Graduate Program

Course Credit: 3

Introduction : An introduction to methods of analysing correlated time-to-event data is provided in this course. Some commonly used methods for analysing univariate time-to-event data, e.g. Kaplan-Meier estimate of survivor functions, Cox's proportional hazards models, etc., are reviewed using counting processes notations. Objectives: The objectives of the course are to teach the theoretical basis of di erent methods related to analysing correlated time-to-event data and competing risks model and to apply statistical softwares to analyse data using such models. Learning Outcomes: At the end of the course, students are expected i) to understand the theoretical basis of di erent methods related to analysing correlated time-to-event data and competing risks model ii) to use a statistical software (e.g. related R packages) to analyse data using such models iii) to interpret the results and write scienti c publication. Contents: Estimating the Survival and Hazard Functions: Introduction and notation, the Nelson-Aalen and Kaplan-Meier estimators, counting process and martingals, properties of Nelson-Aalen estimator. Semiparametric Multiplicative Hazards Regression Model: Introduction, estimation of param- eters, inclusion of strata, handling ties, sample size determinations, counting process form of a Cox model, time-dependent covariates, di erent types of residuals for Cox models, checking proportionality assumption. Multiple Modes of Failure: Basic characteristics of model speci cation, likelihood function formulation, nonparametric methods, parametric methods, semiparametric methods for mul- tiplicative hazards model. Analysis of Correlated Lifetime Data: Introduction, regression models for correlated lifetime data, representation and estimation of bivariate survivor function.

Course Credit: 3

Introduction : Spatial statistics encompasses diversi ed statistical methods for analyzing data obtained from stochastic process indexed by the space. This branch is enrich enough to gain insight from data exploiting the dependence over space. Its myriad applications caught profound attention of people from both academia and practitioners. Objectives: Technology is indispensable for modern life, and its advances in di erent aspects of our life made several things possible. Now a days data have been collected along with extensive additional information. Spatial data is one of such examples. In recent years, analysis of spatial data receives great attention over the world. As a result, several theories have been developed for di erent types of spatial data analysis. This course is designed to introduce the graduate student with few of such theories so that they can develop their skill in spatial data analysis. To comprehend this course, students need a sound knowledge of Mathematical statistics, particularly the concepts of stochastic process. It is expected that the student will be able to analyze di erent spatial data from diverse elds after successful completion of the course. Learning Outcomes After completing this course students are expected to have knowledge about i) spatial and non spatial data, ii) geostatistical data and analysis, iii) spatial interpolation, iv) apply auto regressive model to areal data, v) point pattern data analysis. 13 Contents Review of non-spatial statistics and stochastic process, overview of di erent types of spatial data; random eld and spatial process - geostatistical/point reference process, areal/lattice process and point process; spatial data concern. Geostatistical data: real data examples, measure of spatial dependence- variogram and co- variance, stationarity and isotropic, variograms and covariance functions, tting the vari- ograms functions; Kriging, linear geostatistical model - formulation, simulation, estimation and prediction, generalized linear geostatistical model - formulation, simulations, estimation and prediction. Areal data: neighborhoods, testing for spatial association, autoregressive models (CAR, SAR), estimation/inference; grids and image analysis, disease mapping. Point pattern data: locations of events versus counts of events, types of spatial patterns, CSR and tests - quadrat and nearest neighbor methods, K-functions and L-functions, point process models- estimation and inference, health event clustering. Special topics in spatial modeling: Hierarchical models, Bayesian methods for spatial statis- tics, Bayesian disease mapping, Spatio-temporal modeling, more on stationarity. Use of R and GIS software to give emphasis on analysis of real data from the environmental, geological and agricultural sciences.

Course Credit: 3

Introduction The course provides an introduction to causal inference with a cohesive presentation of con- cepts of, and methods for, causal inference. Objectives The aim of the course is to facilitate students to de ne causation in biomedical research, describe methods to make causal inferences in epidemiology and health services research, and demonstrate the practical application of these methods. Learning Outcomes Upon completion of the course, students are expected i) to be familiar with the concepts of causality and counterfactuals ii) to learn di erent methods for estimating causual e ects in the 14 context of biomedical research iii) to apply the methods to data obtained from observational studies and to interpret the results. Contents Causal e ects: individual causal e ects, average causal e ects, causation versus association. Randomized experiments: randomization, conditional randomization, standardization, and inverse probability weighting. Observational studies: identi ability conditions, exchangeability, positivity, and consistency. E ect modi cation: strati cation, matching, and adjustment methods. Interaction: identifying interaction, counterfactual response type and interaction. Directed acyclic graphs (DAGs): complete and incomplete DAGs, statistical DAGs, DAGs and models, paths, chains and forks, colliders, d-seperation. Unconfounded treatment assignment: balancing scores and the propensity score; Estimating propensity scores: selecting covariates and interactions, constructing propensity score strata, assessing balance conditional on estimated propensity score; Assessing overlap in covariate distributions; Matching to improve balance in covariate distributions: selecting subsample of controls to improve balance, theoretical properties of matching procedures; Subclassi ca- tion on propensity scores: weighting estimators and subclassi cation; Matching estimators: matching estimators of ATE; A general method for estimating sampling variances for standard estimators for average causal e ects. Longitudinal causal inference: g-formula and marginal structural models. Mediation analysis: traditional approaches (direct and indirect e ects), counterfactual de ni- tions of direct and indirect e ects, regression for causal mediation analysis, sensitivity analysis

Course Credit: 3

Introduction Longitudinal data arise when multiple measurements of a response are collected over time for each individual in the study and hence are likely to be correlated, which presents substantial 15 challenge in analyzing such data. This course covers topics related to statistical methods and models for drawing scienti c inferences from longitudinal data. Objectives The objectives of the course are to teach students to understand the unique features of and the methodological implication of analyzing the data from longitudinal studies, as compared to the data from traditional studies, to understand statistical methods/models, particularly linear/generalized linear mixed models, GEE approaches for analyzing longitudinal data. It is also expected that students will be familiar with the proper implementation and interpreta- tion of the statistical methods/models for analyzing longitudinal data and software packages analyzing such data. Learning Outcomes Upon completion of the course, students will achieve skills i) to understand the nature of longitudinal/clustered data ii) to understand the models and methods for analysing longitu- dinal/clustered data iii) to analyse such data and interpret the results iv) to understand and interpret research ndings of the published reports and articles. Contents Longitudinal data: Concepts, examples, objectives of analysis, problems related to one sample and multiple samples, sources of correlation in longitudinal data, exploring longitudinal data. Linear model for longitudinal data: Introduction, notation and distributional assumptions, simple descriptive methods of analysis, modelling the mean, modelling the covariance, esti- mation and statistical inference. ANOVA for longitudinal data: Fundamental model, one sample model, sphericity condition; multiple samples models. Linear mixed e ects models: Introduction, random e ects covariance structure, prediction of random e ects, residual analysis and diagnostics. Extension of GLM for longitudinal data: Review of univariate generalized linear models, quasi- likelihood, marginal models, random e ects models, transition models, comparison between these approaches; the GEE methods: methodology, hypothesis tests using wald statistics, assessing model adequacy; GEE1 and GEE2. Introduction to the concept of conditional models, joint models, their applications to bivariate binary and count data. Estimation, inference and test of independence. Generalized Linear Mixed Models (GLMM): Introduction, estimation procedures{Laplace transformation, penalized quasi-likelihood (PQL), marginal quasi-likelihood (MQL). Numerical integration: Gaussian quadrature, adaptive gaussian quadrature, Monte Carlo integration; markov chain Monte Carlo sampling; comparison between these methods. Statistical analysis with missing data: Missing data, missing data pattern, missing data mech- anism, imputation procedures, mean imputation, hot deck imputation. estimation of sampling variance in the presence of non-response, likelihood based estimation and tests for both com- plete and incomplete cases, regression models with missing covariate values, applications for longitudinal data. 16

Course Credit: 3

Introduction Much of the information around us can be described as signals. Statistical signal process- ing uses stochastic processes, statistical inference and mathematical techniques to describe, transform, and analyze signals in order to extract information from them. Objectives This course is designed to provide statistics graduate students with an overview of di erent types of signals, their representations and the use of statistical methods, such as estimation and hypothesis testing, to extract information from signals. Its objective is to introduce students to real life applications demonstrating the use of statistics in signal analysis. Learning Outcomes At the end of this course a student should be able to : (i) understand basic concepts of signals, signal properties and their representations in time and frequency domains (ii) Apply well-known statistical estimation techniques to estimate signal parameters from noisy signal measurements (iii) Apply well-known statistical decision theory methods to detect signals in Gaussian noise and assess the performance of these methods (iv) Understand and appreciate the importance of statistics in solving real life problems occurring in the domain of signal processing and communication. Contents Introduction to signals: Signals and their classi cation; real world analog signals: audio, video, biomedical (EEG, ECG, MRI, PET, CT, US), SAR, microarray, etc; digital representation of analog signals; role of transformation in signal processing. Orthogonal representation of signals. Review of exponential Fourier series and its properties. Signal estimation theory: Estimation of signal parameters using ML, EM algorithm, minimum variance unbiased estimators (Rao-Blackwell theorem, CRLB, BLUE), Bayesian estimators (MAP, MMSE, MAE), linear Bayesian estimators. 17 Signal detection theory: Detection of DC signals in Gaussian noise: detection criteria (Bayes risk, Probability of error, Neyman-Pearson), LRT; detection of known signals in Gaussian noise: matched lter and its performance, minimum distance receiver; detection of random signals in Gaussian noise: the estimator correlator. Applications: Scalar quantization, image compression, pattern recognition, histogram equal- ization, segmentation, application of signal estimation and detection theory to signal commu- nication, signal recovery from various types of linear and nonlinear degradations, copyright protection, enhancement, etc.

Course Credit: 3

Introduction Meta-analysis refers to the quantitative analysis of study outcomes. Meta-analysis consists of a collection of techniques that attempt to analyze and integrate results that accrue from research studies. This course provides an overview of systematic review and meta-analysis from a statistician's point of view. Objectives The main objectives are to introduce students with the merits of meta-analysis and how it can form an important and informative part of a systematic review, with the most common statistical methods for conducting a meta-analysis, and with how to analyze and interpret the results. Learning Outcomes At the end of the course, students should be familiar with i) the research synthesis, ii) sys- tematic review of the existing research iii) data extraction from systematic review, iv) models and methods for analyzing meta-data for new ndings and publication. 18 Contents Introduction to systematic review and meta analysis: Motivation, strengths and weakness of meta-analysis, problem formulation (why study meta analysis), systematic review process. Types of results to summarize; overview of e ect size; e ect size calculation for both contin- uous and discrete data. Combining e ect size from multiple studies; xed e ect and random e ects models and their estimation; heterogeneity between studies and its estimation techniques; test of homogeneity in meta analysis; prediction intervals; subgroup analysis, Meta regression: random e ect meta regression, baseline risk regression. Publication bias in meta analysis; Power analysis for meta analysis; e ect size rather than P- values; Meta analysis based on direction and P-values, reporting the results of meta analysis. Introduction to Bayesian approach to meta analysis; Meta analysis for multivariate/longitudinal data; network meta analysis.

Course Credit: 3

Introduction The clinical trial is \the most de nitive tool for evaluation of the applicability of clinical research". It represents \a key research activity with the potential to improve the quality of health care and control costs through careful comparison of alternative treatments". The course is designed to give an overall idea of clinical trial studies. It will provide an introduction to the statistical and ethical aspects of clinical trials research. Objectives The main objectve o fthe course is to teach students on the topics include design, implemen- tation, and analysis of trials, including rst-in- human studies, phase II and phase III studies. The course will enable applying existing methodologies in designing clinical trials and will also foster research in this area. 19 Learning Outcomes Upon completion of the course, students will achieve skills i) to understand, design a trial for assess the e ectiveness of a drug, and ii) to implement and analysis data from such trials and interpret the results. Contents Statistical approaches for clinical trials: Introduction, comparison between Bayesian and fre- quentist approaches and adaptivity in clinical trials. Phases of clinical trials, pharmacokinet- ics (PK) and pharmacodynamics (PD) of a drug, dose-concentration-e ect relationship and compartmental models in pharmacokinetic studies. Phase I studies: Determining the starting dose from preclinical studies. Rule-based designs: 3+3 design, Storer's up-and-down designs, pharmacologically-guided dose escalation and de- sign using isotonic regression. Model-based designs: continual reassessment method and its variations, escalation with overdose control and PK guided designs. Phase II studies: Gehan and Simon's two-stage designs. Seamless phase I/II clinical trials: TriCRM, E Tox and penalised D-optimum designs for optimum dose selection. Phase III studies: Randomised controlled clinical trial, group sequential design and multi-arm multi-stage trials in connection with con rmatory studies. Text Books 1. Berry SM, Carlin BP, Lee JJ, and Muller P (2010). Bayesian Adaptive Methods for Clinical Trials. CRC press. 2. Rosenbaum SE (2012). Basic Pharmacokinetics and Pharmacodynamics: An Integrated Textbook and Computer Simulations. John Wiley & Sons.

Course Credit: 3

Introduction The course provides a broad but thorough introduction to the methods and practice of sta- tistical machine learning and its core models and algorithms. Objectives The aim of the course is to provide students of statistics with detailed knowledge of how Ma- chine Learning methods work and how statistical models can be brought to bear in computer systems not only to analyze large data sets, but also to let computers perform tasks, that traditional methods of computer science are unable to address. 20 Learning Outcomes After completing the course, students will have the knowledge and skills to: i) Describe a num- ber of models for supervised, unsupervised, and reinforcement machine learning, ii) Assess the strength and weakness of each of these models, iii) Know the underlying mathematical rela- tionships within and across statistical learning algorithms, iv) Identify appropriate statistical tools for a data analysis problems in the real world based on reasoned arguments, v) Develop and implement optimisation methods for training of statistical models, vi) Design decision and optimal control problems to improve performance of statistical learning algorithms, vii) Design and implement various statistical machine learning algorithms in real-world applica- tions, viii) Evaluate the performance of various statistical machine learning algorithms, ix) Demonstrate a working knowledge of dimension reduction techniques. Identify and implement advanced computational methods in machine learning. Contents Statistical learning: Statistical learning and regression, curse of dimensionality and parametric models, assessing model accuracy and bias-variance trade-o , classi cation problems and K- nearest neighbors. Linear regression: Model selection and qualitative predictors, interactions and nonlinearity. Classi cation: Introduction to classi cation, logistic regression and maximum likelihood, mul- tivariate logistic regression and confounding, case-control sampling and multiclass logistic regressionl, linear discriminant analysis and Bayes theorem, univariate linear discriminant analysis, multivariate linear discriminant analysis and ROC curves, quadratic discriminant analysis and naive bayes. Resampling methods: Estimating prediction error and validation set approach, k-fold cross- validation, cross-validation- the right and wrong ways, the bootstrap, more on the bootstrap. Linear model selection and regularization: Linear model selection and best subset selection, forward stepwise selection, backward stepwise selection, estimating test error using mallow's Cp, AIC, BIC, adjusted R-squared, estimating test error using cross-validation, shrinkage methods and ridge regression, the Lasso, the elastic net, tuning parameter selection for ridge regression and lasso, dimension reduction, principal components regression and partial least squares. Moving beyond linearity: Polynomial regression and step functions, piecewise polynomials and splines, smoothing splines, local regression and generalized additive models. Tree-based methods: Decision trees, pruning a decision tree, classi cation trees and compar- ison with linear models, bootstrap aggregation (Bagging) and random forests, boosting and variable importance. Support vector machines: Maximal margin classi er, support vector classi er, kernels and support vector machines, example and comparison with logistic regression.

Course Credit: 4

Introduction Bayesian statistics refers to practical inferential methods that use probability models for both observable and unobservable quantities. The exibility and generality of these methods allow them to address complex real-life problems that are not amenable to other techniques. This course will provide a pragmatic introduction to Bayesian data analysis and its powerful applications. Objectives Acquire basic understanding in the principles and techniques of Bayesian data analysis. Ap- ply Bayesian methodology to solve real-life problems. Utilize R for Bayesian computation, visualization, and analysis of data. Learning Outcomes Upon completion of the course, students will learn about i) the concept of the Bayesian statistical methods, ii) formulation and derivation of prior and posterior distributions, iii) estimation of the model using di erent MCMC methods, and iv) application of the Bayesian methods to analyze data and interpret and compare results with the frequentist approach. Contents Bayesian thinking: background, bene ts and implementations; Bayes theorem, components of Bayes theorem - likelihood, prior and posterior; informative and non-informative priors; proper and improper priors; discrete priors; conjugate priors; semi-conjugate priors; exponen- tial families and conjugate priors; credible interval; Bayesian hypothesis testing; building a predictive model. Bayesian inference and prediction: single parameter models - binomial model, Poisson model, normal with known variance, normal with known mean; multi-parameter models - concepts of nuisance parameters, normal model with a non-informative, conjugate, and semi-conjugate priors, multinomial model with Dirichlet prior, multivariate normal model; posterior inference for arbitrary functions; methods of prior speci cation; method of evaluating Bayes estimator. Summarizing posterior distributions: introduction; approximate methods: numerical inte- gration method, Bayesian central limit theorem; simulation method: direct sampling and rejection sampling, importance sampling; Markov Chain Monte Carlo (MCMC) methods - 11 Gibbs sampler, general properties of the Gibbs sampler, Metropolis algorithm, Metropolis- Hastings (MH) sampling, relationship between Gibbs and MH sampling, MCMC diagnostics - assessing convergence, acceptance rates of the MH algorithm, autocorrelation; evaluating tted model - sampling from predictive distributions, posterior predictive model checking. Linear model: introduction, classical and Bayesian inference and prediction in the linear mod- els, hierarchical linear models - Bayesian inference and prediction, empirical Bayes estimation; generalized linear model - Bayesian inference and prediction (logit model, probit model, count data model); model selection - Bayesian model comparison. Nonparametric and Semiparametric Bayesian models.

Course Credit: 2

Introduction This course focuses the application of Bayesian statistics in real life situations. It will also enable students to apply machine learning algorithms to manage large datasets originated from various sources. Objectives The successful completion of the course will help a student to apply methods related to Bayesian statistics and machine learning in various situations. Learning Outcomes After completing the course, students will be familiar i) to handle any compution related to Bayesian models and Statistical machione learning using R software ii) with several Bayesian models including Bayesian linear regression, MCMC techniques would be covered, iii) with di erent machine learning tools including classi cation problems, boosting, random forest along with unsupervised learning techniques would be taught. Students will also be able to implement machine learning tools to any raw datasets. Contents Computing problems related to AST 501: Bayesian Statistics, and AST 525: Statistical Machine Learning

Course Credit: 2

Introduction The course focuses on applications related to longitudinal data, sptial statistics, and causal inference. Objectives The course intends to train students with modern tools of analysing longitudinal data and making causal conclusion from real life data and small area estimation techniques. Learning Outcomes At the end of the course students will achieve skills on i) how to use statistical software and packages to analyse longitudinal/cluster/spatial data ii) to t models and interpret the results 22 ii) to compare and identify the appropriate models required for the data iv) to analyse data for estimating average treatment e ect for casual inference v) to write report with statistical results for scienti c publications. Contents Computing problems related to AST 511: Environmental and Spatial Statistics, AST 518: Introduction to Causal Inference, and AST 519: Analysis of Longitudinal Data.

Course Credit: 2

Introduction This course focuses on essential techniques for statistical data analysis, more emphasise will provide in preparing report/article for scienti c publications. Objectives This course will help the students to write statistical ndings for scienti c publications. Learning Outcomes At the end of the course students will achieve skills on i) using statistical softwares and packages to analyse real-life data ii) tting models and interpret the results iii) to write report with statistical results for scienti c publications. Contents This is a comprehensive statistical data analysis course, which will cover computing problems related to all the courses studied in the academic year. It is requred to submit a report for this course. For this course, 30% weight will be alloted to the in-course examinations, 10% weight will be allotted to the lab-based assignments and report, and remaining 60% w

Course Credit: 2

Each student (Group A and Group B) must be examined orally by a committee of selected members at the end of the academic year.

Course Credit:

Assessment of the non-credit seminar course will be either \Satisfactory" or \Non-satisfactory". To get a "Satisfactory" grade, each student needs to attend at least 70% of the seminars or- ganised by the institute during the academic year. 23

Course Credit: 3

Each student should be either in the project report group or in the internship group that the student will decide after discussing with the respective assigned supervisor. Students must submit their project report or internship report within two months of completing the nal examination. The internal members of the examination committee will evaluate the performance of the students in the seminars and the project report or internship report will be evaluated by two examiners nominated by the examination committee. The supervisor cannot evaluate the project or internship report that s/he has supervised. For this course, 50% weight of the course will be allotted to report, 10% weight for supervisor, and the remaining 40% weight will be for a seminar presentation.

Course Credit: 6

After three months of the start, each student of Group B will submit and present a short thesis proposal (length 3-5 A4 pages excluding references, 1.5 line spacing). The proposal will be evaluated by the internal members of the examination committee. Written evaluation on the proposal will be provided to the students explaining the possible improvement and in case of \Not satisfactory" proposals, the reasons for \Not satisfactory" performance will be stated in the written evaluation. In case of \Not satisfactory" performance the examination committee may give the student a second opportunity for proposal presentation. Students with \Not-Satisfactory" performance will be transferred to the Group A. The nal submission of the thesis will be required within 4 months of the completion of nal exam. Thesis will be examined by two external (outside the institute) examiners. Should it be required, the examination committee may consider one internal and one external examiner. Submitted thesis has to be defended at a presentation evaluated by the members of the examination committee. Forty percent (40%) weight will be allotted for thesis defense and remaining 60% weight for thesis itself.