A Course in Statistics with R
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"Integrates the theory and applications of statistics using R the book has been written to bridge the gap between theory and applications and explain how mathematical expressions are converted into R programs. The book has been primarily designed as a useful companion for a Masters student during each semester of the course, but will also help applied statisticians in revisiting the underpinnings of the subject." (Zentralblatt MATH 2016)More details
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Prabhanjan Tattar , Business Analysis Senior Advisor at Dell International Services, Bangalore, India. Professor Tattar is a statistician providing analytical solutions to business problems inclusive of statistical models and machine learning as appropriate.
Suresh Ramaiah, Assistant Professor of Statistics at Dharwad University, Dharwad, India.
B G Manjunath, Business Analysis Advisor at Dell International Services, Bangalore, India
Content
List of Figures xvii
List of Tables xxi
Preface xxiii
Acknowledgments xxv
Part I THE PRELIMINARIES
1 WhyR? 3
1.1 Why R? 3
1.2 R Installation 5
1.3 There is Nothing such as PRACTICALS 5
1.4 Datasets in R and Internet 6
1.4.1 List of Web-sites containing DATASETS 7
1.4.2 Antique Datasets 8
1.5 http://cran.r-project.org 9
1.5.1 http://r-project.org 10
1.5.2 http://www.cran.r-project.org/web/views/ 10
1.5.3 Is subscribing to R-Mailing List useful? 10
1.6 R and its Interface with other Software 11
1.7 help and/or? 11
1.8 R Books 12
1.9 A Road Map 13
2 The R Basics 15
2.1 Introduction 15
2.2 Simple Arithmetics and a Little Beyond 16
2.2.1 Absolute Values, Remainders, etc. 16
2.2.2 round, floor, etc. 17
2.2.3 Summary Functions 18
2.2.4 Trigonometric Functions 18
2.2.5 Complex Numbers 19
2.2.6 Special Mathematical Functions 21
2.3 Some Basic R Functions 22
2.3.1 Summary Statistics 23
2.3.2 is, as, is.na, etc. 25
2.3.3 factors, levels, etc. 26
2.3.4 Control Programming 27
2.3.5 Other Useful Functions 29
2.3.6 Calculus* 31
2.4 Vectors and Matrices in R 33
2.4.1 Vectors 33
2.4.2 Matrices 36
2.5 Data Entering and Reading from Files 41
2.5.1 Data Entering 41
2.5.2 Reading Data from External Files 43
2.6 Working with Packages 44
2.7 R Session Management 45
2.8 Further Reading 46
2.9 Complements, Problems, and Programs 46
3 Data Preparation and Other Tricks 49
3.1 Introduction 49
3.2 Manipulation with Complex Format Files 50
3.3 Reading Datasets of Foreign Formats 55
3.4 Displaying R Objects 56
3.5 Manipulation Using R Functions 57
3.6 Working with Time and Date 59
3.7 Text Manipulations 62
3.8 Scripts and Text Editors for R 64
3.8.1 Text Editors for Linuxians 64
3.9 Further Reading 65
3.10 Complements, Problems, and Programs 65
4 Exploratory Data Analysis 67
4.1 Introduction: The Tukey's School of Statistics 67
4.2 Essential Summaries of EDA 68
4.3 Graphical Techniques in EDA 71
4.3.1 Boxplot 71
4.3.2 Histogram 76
4.3.3 Histogram Extensions and the Rootogram 79
4.3.4 Pareto Chart 81
4.3.5 Stem-and-Leaf Plot 84
4.3.6 Run Chart 88
4.3.7 Scatter Plot 89
4.4 Quantitative Techniques in EDA 91
4.4.1 Trimean 91
4.4.2 Letter Values 92
4.5 Exploratory Regression Models 95
4.5.1 Resistant Line 95
4.5.2 Median Polish 98
4.6 Further Reading 99
4.7 Complements, Problems, and Programs 100
Part II PROBABILITY AND INFERENCE
5 Probability Theory 105
5.1 Introduction 105
5.2 Sample Space, Set Algebra, and Elementary Probability 106
5.3 Counting Methods 113
5.3.1 Sampling: The Diverse Ways 114
5.3.2 The Binomial Coefficients and the Pascals Triangle 118
5.3.3 Some Problems Based on Combinatorics 119
5.4 Probability: A Definition 122
5.4.1 The Prerequisites 122
5.4.2 The Kolmogorov Definition 127
5.5 Conditional Probability and Independence 130
5.6 Bayes Formula 132
5.7 Random Variables, Expectations, and Moments 133
5.7.1 The Definition 133
5.7.2 Expectation of Random Variables 136
5.8 Distribution Function, Characteristic Function, and Moment Generation Function 143
5.9 Inequalities 145
5.9.1 The Markov Inequality 145
5.9.2 The Jensen's Inequality 145
5.9.3 The Chebyshev Inequality 146
5.10 Convergence of Random Variables 146
5.10.1 Convergence in Distributions 147
5.10.2 Convergence in Probability 150
5.10.3 Convergence in rth Mean 150
5.10.4 Almost Sure Convergence 151
5.11 The Law of Large Numbers 152
5.11.1 The Weak Law of Large Numbers 152
5.12 The Central Limit Theorem 153
5.12.1 The de Moivre-Laplace Central Limit Theorem 153
5.12.2 CLT for iid Case 154
5.12.3 The Lindeberg-Feller CLT 157
5.12.4 The Liapounov CLT 162
5.13 Further Reading 165
5.13.1 Intuitive, Elementary, and First Course Source 165
5.13.2 The Classics and Second Course Source 166
5.13.3 The Problem Books 167
5.13.4 Other Useful Sources 167
5.13.5 R for Probability 167
5.14 Complements, Problems, and Programs 167
6 Probability and Sampling Distributions 171
6.1 Introduction 171
6.2 Discrete Univariate Distributions 172
6.2.1 The Discrete Uniform Distribution 172
6.2.2 The Binomial Distribution 173
6.2.3 The Geometric Distribution 176
6.2.4 The Negative Binomial Distribution 178
6.2.5 Poisson Distribution 179
6.2.6 The Hypergeometric Distribution 182
6.3 Continuous Univariate Distributions 184
6.3.1 The Uniform Distribution 184
6.3.2 The Beta Distribution 186
6.3.3 The Exponential Distribution 187
6.3.4 The Gamma Distribution 188
6.3.5 The Normal Distribution 189
6.3.6 The Cauchy Distribution 191
6.3.7 The t-Distribution 193
6.3.8 The Chi-square Distribution 193
6.3.9 The F-Distribution 194
6.4 Multivariate Probability Distributions 194
6.4.1 The Multinomial Distribution 194
6.4.2 Dirichlet Distribution 195
6.4.3 The Multivariate Normal Distribution 195
6.4.4 The Multivariate t Distribution 196
6.5 Populations and Samples 196
6.6 Sampling from the Normal Distributions 197
6.7 Some Finer Aspects of Sampling Distributions 201
6.7.1 Sampling Distribution of Median 201
6.7.2 Sampling Distribution of Mean of Standard Distributions 201
6.8 Multivariate Sampling Distributions 203
6.8.1 Noncentral Univariate Chi-square, t, and F Distributions 203
6.8.2 Wishart Distribution 205
6.8.3 Hotellings T2 Distribution 206
6.9 Bayesian Sampling Distributions 206
6.10 Further Reading 207
6.11 Complements, Problems, and Programs 208
7 Parametric Inference 209
7.1 Introduction 209
7.2 Families of Distribution 210
7.2.1 The Exponential Family 212
7.2.2 Pitman Family 213
7.3 Loss Functions 214
7.4 Data Reduction 216
7.4.1 Sufficiency 217
7.4.2 Minimal Sufficiency 219
7.5 Likelihood and Information 220
7.5.1 The Likelihood Principle 220
7.5.2 The Fisher Information 226
7.6 Point Estimation 231
7.6.1 Maximum Likelihood Estimation 231
7.6.2 Method of Moments Estimator 239
7.7 Comparison of Estimators 241
7.7.1 Unbiased Estimators 241
7.7.2 Improving Unbiased Estimators 243
7.8 Confidence Intervals 245
7.9 Testing Statistical Hypotheses-The Preliminaries 246
7.10 The Neyman-Pearson Lemma 251
7.11 Uniformly Most Powerful Tests 256
7.12 Uniformly Most Powerful Unbiased Tests 260
7.12.1 Tests for the Means: One- and Two-Sample t-Test 263
7.13 Likelihood Ratio Tests 265
7.13.1 Normal Distribution: One-Sample Problems 266
7.13.2 Normal Distribution: Two-Sample Problem for the Mean 269
7.14 Behrens-Fisher Problem 270
7.15 Multiple Comparison Tests 271
7.15.1 Bonferroni's Method 272
7.15.2 Holm's Method 273
7.16 The EM Algorithm* 274
7.16.1 Introduction 274
7.16.2 The Algorithm 274
7.16.3 Introductory Applications 275
7.17 Further Reading 280
7.17.1 Early Classics 280
7.17.2 Texts from the Last 30 Years 281
7.18 Complements, Problems, and Programs 281
8 Nonparametric Inference 283
8.1 Introduction 283
8.2 Empirical Distribution Function and Its Applications 283
8.2.1 Statistical Functionals 285
8.3 The Jackknife and Bootstrap Methods 288
8.3.1 The Jackknife 288
8.3.2 The Bootstrap 289
8.3.3 Bootstrapping Simple Linear Model* 292
8.4 Non-parametric Smoothing 294
8.4.1 Histogram Smoothing 294
8.4.2 Kernel Smoothing 297
8.4.3 Nonparametric Regression Models* 300
8.5 Non-parametric Tests 304
8.5.1 The Wilcoxon Signed-Ranks Test 305
8.5.2 The Mann-Whitney test 308
8.5.3 The Siegel-Tukey Test 309
8.5.4 The Wald-Wolfowitz Run Test 311
8.5.5 The Kolmogorov-Smirnov Test 312
8.5.6 Kruskal-Wallis Test* 314
8.6 Further Reading 315
8.7 Complements, Problems, and Programs 316
9 Bayesian Inference 317
9.1 Introduction 317
9.2 Bayesian Probabilities 317
9.3 The Bayesian Paradigm for Statistical Inference 321
9.3.1 Bayesian Sufficiency and the Principle 321
9.3.2 Bayesian Analysis and Likelihood Principle 322
9.3.3 Informative and Conjugate Prior 322
9.3.4 Non-informative Prior 323
9.4 Bayesian Estimation 323
9.4.1 Inference for Binomial Distribution 323
9.4.2 Inference for the Poisson Distribution 326
9.4.3 Inference for Uniform Distribution 327
9.4.4 Inference for Exponential Distribution 328
9.4.5 Inference for Normal Distributions 329
9.5 The Credible Intervals 332
9.6 Bayes Factors for Testing Problems 333
9.7 Further Reading 334
9.8 Complements, Problems, and Programs 335
Part III STOCHASTIC PROCESSES AND MONTE CARLO
10 Stochastic Processes 339
10.1 Introduction 339
10.2 Kolmogorov's Consistency Theorem 340
10.3 Markov Chains 341
10.3.1 The m-Step TPM 344
10.3.2 Classification of States 345
10.3.3 Canonical Decomposition of an Absorbing Markov Chain 347
10.3.4 Stationary Distribution and Mean First Passage Time of an Ergodic Markov Chain 350
10.3.5 Time Reversible Markov Chain 352
10.4 Application of Markov Chains in Computational Statistics 352
10.4.1 The Metropolis-Hastings Algorithm 353
10.4.2 Gibbs Sampler 354
10.4.3 Illustrative Examples 355
10.5 Further Reading 361
10.6 Complements, Problems, and Programs 361
11 Monte Carlo Computations 363
11.1 Introduction 363
11.2 Generating the (Pseudo-) Random Numbers 364
11.2.1 Useful Random Generators 364
11.2.2 Probability Through Simulation 366
11.3 Simulation from Probability Distributions and Some Limit Theorems 373
11.3.1 Simulation from Discrete Distributions 373
11.3.2 Simulation from Continuous Distributions 380
11.3.3 Understanding Limit Theorems through Simulation 383
11.3.4 Understanding The Central Limit Theorem 386
11.4 Monte Carlo Integration 388
11.5 The Accept-Reject Technique 390
11.6 Application to Bayesian Inference 394
11.7 Further Reading 397
11.8 Complements, Problems, and Programs 397
Part IV LINEAR MODELS
12 Linear Regression Models 401
12.1 Introduction 401
12.2 Simple Linear Regression Model 402
12.2.1 Fitting a Linear Model 403
12.2.2 Confidence Intervals 405
12.2.3 The Analysis of Variance (ANOVA) 407
12.2.4 The Coefficient of Determination 409
12.2.5 The "lm" Function from R 410
12.2.6 Residuals for Validation of the Model Assumptions 412
12.2.7 Prediction for the Simple Regression Model 416
12.2.8 Regression through the Origin 417
12.3 The Anscombe Warnings and Regression Abuse 418
12.4 Multiple Linear Regression Model 421
12.4.1 Scatter Plots: A First Look 422
12.4.2 Other Useful Graphical Methods 423
12.4.3 Fitting a Multiple Linear Regression Model 427
12.4.4 Testing Hypotheses and Confidence Intervals 429
12.5 Model Diagnostics for the Multiple Regression Model 433
12.5.1 Residuals 433
12.5.2 Influence and Leverage Diagnostics 436
12.6 Multicollinearity 441
12.6.1 Variance Inflation Factor 442
12.6.2 Eigen System Analysis 443
12.7 Data Transformations 445
12.7.1 Linearization 445
12.7.2 Variance Stabilization 447
12.7.3 Power Transformation 449
12.8 Model Selection 451
12.8.1 Backward Elimination 453
12.8.2 Forward and Stepwise Selection 456
12.9 Further Reading 458
12.9.1 Early Classics 458
12.9.2 Industrial Applications 458
12.9.3 Regression Details 458
12.9.4 Modern Regression Texts 458
12.9.5 R for Regression 458
12.10 Complements, Problems, and Programs 458
13 Experimental Designs 461
13.1 Introduction 461
13.2 Principles of Experimental Design 461
13.3 Completely Randomized Designs 462
13.3.1 The CRD Model 462
13.3.2 Randomization in CRD 463
13.3.3 Inference for the CRD Models 465
13.3.4 Validation of Model Assumptions 470
13.3.5 Contrasts and Multiple Testing for the CRD Model 472
13.4 Block Designs 477
13.4.1 Randomization and Analysis of Balanced Block Designs 477
13.4.2 Incomplete Block Designs 481
13.4.3 Latin Square Design 484
13.4.4 Graeco Latin Square Design 487
13.5 Factorial Designs 490
13.5.1 Two Factorial Experiment 491
13.5.2 Three-Factorial Experiment 496
13.5.3 Blocking in Factorial Experiments 502
13.6 Further Reading 504
13.7 Complements, Problems, and Programs 504
14 Multivariate Statistical Analysis - I 507
14.1 Introduction 507
14.2 Graphical Plots for Multivariate Data 507
14.3 Definitions, Notations, and Summary Statistics for Multivariate Data 511
14.3.1 Definitions and Data Visualization 511
14.3.2 Early Outlier Detection 517
14.4 Testing for Mean Vectors : One Sample 520
14.4.1 Testing for Mean Vector with Known Variance-Covariance Matrix 520
14.4.2 Testing for Mean Vectors with Unknown Variance-Covariance Matrix 521
14.5 Testing for Mean Vectors : Two-Samples 523
14.6 Multivariate Analysis of Variance 526
14.6.1 Wilks Test Statistic 526
14.6.2 Roy's Test 528
14.6.3 Pillai's Test Statistic 529
14.6.4 The Lawley-Hotelling Test Statistic 529
14.7 Testing for Variance-Covariance Matrix: One Sample 531
14.7.1 Testing for Sphericity 532
14.8 Testing for Variance-Covariance Matrix: k-Samples 533
14.9 Testing for Independence of Sub-vectors 536
14.10 Further Reading 538
14.11 Complements, Problems, and Programs 538
15 Multivariate Statistical Analysis - II 541
15.1 Introduction 541
15.2 Classification and Discriminant Analysis 541
15.2.1 Discrimination Analysis 542
15.2.2 Classification 543
15.3 Canonical Correlations 544
15.4 Principal Component Analysis - Theory and Illustration 547
15.4.1 The Theory 547
15.4.2 Illustration Through a Dataset 549
15.5 Applications of Principal Component Analysis 553
15.5.1 PCA for Linear Regression 553
15.5.2 Biplots 556
15.6 Factor Analysis 560
15.6.1 The Orthogonal Factor Analysis Model 561
15.6.2 Estimation of Loadings and Communalities 562
15.7 Further Reading 568
15.7.1 The Classics and Applied Perspectives 568
15.7.2 Multivariate Analysis and Software 568
15.8 Complements, Problems, and Programs 569
16 Categorical Data Analysis 571
16.1 Introduction 571
16.2 Graphical Methods for CDA 572
16.2.1 Bar and Stacked Bar Plots 572
16.2.2 Spine Plots 575
16.2.3 Mosaic Plots 577
16.2.4 Pie Charts and Dot Charts 580
16.2.5 Four-Fold Plots 583
16.3 The Odds Ratio 586
16.4 The Simpson's Paradox 588
16.5 The Binomial, Multinomial, and Poisson Models 589
16.5.1 The Binomial Model 589
16.5.2 The Multinomial Model 590
16.5.3 The Poisson Model 591
16.6 The Problem of Overdispersion 593
16.7 The ;;2- Tests of Independence 593
16.8 Further Reading 595
16.9 Complements, Problems, and Programs 595
17 Generalized Linear Models 597
17.1 Introduction 597
17.2 Regression Problems in Count/Discrete Data 597
17.3 Exponential Family and the GLM 600
17.4 The Logistic Regression Model 601
17.5 Inference for the Logistic Regression Model 602
17.5.1 Estimation of the Regression Coefficients and Related Parameters 602
17.5.2 Estimation of the Variance-Covariance Matrix of ;;^ 606
17.5.3 Confidence Intervals and Hypotheses Testing for the Regression Coefficients 607
17.5.4 Residuals for the Logistic Regression Model 608
17.5.5 Deviance Test and Hosmer-Lemeshow Goodness-of-Fit Test 611
17.6 Model Selection in Logistic Regression Models 613
17.7 Probit Regression 618
17.8 Poisson Regression Model 621
17.9 Further Reading 625
17.10 Complements, Problems, and Programs 626
Appendix A Open Source Software-An Epilogue 627
Appendix B The Statistical Tables 631
Bibliography 633
Author Index 643
Subject Index 649
R Codes 659
Chapter 1
Why R?
Package(s): UsingR
Dataset(s): +AD1-9
1.1 Why R?
Welcome to the world of Statistical Computing! During the first quartile of the previous century Statistics started growing at a great speed under the schools led by Sir R.A. Fisher and Karl Pearson. Statistical computing replicated similar growth during the last quartile of that century. The first part laid the foundations and the second part made the founders proud of their work. Interestingly, the beginning of this century is also witnessing a mini revolution of its own. The R Statistical Software, developed and maintained by the R Core Team, may be considered as a powerful tool for the statistical community. The software being a Free Open Source Software is simply icing on the cake.
R is evolving as the preferred companion of the Statistician. The reasons are aplenty. To begin with, this software has been developed by a team of Statisticians. Ross Ihaka and Robert Gentleman laid the basic framework for R, and later a group was formed who are responsible for the current growth and state of it. R is a command-line software and thus powerful with a lot of options for the user.
The legendary Prasanta Chandra Mahalanobis delivered one of the important essays in the annals of Statistics, namely, "Why Statistics?" It appears that Indian mathematicians were skeptical to the thought of including Statistics as a legitimate branch of science in general, and mathematics in particular. This essay addresses some of those concerns and establishes the scientific reasoning through the concepts of random samples, importance of random sampling, etc.
Naturally, we ask ourselves the question "Why R?" Of course, the magnitude of the question is oriented in a completely different and (probably) insignificant way, and we hope the reader will excuse us for this idiosyncrasy. The most important reason for the choice of R is that it is an open source software. This translates to the fact that the functioning of the software can be understood to the first line of code which steam rolls into powerful utilities. As an example, we can trace how exactly the important mean function works.
# File src/library/base/R/mean.R # Part of the R package, http://www.R-project.org # # A copy of the GNU General Public License is available at # http://www.r-project.org/Licenses/ mean <- function(x, ...) UseMethod("mean") mean.default <- function(x, trim = 0, na.rm = FALSE, ...) { if(!is.numeric(x) && !is.complex(x) && !is.logical(x)) { warning("argument is not numeric or logical: returning NA") return(NA_real_) } if (na.rm) x <- x[!is.na(x)] if(!is.numeric(trim) || length(trim) != 1) stop("'trim' must be numeric of length one") n <- length(x) if(trim > 0 && n > 0) { if(is.complex(x)) stop("trimmed means are not defined for complex data") if(trim >= 0.5) return(stats::median(x, na.rm=FALSE)) lo <- floor(n*trim)+1 hi <- n+1-lo x <- sort.int(x, partial=unique(c(lo, hi)))[lo:hi] } .Internal(mean(x)) } mean.data.frame <- function(x, ...) sapply(x, mean, ...) Note that there is information about the address of the mean function, src/library/base/R/mean.R. The user can go to that address and open mean.R in any text editor. Now, if you find that the mean function does not work according to your requirement, modifications and new functions can be defined easily. For instance the default setting of the mean function is na.rm=FALSE, that is, if there are missing observations in a vector, see Section 2.3, the mean function will return NA as the answer. It is very simple to define a modified function whose default setting is na.rm=TRUE.
> x <- c(10,11,NA,13,14) > mean(x) [1] NA > mean_new <- function(...,na.rm=TRUE) mean(...,na.rm=TRUE) > mean_new(x) [1] 12 > mean(x,na.rm=TRUE) [1] 12 This is as simple as that. Thus, there are no restrictions imposed by the software on the user. The authors strongly believe that this freedom is priceless. If the decision to acquire the software is dictated by economic considerations, it is convenient that R comes freely.
Computation complexity is a reason for the need of software. As the modern statistical methods are embedded with complexity, it becomes a challenge for the developers of the methodology to complement the applications with appropriate computer programs. It has been our observation that many statisticians tend to address this dimension with relevant R packages. Venables and Ripley (2002) developed a very useful package MASS, an abbreviation for the title of their book Modern Applied Statistics with S. This package is shipped along with the software and is "recommended" as a priority package. In Section 1.8 we will see how many statisticians have adopted R as the language of their statistical computations.
1.2 R Installation
The website http://cran.r-project.org/ consists of all versions of R available for a variety of Operating Systems. CRAN is an abbreviation for Comprehensive R Archive Network. An incidental fact is that R had been developed on the Internet only.
The R software can be installed on a variety of platforms such as Linux, Windows, and Macintosh, among others. There is also an option of choosing 32- or 64-bit versions of the software. For a Linuxian, under appropriate privileges, R may be easily installed from the terminal using the command sudo apt-get install r-base. Ubuntu operating system users can find more help regarding R installation at the link http://ubuntuforums.org/showthread.php?t=639710.
After the installation is complete, the user can start the software by simply keying in R at the terminal. If the user is a beginner and not too familiar with the Linux environments, it is a possibility that she may be disappointed with its appearance as she cannot find much help there. Furthermore, the Linux expert may find this too trivial to explain/help a beginner. Some help for the beginner is available at http://freshmeat.net/articles/view/2237/.
A user of Windows first needs to download the recent versions executable file, currently R-3.0.2-win32.exe, and then merely double-click her way to completing the installation process. Similarly, Macintosh users can easily find the related files and methods for installation. The web links "R MacOS X FAQ" and "R Windows FAQ" should further be useful to the reader. The authors have developed the R codes used in this book and verified them for Linux and Windows versions. We are confident that they will compile without errors on Macintosh too.
1.3 There is Nothing such as PRACTICALS
The reader is absolutely free to differ from our point of view that "There is nothing such as PRACTICALS" and may skip this section altogether. There are two points of view from the authors which will be put forward here. First, with the decreasing cost of computers and availability of Open Source Software, OSS, see Appendix A, there is no need for calculator-based practicals. Also within the purview of a computer lab, a Statistics student/expertise needs to be more familiar with software such as R and SAS among others. Our second point of view is that the integration of theory with applications can be seamlessly achieved using the software modules.
It is apparently clear with the exponential growth of technology that the days of separate sessions for practicals of are a bygone era, and it's not an intelligent proposition to hang onto a weak rope, and blame it for our fall. It has been observed that in many of the developed Departments of the subject, calculator-based computations/practicals session have been done away with altogether. It is also noticed that many Statistical institutes do not teach C++/Fortran programming languages even at a graduate course, and a reason for this may be that statisticians need not necessarily be software programmers. There are many additional reasons for this reluctance. A practical reason is that computers have become very much cheaper, and if not within the financial reach of the students (especially in the developing countries), computing machines are easily available in most of their institutes. It is more often the case that the student has access to at least a couple of hours per week at her institute.
The availability of subject-specific interpretative software has also minimized the need of writing explicit programs for most of the standard practical methods in that subject. For example, in our Statistics subject, there are many software packages such as SAS, SYSTAT, STATISTICA, etc. Each of these contains inbuilt modules/menus which enable the user to perform most of these standard computations in a jiffy, and as such the user need not develop the programs for the statistical techniques in the applied area such as Linear Regression Analysis, Multivariate Statistics, among other topics of the subject.
It is true that one of the driving themes of this book is to convey as many ideas and concepts, both theoretical and practical, through a mixture of software programs and mathematical rigor. This aspect will become clear as the reader goes deeper into the book and especially through the asterisked sections or subsections. In short, this...
