Assumption Of Normal Distribution Research Articles

Combining Biomarkers to Improve Diagnostic Accuracy in Detecting Diseases With Group-Tested Data.

We consider the problem of combining multiple biomarkers to improve the diagnostic accuracy of detecting a disease when only group-tested data on the disease status are available. There are several challenges in addressing this problem, including unavailable individual disease statuses, differential misclassification depending on group size and number of diseased individuals in the group, and extensive computation due to a large number of possible combinations of multiple biomarkers. To tackle these issues, we propose a pairwise model fitting approach to estimating the distribution of the optimal linear combination of biomarkers and its diagnostic accuracy under the assumption of a multivariate normal distribution. The approach is evaluated in simulation studies and applied to data on chlamydia detection and COVID-19 diagnosis.

493 Predicting pellet quality using multiple linear regression with Principal Component Analysis (PCA)

Abstract Pellet quality is a crucial key performance indicator (KPI) for commercial feed manufacturing, which influences both the efficiency of the feed mill and downstream performance of animals fed these diets. However, due to the complexity of feed manufacturing and the large number of factors involved in the manufacturing process, controlling pellet quality is an ongoing challenge for the feed industry. Previous studies have mainly explored the impact of a few factors on pellet quality under experimental settings, and empirical equations have been seldomly developed to reflect the relationship between the factors and pellet quality under the commercial feed mill settings. This study aimed to establish a relationship between pellet quality and factors collected under the settings of a commercial feed mill. The data were collected from Trouw Nutrition Canada’s feed mill located in St. Marys, Ontario (Plant 2), between December 15, 2021, and December 6, 2022. During this period, 2,691 observations were collected, with each observation representing an individual batch of pelleted feed. A total of 75 factors were recorded, including 4 factors associated with the general information of each batch, 10 manufacturing parameters, 41 feed ingredients, 8 factors regarding the nutrient composition of each diet, and 12 environmental factors. Pellet Durability Index (PDI), which was the response variable, was determined for each batch using the Holmen method. The data were randomly split into an 80% subset for training and a 20% subset for testing. The training subset was used to construct the model via a 5-fold cross-validation, while the testing subset was withheld as an independent dataset to evaluate the generalization performance of the model. The response variable (PDI) was transformed (tPDI) using the Box-Cox method to meet a normal distribution assumption. To avoid multicollinearity, Principal Component Analysis (PCA) was used to reduce the dimensionality of the numeric factors before building the multiple linear regression model. The model prediction performance was evaluated on both the training subset (using 5-fold cross-validation) and the testing subset, and the prediction performance metrics were consistent between the two subsets (Mean Absolute Error = 1.94 ± 0.102 vs. 2.02; Root Mean Square Prediction Error = 2.47 ± 0.111 vs. 2.58; Mean Square Prediction Error = 6.12 ± 0.538 vs. 6.68; Concordance Correlation Coefficient = 0.538 ± 0.0231 vs. 0.490; Pearson Correlation Coefficient = 0.606 ± 0.0247 vs. 0.553, respectively). Most feed ingredients and nutrient compositions showed either positive or negative loadings on Component 1 (17.87% of total variance), and outdoor/indoor environmental factors were positively loaded on Component 2 (14.21% of total variance). The model developed in this study could help commercial feed mills better understand how various factors impact pellet quality and optimize the manufacturing processes of pelleted feeds.

A structurally informed data assimilation approach for nonlinear partial differential equations

Ensemble-based Kalman filtering data assimilation is often used to combine available observations with numerical simulations to obtain statistically accurate and reliable state representations in dynamical systems. However, it is well known that the commonly used Gaussian distribution assumption introduces biases for state variables that admit discontinuous profiles, which are prevalent in nonlinear partial differential equations. This investigation designs a new structurally informed prior that exploits statistical information from the simulated state variables. In particular, based on the second moment information of the state variable gradient, we construct a new weighting matrix for the numerical simulation contribution in the data assimilation objective function. This replaces the typical prior covariance matrix used for this purpose. We further adapt our weighting matrix to include information in discontinuity regions via a clustering technique. Our numerical experiments demonstrate that this new approach yields more accurate estimates than those obtained using standard ensemble-based Kalman filtering on shallow water equations, even when it is enhanced with inflation and localization techniques.

UNCERTAINTY EVALUATION USING LAW OF PROPAGATION AND MONTE CARLO SIMULATION METHODS WITH THE AUTORFPOWER MEASUREMENT SOFTWARE

RF power measurement is essential in RF and microwave metrology. For reliable and accurate power measurement, automatic measurement is preferred. A software application in C#, named AutoRFPower, was developed for automatic RF power measurement and uncertainty calculations at this study. According to the GUM document, this application is enhanced for uncertainty calculations by utilizing the Law of Propagation method and the Monte Carlo Simulation method. Trial measurements were performed at different RF power levels and frequencies between 50 MHz and 18 GHz using the AutoRFPower software. Law of Propagation and Monte Carlo Simulation uncertainty calculations were carried out by AutoRFPower based on the trial measurements and by the Oracle Crystal Ball simulation application. All measurements and their uncertainty calculations were compared with each other, and this study validated the uncertainty calculation of AutoRFPower. In addition, it was observed that in the Monte Carlo Simulation, uncertainty calculation results were non-symmetrical normal distribution, contrary to the assumption of symmetrical normal distribution according to the Low of Propagation method. Moreover, it has been observed that the statistical distribution of uncertainty changes depending on the dominant component of the parameters in the model function used for the uncertainty calculation with the Monte Carlo Simulation method.

Iterated rational quadratic kernel - High-order unscented Kalman filtering algorithm for spacecraft tracking

The high-speed development of space defense technology demands a high state estimation capacity for spacecraft tracking methods. However, reentry flight is accompanied by complex flight environments, which brings to the uncertain, complex, and strongly coupled non-Gaussian detection noise. As a result, there are several intractable considerations on the problem of state estimation tasks corrupted by complex non-Gaussian outliers for non-linear dynamics systems in practical application. To address these issues, a new iterated rational quadratic (RQ) kernel high-order unscented Kalman filtering (IRQ-HUKF) algorithm via capturing the statistics to break through the limitations of the Gaussian assumption is proposed. Firstly, the characteristic analysis of the RQ kernel is investigated in detail, which is the first attempt to carry out an exploration of the heavy-tailed characteristic and the ability on capturing high-order moments of the RQ kernel. Subsequently, the RQ kernel method is first introduced into the UKF algorithm as an error optimization criterion, termed the iterated RQ kernel-UKF (RQ-UKF) algorithm by derived analytically, which not only retains the high-order moments propagation process but also enhances the approximation capacity in the non-Gaussian noise problem for its ability in capturing high-order moments and heavy-tailed characteristics. Meanwhile, to tackle the limitations of the Gaussian distribution assumption in the linearization process of the non-linear systems, the high-order Sigma Points (SP) as a subsidiary role in propagating the state high-order statistics is devised by the moments matching method to improve the RQ-UKF. Finally, to further improve the flexibility of the IRQ-HUKF algorithm in practical application, an adaptive kernel parameter is derived analytically grounded in the Kullback-Leibler divergence (KLD) method and parametric sensitivity analysis of the RQ kernel. The simulation results demonstrate that the novel IRQ-HUKF algorithm is more robust and outperforms the existing advanced UKF with respect to the kernel method in reentry vehicle tracking scenarios under various noise environments.

Performance comparison of alternative volatility models applied to economic indexes: the role of asymmetric effects

ABSTRACT This article examines the empirical relationship between economic indexes – namely the industrial production and the consumer confidence – and their volatility by using the United States data over the periods 1919–2023 and 1960–2023, respectively. To address this topic, several GARCH-family models including volatility feedback and the asymmetric effects are estimated. We find that (i) the industrial production index exhibits asymmetric effects, with significant fitting performance over the standard GARCH model; (ii) the inclusion of t -distributed innovations in the error term improves the fitting performance of the competing models against their counterparts with a normal distribution assumption; (iii) both models tested for the consumer confidence index exhibit similar performance to fit with the data; (iv) by excluding the months during and after the COVID-19 pandemics and the Russia–Ukraine conflict, it leads to the same results and conclusions for both economic indexes. Conditional heteroscedasticity, economic growth, GARCH models, asymmetric effects, volatility feedback

BIOMECHANICAL INFLUENCE OF RUNNING SHOES WITH MIDSOLE HOLLOW STRUCTURE ON LOWER LIMBS DURING RUNNING

The objective was to investigate the effects of running shoes with midsole hollow structure span and height on the biomechanics of the lower limbs during running. We collected 21 adults with running habits who wore two pairs of running shoes with different midsole hollow structures and ran at a speed of 3.3[Formula: see text]m/s on a force-measuring treadmill. The lower limb kinematics, ground reaction force (GRF) and lower limb muscle activation characteristics were simultaneously captured by a motion capture system, a 3D force treadmill, and a surface electromyography (sEMG) system. Paired t-tests were performed on data for the two shoe conditions that fit the normal distribution assumptions; otherwise, Wilcoxon signed-rank tests were used. The statistical parameter mapping (SPM) technology was used for the analysis of 1D parameters of kinematic, dynamic, and sEMG activation characteristics. The result showed that the time to the peak impact force at touchdown of Hollow shoe2 was significantly increased ([Formula: see text]), the maximum loading rate ([Formula: see text]) and average loading rate ([Formula: see text]) were significantly reduced, braking time ([Formula: see text]), push time ([Formula: see text]), contact time ([Formula: see text]) of Hollow shoe2 were significantly increased compared with Hollow shoe1. Hollow shoe2 push phase of the tibialis anterior muscle activation characteristics was significantly lower (SPM, [Formula: see text]) than Hollow shoe1. Our conclusion is that running shoes offer the solution as they have the advantage of the complex structure of the hollow midsole.

Small‐sample properties of robust willingness‐to‐pay estimators

AbstractThe small‐sample properties of robust, binary choice willingness‐to‐pay (WTP) estimators are analyzed. A Monte Carlo simulation compares the bias and mean‐squared error of marginal and expected WTP estimates from probit, zero‐centered (“robit”), logit, and a generalized Tukey lambda distribution called a “pregibit” regression under normal and nonnormal distributional assumptions. Robust binary choice estimators allow for variation in tail thickness (the robit and pregibit) or tail asymmetry (the pregibit). No previous studies have compared the performance of these WTP estimators. The findings will interest researchers who use contingent validation methods to estimate WTP for nonmarket or hypothetical goods.

Acausal relationship between hypothyroidism and rheumatoid arthritis, but not hyperthyroidism: evidence from the mendelian randomization study.

To investigate the genetic level causal association among hyperthyroidism, hypothyroidism, and rheumatoid arthritis (RA). We utilized the genome-wide association studies (GWAS) summary data for exposure (hyperthyroidism and hypothyroidism) and outcome (RA) from the IEU OpenGWAS database. We used two different sets of data (test cohort and validation cohort) for causal assessment of exposure and outcome. To establish acausal relationship between these conditions, we conducted atwo-sample Mendelian randomization (MR) analysis. Subsequently, we evaluated the MR analysis results for heterogeneity, horizontal pleiotropy, and outliers, aiming to assess the validity and reliability of the findings. Moreover, we conducted additional analyses to examine the robustness of the MR results, including a"Leave one out" analysis and the MR robust adjusted profile score (MR-RAPS) method, ensuring the robustness and adherence to normal distribution assumptions. The findings from the test cohort indicated that hyperthyroidism did not exhibit agenetic causal association with RA (P = 0.702, odds ratio [OR] 95% confidence interval [CI] = 1.021 [0.918-1.135]). Conversely, hypothyroidism displayed apositive genetic causal relationship with RA (P < 0.001, OR 95% CI = 1.239 [1.140-1.347]). The analysis results of the validation cohort are consistent with those of the test cohort. Notably, our MR analysis results demonstrated no evidence of heterogeneity, horizontal pleiotropy, or outliers. Furthermore, our MR analysis results remained unaffected by any single nucleotide polymorphism (SNP) and exhibited anormal distribution. The results of this study showed that hypothyroidism was positively correlated with RA, while hyperthyroidism was not causally correlated with RA. Hypothyroidism may as arisk factor of RA should be paid attention to in clinical work. Future studies are needed to further confirm this finding.

Application of latent variable evolution for channel reservoir characterization using generative adversarial networks and particle swarm optimization

When characterizing channel reservoirs, ensemble-based methods may compromise channel features due to the conflict between the Gaussian distribution assumption of the methods and bimodality of channel reservoirs. To overcome this challenge, one effective approach involves leveraging generative models. Specifically, a generative model trained with channel reservoir models has ability to generate bimodally distributed values from latent vectors whose elements are intentionally designed to conform a Gaussian distribution. Therefore, it resolves the conflict, allowing the ensemble-based methods to update the latent vectors.However, generating initial latent vectors randomly poses an additional challenge and can lead to a suboptimal quality of the initial ensemble. They can significantly influence the unreliability of characterizations, primarily because their impact is strongly tied to the quality of the initial models in the ensemble-based methods.To address this challenge, we propose a novel scheme for channel reservoir characterization using Latent Variable Evolution. In this study, we adopt Generative Adversarial Networks (GAN) and Particle Swarm Optimization (PSO). PSO explores the latent space of GAN trained with channel reservoirs in order to find ones that generate models resembling a true reservoir. Validation on simple and complex reservoir cases demonstrates our proposed method can efficiently provide reliable channel reservoir characterization regardless of the quality of the initial latent vectors. These results are compared to Ensemble Smoother with Multiple Data Assimilation with GAN.

Evaluation nutrients of Turkish Snack Bars Based on Labeling and Web Page Information: A Qualitative Research

Background: Snack bars, a globally popular food category, contain a diverse variety of ingredients and appeal to different dietary preferences. Even though snack alternatives are gaining prominence as viable options for promoting healthy dietary patterns, the composition of certain snacks may pose sustainability-related concerns and effects on environment. This study investigates the energy and nutrient composition of fruit-based snack bars, with a focus on sustainable practices, and the significant inclusion of dates and nuts in production. Methods: Between 10 and 25 July 2023, 49 healthy bars from 12 companies, accessible in five supermarkets, underwent a comprehensive analysis. The researchers visited selected supermarkets, identified the products on the shelves, and collected information on the contents from the labels. The label information was then cross-referenced with the brands' websites. The average values of energy, protein, total fat, saturated fat, carbohydrate, fiber, sugar, and salt contents of 100 g samples of the products were assessed. Group proportions were investigated through the Chi-square test. As comparisons involving numerical variables failed to meet the assumption of normal distribution, independent comparisons were performed using the Kruskal-Wallis test for more than two groups and Mann-Whitney U test for two groups. The significance level was set at p&lt;0.05. Results: Dried fruits, particularly date and their derivatives, are the most prominently featured ingredients (87.8%) of the nutritious snacks, followed by other dried fruits containing apple, cherry, and orange. Oily seeds are the most preferred additional ingredient (75.5%), followed by cocoa (57.1%), spices (57.1%), sweeteners, and coconut. Almonds are the most preferred variety among nuts (36.7%). Other ingredients frequently utilized in bars, such as chickpea flour and chicory root fiber. It is noteworthy that, several snack bars were including multiple ingredients simultaneously whereas others lacked certain ingredients entirely. Furthermore, combination of those ingredients were discovered to be prevalent in various snack bars. Conclusion: In conclusion, this research provides valuable insights for consumers and industry stakeholders, guiding them towards choices aligned with nutritional preferences and supporting environmental and economic sustainability.

Examining the Distributional Characteristics of Daily Returns of Nifty 50: Normality Assessment and Implications

This paper investigates the distributional characteristics of daily returns of the Nifty 50 index, a benchmark index comprising 50 large-cap stocks traded on the National Stock Exchange of India. Utilizing historical data spanning a specified time period, we conduct normality testing to assess the adequacy of the normal distribution assumption underlying many financial models. Our analysis provides insights into the departure from normality. As a result of departure from Normality, it may affect the tail behaviour, and volatility dynamics of Nifty 50 daily returns, offering implications for risk management, option pricing, portfolio management, and market efficiency. By synthesizing empirical findings with theoretical considerations, this research contributes to a deeper understanding of the statistical properties of Nifty 50 daily returns and informs practitioners in finance about the challenges associated with Modeling and analysing stock market data.

Evaluation of neutron cross sections of 48Ti based on the Unified-Monte-Carlo-B method* *Supported by the National Key Research and Development (R&D) Program (2022YFA1602403) and Continuous Support Basic Scientific Research Project BJ010261223282

A cross section evaluation of neutron induced reactions on 48Ti is undertaken using the Unified Monte Carlo-B (UMC-B) approach. The evaluation concentrates on estimating the covariance and the use of the UMC-B allows avoiding the deficiencies of linear regression brought by the traditional least squares method. Eight main neutron and charged particle emission reactions from n+48Ti in the fast neutron energy region below 20 MeV are studied in this work. The posterior probability density function (PDF) of each neutron cross section is obtained in a UMC-B Bayesian approach by convoluting the model PDFs sampled based on model parameters and the likelihood functions for the experimental data. Nineteen model parameters including level density, pair corrections, optical model and Kalbach matrix element parameter are stochastically sampled with the assumption of normal distributions to estimate the model uncertainty. The Cholesky factorization approach is applied to consider potential parameter correlations. Finally, the posterior covariance matrices are generated using the UMC-B generated weights. The new evaluated results are compared with the CENDL-3.2, ENDF/B-VIII.0, JEFF-3.3, TENDL-2021 and JENDL-5 evaluations and differences are discussed.

Travel Time Reliability Estimation in Urban Road Networks: Utilization of Statistics Distribution and Tensor Decomposition

The travel time reliability (TTR) is crucial for evaluating the reliability of road networks, but real traffic data is often incomplete and sparse. This study validates that road network TTR conforms to a normal distribution and devises a quantification approach for road network TTR. Two reliability estimation methods are tailored for two data sources: section detectors and mobile detectors. Simulation experiments have confirmed the effectiveness of these methods. The study emphasizes that the TTR estimation method using traffic section data (S-TTR), which is based on the verified normal distribution assumption, maintains average absolute errors below 10%. On the other hand, the TTR estimation method that utilizes sparse trajectory data (T-TTR), which relies on tensor decomposition, proficiently fills in all missing data with an average error of 0.0059.

Strengthening resilience in supply chains: the role of multi-layer flexibility, supply chain risks and environmental dynamism

PurposeThis study examines the relationship between multi-layer supply chain flexibility (MSCF) and Supply chain resilience (SCR). Further, it looks at the moderating effect of environmental dynamism (ED) and supply chain risks (SCRI) on the relationship between MSCF and SCR.Design/methodology/approachExecutives from the pharmaceutical, agri-food, electronics, automobile and textile industries were invited to complete a self-administered questionnaire. We received feedback from a total of 302 participants. Prior to conducting the primary analysis, we addressed the potential for nonresponse bias and verified the assumptions of homoscedasticity and normal distribution of the data. The reliability and validity of the constructs were established through confirmatory factor analysis. Structural equation modelling is employed for the purpose of conducting hypothesis testing.FindingsThe results demonstrate a notable influence of MSCF on SCR, particularly in settings characterized by high levels of ED and SCRI. The study highlights the importance of flexibility in multiple aspects of the supply chain to build resilience against a range of disruptions and uncertainties.Originality/valueThe study presents the fundamental role of Multi-Layer Flexibility in building up SCR. The results of this study reinforce the existing literature and offers empirical evidence for how ED, SCRI moderates the influence between MSCF to SCR. These results offer valuable information to both supply chain specialists and researchers for building comprehensive strategy to bring resilience in supply chains.

Symmetric Q-learning: Reducing Skewness of Bellman Error in Online Reinforcement Learning

In deep reinforcement learning, estimating the value function to evaluate the quality of states and actions is essential. The value function is often trained using the least squares method, which implicitly assumes a Gaussian error distribution. However, a recent study suggested that the error distribution for training the value function is often skewed because of the properties of the Bellman operator, and violates the implicit assumption of normal error distribution in the least squares method. To address this, we proposed a method called Symmetric Q-learning, in which the synthetic noise generated from a zero-mean distribution is added to the target values to generate a Gaussian error distribution. We evaluated the proposed method on continuous control benchmark tasks in MuJoCo. It improved the sample efficiency of a state-of-the-art reinforcement learning method by reducing the skewness of the error distribution.

Bayesian model updating of super high‐rise building for construction simulation

SummaryA finite element model was established using SAP2000 software for the C1 tower, a super high‐rise building in the second phase of the Nanjing Financial City project, and the construction process of the tower was simulated. The C1 tower adopts a frame core tube extension arm and ring truss structure system, with 87 floors above ground and five floors underground. The roof structure has an elevation of 416.6 m. Precise measurements of inter‐story compression deformation were conducted using advanced surveying equipment. Sensitivity analysis, based on the finite difference method, identified the shear wall elastic modulus, frame column elastic modulus, steel beam elastic modulus, and shear wall unit weight as four highly influential parameters. Employing the Bayesian principle, the Markov Chain Monte Carlo (MCMC) method was applied to determine the posterior density probability function of the parameters targeted for modification. Subsequently, the Metropolis–Hastings (MH) sampling algorithm was employed to refine the C1 Tower model. This refinement significantly reduced the root mean square error between the measured and simulated vertical displacements, achieving an error reduction of approximately 10% from 6.082 to around 2.160. The modified material parameters, for the most part, adhered to a normal distribution assumption and exhibited mean values in the posterior probability density functions for the elastic modulus of Q345 steel beams, C70 frame columns, and C60 shear walls. Compared to the initial finite element parameters, the variation range was approximately 13% to 17%. These results serve as a validation of the effectiveness of the proposed method.

Probabilistic modeling of double linear damage accumulation and its application in fatigue reliability analysis

AbstractIn this paper, a probabilistic double linear damage accumulation model is proposed to characterize the uncertainty of damage accumulation using the statistics of material fatigue life. The probabilistic modeling framework is performed under normal distribution assumption of fatigue life. By applying a one‐to‐one probability density function transformation method, the probability distribution of critical damage and cumulative damage can be extracted and derived. The proposed probabilistic damage accumulation model is then used for fatigue reliability analysis by building the reliability formulas with respect to the cumulative damage and the critical damage under both constant and variable amplitude loadings. It treats the mean and variance of cumulative damage as time‐dependent values that dynamically vary with increase in usage life. The reliability can be easily calculated by limited data necessary from fatigue life samples. The applicability of the proposed method is demonstrated using the experimental data of two metallic materials under constant amplitude and two‐block loadings. The results show that the model predictions are in good agreement with the experimental observations under constant amplitude loading and L‐H loading sequence. In addition, the proposed method provides better predictive ability than the Miner rule under constant amplitude loading and H‐L loading sequence. Moreover, the predicted reliability plots can typically present a two‐stage fatigue characteristic during crack initiation and crack propagation periods.

대학전공선택검사의 성차에 의한 차별기능문항 탐색

Objectives Paying attention to the phenomenon of a gender gap in the choice of a major in the engineering de-partment, this study set out to examine whether there was differential item functioning by gender and what caused it by analyzing the items related to engineering in the Korean College Major Interest Inventory, 2nd Edition(KCMII-2).&#x0D; Methods For this purpose, the study used the data of 9,069 respondents who answered the activity preference items and the data of 9,040 respondents who answered the vocational preference items in the engineering sec-tion of the inventory in its analysis before checking if the data met the normal distribution and unidimensionality assumption requirements. Differential item functioning was extracted with the ‘lordif’ package of R and the PARSCALE program. The items whose differential function was found both in the programs were chosen as the final differential item functioning.&#x0D; Results The results of testing the assumption of unidimensionality suggested the possibility of other dimensions of latent trait beyond activity and career preferences. Subsequent analysis conducted through differential item functioning analysis revealed that nine out of the 30 activity preference items and seven out of the 30 vocational preference items were extracted as differential item functioning. All these items except for one activity preference item were favorable to female students. The activity preference items whose differential function included majors in life science and biotechnology, computer software, environmental studies and energy, and chemistry and chemical engineering. In terms of career preference, the differential functioning items encompassed majors in life science and biotechnology, chemistry and chemical engineering, and environmental studies and energy.&#x0D; Conclusions The study then inferred the causes of the differential item function extracted based on the findings and discussed the future direction of career education to help high school students overcome the gender gap in the choice of an engineering career and make a reasonable and sustainable career and occupation choice.

Statistical Methods for Selective Biomarker Testing

AbstractBiomarkers are critically important tools in modern clinical diagnosis, prognosis, and classification/prediction. However, there are fiscal and analytical barriers to biomarker research. Selective Genotyping is an approach to increasing study power and efficiency where individuals with the most extreme phenotype (response) are chosen for genotyping (exposure) in order to maximize the information in the sample. In this article, we describe an analogous procedure in the biomarker testing landscape where both response and biomarker (exposure) are continuous. We propose an intuitive reverse-regression least squares estimator for the parameters relating biomarker value to response. An expression for robust standard error and corresponding confidence interval are derived. A simulation study is used to demonstrate that this method is unbiased and efficient relative to estimates from random sampling when the joint normal distribution assumption is met, and to compare the estimator to an alternative under a related sampling design. We illustrate application of proposed methods on data from a chronic pain clinical trial.