New Developments in Statistical Computing: New Bias Evaluation Features of EXPLOR—A Program for Assessing Experimental Design Properties

Longitudinal investigations play an increasingly prominent role in biomedical research. Much of the literature on specifying and fitting linear models for serial measurements uses methods based on the standard multivariate linear model. This article proposes a more flexible approach that permits specification of the expected response as an arbitrary linear function of fixed and time-varying covariates so that mean-value functions can be derived from subject matter considerations rather than methodological constraints. Three families of models for the covariance function are discussed: multivariate, autoregressive, and random effects. Illustrations demonstrate the flexibility and utility of the proposed approach to longitudinal analysis.     

Understanding Canonical Correlation through the General Linear Model and Principal Components

Canonical correlation has been little used and little understood, even by otherwise sophisticated analysts. An alternative approach to canonical correlation, based on a general linear multivariate model, is presented. Properties of principal component analysis are used to help explain the method. Standard computational methods for full rank canonical correlation, techniques for canonical correlation on component scores, and canonical correlation with less than full rank are discussed. They are seen to be essentially equivalent when the model equation for canonical correlation on component scores is presented. The two approaches to less than full rank situations are equivalent in some senses, but quite different in usefulness, depending on the application. An example dataset is analyzed in detail to help demonstrate the conclusions.     

Pooled Cross-Sectional and Time Series Data: A Survey of Current Statistical Methodology

A data base that provides a multivariate statistical history for each of a number of individual entities is called a pooled cross-sectional and time series data base in the econometrics literature. In marketing and survey literature the terms panel data or longitudinal data are often used. In management science a convenient term might be management data base. Such a data base provides a particularly rich environment for statistical analysis. This article reviews methods for estimating multivariate relationships particular to each individual entity and for summarizing these relationships for a number of individuals. Inference to a larger population when the data base is viewed as a sample is also considered.     

Simultaneous Confidence Regions for Predictions

After observing n independent responses at n corresponding design points in a linear regression setting, one wishes to make a confidence statement about future responses that will apply simultaneously to all possible design points. Two appropriate prediction regions are derived using normal theory.     

Frequency of Selecting Noise Variables in Subset Regression Analysis: A Simulation Study

This article presents the results of a simulation study of variable selection in a multiple regression context that evaluates the frequency of selecting noise variables and the bias of the adjusted R ² of the selected variables when some of the candidate variables are authentic. It is demonstrated that for most samples a large percentage of the selected variables is noise, particularly when the number of candidate variables is large relative to the number of observations. The adjusted R ² of the selected variables is highly inflated.     

Hierarchical Variable Selection in Polynomial Regression Models

Significance tests on coefficients of lower-order terms in polynomial regression models are affected by linear transformations. For this reason, a polynomial regression model that excludes hierarchically inferior predictors (i.e., lower-order terms) is considered to be not well formulated. Existing variable-selection algorithms do not take into account the hierarchy of predictors and often select as “best” a model that is not hierarchically well formulated. This article proposes a theory of the hierarchical ordering of the predictors of an arbitrary polynomial regression model in m variables, where m is any arbitrary positive integer. Ways of modifying existing algorithms to restrict their search to well-formulated models are suggested. An algorithm that generates all possible well-formulated models is presented.     

Interpreting an Inequality in Multiple Regression

This article provides a method of interpreting a surprising inequality in multiple linear regression: the squared multiple correlation can be greater than the sum of the simple squared correlations between the response variable and each of the predictor variables. The interpretation is obtained via principal component analysis by studying the influence of some components with small variance on the response variable. One example is used as an illustration and some conclusions are derived.     

Bayesian Binomial Regression: Predicting Survival at a Trauma Center

Standard methods for analyzing binomial regression data rely on asymptotic inferences. Bayesian methods can be performed using simple computations, and they apply for any sample size. We provide a relatively complete discussion of Bayesian inferences for binomial regression with emphasis on inferences for the probability of “success.” Furthermore, we illustrate diagnostic tools, perform model selection among nonnested models, and examine the sensitivity of the Bayesian methods.     

On a Relation between Principal Components and Regression Analysis

A regression approach to principal component analysis is presented in this note. We provide an alternative interpretation of principal components that illustrates the relation between the extra sum of squares in regression analysis and the eigenvalues associated with the principal components.     

Asymptotic properties of a rank estimate in linear regression with symmetric non-identically distributed errors

In this article, a simple linear regression model with independent and symmetric but non-identically distributed errors is considered. Asymptotic properties of the rank regression estimate defined in Jaeckel [Estimating regression coefficients by minimizing the dispersion of the residuals, Ann. Math. Statist. 43 (1972), pp. 1449–1458] are studied. We show that the studied estimator is consistent and asymptotically normally distributed. The cases of bounded and unbounded score functions are examined separately. The regularity conditions of the article are exemplified for finite mixture distributions.