The Prediction Sum of Squares as a General Measure for Regression Diagnostics

Statistics that usually accompany the regression model do not provide insight into the quality of the data or the potential influence of the individual observations on the estimates. In this study, the Q² statistic is used as a criterion for detecting influential observations or outliers. The statistic is derived from the jackknifed residuals, the squared sum of which is generally known as the prediction sum of squares or PRESS. This article compares R ² with Q² and suggests that the latter be used as part of the data-quality check. It is shown, for two separate data sets obtained from regional cost of living and U.S. food industry studies, that in the presence of outliers the Q² statistic can be negative, because it is sensitive to the choice of regressors and the inclusion of influential observations. Once the outliers are dropped from the sample, the discrepancy between Q² and R ² values is negligible.     

Phase I Analysis of Multiple Linear Regression Profiles

This article considers the Phase I analysis of data when the quality of a process or product is characterized by a multiple linear regression model. This is usually referred to as the analysis of linear profiles in the statistical quality control literature. The literature includes several approaches for the analysis of simple linear regression profiles. Little work, however, has been done in the analysis of multiple linear regression profiles. This article proposes a new approach for the analysis of Phase I multiple linear regression profiles. Using this approach, regardless of the number of explanatory variables used to describe it, the profile response is monitored using only three parameters, an intercept, a slope, and a variance. Using simulation, the performance of the proposed method is compared to that of the existing methods for monitoring multiple linear profiles data in terms of the probability of a signal. The advantage of the proposed method over the existing methods is greatly improved detection of changes in the process parameters of linear profiles with high-dimensional space. The article also proposes useful diagnostic aids based on F-statistics to help in identifying the source of profile variation and the locations of out-of-control samples. Finally, the use of multiple linear profile methods is illustrated by a data set from a calibration application at National Aeronautics and Space Administration (NASA) Langley Research Center.     

Diagnosis of Multivariate Control Chart Signal Based on Dummy Variable Regression Technique

Abstract

It is common to monitor several correlated quality characteristics using the Hotelling's T ² statistic. However, T ² confounds the location shift with scale shift and consequently it is often difficult to determine the factors responsible for out of control signal in terms of the process mean vector and/or process covariance matrix. In this paper, we propose a diagnostic procedure called ‘D-technique’ to detect the nature of shift. For this purpose, two sets of regression equations, each consisting of regression of a variable on the remaining variables, are used to characterize the ‘structure’ of the ‘in control’ process and that of ‘current’ process. To determine the sources responsible for an out of control state, it is shown that it is enough to compare these two structures using the dummy variable multiple regression equation. The proposed method is operationally simpler and computationally advantageous over existing diagnostic tools. The technique is illustrated with various examples.     

On Two Types of Breakdown Points of Weighted L 2-median

Zuo (2004) investigated the simplified replacement finite sample breakdown point of weighted L ^p -depth and L ^p -median for some appropriate weight functions. The addition breakdown point of weighted L ^p -depth functions is studied firstly in this article. In addition, for some other weight functions different from those in Zuo (2004 Zuo , Y. ( 2004 ). Robustness of weighted L ^p -depth and L ^p -median . Allgemeines Statistics Archiv. 88 : 215 – 234 . [Google Scholar]), we establish the lower bounds of these two types of breakdown point of weighted L ²-median.     

Things that make us different: analysis of deviance with time-use data

The constrained, non-normal nature of time-use data poses a challenge to ordinary analysis of variance. This paper investigates a computationally simple variance decomposition technique suitable for those data. As a by-product of the analysis, a measure of fit for systems of time-demand equations is proposed that possesses several useful properties.     

The coefficient of determination and its adjusted version in linear regression models

This article presents a comparative study of the efficiency properties of the coefficient of determination and its adjusted version in linear regression models when disturbances are not necessarily normal.     

Comparison of Goodness-of-Fit Measures in Probit Regression Model

This article examines several goodness-of-fit measures in the binary probit regression model. Existing pseudo-R ² measures are reviewed, two modified and one new pseudo-R ² measure are proposed. For the probit regression model, empirical comparisons are made for different goodness-of-fit measures with the squared sample correlation coefficient of the observed response and the predicted probabilities. As an illustration, the goodness-of-fit measures are applied to a “paid labor force” data set.     

O(N 2)-Operation Approximation of Covariance Matrix Inverse in Gaussian Process Regression Based on Quasi-Newton BFGS Method

Gaussian process (GP) is a Bayesian nonparametric regression model, showing good performance in various applications. However, during its model-tuning procedure, the GP implementation suffers from numerous covariance-matrix inversions of expensive O(N³) operations, where N is the matrix dimension. In this article, we propose using the quasi-Newton BFGS O(N²)-operation formula to approximate/replace recursively the inverse of covariance matrix at every iteration. The implementation accuracy is guaranteed carefully by a matrix-trace criterion and by the restarts technique to generate good initial guesses. A number of numerical tests are then performed based on the sinusoidal regression example and the Wiener–Hammerstein identification example. It is shown that by using the proposed implementation, more than 80% O(N³) operations could be eliminated, and a typical speedup of 5–9 could be achieved as compared to the standard maximum-likelihood-estimation (MLE) implementation commonly used in Gaussian process regression.     

On the distribution of hotelling's t2 and multiple correlation r2when sampling from a mixture of two normals

In this paper the non-null distribution of Hotelling's T² and the null distribution of multiple correlation R² are derived when the sample is taken from a mixture of two p-component multivariate normal distributions with mean vectors μ₁ and μ₂ respectively and common covariance matrix ∑, ∑. In a special case the non-null distribution of R² is a l s o given, while the general noncentral distribution is given i n Awan (1981). These results have been used to study the robustness of T² and R² tests by Srivastava and Awan (1982), and Awan and Srivastava (1982) respectively.     

Estimators of the multiple correlation coefficient: Local robustness and confidence intervals

Many robust regression estimators are defined by minimizing a measure of spread of the residuals. An accompanying R ²-measure, or multiple correlation coefficient, is then easily obtained. In this paper, local robustness properties of these robust R ²-coefficients are investigated. It is also shown how confidence intervals for the population multiple correlation coefficient can be constructed in the case of multivariate normality.     

Diagnosis Aids in Multivariate Multiple Linear Regression Profiles Monitoring

Diagnosis aids in addition to detecting the out-of-control state is an important issue in multivariate multiple linear regression profiles monitoring; because a large number of parameters and profiles in this structure are involved. In this paper, we specifically concentrate on identification of profile(s) and parameter(s) which have changed during the process in multivariate multiple linear regression profiles structure in Phase II. We demonstrate the effectiveness of our proposed approaches through Monte Carlo simulations and a real case study in terms of accuracy percent.     

On the behaviour of some transforms of the sample correlation coefficient in samples from the bivariate t and the bivariate X2distribution

The present paper studies the normality of five transformations suggested in the literature to normalize the sample correlation coefficient. The parent populations are the bivariate t and the bivariate X ²The results in the previous work of Subrahmaniam and Gajjar are exploited to assess their performance. The density estimation procedure of Tarter and Kronmal is used to provide empiric support to the asymptotic results     

On Rao score and PearsonX 2 statistics in generalized linear models

The identity of the Rao score and PearsonX ² statistics is well known in the areas where the latter was first introduced: goodness-of-fit in contingency tables and binary responses. We show in this paper that the same identity holds when the two statistics are used for testing goodness-of-fit of Generalized Linear Models. We also highlight the connections that exist between the two statistics when they are used for the comparison of nested models. Finally, we discuss some merits of these unifying results. Work financially supported by cofin. MIUR grants 2000 and 2002.     

C434. Local bounds on power of T2 in scale families

In this paper, we have developed a likelihood ratio factorization technique for variable selection for multiple observations model. The asymptotic distribution of the selection criterion is also given. This technique has been applied to a marketing data set for illustration of the technique developed.     

Group sequential x2 statistic for testing the difference of two mean vectors in clinical trials

In this study we discuss the group sequential procedures for comparing two treatments based on multivariate observations in clinical trials. Also we suppose that a response vector on each of two treatments has a multivariate normal distribution with unknown covariance matrix. Then we propose a group sequential x² statistic in order to carry out repeated significance test for hypothesis of no difference between two population mean vectors. In order to realize the group sequential test where average sample number is reduced, we propose another modified group sequential x² statistic by extension of Jennison and Turnbull ( 1991 ). After construction of repeated confidence boundaries for making the repeated significance test, we compare two group sequential procedures based on two statistics regarding the average sample number and the power of the test in the simulations.     

Bivariate transition model for analysing ordinal and nominal categorical responses: an application to the Labour Force Survey data

In many panel studies, bivariate ordinal–nominal responses are measured and the aim is to investigate the effects of explanatory variables on these responses. A regression analysis for these types of data must allow for the correlation among responses of the same individual. To analyse such ordinal–nominal responses using a proper weighting approach, an ordinal–nominal bivariate transition model is proposed and maximum likelihood is used to find the parameter estimates. We propose a method in which the likelihood function can be partitioned to make possible the use of existing software. The approach is applied to the Labour Force Survey data in Iran, where the ordinal response, at the first period, is the duration of unemployment for unemployed people and the nominal response, in the second period, is economic activity status of these individuals. The interest is to find the reasons for staying unemployed or moving to another status of economic activity.