture SOLARC coronagraph atop Haleakalā. Using its fiber-based integral field spectropolarimeter, we derive observations sampled at radial intervals of 0.05 (i.e., ∼50″) with a spectral resolving power of ≈36,000. Results for both lines, which represent averages over different active and nonactive regions of the corona, indicate a relatively flat radial variation for the line widths and line centers and a factor of ≈2–3 decrease in polarized brightness between 1.05 and 1.45 . Averaging over all the measurements the mean and standard deviations of line properties for Si x 14301 Å and Fe xiii 10747 Å are, respectively, FWHM of 3.0 ± 0.4 Å and 1.6 ± 0.1 Å, line-integrated polarized brightness of 0.07 ± 0.03 and 0.3 ± 0.3 erg s−2 cm−2 sr−1, where the uncertainty quoted reflects a large sample variance, and line center wavelengths 14300.7 ± 0.2 Å and 10746.3 ± 0.1 Å. The polarized brightness for both lines may be underestimated by up to a factor of 5 due to limitations in the photometric calibration. When accounting for this uncertainty we find consistency between our observations and previous measurements of the two lines as well as theoretical calculations and affirm the potential of the Si x line as a polarized diagnostic of the solar corona.

Hubert's multi-rater kappa revisited.

There is a frequent need to measure the degree of agreement among R observers who independently classify n subjects within K nominal or ordinal categories. The most popular methods are usually kappa-type measurements. When R=2, Cohen's kappa coefficient (weighted or not) is well known. When defined in the ordinal case while assuming quadratic weights, Cohen's kappa has the advantage of coinciding with the intraclass and concordance correlation coefficients. When R>2, there are more discrepancies because the definition of the kappa coefficient depends on how the phrase 'an agreement has occurred' is interpreted. In this paper, Hubert's interpretation, that 'an agreement occurs if and only if all raters agree on the categorization of an object', is used, which leads to Hubert's (nominal) and Schuster and Smith's (ordinal) kappa coefficients. Formulae for the large-sample variances for the estimators of all these coefficients are given, allowing the latter to illustrate the different ways of carrying out inference and, with the use of simulation, to select the optimal procedure. In addition, it is shown that Schuster and Smith's kappa coefficient coincides with the intraclass and concordance correlation coefficients if the first coefficient is also defined assuming quadratic weights.

Strategic allocation of test units in an accelerated degradation test plan

Degradation is often defined in terms of the change of a key performance characteristic over time. When the degradation is slow, accelerated degradation tests (ADTs) that apply harsh test conditions are often used to obtain reliability information in a timely manner. It is common to see that the initial performance of the test units varies and it is strongly correlated with the degradation rate. Motivated by a real application in the semiconductor sensor industry, this study advocates an allocation strategy in ADT planning by capitalizing on the correlation information. In the proposed strategy, the initial degradation levels of the test units are measured and the measurements are ranked. The ranking information is used to allocate the test units to different factor levels of the accelerating variable. More specifically, we may prefer to allocate units with lower degradation rates to a higher factor level in order to hasten the degradation process. The allocation strategy is first demonstrated using a cumulative-exposure degradation model. Likelihood inference for the model is developed. The optimum test plan is obtained by minimizing the large sample variance of a lifetime quantile at nominal use conditions. Various compromise plans are discussed. A comparison of the results with those from traditional ADTs with random allocation reveals the value of the proposed allocation rule. To demonstrate the broad applicability, we further apply the allocation strategy to two more degradation models which are variants of the cumulative-exposure model.

On the Index of Repeatability: Estimation and Sample Size Requirements

Background: Repeatability is a statement on the magnitude of measurement error. When biomarkers are used for disease diagnoses, they should be measured accurately. Objectives: We derive an index of repeatability based on the ratio of two variance components. Estimation of the index is derived from the one-way Analysis of Variance table based on the one-way random effects model. We estimate the large sample variance of the estimator and assess its adequacy using bootstrap methods. An important requirement for valid estimation of repeatability is the availability of multiple observations on each subject taken by the same rater and under the same conditions. Methods: We use the delta method to derive the large sample variance of the estimate of repeatability index. The question related to the number of required repeats per subjects is answered by two methods. In first methods we estimate the number of repeats that minimizes the variance of the estimated repeatability index, and the second determine the number of repeats needed under cost-constraints. Results and Novel Contribution: The situation when the measurements do not follow Gaussian distribution will be dealt with. It is shown that the required sample size is quite sensitive to the relative cost. We illustrate the methodologies on the Serum Alanine-aminotransferase (ALT) available from hospital registry data for samples of males and females. Repeatability is higher among females in comparison to males.

Tobit regression for modeling mean survival time using data subject to multiple sources of censoring.

Mean survival time is often of inherent interest in medical and epidemiologic studies. In the presence of censoring and when covariate effects are of interest, Cox regression is the strong default, but mostly due to convenience and familiarity. When survival times are uncensored, covariate effects can be estimated as differences in mean survival through linear regression. Tobit regression can validly be performed through maximum likelihood when the censoring times are fixed (ie, known for each subject, even in cases where the outcome is observed). However, Tobit regression is generally inapplicable when the response is subject to random right censoring. We propose Tobit regression methods based on weighted maximum likelihood which are applicable to survival times subject to both fixed and random censoring times. Under the proposed approach, known right censoring is handled naturally through the Tobit model, with inverse probability of censoring weighting used to overcome random censoring. Essentially, the re-weighting data are intended to represent those that would have been observed in the absence of random censoring. We develop methods for estimating the Tobit regression parameter, then the population mean survival time. A closed form large-sample variance estimator is proposed for the regression parameter estimator, with a semiparametric bootstrap standard error estimator derived for the population mean. The proposed methods are easily implementable using standard software. Finite-sample properties are assessed through simulation. The methods are applied to a large cohort of patients wait-listed for kidney transplantation.

A Generalized Smoother for Linear Ordinary Differential Equations

ABSTRACTOrdinary differential equations (ODEs) are equalities involving a function and its derivatives that define the evolution of the function over a prespecified domain. The applications of ODEs range from simulation and prediction to control and diagnosis in diverse fields such as engineering, physics, medicine, and finance. Parameter estimation is often required to calibrate these theoretical models to data. While there are many methods for estimating ODE parameters from partially observed data, they are invariably subject to several problems including high computational cost, complex estimation procedures, biased estimates, and large sampling variance. We propose a method that overcomes these issues and produces estimates of the ODE parameters that have less bias, a smaller sampling variance, and a 10-fold improvement in computational efficiency. The package GenPen containing the Matlab code to perform the methods described in this article is available online.