Multivariate Statistical Inference and Applications

Size: px
Start display at page:

Download "Multivariate Statistical Inference and Applications"

Transcription

1 Multivariate Statistical Inference and Applications ALVIN C. RENCHER Department of Statistics Brigham Young University A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

2 Contents Some Properties of Random Vectors and Matrices Introduction, Univariate and Bivariate Random Variables, Univariate Random Variables, Bivariate Random Variables, Mean Vectors and Covariance Matrices for Random Vectors, Correlation Matrices, Partitioned Mean Vectors and Covariance Matrices, Linear Functions of Random Variables, Sample Means, Variances, and Covariances, Population Means, Variances, and Covariances, Measuring Intercorrelation, Mahalanobis Distance, MissingData, Robust Estimators of fi and 2, 27 The Multivariate Normal Distribution Univariate and Multivariate Normal Density Functions, Univariate Normal, Multivariate Normal, Constant Density Ellipsoids, Generating Multivariate Normal Data, Moments, Properties of Multivariate Normal Random Vectors, Estimation of Parameters in the Multivariate Normal Distribution, 49 v

3 vi CONTENTS Maximum Likelihood Method, PropertiesofyandS, Wishart Distribution, Additional Topics, Hotelling's T 2 -Tests Introduction, Test for HQ: fi = fi 0 with 2 Known, Hotelling's T 2 -test for // 0 : fi = /x 0 with 2 Unknown, Univariate f-test for Ho: ix IM> with a 2 Unknown, Likelihood Ratio Method of Test Construction, One-Sample r 2 -Test, Formal Definition of T 2 and Relationship to F, Effect on 7 2 of Adding a Variable, Propertiesofthe7 2 -Test, Likelihood Ratio Test, Union-Intersection Test, Confidence Intervals and Tests for Linear Functions of fi, Confidence Region for ft, Confidence Interval for a Single Linear Combination a'fi, Simultaneous Confidence Intervals for IJLJ and a'/m, Bonferroni Confidence Intervals for /x, and a'/u, Tests for H 0 : a'/u, = a'jto and H 0 : ju, 7 = /xo ; -, Tests for H 0 : Cp = 0, Tests of H 0 : fx { = ft 2 Assuming 21 = 2 2, Review of Univariate Likelihood Ratio Test for H 0 : ix\ n>2 When a 2 = er 2, Test for H 0 : (JL { = fi 2 When 21 = 2 2, Effect on T 2 of Adding a Variable, Properties of thetwo-sample r 2 -Statistic, Likelihood Ratio and Union-Intersection Tests, Confidence Intervals and Tests for Linear Functions of Two Mean Vectors, Confidence Region for fn x fi, 2, Simultaneous Confidence Intervals for a'ipi ~~ M2) anc^ Mi; - M2;, Bonferroni Confidence Intervals for a'(/*i _ M2) anc * Mi; _ M2;, 94

4 CONTENTS vii Tests for H 0 :a'(fi 1 fi 2 ) = a '^o and H 0j : tnj ~ Mj = 0, Test for H 0 : C(/A, - fi 2 ) = 0, Robustness of the r 2 -test, Robustness to 2, + X 2, Robustness to Nonnormality, Paired Observation Test, Testing H 0 : Mi = M 2 When 21 = X 2, Univariate Case, Multivariate Case, Power and Sample Size, Tests on a Subvector, Two-Sample Case, Step-Down Test, Selectionof Variables, One-Sample Case, Nonnormal Approaches to Hypothesis Testing, Elliptically Contoured Distributions, Nonparametric Tests, Robust Versions of T 2, Application of T 2 In Multivariate Quality Control, Multivariate Analysis of Variance One-Way Classification, Model for One-Way Multivariate Analysis of Variance, Wilks' Likelihood Ratio Test, Roy's Union-Intersection Test, The Pillai and Lawley-Hotelling Test Statistics, Summary of the Four Test Statistics, Effect of an Additional Variable on Wilks' A, Tests on Individual Variables, Power and Robustness Comparisons for the Four MANOVA Test Statistics, Tests for Equality of Covariance Matrices, Power and Sample Size for the Four MANOVA Tests, Contrasts Among Mean Vectors, Univariate Contrasts, Multivariate Contrasts, 145

5 viii CONTENTS 4.6. Two-Way Multivariate Analysis of Variance, Higher Order Models, Unbalanced Data, Introduction, Univariate One-Way Model, Multivariate One-Way Model, Univariate Two-Way Model, Multivariate Two-Way Model, Tests on a Subvector, Testing a Single Subvector, Step-Down Test, Stepwise Selection of Variables, Multivariate Analysis of Covariance, Introduction, Univariate Analysis of Covariance: One-Way Model with One Covariate, Univariate Analysis of Covariance: Two-Way Model with One Covariate, Additional Topics in Univariate Analysis of Covariance, Multivariate Analysis of Covariance, Alternative Approaches to Testing Ho-t*>i ~ J*2 = - Pk> Discriminant Functions for Descriptive Group Separation Introduction, TwoGroups, Several Groups, Discriminant Functions, Assumptions, Standardized Coefficients, Tests of Hypotheses, TwoGroups, Several Groups, Discriminant Analysis for Higher Order Designs, Interpretation of Discriminant Functions, Standardized Coefficients and Partial F-Values, Correlations between Variables and Discriminant Functions, 211

6 CONTENTS ix Other Approaches, Confidence Intervals, Subset Selection, Discriminant Function Approach to Selection, Stepwise Selection, All Possible Subsets, Selection in Higher Order Designs, Bias in Subset Selection, Other Estimators of Discriminant Functions, Ridge Discriminant Analysis and Related Techniques, Robust Discriminant Analysis, Classification of Observations into Groups Introduction, Two Groups, Equal Population Covariance Matrices, Unequal Population Covariance Matrices, Unequal Costs of Misclassification, Posterior Probability Approach, Robustness to Departures from the Assumptions, Robust Procedures, Several Groups, Equal Population Covariance Matrices, Unequal Population Covariance Matrices, Use of Linear Discriminant Functions for Classification, Estimation of Error Rates, Correcting for Bias in the Apparent Error Rate, Partitioning the Sample, Holdout Method, Bootstrap Estimator, Comparison of Error Estimators, Subset Selection, Selection Based on Separation of Groups, Selection Based on Allocation, Selection in the Heteroscedastic Case, Bias in Stepwise Classification Analysis, Logistic and Probit Classification, 254

7 X CONTENTS The Logistic Model for Two Groups with 2, = 2 2 > Comparison of Logistic Classification with Linear Classification Functions, Quadratic Logistic Functions When X i = X2, Logistic Classification for Several Groups, Additional Topics in Logistic Classification, Probit Classification, Additional Topics in Classification, Multivariate Regression Introduction, Multiple Regression: Fixedx's, Least Squares Estimators and Properties, An Estimator for er 2, The Model in Centered Form, Hypothesis Tests and Confidence Intervals, R 2 in Fixed-Jt Regression, Model Validation, Multiple Regression: Random x's, Model for Random x's, Estimation of ßo> ß\, and CT 2, R 2 in Random-x Regression, Tests and Confidence Intervals, Estimation in the Multivariate Multiple Regression Model: Fixedx's, The Multivariate Model, Least Squares Estimator for B, Properties of B, An Estimator for X, Normal Model for the y,'s, The Multivariate Model in Centered Form, Measures of Multivariate Association, Hypothesis Tests in the Multivariate Multiple Regression Model: Fixedx's, Test for Significance of Regression, Test onasubsetof the RowsofB, General Linear Hypotheses CB = O and CBM = O, Tests and Confidence Intervals for a Single ßß and a Bilinear Function a'bb, 297

8 CONTENTS xi Simultaneous Tests and Confidence Intervals for the ß jk 's and Bilinear Functions a'bb, Tests in the Presence of Missing Data, Multivariate Model Validation: Fixedx's, Lack-of-Fit Tests, Residuais, Influence and Outliers, Measurement Errors, Multivariate Regression: Random x's, Multivariate Normal Model for Random x's, Estimationofßo, Bi.andX, Tests and Confidence Intervals in the Multivariate Random-x Case, Additional Topics, Correlated Response Methods, Categorical Data, Subset Selection, Other Topics, Canonical Correlation Introduction, Canonical Correlations and Canonical Variates, Properties of Canonical Correlations and Variates, Properties of Canonical Correlations, Properties of Canonical Variates, Tests of Significance for Canonical Correlations, Tests of Independence of y and x, Test of Dimension of Relationship between the y's and the x's, Validation, Interpretation of Canonical Variates, Standardized Coefficients, Rotation of Canonical Variate Coefficients, Correlations between Variables and Canonical Variates, Redundancy Analysis, Additional Topics, 333

9 xii CONTENTS 9. Principal Component Analysis Introduction, Definition and Properties of Principal Components, Maximum Variance Property, Principal Components as Projections, Properties of Principal Components, Principal Components as a Rotation of Axes, Principal Components from the Correlation Matrix, Methods for Discarding Components, Percent of Variance, Average Eigenvalue, Scree Graph, Significance Tests, Other Methods, Information in the Last Few Principal Components, Interpretation of Principal Components, Special Patterns in S or R, Testing H 0 : X = er 2 [(1 - p)i + pj] and P p = (1 - p)i + pj, Additional Rotation, Correlations between Variables and Principal Components, Relationship Between Principal Components and Regression, Principal Component Regression, Latent Root Regression, Principal Component Analysis with Grouped Data, Additional Topics, Factor Analysis Introduction, Basic Factor Model, Model and Assumptions, Scale Invariance of the Model, Rotation of Factor Loadings in the Model, Estimation of Loadings and Communalities, Principal Component Method, Principal Factor Method, Iterated Principal Factor Method, 384

10 CONTENTS xiii Maximum Likelihood Method, Other Methods, Comparison of Methods, Determining the Number of Factors, m, Rotation of Factor Loadings, Introduction, Orthogonal Rotation, Oblique Rotations, Interpretation of the Factors, Factor Scores, Applicability of the Factor Analysis Model, Factor Analysis and Grouped Data, Additional Topics, 394 Appendix A. Review of Matrix Algebra 399 A.l. Introduction, 399 A.l.l. Basic Definitions, 399 A.1.2. Matrices with Special Patterns, 400 A.2. Properties of Matrix Addition and Multiphcation, 401 A.3. Partitioned Matrices, 404 A.4. Rank of Matrices, 406 A.5. Inverse Matrices, 407 A.6. Positive Definite and Positive Semidefinite Matrices, 408 A.7. Determinants, 409 A.8. Traceofa Matrix, 410 A.9. Orthogonal Vectors and Matrices, 410 A.10. Eigenvalues and Eigenvectors, 411 A. 11. Eigenstructure of Symmetrie and Positive Definite Matrices, 412 A.12. Idempotent Matrices, 414 A.B. Differentiation, 414 Appendix B. Tables 417 Appendix C. Answers and Hints to Selected Problems 449 Appendix D. About the Diskette 505 Bibliography 507 Index 549

Applied Multivariate Analysis

Applied Multivariate Analysis Neil H. Timm Applied Multivariate Analysis With 42 Figures Springer Contents Preface Acknowledgments List of Tables List of Figures vii ix xix xxiii 1 Introduction 1 1.1 Overview 1 1.2 Multivariate Models

More information

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

More information

Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

More information

Computer-Aided Multivariate Analysis

Computer-Aided Multivariate Analysis Computer-Aided Multivariate Analysis FOURTH EDITION Abdelmonem Af if i Virginia A. Clark and Susanne May CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C Contents Preface

More information

Applied Regression Analysis and Other Multivariable Methods

Applied Regression Analysis and Other Multivariable Methods THIRD EDITION Applied Regression Analysis and Other Multivariable Methods David G. Kleinbaum Emory University Lawrence L. Kupper University of North Carolina, Chapel Hill Keith E. Muller University of

More information

Statistics Graduate Courses

Statistics Graduate Courses Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.

More information

Multivariate Normal Distribution

Multivariate Normal Distribution Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

More information

DISCRIMINANT FUNCTION ANALYSIS (DA)

DISCRIMINANT FUNCTION ANALYSIS (DA) DISCRIMINANT FUNCTION ANALYSIS (DA) John Poulsen and Aaron French Key words: assumptions, further reading, computations, standardized coefficents, structure matrix, tests of signficance Introduction Discriminant

More information

Least Squares Estimation

Least Squares Estimation Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

More information

Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences

Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences Third Edition Jacob Cohen (deceased) New York University Patricia Cohen New York State Psychiatric Institute and Columbia University

More information

CONTENTS PREFACE 1 INTRODUCTION 1 2 DATA VISUALIZATION 19

CONTENTS PREFACE 1 INTRODUCTION 1 2 DATA VISUALIZATION 19 PREFACE xi 1 INTRODUCTION 1 1.1 Overview 1 1.2 Definition 1 1.3 Preparation 2 1.3.1 Overview 2 1.3.2 Accessing Tabular Data 3 1.3.3 Accessing Unstructured Data 3 1.3.4 Understanding the Variables and Observations

More information

INTRODUCTORY STATISTICS

INTRODUCTORY STATISTICS INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

More information

Multivariate Analysis of Variance (MANOVA)

Multivariate Analysis of Variance (MANOVA) Chapter 415 Multivariate Analysis of Variance (MANOVA) Introduction Multivariate analysis of variance (MANOVA) is an extension of common analysis of variance (ANOVA). In ANOVA, differences among various

More information

Multivariate Analysis of Variance (MANOVA)

Multivariate Analysis of Variance (MANOVA) Multivariate Analysis of Variance (MANOVA) Aaron French, Marcelo Macedo, John Poulsen, Tyler Waterson and Angela Yu Keywords: MANCOVA, special cases, assumptions, further reading, computations Introduction

More information

Service courses for graduate students in degree programs other than the MS or PhD programs in Biostatistics.

Service courses for graduate students in degree programs other than the MS or PhD programs in Biostatistics. Course Catalog In order to be assured that all prerequisites are met, students must acquire a permission number from the education coordinator prior to enrolling in any Biostatistics course. Courses are

More information

MATHEMATICAL METHODS OF STATISTICS

MATHEMATICAL METHODS OF STATISTICS MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS

More information

Dimensionality Reduction: Principal Components Analysis

Dimensionality Reduction: Principal Components Analysis Dimensionality Reduction: Principal Components Analysis In data mining one often encounters situations where there are a large number of variables in the database. In such situations it is very likely

More information

Advanced Linear Modeling

Advanced Linear Modeling Ronald Christensen Advanced Linear Modeling Multivariate, Time Series, and Spatial Data; Nonparametric Regression and Response Surface Maximization Second Edition Springer Preface to the Second Edition

More information

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not. Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C

More information

STATISTICA Formula Guide: Logistic Regression. Table of Contents

STATISTICA Formula Guide: Logistic Regression. Table of Contents : Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

More information

Statistical Machine Learning

Statistical Machine Learning Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

More information

Multivariate Analysis of Ecological Data

Multivariate Analysis of Ecological Data Multivariate Analysis of Ecological Data MICHAEL GREENACRE Professor of Statistics at the Pompeu Fabra University in Barcelona, Spain RAUL PRIMICERIO Associate Professor of Ecology, Evolutionary Biology

More information

SPSS ADVANCED ANALYSIS WENDIANN SETHI SPRING 2011

SPSS ADVANCED ANALYSIS WENDIANN SETHI SPRING 2011 SPSS ADVANCED ANALYSIS WENDIANN SETHI SPRING 2011 Statistical techniques to be covered Explore relationships among variables Correlation Regression/Multiple regression Logistic regression Factor analysis

More information

Econometric Analysis of Cross Section and Panel Data Second Edition. Jeffrey M. Wooldridge. The MIT Press Cambridge, Massachusetts London, England

Econometric Analysis of Cross Section and Panel Data Second Edition. Jeffrey M. Wooldridge. The MIT Press Cambridge, Massachusetts London, England Econometric Analysis of Cross Section and Panel Data Second Edition Jeffrey M. Wooldridge The MIT Press Cambridge, Massachusetts London, England Preface Acknowledgments xxi xxix I INTRODUCTION AND BACKGROUND

More information

Data analysis process

Data analysis process Data analysis process Data collection and preparation Collect data Prepare codebook Set up structure of data Enter data Screen data for errors Exploration of data Descriptive Statistics Graphs Analysis

More information

Introduction to Principal Components and FactorAnalysis

Introduction to Principal Components and FactorAnalysis Introduction to Principal Components and FactorAnalysis Multivariate Analysis often starts out with data involving a substantial number of correlated variables. Principal Component Analysis (PCA) is a

More information

3. The Multivariate Normal Distribution

3. The Multivariate Normal Distribution 3. The Multivariate Normal Distribution 3.1 Introduction A generalization of the familiar bell shaped normal density to several dimensions plays a fundamental role in multivariate analysis While real data

More information

Multivariate normal distribution and testing for means (see MKB Ch 3)

Multivariate normal distribution and testing for means (see MKB Ch 3) Multivariate normal distribution and testing for means (see MKB Ch 3) Where are we going? 2 One-sample t-test (univariate).................................................. 3 Two-sample t-test (univariate).................................................

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

More information

Multivariate Analysis of Variance (MANOVA): I. Theory

Multivariate Analysis of Variance (MANOVA): I. Theory Gregory Carey, 1998 MANOVA: I - 1 Multivariate Analysis of Variance (MANOVA): I. Theory Introduction The purpose of a t test is to assess the likelihood that the means for two groups are sampled from the

More information

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components

Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they

More information

Chapter 14: Analyzing Relationships Between Variables

Chapter 14: Analyzing Relationships Between Variables Chapter Outlines for: Frey, L., Botan, C., & Kreps, G. (1999). Investigating communication: An introduction to research methods. (2nd ed.) Boston: Allyn & Bacon. Chapter 14: Analyzing Relationships Between

More information

Univariate and Multivariate Methods PEARSON. Addison Wesley

Univariate and Multivariate Methods PEARSON. Addison Wesley Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston

More information

Handling attrition and non-response in longitudinal data

Handling attrition and non-response in longitudinal data Longitudinal and Life Course Studies 2009 Volume 1 Issue 1 Pp 63-72 Handling attrition and non-response in longitudinal data Harvey Goldstein University of Bristol Correspondence. Professor H. Goldstein

More information

Empirical Model-Building and Response Surfaces

Empirical Model-Building and Response Surfaces Empirical Model-Building and Response Surfaces GEORGE E. P. BOX NORMAN R. DRAPER Technische Universitat Darmstadt FACHBEREICH INFORMATIK BIBLIOTHEK Invortar-Nf.-. Sachgsbiete: Standort: New York John Wiley

More information

Factor Analysis. Principal components factor analysis. Use of extracted factors in multivariate dependency models

Factor Analysis. Principal components factor analysis. Use of extracted factors in multivariate dependency models Factor Analysis Principal components factor analysis Use of extracted factors in multivariate dependency models 2 KEY CONCEPTS ***** Factor Analysis Interdependency technique Assumptions of factor analysis

More information

Random Vectors and the Variance Covariance Matrix

Random Vectors and the Variance Covariance Matrix Random Vectors and the Variance Covariance Matrix Definition 1. A random vector X is a vector (X 1, X 2,..., X p ) of jointly distributed random variables. As is customary in linear algebra, we will write

More information

Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

More information

D-optimal plans in observational studies

D-optimal plans in observational studies D-optimal plans in observational studies Constanze Pumplün Stefan Rüping Katharina Morik Claus Weihs October 11, 2005 Abstract This paper investigates the use of Design of Experiments in observational

More information

Teaching Multivariate Analysis to Business-Major Students

Teaching Multivariate Analysis to Business-Major Students Teaching Multivariate Analysis to Business-Major Students Wing-Keung Wong and Teck-Wong Soon - Kent Ridge, Singapore 1. Introduction During the last two or three decades, multivariate statistical analysis

More information

Statistical Analysis with Missing Data

Statistical Analysis with Missing Data Statistical Analysis with Missing Data Second Edition RODERICK J. A. LITTLE DONALD B. RUBIN WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION Contents Preface PARTI OVERVIEW AND BASIC APPROACHES

More information

Canonical Correlation Analysis

Canonical Correlation Analysis Canonical Correlation Analysis LEARNING OBJECTIVES Upon completing this chapter, you should be able to do the following: State the similarities and differences between multiple regression, factor analysis,

More information

Principal Component Analysis

Principal Component Analysis Principal Component Analysis Principle Component Analysis: A statistical technique used to examine the interrelations among a set of variables in order to identify the underlying structure of those variables.

More information

Factor analysis. Angela Montanari

Factor analysis. Angela Montanari Factor analysis Angela Montanari 1 Introduction Factor analysis is a statistical model that allows to explain the correlations between a large number of observed correlated variables through a small number

More information

Linear Threshold Units

Linear Threshold Units Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear

More information

Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy. Blue vs. Orange. Review Jeopardy Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

More information

Application of discriminant analysis to predict the class of degree for graduating students in a university system

Application of discriminant analysis to predict the class of degree for graduating students in a university system International Journal of Physical Sciences Vol. 4 (), pp. 06-0, January, 009 Available online at http://www.academicjournals.org/ijps ISSN 99-950 009 Academic Journals Full Length Research Paper Application

More information

Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm

Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm

More information

T-test & factor analysis

T-test & factor analysis Parametric tests T-test & factor analysis Better than non parametric tests Stringent assumptions More strings attached Assumes population distribution of sample is normal Major problem Alternatives Continue

More information

Regression Modeling Strategies

Regression Modeling Strategies Frank E. Harrell, Jr. Regression Modeling Strategies With Applications to Linear Models, Logistic Regression, and Survival Analysis With 141 Figures Springer Contents Preface Typographical Conventions

More information

Business Analytics. Methods, Models, and Decisions. James R. Evans : University of Cincinnati PEARSON

Business Analytics. Methods, Models, and Decisions. James R. Evans : University of Cincinnati PEARSON Business Analytics Methods, Models, and Decisions James R. Evans : University of Cincinnati PEARSON Boston Columbus Indianapolis New York San Francisco Upper Saddle River Amsterdam Cape Town Dubai London

More information

Methods for Meta-analysis in Medical Research

Methods for Meta-analysis in Medical Research Methods for Meta-analysis in Medical Research Alex J. Sutton University of Leicester, UK Keith R. Abrams University of Leicester, UK David R. Jones University of Leicester, UK Trevor A. Sheldon University

More information

Partial Least Squares (PLS) Regression.

Partial Least Squares (PLS) Regression. Partial Least Squares (PLS) Regression. Hervé Abdi 1 The University of Texas at Dallas Introduction Pls regression is a recent technique that generalizes and combines features from principal component

More information

Statistical Rules of Thumb

Statistical Rules of Thumb Statistical Rules of Thumb Second Edition Gerald van Belle University of Washington Department of Biostatistics and Department of Environmental and Occupational Health Sciences Seattle, WA WILEY AJOHN

More information

Multiple Regression: What Is It?

Multiple Regression: What Is It? Multiple Regression Multiple Regression: What Is It? Multiple regression is a collection of techniques in which there are multiple predictors of varying kinds and a single outcome We are interested in

More information

1 Introduction to Matrices

1 Introduction to Matrices 1 Introduction to Matrices In this section, important definitions and results from matrix algebra that are useful in regression analysis are introduced. While all statements below regarding the columns

More information

Multivariate Analysis. Overview

Multivariate Analysis. Overview Multivariate Analysis Overview Introduction Multivariate thinking Body of thought processes that illuminate the interrelatedness between and within sets of variables. The essence of multivariate thinking

More information

Chapter 7 Factor Analysis SPSS

Chapter 7 Factor Analysis SPSS Chapter 7 Factor Analysis SPSS Factor analysis attempts to identify underlying variables, or factors, that explain the pattern of correlations within a set of observed variables. Factor analysis is often

More information

Multivariate Analysis of Variance. The general purpose of multivariate analysis of variance (MANOVA) is to determine

Multivariate Analysis of Variance. The general purpose of multivariate analysis of variance (MANOVA) is to determine 2 - Manova 4.3.05 25 Multivariate Analysis of Variance What Multivariate Analysis of Variance is The general purpose of multivariate analysis of variance (MANOVA) is to determine whether multiple levels

More information

Factor Analysis. Chapter 420. Introduction

Factor Analysis. Chapter 420. Introduction Chapter 420 Introduction (FA) is an exploratory technique applied to a set of observed variables that seeks to find underlying factors (subsets of variables) from which the observed variables were generated.

More information

Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus

Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus Auxiliary Variables in Mixture Modeling: 3-Step Approaches Using Mplus Tihomir Asparouhov and Bengt Muthén Mplus Web Notes: No. 15 Version 8, August 5, 2014 1 Abstract This paper discusses alternatives

More information

Probability and Statistics

Probability and Statistics Probability and Statistics Syllabus for the TEMPUS SEE PhD Course (Podgorica, April 4 29, 2011) Franz Kappel 1 Institute for Mathematics and Scientific Computing University of Graz Žaneta Popeska 2 Faculty

More information

Algebra 1 Course Information

Algebra 1 Course Information Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

More information

Sections 2.11 and 5.8

Sections 2.11 and 5.8 Sections 211 and 58 Timothy Hanson Department of Statistics, University of South Carolina Stat 704: Data Analysis I 1/25 Gesell data Let X be the age in in months a child speaks his/her first word and

More information

Multivariate analyses

Multivariate analyses 14 Multivariate analyses Learning objectives By the end of this chapter you should be able to: Recognise when it is appropriate to use multivariate analyses (MANOVA) and which test to use (traditional

More information

Elements of statistics (MATH0487-1)

Elements of statistics (MATH0487-1) Elements of statistics (MATH0487-1) Prof. Dr. Dr. K. Van Steen University of Liège, Belgium December 10, 2012 Introduction to Statistics Basic Probability Revisited Sampling Exploratory Data Analysis -

More information

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( ) Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

More information

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics. Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

More information

business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar

business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel

More information

A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS

A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS Eusebio GÓMEZ, Miguel A. GÓMEZ-VILLEGAS and J. Miguel MARÍN Abstract In this paper it is taken up a revision and characterization of the class of

More information

Graduate Programs in Statistics

Graduate Programs in Statistics Graduate Programs in Statistics Course Titles STAT 100 CALCULUS AND MATR IX ALGEBRA FOR STATISTICS. Differential and integral calculus; infinite series; matrix algebra STAT 195 INTRODUCTION TO MATHEMATICAL

More information

THE MULTIVARIATE ANALYSIS RESEARCH GROUP. Carles M Cuadras Departament d Estadística Facultat de Biologia Universitat de Barcelona

THE MULTIVARIATE ANALYSIS RESEARCH GROUP. Carles M Cuadras Departament d Estadística Facultat de Biologia Universitat de Barcelona THE MULTIVARIATE ANALYSIS RESEARCH GROUP Carles M Cuadras Departament d Estadística Facultat de Biologia Universitat de Barcelona The set of statistical methods known as Multivariate Analysis covers a

More information

Analysis of Variance. MINITAB User s Guide 2 3-1

Analysis of Variance. MINITAB User s Guide 2 3-1 3 Analysis of Variance Analysis of Variance Overview, 3-2 One-Way Analysis of Variance, 3-5 Two-Way Analysis of Variance, 3-11 Analysis of Means, 3-13 Overview of Balanced ANOVA and GLM, 3-18 Balanced

More information

SAS Certificate Applied Statistics and SAS Programming

SAS Certificate Applied Statistics and SAS Programming SAS Certificate Applied Statistics and SAS Programming SAS Certificate Applied Statistics and Advanced SAS Programming Brigham Young University Department of Statistics offers an Applied Statistics and

More information

Advanced Signal Processing and Digital Noise Reduction

Advanced Signal Processing and Digital Noise Reduction Advanced Signal Processing and Digital Noise Reduction Saeed V. Vaseghi Queen's University of Belfast UK WILEY HTEUBNER A Partnership between John Wiley & Sons and B. G. Teubner Publishers Chichester New

More information

STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct

More information

Didacticiel - Études de cas

Didacticiel - Études de cas 1 Topic Linear Discriminant Analysis Data Mining Tools Comparison (Tanagra, R, SAS and SPSS). Linear discriminant analysis is a popular method in domains of statistics, machine learning and pattern recognition.

More information

MEU. INSTITUTE OF HEALTH SCIENCES COURSE SYLLABUS. Biostatistics

MEU. INSTITUTE OF HEALTH SCIENCES COURSE SYLLABUS. Biostatistics MEU. INSTITUTE OF HEALTH SCIENCES COURSE SYLLABUS title- course code: Program name: Contingency Tables and Log Linear Models Level Biostatistics Hours/week Ther. Recite. Lab. Others Total Master of Sci.

More information

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance

More information

11 Linear and Quadratic Discriminant Analysis, Logistic Regression, and Partial Least Squares Regression

11 Linear and Quadratic Discriminant Analysis, Logistic Regression, and Partial Least Squares Regression Frank C Porter and Ilya Narsky: Statistical Analysis Techniques in Particle Physics Chap. c11 2013/9/9 page 221 le-tex 221 11 Linear and Quadratic Discriminant Analysis, Logistic Regression, and Partial

More information

Chapter 6: Multivariate Cointegration Analysis

Chapter 6: Multivariate Cointegration Analysis Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration

More information

MTH 140 Statistics Videos

MTH 140 Statistics Videos MTH 140 Statistics Videos Chapter 1 Picturing Distributions with Graphs Individuals and Variables Categorical Variables: Pie Charts and Bar Graphs Categorical Variables: Pie Charts and Bar Graphs Quantitative

More information

APPLIED MISSING DATA ANALYSIS

APPLIED MISSING DATA ANALYSIS APPLIED MISSING DATA ANALYSIS Craig K. Enders Series Editor's Note by Todd D. little THE GUILFORD PRESS New York London Contents 1 An Introduction to Missing Data 1 1.1 Introduction 1 1.2 Chapter Overview

More information

PROBABILITY AND STATISTICS. Ma 527. 1. To teach a knowledge of combinatorial reasoning.

PROBABILITY AND STATISTICS. Ma 527. 1. To teach a knowledge of combinatorial reasoning. PROBABILITY AND STATISTICS Ma 527 Course Description Prefaced by a study of the foundations of probability and statistics, this course is an extension of the elements of probability and statistics introduced

More information

Section 13, Part 1 ANOVA. Analysis Of Variance

Section 13, Part 1 ANOVA. Analysis Of Variance Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability

More information

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

More information

Some probability and statistics

Some probability and statistics Appendix A Some probability and statistics A Probabilities, random variables and their distribution We summarize a few of the basic concepts of random variables, usually denoted by capital letters, X,Y,

More information

Section Format Day Begin End Building Rm# Instructor. 001 Lecture Tue 6:45 PM 8:40 PM Silver 401 Ballerini

Section Format Day Begin End Building Rm# Instructor. 001 Lecture Tue 6:45 PM 8:40 PM Silver 401 Ballerini NEW YORK UNIVERSITY ROBERT F. WAGNER GRADUATE SCHOOL OF PUBLIC SERVICE Course Syllabus Spring 2016 Statistical Methods for Public, Nonprofit, and Health Management Section Format Day Begin End Building

More information

Imputing Values to Missing Data

Imputing Values to Missing Data Imputing Values to Missing Data In federated data, between 30%-70% of the data points will have at least one missing attribute - data wastage if we ignore all records with a missing value Remaining data

More information

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

More information

QUALITY ENGINEERING PROGRAM

QUALITY ENGINEERING PROGRAM QUALITY ENGINEERING PROGRAM Production engineering deals with the practical engineering problems that occur in manufacturing planning, manufacturing processes and in the integration of the facilities and

More information

1. Complete the sentence with the correct word or phrase. 2. Fill in blanks in a source table with the correct formuli for df, MS, and F.

1. Complete the sentence with the correct word or phrase. 2. Fill in blanks in a source table with the correct formuli for df, MS, and F. Final Exam 1. Complete the sentence with the correct word or phrase. 2. Fill in blanks in a source table with the correct formuli for df, MS, and F. 3. Identify the graphic form and nature of the source

More information

How to report the percentage of explained common variance in exploratory factor analysis

How to report the percentage of explained common variance in exploratory factor analysis UNIVERSITAT ROVIRA I VIRGILI How to report the percentage of explained common variance in exploratory factor analysis Tarragona 2013 Please reference this document as: Lorenzo-Seva, U. (2013). How to report

More information

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical

More information

Lecture 3: Linear methods for classification

Lecture 3: Linear methods for classification Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,

More information

SPSS Advanced Statistics 17.0

SPSS Advanced Statistics 17.0 i SPSS Advanced Statistics 17.0 For more information about SPSS Inc. software products, please visit our Web site at http://www.spss.com or contact SPSS Inc. 233 South Wacker Drive, 11th Floor Chicago,

More information

A THEORETICAL COMPARISON OF DATA MASKING TECHNIQUES FOR NUMERICAL MICRODATA

A THEORETICAL COMPARISON OF DATA MASKING TECHNIQUES FOR NUMERICAL MICRODATA A THEORETICAL COMPARISON OF DATA MASKING TECHNIQUES FOR NUMERICAL MICRODATA Krish Muralidhar University of Kentucky Rathindra Sarathy Oklahoma State University Agency Internal User Unmasked Result Subjects

More information

STATISTICAL PACKAGE FOR THE SOCIAL SCIENCES

STATISTICAL PACKAGE FOR THE SOCIAL SCIENCES STATISTICAL PACKAGE FOR THE SOCIAL SCIENCES SECOND EDITION NORMAN H. NIE Department of Political Science and National Opinion Research Center University Of Chicago C. HADLAI HULL Computation Center University

More information

Sun Li Centre for Academic Computing lsun@smu.edu.sg

Sun Li Centre for Academic Computing lsun@smu.edu.sg Sun Li Centre for Academic Computing lsun@smu.edu.sg Elementary Data Analysis Group Comparison & One-way ANOVA Non-parametric Tests Correlations General Linear Regression Logistic Models Binary Logistic

More information