First of a two-semester sequence in probability and statistics taught at a calculus-based level. Probability: discrete and continuous distributions, expected values, transformations of random variables, sampling distributions. Credit not given for both ST 701 and ST 501. Note: this course will be offered in person (Fall) and online (Summer).

Prerequisite: MA 242 or equivalent

Typically offered in Fall and Summer

Second of a two-semester sequence in probability and statistics taught at a calculus-based level. Statistical inference: methods of construction and evaluation of estimators, hypothesis tests, and interval estimators, including maximum likelihood. Credit not given for both ST 702 and ST 502. Note: this course will be offered in person (Spring) and online (Fall).

Prerequisite: ST 501

Typically offered in Fall and Spring

Estimation and testing in full and non-full rank linear models. Normal theory distributional properties. Least squares principle and the Gauss-Markov theorem. Estimability, analysis of variance and co variance in a unified manner. Practical model-building in linear regression including residual analysis, regression diagnostics, and variable selection. Emphasis on use of the computer to apply methods with data sets. Credit not given for both ST 705 and ST 503. Note: this course will be offered in person (Spring) and online (Summer).

P: ST 501 and MA 405 or equivalent (Linear Algebra); C: ST 502

Typically offered in Spring and Summer

Statistical methods requiring relatively mild assumptions about the form of the population distribution. Classical nonparametric hypothesis testing methods, Spearman and Kendall correlation coefficients, permutation tests, bootstrap methods, and nonparametric regressions will be covered.

Prerequisite: ST 508 or ST 512 or ST 514 or ST 516

Typically offered in Fall only

A general introduction to the use of descriptive and inferential statistics in behavioral science research. Methods for describing and summarizing data presented, followed by procedures for estimating population parameters and testing hypotheses concerning summarized data.

Prerequisite: Graduate standing

Typically offered in Fall and Spring

Basic concepts of statistical models and use of samples; variation, statistical measures, distributions, tests of significance, analysis of variance and elementary experimental design, regression and correlation, chi-square.

Prerequisite: Graduate Standing

Typically offered in Fall, Spring, and Summer

Covariance, multiple regression, curvilinear regression, concepts of experimental design, factorial experiments, confounded factorials, individual degrees of freedom and split-plot experiments. Computing laboratory addressing computational issues and use of statistical software.

Prerequisite: ST 511 or ST 513 or ST 517

Typically offered in Fall, Spring, and Summer

This course introduces important ideas about collecting high quality data and summarizing that data appropriately both numerically and graphically. We explore the use of probability distributions to model data and find probabilities. Estimation of parameters and properties of estimators are discussed. Construction and interpretation of commonly used confidence intervals and hypothesis tests are investigated. Students will gain considerable experience working with data. Software is used throughout the course with the expectation of students being able to produce their own analyses.

Prerequisite: Graduate standing

Typically offered in Fall and Spring

This course provides an in-depth study of building, validating, and predicting using regression models. Topics include multiple linear regression models with both continuous and categorical predictors, model selection techniques, and residual diagnostics. Bayesian regression models are also explored. Categorical data analysis is covered including contingency table analysis and logistic regression models. Students will gain considerable experience working with data. Software is used throughout the course with the expectation of students being able to produce their own analyses.

P: ST 511, ST 513, ST 517, or equivalent

Typically offered in Spring and Summer

An introduction to the foundations of probability theory and mathematical statistics useful for research in engineering. Topics include descriptive statistics, probability, discrete and continuous random variables and probability distributions, joint probability distributions and random samples, point estimation, confidence intervals, hypothesis testing, and analysis of variance.

Prerequisite: Graduate standing

Typically offered in Fall and Spring

This course is intended to give students a background in the methods of statistical analysis and design of experiments that will assist them in conducting research and analyzing data in engineering. Concentration in this course will be on principles of the design of experiments and analysis of variance and regression including post-hoc tests, inference for simple regression, multiple regression, and curvilinear regression.

Prerequisite: ST 515

Typically offered in Fall and Spring

Course covers basic methods for summarizing and describing data, accounting for variability in data, and techniques for inference. Topics include basic exploratory data analysis, probability distributions, confidence intervals, hypothesis testing, and regression analysis. This is a calculus-based course. Statistical software is used; however, there is no lab associated with the course. Credit not given for this course and ST 511 or ST 513 or ST 515. This course does NOT count as an elective towards a degree or a minor in Statistics. Note: the course will be offered in person (Fall) and online (Fall and Summer).

Prerequisites: MA 241 or equivalent (Calculus II) and MA 405 or equivalent (Linear Algebra)

Typically offered in Fall and Summer

This second course in statistics for graduate students is intended to further expand students' background in the statistical methods that will assist them in the analysis of data. Course covers many fundamental analysis methods currently used to analyze a wide array of data, mostly arising from designed experiments. Topics include multiple regression models, factorial effects models, general linear models, mixed effect models, logistic regression analysis, and basic repeated measures analysis. This is a calculus-based course. Statistical software is used, however, there is no lab associated with the course. Credit not given for this course and ST 512 or ST 514 or ST 516. Note: this course will be offered in person (Spring) and online (Fall and Spring).

Prerequisite: ST 517

Typically offered in Fall and Spring

This course is designed to bridge theory and practice on how students develop understandings of key concepts in data analysis, statistics, and probability. Discussion of students' understandings, teaching strategies and the use of manipulatives and technology tools. Topics include distribution, measures of center and spread, sampling, sampling distribution, randomness, and law of large numbers. Must complete a first level graduate statistics course ( ST 507, ST 511, or equivalent) before enrolling.

Prerequisite: ST 507 or ST 511

Typically offered in Spring only

This course is offered alternate even years

Statistical methods for design and analysis of clinical trials and epidemiological studies. Phase I, II, and III clinical trials. Principle of Intention-to-Treat, effects of non-compliance, drop-outs. Interim monitoring of clinical trials and data safety monitoring boards. Introduction to meta-analysis. There is also discussion of Epidemiological methods time permitting.

Corequisite: ST 501 or ST 521 or ST 701

Typically offered in Fall only

The fundamentals of designed experiments, analysis of variance, and regression modeling. Categorical data analysis including logistic regression will be covered. Regular access to a computer for homework, class exercises, and statistical computing is required. The emphasis in this class is on the practical aspects of statistical modeling. Assignments will concentrate on problem solving rather than formal proofs and derivations.

P: ST 511 or equivalent

Typically offered in Fall and Spring

Overview and comparison of observational studies and designed experiments followed by a thorough discussion of design principles. Review of estimation and inference for regression and ANOVA models from an experimental design perspective. Review of design and analysis for completely randomized, randomized complete block, and Latin square designs. Designs and analysis methods for factorial experiments, general blocking structures, incomplete block designs, confounded factorials, split-plot experiments, and fractional factorial designs. Examples used to illustrate application and analysis of these designs.

Prerequisite: ST 512, or ST 515, or ST 516, or ST 517, or ST 703

Typically offered in Fall only

Introduction to statistical models and methods for analyzing various types of spatially referenced data. The focus is on applications with real data and their analysis with statistical programs such as R and SAS. Students are required to write, modify, and run computer code in order to complete homework assignments and final projects.

P: ST 422 and ST 430

Typically offered in Spring only

Statistical models and methods for the analysis of time series data using both time domain and frequency domain approaches. An introduction and review of necessary statistical concepts will be given, and a statistical computing package will be introduced. Analyses of real data sets using statistical software will be emphasized.

Prerequisite: ST 422 and ST 430

Typically offered in Fall only

Use of statistics for quality control and productivity improvement. Control chart calculations and graphing, process control and specification; sampling plans; and reliability. Computer use will be stressed for performing calculations and graphing.

Prerequisite: (ST 305 or ST 312 or ST 372) and ST 307

Typically offered in Spring only

An introduction to use of statistical methods for analyzing multivariate and longitudinal data collected in experiments and surveys. Topics covered include multivariate analysis of variance, discriminant analysis, principal components analysis, factor analysis, covariance modeling, and mixed effects models such as growth curves and random coefficient models. Emphasis is on use of a computer to perform statistical analysis of multivariate and longitudinal data.

Prerequisite: ST 422 and ST 430

Typically offered in Fall and Spring

Introduction to Bayesian concepts of statistical inference; Bayesian learning; Markov chain Monte Carlo methods using existing software (SAS and OpenBUGS); linear and hierarchical models; model selection and diagnostics.

Prerequisite: ST 422 and ST 430

Typically offered in Spring only

This course will provide a discussion-based introduction to statistical practice geared towards students in the final semester of their Master of Statistics degree. Note: the course will be offered in person (Fall) and online (Spring and Summer).

Prerequisite: (ST 512 or ST 514 or ST 516 or ST 518) and (ST 502 or ST 522 or ST 702)

Typically offered in Fall, Spring, and Summer

This course focuses on the concepts, methods, and models used to analyze categorical data, particularly contingency tables, count data and binary/binomial type of data. The topics covered include Pearson Chi-squared independence test for contingency tables, measures of marginal and conditional associations, small-sample inference, logistic regression models for independent binary/binomial data and many extended models for correlated binary/binomial data including matched data and longitudinal data. The course emphasizes the implementation of methods/models using SAS and the interpretation of the results from the output.

Prerequisite: ST 512 or ST 514 or ST 515 or ST 516

Typically offered in Fall only

Modern introduction to Probability Theory and Stochastic Processes. The choice of material is motivated by applications to problems such as queueing networks, filtering and financial mathematics. Topics include: review of discrete probability and continuous random variables, random walks, markov chains, martingales, stopping times, erodicity, conditional expectations, continuous-time Markov chains, laws of large numbers, central limit theorem and large deviations.

Prerequisite: MA 421 and MA 425 or MA 511

Typically offered in Fall only

An introduction to programming and data management using SAS, the industry standard for statistical practice. Detailed discussion of the program data vector and data handling techniques that are required to apply statistical methods. Topics are based on the current content of the Base SAS Certification Exam and typically include: importing, validating, and exporting of data files; manipulating, subsetting, and grouping data; merging and appending data sets; basic detail and summary reporting; and code debugging. Additional topics with practical applications are also introduced, such as graphics and advanced reporting. Statistical methods for analyzing data are not covered in this course. Regular access to a computer for homework and class exercises is required. Previous exposure to SAS is not expected.

Prerequisite: Graduate standing

Typically offered in Fall, Spring, and Summer

Statistical procedures for importing/managing complex data structures using SQL, automated analysis using macro programming, basic simulation methods and text parsing/analysis procedures. Students learn SAS, the industry standard for statistical practice. Regular access to a computer for homework and class exercises is required.

P: ST 555 or Base SAS Certification

Typically offered in Spring and Summer

Methods for reading, manipulating, and combining data sources including databases. Custom functions, visualizations, and summaries. Common analyses done by data scientists. Methods for communicating results including dashboards. Regular access to a computer for homework and class exercises is required.

Prerequisites: (ST 511, ST 513, ST 517, or equivalent) and (ST 555 or moderate computer programming experience)

Typically offered in Fall and Summer

Introduction and application of econometrics methods for analyzing cross-sectional data in economics, and other social science disciplines, such as OLS, IV regressions, and simultaneous equations models. Students should have had a statistical methods course at the 300 level or above as well as Calculus I and II.

Typically offered in Fall only

This is a hands-on course using modeling techniques designed mostly for large observational studies. Estimation topics include recursive splitting, ordinary and logistic regression, neural networks, and discriminant analysis. Clustering and association analysis are covered under the topic "unsupervised learning," and the use of training and validation data sets is emphasized. Model evaluation alternatives to statistical significance include lift charts and receiver operating characteristic curves. SAS Enterprise Miner is used in the demonstrations, and some knowledge of basic SAS programming is helpful.

Prerequisite: ST 512 or ST 514 or ST 515 or ST 516 or ST 517

Typically offered in Spring only

This course will introduce common statistical learning methods for supervised and unsupervised predictive learning in both the regression and classification settings. Topics covered will include linear and polynomial regression, logistic regression and discriminant analysis, cross-validation and the bootstrap, model selection and regularization methods, splines and generalized additive models, principal components, hierarchical clustering, nearest neighbor, kernel, and tree-based methods, ensemble methods, boosting, and support-vector machines.

Prerequisite: ST 512 or ST 514 or ST 515 or ST 517

Typically offered in Summer only

Typically offered in Fall, Spring, and Summer

Typically offered in Fall, Spring, and Summer

Instruction in research and research under the mentorship of a member of the Graduate Faculty.

Prerequisite: Master's student

Typically offered in Fall, Spring, and Summer

Thesis Research

Prerequisite: Master's student

Typically offered in Fall, Spring, and Summer

Probability tools for statistics: description of discrete and absolutely continuous distributions, expected values, moments, moment generating functions, transformation of random variables, marginal and conditional distributions, independence, orderstatistics, multivariate distributions, concept of random sample, derivation of many sampling distributions.

Typically offered in Fall only

General framework for statistical inference. Point estimators: biased and unbiased, minimum variance unbiased, least mean square error, maximum likelihood and least squares, asymptotic properties. Interval estimators and tests of hypotheses: confidence intervals, power functions, Neyman-Pearson lemma, likelihood ratio tests, unbiasedness, efficiency and sufficiency.

Prerequisite: ST 701

Typically offered in Spring only

Introduction of statistical methods. Examples include multiple linear regression, concepts of experimental design, factorial experiments, and random-effects modeling. A computing laboratory addresses computational issues and use of statistical software. This course is a prerequisite for most advanced courses in statistics. This section is restricted to statistics and closely related majors.

R: 17STPHD Students Only

Typically offered in Fall only

This course will introduce many methods that are commonly used in applications. Examples include: model generation, selection, assessment, and diagnostics in the context of multiple linear regression (including penalized regression); linear mixed models; generalized linear models; generalized linear mixed models; nonparametric regression and smoothing; and finite-population sampling basics. Coverage will include some theory, plus implementation using SAS and/or R.

Prerequisite:ST 703; Corequisites: ST 702 and ST 705

Typically offered in Spring only

Theory of estimation and testing in full and non-full rank linear models. Normal theory distributional properties. Least squares principle and the Gauss-Markoff theorem. Estimability and properties of best linear unbiased estimators. General linear hypothesis. Application of dummy variable methods to elementary classification models for balanced and unbalanced data. Analysis of covariance. Variance components estimation for balanced data.

Corequisite: ST 702

Typically offered in Spring only

An advanced mathematical treatment of analytical and algorithmic aspects of finite dimensional nonlinear programming. Including an examination of structure and effectiveness of computational methods for unconstrained and constrained minimization. Special attention directed toward current research and recent developments in the field.

Prerequisite: OR(IE,MA) 505 and MA 425

Typically offered in Spring only

Principles for interpretation and design of sample surveys. Estimator biases, variances and comparative costs. Simple random sample, cluster sample, ratio estimation, stratification, varying probabilities of selection. Multi-stage, systematic and double sampling. Response errors.

Prerequisite: ST 422, ST 512

Typically offered in Fall only

The course aims to provide students with the relevant background knowledge and quantitative skills for conducting genetic data analysis to evaluate the genetic effects of complex traits. The course will focus on statistical methodologies and analytical strategies for population-based association studies with genotype and sequencing data collected from whole genome and exome. The specific topics include genetic variants; genetic identity coefficients and its applications; heritability; Hardy-Weinberg disequilibrium; recombination; linkage disequilibrium and association mapping; genome-wide association studies (GWAS); population substructures; multiple testing; single-variant and multi-variant association methods; next-generation sequencing (NGS) data and rare variant analysis; copy number variant analysis; analysis using summary statistics.

Prerequisite: ST 511 or equivalent

Typically offered in Fall only

This course is offered alternate years

Introduction to modeling longitudinal data; Population-averaged vs. subject-specific modeling; Classical repeated measures analysis of variance methods and drawbacks; Review of estimating equations; Population-averaged linear models; Linear mixed effects models; Maximum likelihood, restricted maximum likelihood, and large sample theory; Review of nonlinear and generalized linear regression models; Population-averaged models and generalized estimating equations; Nonlinear and generalized linear mixed effects models; Implications of missing data; Advanced topics (including Bayesian framework, complex nonlinear models, multi-level hierarchical models, relaxing assumptions on random effects in mixed effects models, among others). Implementation in SAS and R.

Prerequisites: ST 702 and ST 705

Typically offered in Spring only

Introduction to the theory and methods of spatial data analysis including: visualization; Gaussian processes; spectral representation; variograms; kriging; computationally-efficient methods; nonstationary processes; spatiotemporal and multivariate models.

Prerequisite: ST 705

Typically offered in Spring only

Introduction to Bayesian inference; specifying prior distributions; conjugate priors, summarizing posterior information, predictive distributions, hierachical models, asymptotic consistency and asymptotic normality. Markov Chain Monte Carlo (MCMC) methods and the use of exising software(e.g., WinBUGS).

Prerequisite: ST 702

Typically offered in Fall only

Statistical methods for analysis of time-to-event data, with application to situations with data subject to right-censoring and staggered entry, including clinical trials. Survival distribution and hazard rate; Kaplan-Meier estimator for survival distribution and Greenwood's formula; log-rank and weighted long-rank tests; design issues in clinical trials. Regression models, including accelerated failure time and proportional hazards; partial likelihood; diagnostics.

Prerequisite: ST 502 or ST 702

Typically offered in Spring only

Markov chains and Markov processes, Poisson process, birth and death processes, queuing theory, renewal theory, stationary processes, Brownian motion.

Typically offered in Spring only

Fundamental mathematical results of probabilistic measure theory needed for advanced applications in stochastic processes. Probability measures, sigma-algebras, random variables, Lebesgue integration, expectation and conditional expectations w.r.t.sigma algebras, characteristic functions, notions of convergence of sequences of random variables, weak convergence of measures, Gaussian systems, Poisson processes, mixing properties, discrete-time martingales, continuous-time markov chains.

Prerequisite: MA(ST) 546

Typically offered in Spring only

Theory of stochastic differential equations driven by Brownian motions. Current techniques in filtering and financial mathematics. Construction and properties of Brownian motion, wiener measure, Ito's integrals, martingale representation theorem, stochastic differential equations and diffusion processes, Girsanov's theorem, relation to partial differential equations, the Feynman-Kac formula.

Prerequisite: MA(ST) 747

Typically offered in Fall only

Introduction to principles of estimation of linear regression models, such as ordinary least squares and generalized least squares. Extensions to time series and panel data. Consideration of endogeneity and instrumental variables estimation. Limited dependent variable and sample selection models. Attention to implementation of econometric methods using a statistical package and microeconomic and macroeconomic data sets.

Prerequisite: ST 421; Corequisite: ST 422

Typically offered in Spring only

Introduction to important econometric methods of estimation such as Least Squares, instrumentatl Variables, Maximum Likelihood, and Generalized Method of Moments and their application to the estimation of linear models for cross-sectional ecomomic data. Discussion of important concepts in the asymptotic statistical analysis of vector process with application to the inference procedures based on the aforementioned estimation methods.

Prerequisite: ST 421, ST 422

Typically offered in Fall only

The characteristics of macroeconomic and financial time series data. Discussion of stationarity and non-stationarity as they relate to economic time series. Linear models for stationary economic time series: autoregressive moving average (ARMA) models; vector autoregressive (VAR) models. Linear models for nonstationary data: deterministic and stochastic trends; cointegration. Methods for capturing volatility of financial time series such as autoregressive conditional heteroscedasticity (ARCH) models. Generalized Method of Moments estimation of nonlinear dynamic models.

Prerequisite: ECG(ST) 751

Typically offered in Spring only

The characteristics of microeconomic data. Limited dependent variable models for cross-sectional microeconomic data: logit/probit models; tobit models; methods for accounting for sample selection; count data models; duration analysis; non-parametricmethods. Panel data models: balanced and unbalanced panels; fixed and random effects; dynamic panel data models; limited dependent variables and panel data analysis.

Prerequisite: ECG 751

Typically offered in Spring only

Phylogenetic analyses of nucleotide and protein sequence data. Sequence alignment, phylogeny reconstruction and relevant computer software. Prediction of protein secondary structure, database searching, bioinformatics and related topics. Project required.

Prerequisite: GN 311 and ST 511

Typically offered in Fall only

This course is offered alternate years

The essence of quantitative genetics is to study multiple genes and their relationship to phenotypes. How to study and interpret the relationship between phenotypes and whole genome genotypes in a cohesive framework is the focus of this course. We discuss how to use genomic tools to map quantitative trait loci, how to study epistasis, how to study genetic correlations and genotype-by-environment interactions. We put special emphasis in using genomic data to study and interpret general biological problems, such as adaptation and heterosis. The course is targeted for advanced graduate students interested in using genomic information to study a variety of problems in quantitative genetics.

Prerequisite: ST 511

Typically offered in Fall only

This course is offered alternate even years

Computational tools for research in statistics, including applications of numerical linear algebra, optimization and random number generation, using the statistical language R. A project encompassing a simulation experiment will be required.

Prerequisite: ST 702 and ST 705

Typically offered in Fall only

Role of theory construction and model building in development of experimental science. Historical development of mathematical theories and models for growth of one-species populations (logistic and off-shoots), including considerations of age distributions (matrix models, Leslie and Lopez; continuous theory, renewal equation). Some of the more elementary theories on the growth of organisms (von Bertalanffy and others; allometric theories; cultures grown in a chemostat). Mathematical theories oftwo and more species systems (predator-prey, competition, symbosis; leading up to present-day research) and discussion of some similar models for chemical kinetics. Much emphasis on scrutiny of biological concepts as well as of mathematical structureof models in order to uncover both weak and strong points of models discussed. Mathematical treatment of differential equations in models stressing qualitative and graphical aspects, as well as certain aspects of discretization. Difference equation models.

Prerequisite: Advanced calculus, reasonable background in biology

Typically offered in Fall only

Continuation of topics of BMA 771. Some more advanced mathematical techniques concerning nonlinear differential equations of types encountered in BMA 771: several concepts of stability, asymptotic directions, Liapunov functions; different time-scales. Comparison of deterministic and stochastic models for several biological problems including birth and death processes. Discussion of various other applications of mathematics to biology, some recent research.

Prerequisite: BMA 771, elementary probability theory

Typically offered in Spring only

Survey of modeling approaches and analysis methods for data from continuous state random processes. Emphasis on differential and difference equations with noisy input. Doob-Meyer decomposition of process into its signal and noise components. Examples from biological and physical sciences, and engineering. Student project.

Prerequisite: BMA 772 or ST (MA) 746

Typically offered in Spring only

This course is offered alternate years

Sets and classes, sigma-fields and related structures, probability measures and extensions, random variables, expectation and integration, uniform integrability, inequalities, L_p-spaces, product spaces, independence, zero-one laws, convergence notions, characteristic functions, simplest limit theorems, absolute continuity, conditional expectation and conditional probabilities, martingales.

Prerequisite: ST 702

Typically offered in Spring only

Typically offered in Fall, Spring, and Summer

Statistical inference with emphasis on the use of statistical models, construction and use of likelihoods, general estimating equations, and large sample methods. Includes introduction to Bayesian statistics and the jackknife and bootstrap.

Prerequisite: ST 702

Typically offered in Fall only

Typically offered in Fall and Spring

Typically offered in Fall and Spring

Typically offered in Spring only

Participation in regularly scheduled supervised statistical consulting sessions with faculty member and client. Consultant's report written for each session. Regularly scheduled meetings with course instructor and other student consultants to present and discuss consulting experiences.

Prerequisite: ST 512 and ST 702

Typically offered in Spring only

This is an external internship component of an NHLBI-funded training grant of the Statistics Department jointly with Duke for which PhD students, who are appointed to the grant, work on research projects at Duke Clinical Research Institute.

Typically offered in Fall only

Teaching experience under the mentorship of faculty who assist the student in planing for the teaching assignment, observe and provide feedback to the student during the teaching assignment, and evaluate the student upon completion of the assignment.

Prerequisite: Doctoral student

Typically offered in Fall and Spring

Instruction in research and research under the mentorship of a member of the Graduate Faculty.

Prerequisite: Doctoral student

Typically offered in Fall, Spring, and Summer

Dissertation Research

Prerequisite: Doctoral student

Typically offered in Fall, Spring, and Summer

For graduate students whose programs of work specify no formal course work during a summer session and who will be devoting full time to thesis research.

Prerequisite: Doctoral student

Typically offered in Summer only

For students who have completed all credit hour requirements, full-time enrollment, preliminary examination, and residency requirements for the doctoral degree, and are writing and defending their dissertations.

Prerequisite: Doctoral student

Typically offered in Fall, Spring, and Summer