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Full-Text Articles in Statistics and Probability

Ground Weather Radar Signal Characterization Through Application Of Convolutional Neural Networks, Stephen M. Lee Mar 2020

Ground Weather Radar Signal Characterization Through Application Of Convolutional Neural Networks, Stephen M. Lee

Theses and Dissertations

The 45th Weather Squadron supports the space launch efforts out of the Kennedy Space Center and Cape Canaveral Air Force Station for the Department of Defense, NASA, and commercial customers through weather assessments. Their assessment of the Lightning Launch Commit Criteria (LLCC) for avoidance of natural and rocket triggered lightning to launch vehicles is critical in approving space shuttle and rocket launches. The LLCC includes standards for cloud formations, which requires proper cloud identification and characterization methods. Accurate reflectivity measurements for ground weather radar are important to meet the LLCC for rocket triggered lightning. Current linear interpolation methods for ground …


A New Liu Type Of Estimator For The Restricted Sur Estimator, Kristofer Månsson, B. M. Golam Kibria, Ghazi Shukur Mar 2020

A New Liu Type Of Estimator For The Restricted Sur Estimator, Kristofer Månsson, B. M. Golam Kibria, Ghazi Shukur

Journal of Modern Applied Statistical Methods

A new Liu type of estimator for the seemingly unrelated regression (SUR) models is proposed that may be used when estimating the parameters vector in the presence of multicollinearity if the it is suspected to belong to a linear subspace. The dispersion matrices and the mean squared error (MSE) are derived. The new estimator may have a lower MSE than the traditional estimators. It was shown using simulation techniques the new shrinkage estimator outperforms the commonly used estimators in the presence of multicollinearity.


Sampling The Porridge: A Comparison Of Ordered Variable Regression With F And R2 And Multiple Linear Regression With Corrected F And R2 In The Presence Of Multicollinearity, Grayson L. Baird, Stephen L. Bieber Mar 2020

Sampling The Porridge: A Comparison Of Ordered Variable Regression With F And R2 And Multiple Linear Regression With Corrected F And R2 In The Presence Of Multicollinearity, Grayson L. Baird, Stephen L. Bieber

Journal of Modern Applied Statistical Methods

Differences between the multiple linear regression model with Corrected R2 and Corrected F and the ordered variable regression model with R2 and F when intercorrelation is present are illustrated with simulated and real-world data.


A Novel Penalized Log-Likelihood Function For Class Imbalance Problem, Lili Zhang Mar 2020

A Novel Penalized Log-Likelihood Function For Class Imbalance Problem, Lili Zhang

Doctor of Data Science and Analytics Dissertations

The log-likelihood function is the optimization objective in the maximum likelihood method for estimating models (e.g., logistic regression, neural network). However, its formulation is based on assumptions that the target classes are equally distributed and the overall accuracy is maximized, which do not apply to class imbalance problems (e.g., fraud detection, rare disease diagnoses, customer conversion prediction, cybersecurity, predictive maintenance). When trained on imbalanced data, the resulting models tend to be biased towards the majority class (i.e. non-event), which can bring great loss in practice. One strategy for mitigating such bias is to penalize the misclassification costs of observations differently …


Using Medical Claims Database To Develop A Population Disease Progression Model For Leuprorelin-Treated Subjects With Hormone-Sensitive Prostate Cancer, Yixuan Zou, Fei Tang, Jeffery C. Talbert, Chee M. Ng Mar 2020

Using Medical Claims Database To Develop A Population Disease Progression Model For Leuprorelin-Treated Subjects With Hormone-Sensitive Prostate Cancer, Yixuan Zou, Fei Tang, Jeffery C. Talbert, Chee M. Ng

Pharmacy Practice and Science Faculty Publications

Androgen deprivation therapy (ADT) is a widely used treatment for patients with hormone-sensitive prostate cancer (PCa). However, duration of treatment response varies, and most patients eventually experience disease progression despite treatment. Leuprorelin is a luteinizing hormone-releasing hormone (LHRH) agonist, a commonly used form of ADT. Prostate-specific antigen (PSA) is a biomarker for monitoring disease progression and predicting treatment response and survival in PCa. However, time-dependent profile of tumor regression and growth in patients with hormone-sensitive PCa on ADT has never been fully characterized. In this analysis, nationwide medical claims database provided by Humana from 2007 to 2011 was used to …


Predictive Validity Of Standards-Based And Curriculum-Embedded Assessments For Predicting Readiness At Kindergarten Entry, Elizabeth Ashton Decamilla Mar 2020

Predictive Validity Of Standards-Based And Curriculum-Embedded Assessments For Predicting Readiness At Kindergarten Entry, Elizabeth Ashton Decamilla

USF Tampa Graduate Theses and Dissertations

As with traditional K-12 educational settings, early childhood assessments have been a primary source of information determining whether early educational experiences have promoted children’s readiness to start school in kindergarten. The level of use of Kindergarten Entry Assessments (KEAs) has become more wide-spread to establish levels of school readiness at kindergarten entry.

This quantitative, correlational study of children in schools that have blended Head Start/Voluntary Prekindergarten funded programs examined the predictive relationships between the independent variables (i.e., VPK Assessments and Teaching Strategies GOLD) and the dependent variable of kindergarten readiness, as measured by the Work Sampling System™ (WSS). Additionally, the …


The Analytics Managers Ultimate Guide For Working With Universities, Robert J. Mcgrath Mar 2020

The Analytics Managers Ultimate Guide For Working With Universities, Robert J. Mcgrath

Faculty Publications

The challenges organizations are having related to finding (and retaining) deep analytical talent did not materialize out of thin air…or overnight. Analytics and Data science – and the role of the analytics professional – has evolved over the last several decades and has been fueled by our ability to capture and process increasingly larger and more complex variations of data and our desire to gain increasingly granular insights to fuel innovation and creativity. While many organizations recognize that a partnership with a university can be a resource to many of these challenges, the best way to start a conversation with …


A Permutation Test And Spatial Cross-Validation Approach To Assess Models Of Interspecific Competition Between Trees, David Allen, Albert Y. Kim Mar 2020

A Permutation Test And Spatial Cross-Validation Approach To Assess Models Of Interspecific Competition Between Trees, David Allen, Albert Y. Kim

Statistical and Data Sciences: Faculty Publications

Measuring species-specific competitive interactions is key to understanding plant communities. Repeat censused large forest dynamics plots offer an ideal setting to measure these interactions by estimating the species-specific competitive effect on neighboring tree growth. Estimating these interaction values can be difficult, however, because the number of them grows with the square of the number of species. Furthermore, confidence in the estimates can be overestimated if any spatial structure of model errors is not considered. Here we measured these interactions in a forest dynamics plot in a transitional oak-hickory forest. We analytically fit Bayesian linear regression models of annual tree radial …


A Study Verifying The Dimensioning Of A Multivariate Dichotomized Sample In Exploratory Factor Analysis, Rosilei S. Novak, Jair M. Marques Mar 2020

A Study Verifying The Dimensioning Of A Multivariate Dichotomized Sample In Exploratory Factor Analysis, Rosilei S. Novak, Jair M. Marques

Journal of Modern Applied Statistical Methods

The sample size dichotomized was related to the measure of sampling adequacy, considering the explanations provided by factors and commonalities. Monte Carlo simulation generated multivariate normal samples and varying the number of observations, the factor analysis was applied in each sample dichotomized. Results were modeled by polynomial regression based on the sample sizing.


Parametric And Non-Parametric Tests For The Comparison Of Two Samples Which Both Include Paired And Unpaired Observations, Ben Derrick, Paul White, Deirdre Toher Mar 2020

Parametric And Non-Parametric Tests For The Comparison Of Two Samples Which Both Include Paired And Unpaired Observations, Ben Derrick, Paul White, Deirdre Toher

Journal of Modern Applied Statistical Methods

Samples that include both independent and paired observations cause a dilemma for researchers that covers the full breadth of empirical research. Parametric approaches for the comparison of two samples using all available observations are considered, under normality and non-normality. These approaches are compared to naive and newly proposed non-parametric alternatives.


Identification And Description Of Potentially Influential Social Network Members Using The Strategic Player Approach, Miles Q. Ott, Sara G. Balestrieri, Graham Diguiseppi, Melissa A. Clark, Michael Bernstein, Sarah Helseth, Nancy P. Barnett Mar 2020

Identification And Description Of Potentially Influential Social Network Members Using The Strategic Player Approach, Miles Q. Ott, Sara G. Balestrieri, Graham Diguiseppi, Melissa A. Clark, Michael Bernstein, Sarah Helseth, Nancy P. Barnett

Statistical and Data Sciences: Faculty Publications

Background: Diffusion of innovations theory posits that ideas and behaviors can be spread through social network ties. In intervention work, intervening upon certain network members may lead to intervention effects “diffusing” into the network to affect the behavior of network members who did not receive the intervention. The strategic players (SP) method, an extension of Borgatti’s Key Players approach, is used to balance the (sometimes) opposing goals of spreading the intervention to as many members of the target group as possible, while preventing the spread of the intervention to others. Objectives: We sought to test whether members of the SP …


Describing Quasi-Graphic Matroids, Nathan Bowler, Daryl Funk, Dan Slilaty Mar 2020

Describing Quasi-Graphic Matroids, Nathan Bowler, Daryl Funk, Dan Slilaty

Mathematics and Statistics Faculty Publications

The class of quasi-graphic matroids recently introduced by Geelen, Gerards, and Whittle generalises each of the classes of frame matroids and liftedgraphic matroids introduced earlier by Zaslavsky. For each biased graph (G, B) Zaslavsky defined a unique lift matroid L(G, B) and a unique frame matroid F(G, B), each on ground set E(G). We show that in general there may be many quasi-graphic matroids on E(G) and describe them all: for each graph G and partition (B, L, F) of its cycles such that B satisfies the theta property and each cycle in L meets each cycle in F, there …


Inferences For Weibull-Gamma Distribution In Presence Of Partially Accelerated Life Test, Mahmoud Mansour, M A W Mahmoud Prof., Rashad El-Sagheer Mar 2020

Inferences For Weibull-Gamma Distribution In Presence Of Partially Accelerated Life Test, Mahmoud Mansour, M A W Mahmoud Prof., Rashad El-Sagheer

Basic Science Engineering

In this paper, the point at issue is to deliberate point and interval estimations for the parameters of Weibull-Gamma distribution (WGD) using progressively Type-II censored (PROG-II-C) sample under step stress partially accelerated life test (SSPALT) model. The maximum likelihood (ML), Bayes, and four parametric bootstrap methods are used to obtain the point estimations for the distribution parameters and the acceleration factor. Furthermore, the approximate confidence intervals (ACIs), four bootstrap confidence intervals and credible intervals of the estimators have been gotten. The results of Bayes estimators are computed under the squared error loss (SEL) function using Markov Chain Monte Carlo (MCMC) …


Next-Generation Air Force Weather Metrics Via Bayes Cost Analysis, Brandon M. Bailey Mar 2020

Next-Generation Air Force Weather Metrics Via Bayes Cost Analysis, Brandon M. Bailey

Theses and Dissertations

This research proposes a new methodology for U.S. Air Force weather forecast metrics. Military weather forecasters are essentially statistical classifiers. They categorize future conditions into an operationally relevant category based on current data, much like an Artificial Neural Net or Logistic Regression model. There is extensive literature on statistically-based metrics for these types of classifiers. Additionally, in the U.S. Air Force, forecast errors (errors in classification) have quantifiable operational costs and benefits associated with incorrect or correct classification decisions. There is a methodology in the literature, Bayes Cost, which provides a structure for creating statistically rigorous metrics for classification decisions …


Analysis With Dynamic Bayesian Networks Compared To Simulation, Aaron J. Salazar Mar 2020

Analysis With Dynamic Bayesian Networks Compared To Simulation, Aaron J. Salazar

Theses and Dissertations

This research compares simulations to Dynamic Bayesian Networks in analyzing situations. The research applies models that have known output mean and variance. Queueing systems have theoretical values of the steady-state mean and variance for the number of entities in the system. Monte Carlo simulation development is broken down into two separate approaches: discrete-event simulation and time-oriented simulation. The discrete-event simulation uses pseudo-random numbers to schedule and trigger future events (i.e. customer arrivals and services) and is based on the generated objects.The time-oriented simulation utilizes fixed-width time intervals and updates the system state according to a stochastic process for the set …


Characterizing Uncertainty In Correlated Response Variables For Pareto Front Optimization, Peter A. Calhoun Mar 2020

Characterizing Uncertainty In Correlated Response Variables For Pareto Front Optimization, Peter A. Calhoun

Theses and Dissertations

Current research provides a method to incorporate uncertainty into Pareto front optimization by simulating additional response surface model parameters according to a Multivariate Normal Distribution (MVN). This research shows that analogous to the univariate case, the MVN understates uncertainty, leading to overconfident conclusions when variance is not known and there are few observations (less than 25-30 per response). This research builds upon current methods using simulated response surface model parameters that are distributed according to an Multivariate t-Distribution (MVT), which can be shown to produce a more accurate inference when variance is not known. The MVT better addresses uncertainty in …


Analysis And Forecasting Of The 360th Air Force Recruiting Group Goal Distribution, Tyler Spangler Mar 2020

Analysis And Forecasting Of The 360th Air Force Recruiting Group Goal Distribution, Tyler Spangler

Theses and Dissertations

This research utilizes monthly data from 2012-2017 to determine economic or demographic factors that significantly contribute to increased goaling and production potential in areas of the 360th Recruiting Groups. Using regression analysis, a model of recruiting goals and production is built to identify squadrons within the 360 RCGs zone that are capable of producing more or fewer recruits and the factors that contribute to this increased or decreased capability. This research identifies that a zones high school graduation rate, the number of recruiters, and the number of JROTC detachments in a zone are positively correlated with recruiting goals and that …


Some New Results On Stochastic Comparisons Of Coherent Systems Using Signatures, Ebrahim Amini-Seresht, Baha-Eldin Khaledi, Subhash C. Kochar Mar 2020

Some New Results On Stochastic Comparisons Of Coherent Systems Using Signatures, Ebrahim Amini-Seresht, Baha-Eldin Khaledi, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed …


Health Risk Behaviors: Smoking, Alcohol, Drugs, And Dating Among Youths In Rural Central Java, Zahroh Shaluhiyah, Syamsulhuda Budi Musthofa, Ratih Indraswari, Aditya Kusumawati Feb 2020

Health Risk Behaviors: Smoking, Alcohol, Drugs, And Dating Among Youths In Rural Central Java, Zahroh Shaluhiyah, Syamsulhuda Budi Musthofa, Ratih Indraswari, Aditya Kusumawati

Kesmas

Adolescents are more likely to adopt risky health behaviors, such as smoking, alcohol use, and sexual activity. This study examined the links betweensmoking, alcohol use, and risky dating behavior and analyzed how these factors influenced risky dating and other behaviors. It is expected that this studywould be used as a foundation for developing appropriate integrated intervention for multiple risk behaviors among youths. This study was an explanatory research study with a cross-sectional approach. It involved 160 youths aged 15-24 years randomly selected from purposive villages. Participants completedself-administrated questionnaires with an enumerator present. Data were analyzed using univariate, chi-square, and multiple …


Evaluation Of Program For Overcoming Intestinal Worm Infections Among Children, Henny Febriyanti, Haerawati Idris Feb 2020

Evaluation Of Program For Overcoming Intestinal Worm Infections Among Children, Henny Febriyanti, Haerawati Idris

Kesmas

Prevalence of intestinal worm infection in generall is extremely high in Indonesia among the poor population with poor sanitation. One of the government programs to address this problem is the distribution of medicines to prevent intestinal worm infections. However, the coverage of the achievement for this program is still low in several areas of public health centers in Palembang. Therefore, this study was conducted to evaluate the efficacy of the national program for preventing intestinal worm infections. The qualitative research design used evaluation model approach Context, Input, Process, and Product (CIPP) model. This study was conducted in one of health …


Effects Of Diabetes On The Output Of Farmer And Its Policy Implications, Syed Asif Ali Naqvi, Bilal Hussain, Syed Ale Raza Shah, Muhammad Sohail Amjad Makhdum Feb 2020

Effects Of Diabetes On The Output Of Farmer And Its Policy Implications, Syed Asif Ali Naqvi, Bilal Hussain, Syed Ale Raza Shah, Muhammad Sohail Amjad Makhdum

Kesmas

This study investigated the impact of diabetes on work performance of different farming communities from Punjab, Pakistan. This study was based on cross-sectional data. A representative sample of 374 farmers was collected from five selected districts. Three types of respondents were analyzed in the study e.g.,laborer, small and large growers. Poisson and logistic regression techniques were used for the sake of analysis. According to the investigated results for thelabor category, respondents with more age, less qualification, low earning per month (Rupees), and having positive record of family diabetes, would havemore leave per month. In the same way, findings for small …


Qualitative Exploration Of Experiences And Consequences Of Health-Related Stigma Among Indonesians With Hiv, Leprosy, Schizophrenia And Diabetes, Sarju Sing Rai, Irwanto Irwanto, Ruth M.H. Peters, Elena V. Syurina, Annisa Ika Putri, Altana Mikhakhanova, Denise Naniche, Marjolein B.M. Zweekhorst Feb 2020

Qualitative Exploration Of Experiences And Consequences Of Health-Related Stigma Among Indonesians With Hiv, Leprosy, Schizophrenia And Diabetes, Sarju Sing Rai, Irwanto Irwanto, Ruth M.H. Peters, Elena V. Syurina, Annisa Ika Putri, Altana Mikhakhanova, Denise Naniche, Marjolein B.M. Zweekhorst

Kesmas

Health-related stigma causes a negative impact on the lives of affected people and undermines the effectiveness of public health programs. This study aimedto explore experiences and consequences of stigma among people affected by four health conditions relevant in Indonesia– HIV (Human ImmunodeficiencyVirus), leprosy, schizophrenia and diabetes. In this qualitative study 40 people affected by the four health conditions in Jakarta and West Java, Indonesia–,were interviewed between March and June 2018. Data were analyzed thematically by following an integrative inductive-deductive approach. The experiencesand consequences of people with stigma were similar, but such experience were more severe among people affected by HIV, …


Entomological Index And Home Environment Contribution­ ­To Dengue Hemorrhagic Fever In Mataram City, Indonesia, Tri Baskoro Tunggul Satoto, Nur Alvira Pascawati, Tri Wibawa, Roger Frutos, Sylvie Maguin, I Kadek Mulyawan, Ali Wardana Feb 2020

Entomological Index And Home Environment Contribution­ ­To Dengue Hemorrhagic Fever In Mataram City, Indonesia, Tri Baskoro Tunggul Satoto, Nur Alvira Pascawati, Tri Wibawa, Roger Frutos, Sylvie Maguin, I Kadek Mulyawan, Ali Wardana

Kesmas

Indonesia is a member of Southeast Asia Regional Office (SEARO) ranked the first in dengue hemorrhagic fever (DHF) problem based on incidence rate (IR) and case fatality rate (CFR). Several provinces in Indonesia experience an outbreak, one of which is the Mataram City in West Nusa Tenggara Province. Mataram City is an endemic area of DHF because the DHF cases are always found in three consecutive years with the number of cases that fluctuate and tend to increase. This study aimed to obtain factors that could be used to improve early warning systems in controlling DHF. This study used a …


Utilization Of Family Planning Contraceptives Among Women Inthe Coastal Area Of South Buru District, Maluku, 2017, Christiana Rialine Titaley, Ninik Sallatalohy Feb 2020

Utilization Of Family Planning Contraceptives Among Women Inthe Coastal Area Of South Buru District, Maluku, 2017, Christiana Rialine Titaley, Ninik Sallatalohy

Kesmas

Maluku Province is one among provinces in Indonesia with a contraceptive prevalence rate (CPR) lower than the national average. This study aimed toexamine factors associated with the utilization of family planning contraceptives among women of reproductive age living in the coastal area of South BuruDistrict, Maluku, Indonesia. Data were derived from a household health survey conducted in five subdistricts in South Buru, e.g., Namrole, Leksula, Waesama,Kapala Madan and Ambalau Subdistricts on November 2017 by the Faculty of Medicine, Pattimura University in Ambon. Information on contraceptive usewere collected from 390 married women aged 20 - 49 years. Bivariate and multivariate logistic …


Determinants Of Stunted Children In Indonesia: A Multilevel Analysis At The Individual, Household, And Community Levels, Febri Wicaksono, Titik Harsanti Feb 2020

Determinants Of Stunted Children In Indonesia: A Multilevel Analysis At The Individual, Household, And Community Levels, Febri Wicaksono, Titik Harsanti

Kesmas

This study aimed to examine the risk factors of childhood undernutrition in Indonesia. Determinants of childhood stunting were examined by using the 2013 Indonesia Basic Health Research Survey dataset. A total of 76,165 children aged under 5 years were included in this study. The analysis used multivariate multilevel logistic regression to determine adjusted odds ratios (aORs). The prevalence of stunting in the sample population was 36.7%. The odds of stunting increased significantly among the under-five boys, children living in slum area, and the increase of household member (aOR = 1.11, 95 %CI: 1.06–1.15; 1.09, 95%CI: 1.04–1.15; and 1.03, 95%CI: 1.02–1.04 …


Small Area Estimation On Zero-Inflated Data Using Frequentist And Bayesian Approach, Kusman Sadik, Rahma Anisa, Euis Aqmaliyah Feb 2020

Small Area Estimation On Zero-Inflated Data Using Frequentist And Bayesian Approach, Kusman Sadik, Rahma Anisa, Euis Aqmaliyah

Journal of Modern Applied Statistical Methods

The most commonly used method of small area estimation (SAE) is the empirical best linear unbiased prediction method based on a linear mixed model. However, it is not appropriate in the case of the zero-inflated target variable with a mixture of zeros and continuously distributed positive values. Therefore, various model-based SAE methods for zero-inflated data are developed, such as the Frequentist approach and the Bayesian approach. Both approaches are compared with the survey regression (SR) method which ignores the presence of zero-inflation in the data. The results show that the two SAE approaches for zero-inflated data are capable to yield …


The Importance Of Type I Error Rates When Studying Bias In Monte Carlo Studies In Statistics, Michael Harwell Feb 2020

The Importance Of Type I Error Rates When Studying Bias In Monte Carlo Studies In Statistics, Michael Harwell

Journal of Modern Applied Statistical Methods

Two common outcomes of Monte Carlo studies in statistics are bias and Type I error rate. Several versions of bias statistics exist but all employ arbitrary cutoffs for deciding when bias is ignorable or non-ignorable. This article argues Type I error rates should be used when assessing bias.


Dynamic Conditional Correlation Garch: A Multivariate Time Series Novel Using A Bayesian Approach, Diego Nascimento, Cleber Xavier, Israel Felipe, Francisco Louzada Neto Feb 2020

Dynamic Conditional Correlation Garch: A Multivariate Time Series Novel Using A Bayesian Approach, Diego Nascimento, Cleber Xavier, Israel Felipe, Francisco Louzada Neto

Journal of Modern Applied Statistical Methods

The Dynamic Conditional Correlation GARCH (DCC-GARCH) mutation model is considered using a Monte Carlo approach via Markov chains in the estimation of parameters, time-dependence variation is visually demonstrated. Fifteen indices were analyzed from the main financial markets of developed and developing countries from different continents. The performances of indices are similar, with a joint evolution. Most index returns, especially SPX and NDX, evolve over time with a higher positive correlation.


Regression When There Are Two Covariates: Some Practical Reasons For Considering Quantile Grids, Rand Wilcox Feb 2020

Regression When There Are Two Covariates: Some Practical Reasons For Considering Quantile Grids, Rand Wilcox

Journal of Modern Applied Statistical Methods

When dealing with the association between some random variable and two covariates, extensive experience with smoothers indicates that often a linear model poorly reflects the nature of the association. A simple approach via quantile grids that reflects the nature of the association is given. The two main goals are to illustrate this approach can make a practical difference, and to describe R functions for applying it. Included are comments on dealing with more than two covariates.


Bivariate Analogs Of The Wilcoxon–Mann–Whitney Test And The Patel–Hoel Method For Interactions, Rand Wilcox Feb 2020

Bivariate Analogs Of The Wilcoxon–Mann–Whitney Test And The Patel–Hoel Method For Interactions, Rand Wilcox

Journal of Modern Applied Statistical Methods

A fundamental way of characterizing how two independent compares compare is in terms of the probability that a randomly sampled observation from the first group is less than a randomly sampled observation from the second group. The paper suggests a bivariate analog and investigates methods for computing confidence intervals. An interaction for a two-by-two design is investigated as well.