The Annals of Statistics

Maximin clusters for near-replicate regression lack of fit tests

Forrest R. Miller, James W. Neill, and Brian W. Sherfey

Full-text: Open access


To assess the adequacy of a nonreplicated linear regression model, Christensen introduced the concepts of orthogonal between- and within-cluster lack of fit with corresponding optimal tests. However, the properties of these tests depend on the choice of near-replicate clusters. In this paper, a graph theoretic framework is presented to represent candidate clusterings. A clustering is then selected according to a proposed maximin power criterion from among the clusterings consistent with a specified graph on the predictor settings. Examples are given to illustrate the methodology.

Article information

Ann. Statist., Volume 26, Number 4 (1998), 1411-1433.

First available in Project Euclid: 21 June 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J05: Linear regression
Secondary: 62F03: Hypothesis testing

Regression lack of fit nonreplication between clusters within clusters maximin power graph theory


Miller, Forrest R.; Neill, James W.; Sherfey, Brian W. Maximin clusters for near-replicate regression lack of fit tests. Ann. Statist. 26 (1998), no. 4, 1411--1433. doi:10.1214/aos/1024691249.

Export citation


  • ATKINSON, A. C. 1972. Planning experiments to detect inadequate regression models. Biometrika 59 275 293. Z.
  • ATKINSON, A. C. and FEDOROV, V. V. 1975. The design of experiments for discriminating between two rival models. Biometrika 62 57 70. Z.
  • ATWOOD, C. L. and Ry AN, T. A., JR. 1977. A class of tests for lack of fit to a regression model. Unpublished manuscript. Z.
  • BREIMAN, L. and MEISEL, W. S. 1976. General estimates of the intrinsic variability of data in nonlinear regression models. J. Amer. Statist. Assoc. 71 301 307. Z.
  • CHRISTENSEN, R. R. 1989. Lack of fit based on near or exact replicates. Ann. Statist. 17 673 683. Z.
  • CHRISTENSEN, R. R. 1991. Small sample characterizations of near replicate lack of fit tests. J. Amer. Statist. Assoc. 86 752 756. Z.
  • CONSTANTINE, G. M. 1987. Combinatorial Theory and Statistical Design. Wiley, New York. Z.
  • DANIEL, C. and WOOD, F. S. 1980. Fitting Equations to Data, 2nd ed. Wiley, New York. Z.
  • DETTE, H. 1994. Discrimination designs for poly nomial regression on compact intervals. Ann. Statist. 22 890 903. Z.
  • DETTE, H. 1995. Optimal designs for identifying the degree of a poly nomial regression. Ann. Statist. 23 1248 1266. Z.
  • DRAPER, N. R. and SMITH, H. 1981. Applied Regression Analy sis, 2nd. ed. Wiley, New York. Z.
  • FISHER, R. A. 1922. The goodness of fit of regression formulae and the distribution of regression coefficients. J. Roy. Statist. Soc. 85 597 612. Z. 2
  • GHOSH, B. K. 1973. Some monotonicity theorems for, F and t distributions with applications. J. Roy. Statist. Soc. Ser. B 35 480 492. Z.
  • GREEN, J. R. 1971. Testing departure from a regression without using replication. Technometrics 13 609 615. Z.
  • HART, J. D. 1997. Nonparametric Smoothing and Lack of Fit Tests. Springer, New York. Z.
  • JOGLEKAR, G., SCHUENEMEy ER, J. H. and LARICCIA, V. 1989. Lack of fit testing when replicates are not available. Amer. Statist. 43 135 143. Z.
  • JONES, E. R. and MITCHELL, T. J. 1978. Design criteria for detecting model inadequacy. Biometrika 65 541 551. Z.
  • Ly ONS, N. I. and PROCTOR, C. H. 1977. A test for regression function adequacy. Comm. Statist. Theory Methods A6 81 86. Z.
  • NEILL, J. W. and JOHNSON, D. E. 1985. Testing linear regression function adequacy without replication. Ann. Statist. 13 1482 1489. Z.
  • SHELTON, J. H., KHURI, A. I. and CORNELL, J. A. 1983. Selecting check points for testing lack of fit in response surface models. Technometrics 25 357 365. Z.
  • SHILLINGTON, E. R. 1979. Testing for lack of fit in regression without replication. Canad. J. Statist. 7 137 146. Z.
  • SU, J. Q. and WEI, L. J. 1991. A lack of fit test for the mean function in a generalized linear model. J. Amer. Statist. Assoc. 86 420 426. Z.
  • UTTS, J. M. 1982. The rainbow test for lack of fit in regression. Comm. Statist. Theory Methods A11 2801 2815.
  • MANHATTAN, KANSAS 66506-0802 E-MAIL: