Statistical Methods for Analysis of Spatial Data

List of Courses of the Board of Study

Course No	Course Title	Credits
First Semester
ST 5101	Calculus and Matrix Algebra	2
ST 5102	Basic Statistics	2
ST 5103	Data Analysis using Statistical Software	3
ST 5104	Sampling Techniques	2
ST 5105	Time Series Analysis	2
ST 5106	Computer Programming	2
ST 5151	Statistical Theory	3
ST 5152	Exploratory and Robust Data Analysis	2
ST 5153	Modeling Binary Data	2
ST 5154	Statistical Genetics	2
ST 5198	Directed Study	2
ST 5199	Seminar	1
ST 6101	Vector Analysis	2
ST 6102	Measure Theory	2
ST 6103	Group Theory	2
ST 6151	Variance Components Estimation	2
ST 6152	Advanced Designs and Analysis of Experiments	2
Second Semester
ST 5201	Advanced Calculus	2
ST 5202	Design and Analysis of Experiments	2
ST 5203	Regression Analysis	2
ST 5204	Nonparametric Statistics	2
ST 5205	Categorical data Analysis	2
ST 5251	Statistical Methods for Analysis of Spatial Data	3
ST 5252	Design and Analysis of Epidemiological Experiments	2
ST 5253	Crop and Animal Experimentation	3
ST 6201	Linear Models	3
ST 6202	Multivariate Statistical Methods	3
ST 6203	Stochastic processes	2
ST 6251	Statistical Computing	2
ST 6252	Analysis of Repeated Measurements	2

COURSE CAPSULES OF THE BOARD OF STUDY IN BIO-STATISTICS

Courses Offered in the First Semester

            ST 5101.Calculus and Matrix Algebra (2)

            Preliminaries: Number system, Concept of sets, Intervals, Inequalities, relations and functions,        Types of functions. Trigonometry: Basic trigonometric identifies, Trigonometric functions and inverse functions, Additive formulas. Differentiation: Concept of limit, Concept of derivatives,        Techniques of differentiation, Differentiation of algebraic, logarithmic, exponential and        trigonometric functions, Determination of maxima and minima, partial derivatives, Total        differential. Integration: Basic rules of integration, Definite integral, Application of integral          calculus. Matrix algebra: Different forms of matrices, Matrix operations, Determinants,       Conditions for non singularity, Matrix inverse, Solution of set of linear equations, Cramer s rule,      Quadratic forms, Jacobian and Hessian determinants, Eigen vectors and Eigen values.

            ST 5102. Basic Statistics (2)

            Variability in observations, Frequency distributions and histograms, Stem and leaf and Box           plots. Population and sample. Probability structure and cumulative distribution functions.            Expected values and moments. The family of normal distributions. Statistical inference. Point      and interval estimation. Tests of hypothesis. Introduction to Analysis of variance (ANOVA).       Linear regression and correlation. Estimation and tests on proportions. Contingency tables and          test of associations.

            ST 5103. Data Analysis Using Statistical Software (3)

            Introduction to statistical software. MINITAB, basics and features. Data entry, data              manipulation using MINITAB, Histograms, boxplots, Graphs using MINITAB. Analysis of     variance of one-way, two way and factorials. Analysis of blocked and unbalanced designs.   Regression analysis, Simple linear and multiple regression analysis, Macros in Minitab.     Introduction to SAS. Data management in SAS, Advanced features, SAS for the analysis of          blocked and unbalanced designs. Regression analysis in SAS, Simple linear and multiple    regression, Non-linear regression in SAS. Non parametric statistical analysis in SAS.     Multivariate analysis in SAS. Analysis generalized linear models using GLIM, Macro  programming in SAS.

            ST 5104. Sampling Techniques (2)

            Role of sampling, Simple random sampling, Use of random number tables for drawing samples,        Stratified random sampling, Proportional allocation, Equal allocation, Determination of sample             size, Neyman allocation, Precision of estimates under different allocations, Sampling incidence       observations, Sampling in forest experiments, Sample size for proportions, Systematic sampling     and precision under systematic sampling, Other types of sampling methods including multi  

            stage sampling.

            ST 5105. Time Series Analysis (2)

            Pre requisites: ST 5102

            Trend Analysis, Smoothing techniques (Moving averages, Weighted moving averages),       Decomposition techniques, Seasonal adjustments, Stochastic process; Stationary process; white       noise stochastic process, Markov Chain process.  General Linear Process, Autocovariance and autocorrelation functions, Estimation of Autocovariance &  Autocorrelation function,           Estimation of partial autocorrelation, Backshift operation notation, Stationary and Inevitability   conditions for a linear process. Autoregressive (AR) process,  Moving average (MA) process,       Autocovariance generating function of AR and MA  process, stationary and Inevitability  conditions for AR and MA process, Mixed autoregressive moving average process (ARMA and        ARIMA), Time series modeling, and diagnostic checking and forecasting, Fourier Analysis,             Multivariate Time series.

            Practical: Analysis of large data set SAS.

            ST 5106. Computer Programming (2)

            Introduction to computers. Computer terminology. The Structure of BASIC. Input/output    Statements. Arrays, branching and looping. Sub programmes, Sequential and Random files.             Introduction to LOTUS 1-2-3 and SAS (Statistical Analysis System).

            ST 5151. Statistical Theory (3)

            Probability: Properties of random vectors, Conditional probability, independence. Discrete          random variables; Probability mass functions and cumulative distributions. Some common            discrete distributions. Continuous random variables: Expected values and moments, moment           generating and characteristic functions. Marginal and conditional distributions, Bayes Rule.           Expectations and Central Limit Theorem. Sampling from the Normal distribution. Point and   interval estimation. Test of Hypotheses: Simple and composite hypothesis. Maximum likelihood     estimation. Generalized Likelihood Ratio Tests. Tests of means and variances.

            ST 5152. Exploratory and Robust Data Analysis (2)

            i. Exploratory data analysis for the location model Basic data displays, The box-plot, The                     empirical cumulative distribution plot, Some comments on order statistics, Transformation of data, The symmetry plot and probability plotting.

            ii Combining Exploratory and Confirmatory data analysis for the location model Tests for          normality, Review of some concepts of statistical theory, Least squares and weighted least             squares estimation in the location model, Approximate variance of functions of random  variables.

            iii Robust estimation in the location model Parameters and the estimation as functional, The           influence curve, The general concepts of Robustness, Robust efficiency, L-estimators, M   estimators, The Monte Carlo method, The Bootstrap method.

            iv Comparing two or more groups of data : Exploratory, Classical inference and other forms     of analysis One way ANOVA, Transformation in one way ANOVA, The box-cox      transformation, Robust estimation in the SL, Multiple linear and non parametric regression.

            ST 5153. Modeling Binary Data (2)

            Experiments with binary outcomes, The binomial distribution, Negative binomial distribution,    Geometric distribution and Multinomial distribution, Fitting binomial distribution, Testing two        proportions, Methods of estimation of parameters of the Bernoulli and binomial distribution,             Profile and conditional likelihood, Models for binary data, The linear logistics and logistics           regression model, Parameters estimates in linear logistic model, Residuals and model          diagnostics, Over dispersion in binomial responses.

            ST 5154. Statistical Genetics (2)

            Genetic structures, Hardy-Weinberg equilibrium, Estimation of gene frequencies, Sex linked       gene, Inheritance of quantitative characters, Covariance among relatives Coefficient of             inbreeding, Estimation of genetic variance components, Diallel crossing systems, Index         selection, Genotype-environment interactions.

            ST 5198. Directed Study (2)

            ST 5199. Seminar (1)

            ST 6101. Vector Analysis (2)

            Pre-requisite: ST 5101

            Vector Algebra: Scalars and Vectors; Addition and multiplication by a scalar; Unit vectors;           (i,j,k) Components of a vector; Linear dependence and independence; Scalar and vector fields;        Scalar product of two vectors; a.b) Vector product of two vectors; (a X         b) Triple scalar and    vector products; Solution to vector equations. Vector Analysis: Ordinary     derivative of a vector;            Unit tangent vector and principal normal to a space curve; Partial        derivatives of vectors;            Differentials of vectors; Differential geometry and applications; Gradient, Divergence and curl      operators; in rotational and solenoidal fields; Line, surface and volume integrals, Stoke s         theorem and Gauss divergence theorem; Tensors and their    fundamental operations.

            ST 6102. Measure Theory (2)

            Lebesgue measure on the real line and its properties; Abstract measure space and measurable     functions; Lebesgue integral; Fatou s Lemma, the Monotone and Dominated Convergence   Theorems; Lp-space; Models of convergence; convergence almost everywhere, convergence in   norm and in measure; Signed measures; product measures and Fubini-Tonelli Theorems.

            ST6103. Group Theory (2)

            Introduction to Groups, Cyclic Groups, Abelian Groups, Permutation Groups, Normal            Subgroups, Factor Groups, Homomorphism and Isomorphism theorem on Groups, Class of         Groups, Radicals and residuals, Action of Groups on sets, Semi-direct products, Series, Soluble        Groups, Sylow theorems, p Groups, and Nilpotent Groups.

            ST 6151. Variance Component Estimation (2)

            Pre-requisite: ST 6201

             Mixed and random effect models, Properties of quadratic forms, Methods of variance          components estimation; Henderson methods, MIVQUE, EM Algorithm, Maximum likelihood,            Restricted maximum likelihood, Derivative free methods, Bayesian Estimation, Gibbs   sampling. Emphasis on application and computing strategies.

            ST 6152. Advanced Designs and Analysis of Experiments (2)

            Pre-requisite: ST 5202

            Confounding and fractional factorial in 2n, 3n, and pn experiments. Asymmetric factorials.        Construction of designs (BIB and PBIB). Lattice designs. Unbalance designs.

7.6.6.2 Courses Offered in the Second Semester

            ST 5201. Advanced Calculus (2)

            Pre-requisite: ST 5101

             Determination of maxima and minima with two variables, Maclaurin and Taylor series, Nth       derivative test for relative extremum. Indeterminate forms, Curve tracing, Advance integration   techniques, Application of integration. Constrained optimization with grange multipliers.     Differential equations of the 1st order; Definitions, formation of  differential equations,            particular integrals and complementary functions, methods of solution, Clairaut s equations,           Orthogonal trajectories.

            a.  Differential equation of higher orders: Linear equation with constant co-efficient, Linear             differential operators, Simultaneous linear equations with constant coefficient.

            b.  Elementary partial differential equations; Numerical solutions.

            c.   Mathematical modeling for biological systems.

            ST 5202. Design and Analysis of Experiments (2)

            Pre-requisite: ST 5102 and ST 5101

            Principles of experimental design. Completely randomized, Randomized Complete Block, and       Latin square designs. Covariance analysis, Transformation of data, Factorial Experiments. Fixed    effects and random effect models. Sub-sampling, nested factor designs. Confounding in 2n   factorial experiments. Fractional factorials (2n), split-plot designs. Incomplete Block Designs, BIB and PBIB designs.

            ST 5203. Regression Analysis (2)

            Pre-requisite: ST 5102

            Matrix approach to linear regression. Multiple linear regression; General Linear Models, Least             Squares procedure, Inferences in regression, Model selection procedures, Analysis of residuals,     Influence diagnostics, Detecting and combating multicollinearity, Nonstandard conditions,  Violation of assumptions, Transformations. Non-linear regression; Non-linear least squares,      Gauss-Newton procedure for finding estimates, other modifications of Gauss- Newton 

            procedure.

            ST 5204. Non Parametric Statistics (2)

            Scale of measurement. Empirical cumulative distribution functions. Rank tests for comparing       two treatments, Blocked comparison for two treatments, Paired comparison and the one sample     problem. The comparison of more than two treatments. Randomized complete blocks.

            Practical: exercises using SAS & MINITAB.

            ST 5205. Categorical Data Analysis (2)

            Categorical response data, inference for two-way tables; Sampling distributions, Testing            goodness of fit and independence, Large Sample confidence intervals. Log linear models: Direct and indirect models, Iterative maximum likelihood and weighted least squares estimation,        Sufficiency and likelihood for log linear models. Estimating model parameters. Testing     goodness of fit for log linear models. Estimating model parameters. Partitioning chi-square to          compare models, Strategies in model selection. Analysis of residuals. Testing conditional    independence. Logit model for categorical data; Generalized logit, multinomial logit and         cumulative logit models.  Log liner and logit models for ordered categorical data; Testing   independence for ordinal classification and models for ordered variables in multi-dimensional       tables.

            ST 5251. Statistical Methods for Analysis of Spatial Data (3)

            Introduction to spatial data, Co-ordinate / projection systems, Raster data, Vector data,    Dimensions of spatial data, Statistics and spatial data, Probability concepts related to remote        sensing theories, Network data, Quantitative geometry of stream network. Problems of    descriptive statistics for spatial data, Temporal analysis of spatial data, Processing of spatial data      (image data), Enhancement techniques, Spatial sampling techniques, Spatial data classification,          Re-sampling techniques, errors of spatial data, Other applications statistical techniques for             spatial data.

            Practical; Use of Statistical and Spatial information system software, SAS, SPANS, GIS,                 ERDAS

            ST 5252. Design and Analysis of Epidemiological Experiments (2)

            Basic designs for aetiological studies; Cohort and case control studies, Analysis of data from        intervention studies, Fitting models to cohort and clinical trial data, Model selection and  interpreting parameters in linear logistic models, Models with several exposure factors and confounders, Fitting model to data from case control studies, Analysis of bioassay experiments,    Analysis of infectious disease data.

            ST 5253. Crop and Animal Experimentation (3)

            Selection of site, Size, shape and orientation of plots and blocks, Uniformity trials, Systematic      spacing design, Design and analysis of intercropping experiments, Use of control, choosing        levels of a factor, Number of replications, Yield density models, Growth curves, Sequential and          particle S.S., Particle correlation, Step wise regression, Serial correlation, Use of multivariate          techniques in agricultural experimentation, Non linear regression, Experimentation with            vegetables, fruits and tree crops Experimental units (large and small animals), Selection of             animals for experimentation, Adaptation period, Preliminary and sample collection period,    Carry over effects, herd %, year %, Seasonal effect, Choice of covariate in animal     experimentation, Non- linear regression, lactation curves, growth curves etc. Intra class      correlation among full and half sibs, Introduction to sensory evaluation techniques, Exercises             using Statistical Analysis System (SAS).

            ST 6201. Linear Models (3)

            Matrix concepts, Distribution of quadratic forms. General Linear Models (Full rank);              Estimation and Hypothesis testing. Less than full rank models. Methods to combat            Multicollinearity; Ridge regression.

            ST 6202. Multivariate Statistical Methods (3)

            Pre-requisites: ST 5101 and ST 5202

            Introduction, Multivariate normal distribution, Marginal and conditional distributions, Expected          values, Variance and covariance matrices. Descriptive multivariate methods, Multivariate plots      and diagrams, Preliminary analysis of multivariate data. Linear Transformation Basics and           methods of Principal Components Analysis (PCA). Component scores and loading. Optimum    number of components. Interpretation. Practical uses of PCA. Factor analysis. Factor model.             Comparison with PCA. Estimation of the factor loadings. Rotation of factors. Interpretation and       use. Limitation and problem with factor models. Multivariate Analysis of Variance (MANOVA),

            Calculations, test of hypothesis. Canonical, multiple and partial correlation, Cluster analysis.          Measures of similarity and dissimilarity, Dendogram, Different methods of clustering, Use and     limitation of cluster analysis. Discriminant analysis.

            ST 6203. Stochastic Processes (2)

            Pre-requisite: ST 5151

            Markov chains on discrete space in discrete and continuous time (random walks, Poisson           processes, birth and death processes) and their long-term behavior. Branching processes, renewal theory, Brownian motion.

            ST 6251. Statistical Computing (2)

            Pre-requisite: ST 5101 and ST 5102

            S-Plus programming language. Numerical computations and algorithms with application in statistics; Solution of equations, Matrix Computations, Linear algebra applications, Numerical         integration, Linear and nonlinear least squares and regression computations, random number          generation, application of Monte Carlo methods in statistical research. Optimization methods; Newton-Raphson, Linear Search and Gradient Methods, Direct Search: Nelder- Mead Simplex      Algorithm, Gauss-Newton Algorithm, Levenberg-Marquard and other modifications of Gauss-

            Newton. Computations associated with estimation; EM Algorithm, Maximum Likelihood      Estimation: N-R and Scoring, Robust Regression Computations, Re-sampling Methods:     Bootstrapping.

            ST 6252. Analysis of Repeated Measurements (2)

            Identification of repeated measures experiments in different fields. Matrix form of statistical        models, basic notation. Experimental designs involving several sizes of experimental units, split        plot type designs, Repeated measures designs. Variance-Covariance matrix, compound           symmetry, Huynh-Field condition. Partitioning within subject and between subject effects, advantages and disadvantages of RM designs. Comparison of treatments, trends, Analysis of         Repeated measures data using statistical software.

            Quantitative Techniques for Behavioural Science (2:30/00)

Pre requisite: ST5101 and ST 5102

Special issues of Statistics in behavioural sciences; Sampling in behavioural science studies; Applications of common probability distributions: uniform, Bernoulli, binomial, Poisson and normal; Decision theory; Use of linear programming in decision making; Network analysis: activity time estimates, critical path method, network diagrams, analysis of projects; Identifying relationships: applications of simple and multiple linear regression; Identifying direct relationships: path analysis; Testing association between attributes; Use of non parametric procedures in behavioural sciences; Construction of indices using multivariate techniques; Testing reliability and validity of indices; Identifying common factors; Grouping similar responses; Risk analysis and modelling discrete responses; Use of statistical software (SPSS) on implementing above applications.