Richard M. Heiberger

Data and Statistics

Statistics is the field of study whose objective is the transformation of data (usually sets of numbers along with identifying characteristics) into information (usually in the form of tables, graphs, and written and verbal summaries) that can inform sound policy decisions. We give examples of applications of statistics to many fields in Chapter 1 Here we focus on the general concepts describing the collection and arrangement of the numbers themselves.

Design of an S Function for Robust Regression Using Iteratively Reweighted Least Squares

Abstract We develop a set of 5 functions for robust regression using the technique of iteratively reweighted least squares (IRLS). Together with a set of weight functions, function rreg is simple to understand and provides great flexibility for IRLS methods. This article focuses on the programming strategies adopted to achieve the twin goals of power and simplicity.

Emacs Speaks Statistics: A Multiplatform, Multipackage Development Environment for Statistical Analysis

Computer programming is an important component of statistical research and data analysis. It is a necessary skill for using sophisticated statistical packages and for writing custom scripts and software to perform data analysis using modern statistical methods. Emacs Speaks Statistics (ESS) provides an intelligent and consistent interface between the user and statistics software. ESS interfaces with SAS, S-Plus, R, and other statistics packages under the Unix, Microsoft Windows, and Apple Macintosh operating systems. ESS extends the Emacs text editor to streamline the use and creation of statistical software. ESS understands the syntax for numerous data analysis languages, provides consistent display and editing features across packages, and assists in the interactive or batch execution of statements by statistics packages. We describe in detail the features that ESS provides to increase efficiency.

Optimal Data Augmentation Strategies for Additive Models

Abstract Consider an experiment where the factors are measured on a continuous scale, and suppose that the experimenter is permitted to augment the existing observations with one or more new data points. Bondars universal optimality criterion (U optimality) suggests that the problem is best studied in the eigenvector coordinate system. We proceed by showing how to construct new points that first equate and then jointly increase the smaller eigenvalues of the crossproduct of the independent variables. We discuss the limitations of design augmentation strategies based solely on the crossproduct matrix. Our goals are to equate and minimize the variances of the estimated regression coefficients, keep the new points constrained in a prespecified experimental region, use as much of the information in the original points as possible, and keep the number of required new points as small as possible. We offer several data augmentation strategies to meet these requirements. As U optimality subsumes each of the D, A...

Generating Test Data with Independently Controllable Features for Multivariate General Linear Forms

Abstract Experiments to assess the performance characteristics, both of statistical methods and of computer programs that implement them, are best performed using test data with controlled statistical and numerical properties. We describe an algorithm for constructing test data for any multivariate linear model that provides complete control (subject only to the requirement of mutual consistency) over the following factors: regression coefficients; regression and residual sums of squares and products matrices; means, standard deviations, and correlations of the independent and dependent variables, residuals, and predicted values; and canonical correlations or multiple correlation. The algorithm permits aspects of the underlying data structure including high-leverage points, outliers, and the residual distribution, to be controlled by specifying the components of the singular-value decompositions of the independent variables, the dependent variables, or the error space. Some features of the generated test ...

One-Way ANOVA

One-way ANOVA (analysis of variance) is a technique that generalizes the two-sample t-test to three or more samples. We test the hypotheses (specified here for k=6 samples) about population means μ j : H 0:μ 1=μ 2=μ 3=μ 4=μ 5=μ 6 H 1: Not all μ j are equal (j=1:6) The test is based on the observed sample means \(\bar x_j\).

Simple Linear Regression

Linear regression by the least-squares method is a way of fitting a straight-line model to observed data.

Displays for Direct Comparison of ARIMA Models

The series of graphs presented here, based on standard time series diagnostics and display graphs, eases the tasks of identifying and checking an ARIMA model. Each diagnostic display consists of a matrix of plots for a series of ARIMA(p, d, q) models (with p = 1:pmax, q = 1:qmax, and d constant). In this way the identification phase of the analysis is eased because the analyst can directly see the incremental effect of each proposed term. The direct visual comparison of the models is helpful to the experienced analyst because it makes immediate the difference in the explanatory capabilities of the various models. The series of plots is very helpful in presenting time series concepts, particularly the identification phase, to introductory classes. The plots have been implemented using the Trellis system in S-Plus.

Mean–Mean Multiple Comparison Displays for Families of Linear Contrasts

Traditional tabular and graphical displays of results of simultaneous confidence intervals or hypothesis tests are deficient in several respects. Expanding on earlier work, we present new mean–mean multiple comparison graphs that succinctly and compactly display the results of traditional procedures for multiple comparisons of population means or linear contrasts involving means. The MMC plot can be used with unbalanced, multifactor designs with covariates. After reviewing the construction of these displays in the S language (S-Plus and R), we demonstrate their application to four multiple comparison scenarios.

Time Series Analysis

Time series analysis is the technique used to study observations that are measured over time. Examples include natural phenomena (temperature, humidity, wind speed) and business variables (price of commodities, stock market indices) that are measured at regular intervals (hourly, daily).