James W. Neill

Implementation of a maximin power clustering criterion to select near replicates for regression lack-of-fit tests

Abstract In earlier work, we presented a maximin power clustering criterion to partition observations into groups of near replicates. Specifically, the criterion selects near replicate clusters for use with Christensens tests for orthogonal between and within cluster lack of fit. This article further explores implementation of this clustering criterion. In particular, a methodology is developed to determine a collection of candidate groupings to which the maximin power criterion can be applied.

A comparison of some lack of fit tests based on near replicates

Several tests for regression lack of fit proposed by Christensen (1989), Shillington (1979) and Neill and Johnson (1985) are compared. The tests considered are applicable for the case of nonreplication and reduce to the classical lack of fit test when independent replications are available. A simulation study is used to compare the size and power of the test procedures for small sample sizes and various configurations of nonreplication. In addition, each test is shown to be consistent as well as invariant with respect to location and scale changes made on the regressor variables.

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.

Maximin clusters for nonreplicated multiresponse lack of fit tests

Abstract Khuri (Technometrics 27 (1985) 213) and Levy and Neill (Comm. Statist. A 19 (1990) 1987) presented regression lack of fit tests for multiresponse data with replicated observations available at points in the experimental region, thereby extending the classical univariate lack of fit test given by Fisher (J. Roy. Statist. Soc. 85 (1922) 597). In this paper, multivariate tests for lack of fit in a linear multiresponse model are derived for the common circumstance in which replicated observations are not obtained. The tests are based on the union–intersection principle, and provide multiresponse extensions of the univariate tests for between- and within-cluster lack of fit introduced by Christensen (Ann. of Statist. 17 (1989) 673; J. Amer. Statist. Assoc. 86 (1991) 752). Since the properties of these tests depend on the choice of multivariate clusters of the observations, a multiresponse generalization of the maximin power clustering criterion given by Miller, Neill and Sherfey (Ann. of Statist. 26 (1998) 1411; J. Amer. Statist. Assoc. 94 (1999) 610) is also developed.

The distribution of the ratio of independent central wishart determinants

Ratios of independent central Wishart determinants are useful statistics in multivariate analyses, particularly in the study of multivariate linear models. A method based on the inversion of characteristic functions is outlined for deriving new experessions for the probability distribution functions of the logarithms of these statistics. Accurate tables of the percentiles of these distributions have been obtained covering many bivariate and trivariate cases which have been computed by approximating these expression.

Lack of fit tests for linear regression models with many predictor variables using minimal weighted maximal matchings

We develop lack of fit tests for linear regression models with many predictor variables. General alternatives for model comparison are constructed using minimal weighted maximal matchings consistent with graphs on the predictor vectors. The weighted graphs we employ have edges based on model-driven distance thresholds in predictor space, thereby making our testing procedure implementable and computationally efficient in higher dimensional settings. In addition, it is shown that the testing procedure adapts to efficacious maximal matchings. An asymptotic analysis, along with simulation results, demonstrate that our tests are effective against a broad class of lack of fit.

Kansas State University Department of Statistics

The Department of Statistics at Kansas State University came into being under the authorization of the Kansas Board of Regents on July 1, 1959. Holly C. Fryer was named the initial head and there were five faculty members in the department. The department continues to be a vibrant and active entity today, with 14 faculty members, nearly 60 graduate students, and 3 staff positions.

Convergence criteria for maxima with regularly varying normalizing constants

This paper considers domain of attraction criteria for the maxima sequence associated with independent and identically distributed random variables. The criteria are based on regular variation of the normalizing constants and the existence of a real point at which the distribution function of the normalized maxima converges to a value in the open unit interval. Examples are made to specific underlying distributions.

Testing regression function adequacy in nonlinear multiresponse models

The problem of testing a proposed nonlinear multiresponse regression function for lack of fit is considered. With replication and general alternative, pseudo and classical likelihood ratio tests are determined. Asymptotic distributions and consistency properties are derived.