Statistical Methods - IIT Kanpur

1 8 Statistical MethodsRaghu Nandan Sengupta and Debasis Basic Concepts of Data Analysis .. Probability .. Space and Events .. , Interpretations, and Properties of Probability .. -Field, Random Variables, and SomeImportantResults .. Estimation .. of Estimation .. MethodofMomentEstimators .. Estimators .. Linear and Nonlinear Regression Analysis .. Regression Analysis .. Bayesian Inference .. Regression Introduction to Multivariate Analysis .. JointandMarginalDistribution .. MultinomialDistribution .. Multivariate MultivariateExtremeValueDistribution.

2 MLEE stimatesofParameters(RelatedtoMNDOnly) .. Copula theory .. Principal Component Analysis .. Factor Analysis .. Mathematical Formulation of Factor Analysis .. Estimation in Factor Analysis .. Principal Component Method .. Maximum Likelihood Method .. General Working Principle for FA .. Multiple Analysis of Variance and Multiple Analysis of Covariance .. Introduction to Analysis of Variance .. Multiple Analysis of Variance .. ConjointAnalysis .. 475413414 decision Canonical Correlation Analysis .. Formulation of Canonical Correlation Analysis.

3 Standardized Form of CCA .. Correlation between Canonical Variates and Their Component TestingtheTestStatisticsinCCA .. Geometric and Graphical Interpretation of CCA .. Conclusions about CCA .. ClusterAnalysis .. ClusteringAlgorithms .. Multiple Discriminant and Classification Analysis .. Multidimensional Scaling .. StructuralEquationModeling .. FutureAreasofResearch .. 501 References .. 502 ABSTRACTThe chapter of Statistical Methods starts with the basic concepts of data analysisand then leads into the concepts of probability, important properties of probability, limit theorems,and inequalities.

4 The chapter also covers the basic tenets of estimation, desirable properties of esti-mates, before going on to the topic of maximum likelihood estimation, general Methods of moments,Baye s estimation principle. Under linear and nonlinear regression different concepts of regressionsare discussed. After which we discuss few important multivariate distributions and devote sometime on copula theory also. In the later part of the chapter, emphasis is laid on both the theoreticalcontent as well as the practical applications of a variety of multivariate techniques like PrincipleComponent Analysis (PCA), Factor Analysis, Analysis of Variance (ANOVA), Multivariate Analy-sis of Variance (MANOVA), Conjoint Analysis, Canonical Correlation, Cluster Analysis, MultipleDiscriminant Analysis, Multidimensional Scaling, Structural Equation Modeling, etc.

5 Finally, thechapter ends with a good repertoire of information related to softwares, data sets, journals, etc.,related to the topics covered in this IntroductionMany people are familiar with the termstatistics. It denotes recording of numerical facts and figures,for example, the daily prices of selected stocks on a stock exchange, the annual employment andunemployment of a country, the daily rainfall in the monsoon season, etc. However, statistics dealswith situations in which the occurrence of some events cannot be predicted with certainty. It alsoprovides Methods for organizing and summarizing facts and for using information to draw , the wordstatisticsis derived from the Latin wordstatusmeaningstate.

6 For severaldecades, statistics was associated solely with the display of facts and figures pertaining to eco-nomic, demographic, and political situations prevailing in a country. As a subject, statistics nowencompasses concepts and Methods that are of far-reaching importance in all enquires/questionsthat involve planning or designing of the experiment, gathering of data by a process of experimen-tation or observation, and finally making inference or conclusions by analyzing such data, whicheventually helps in making the future finding through the collection of data is not confined to professional researchers.

7 It is apart of the everyday life of all people who strive, consciously or unconsciously, to know mattersof interest concerning society, living conditions, the environment, and the world at large. SourcesDownloaded by [Debasis Kundu] at 16:48 25 January 2017 Statistical Methods415of factual information range from individual experience to reports in the news media, governmentrecords, and articles published in professional journals. Weather forecasts, market reports, costs ofliving indexes, and the results of public opinion are some other examples. Statistical Methods areemployed extensively in the production of such reports.

8 Reports that are based on sound statisticalreasoning and careful interpretation of conclusions are truly informative. However, the deliberate orinadvertent misuse of statistics leads to erroneous conclusions and distortions of Basic Concepts of Data AnalysisIn order to clarify the preceding generalities, a few examples are provided:Socioeconomic surveys:In the interdisciplinary areas of sociology, economics, and politicalscience, such aspects are taken as the economic well-being of different ethnic groups,consumer expenditure patterns of different income levels, and attitudes toward pendinglegislation.

9 Such studies are typically based on data oriented by interviewing or contactinga representative sample of person selected by Statistical process from a large populationthat forms the domain of study. The data are then analyzed and interpretations of the issuein questions are made. See, for example, a recent monograph by Bandyopadhyay et al.(2011) on this diagnosis:Early detection is of paramount importance for the successful surgicaltreatment of many types of fatal diseases, say, for example, cancer or AIDS. Becausefrequent in-hospital checkups are expensive or inconvenient, doctors are searching foreffective diagnosis process that patients can administer themselves.

10 To determine the mer-its of a new process in terms of its rates of success in detecting true cases avoiding falsedetection, the process must be field tested on a large number of persons, who must thenundergo in-hospital diagnostic test for comparison. Therefore, proper planning (designingthe experiments) and data collection are required, which then need to be analyzed for finalconclusions. An extensive survey of the different Statistical Methods used in clinical trialdesign can be found in Chen et al. (2015).Plant breeding:Experiments involving the cross fertilization of different genetic types ofplant species to produce high-yielding hybrids are of considerable interest to agriculturalscientists.