A Note on the Non-Existence of a Best Test for Analyzing Unreplicated Two-Level Factorial Designs

Document Type : Original Article

Authors
Mahmood Kharrati-Kopaei
Mohammad Sadooghi-Alvandi
Department of Statistics Shiraz University, Shiraz, Iran.
10.22034/jirss.2023.2006195.1029
Abstract
The usual t-test or F-test can not be used to analyze unreplicated two-level factorial designs, since all the observations are used to estimate the factor effects and no observation is left to estimate the error variance. To overcome this difficulty, various procedures have been proposed in the literature and several simulation studies have been carried out to compare the performance of these methods. The results of these studies have been inconclusive, and no test is widely accepted as a “best” test. In this paper, we present results that show theoretically that no test has high power against all possible alternatives; i.e. no test can detect all patterns of active effects. Therefore, in the absence of any prior information concerning active and inactive effects, no test can be preferred to any other test based on power and the choice of a test should be based on other considerations, such as ease of use, control of individual or experimental error rate, the purpose of the experiment, etc.
Keywords
Active Effect
EER
Error Rate
IER
Inactive Effect
Power
Subjects
Aboukalam, M. A. F., and Al-Shiha, A. A. (2001), A robust analysis for unreplicated factorial experiments. Computational Statistics & Data Analysis, 36, 31-46.
Al-Shiha, A. A., and Yang, Shie-Shien (1999), A multistage procedure for analyzing unreplicated factorial experiments. Biometrical Journal, 41, 659-670.
Al-Shiha, A. A., and Yang, Shie-Shien (2000), Critical values and some properties of a new test statistic for analyzing unreplicated factorial experiments. Biometrical Journal, 42, 605-616.
Benski, H. C. (1989), Use of a normality test to identify significant effects in factorial designs. Journal of Quality Technology, 21, 174-178.
Benski, C. and Cabau, E. (1995), Unreplicated experimental designs in reliability growth program. IEEE Trans. Reliability, 44, 199-205.
Berk, K. N., and Picard, R. R. (1991), Significance tests for saturated orthogonal arrays. Journal of Quality Technology, 23, 79-89.
Birnbaum, A. (1959), On the analysis of factorial experiments without replication. Technometrics, 1, 343-357.
Birnbaum, A. (1961), A multi-decision procedure related to the analysis of single degrees of freedom. Annals of the Institute of Statistical Mathematics, 12, 227-236.
Bissell, A. F. (1989), Interpreting mean squares in saturated fractional designs. Journal of Applied Statistics, 16, 7-18.
Bissell, A. F. (1992), Mean squares in saturated fractional designs revisited. Journal of Applied Statistics, 19, 351-366.
Box, G. E. P., and Meyer, R. D. (1986), An analysis for unreplicated fractional factorials. Technometrics, 28, 11-18.
Chen, Y. (2003), On the analysis of unreplicated factorial designs. (PH.D. thesis submitted to the Department of Statistics of the University of Dortmund, Dortmund, March 2003).
Chen, Y., and Kunert, J. (2004), A new quantitative method for analysing unreplicated factorial designs. Biometical Journal, 46(1), 125-140.
Daniel, C. (1959), Use of Half-normal plots in interpreting factorial two-level experiments. Technometrics, 1, 311-341.
Dong, F. (1993), On the identification of active contrasts in unreplicated fractional factorials. Statistica Sinica, 3(1), 209-217.
Haaland, P. D., and O’Connell, M. A. (1995), Inference for effect-saturated fractional factorials. Technometrics, 37, 82-93.
Hamada, M., and Balakrishnan, N. (1998), Analyzing unreplicated factorial experiments: Areview with some new proposals (with Comments and Rejoinder). Statistica Sinica, 8(1), 1-41.
Holms, A. G., and Berrettoni, J. N. (1969), Chain-poolingANOVAfor two-level factorial replication-free experiments. Technometrics, 11, 725-746.
Johnson, E. G., and Tukey, J.W. (1987), Graphical exploratory analysis of variance illustrate on a splitting of the Johnson and Tsao data. In Design, Data and Analysis (Edited by C. L. Mallows). John Wiley, New York.
Juan, J., and Pena, D. (1992), A simple method to identify significant effects in unreplicated two-level factorial designs. Communications in Statistics- Theory and Methods, 21, 1383-1403.
Kharrati-Koapei, M., and Sadooghi-Alvandi, S. M. (2007), A new method for testing interaction in unreplicated two-way analysis of variance. Communications in Statistics-Theory and Methods, 36, 2787-2804.
Kharrati-Kopaei, M., and Shenavari, Z. (2023), Analyzing unreplicated two-level factorial designs by combining multiple tests. Communications in Statistics-Theory and Methods, 1-16.
Le, N. D., and Zamar, R. H. (1992), A global test for effects in 2k factorial design without replicates. Journal of Statistical Computation and Simulation, 41, 41-54.
Lenth, R. V. (1989), Quick and easy analysis of unreplicated factorials. Technometrics, 31, 469-473.
Loh,W.Y. (1992), Identification of active contrasts in unreplicated factorial experiments. Computational Statistics and Data Analysis, 14, 135-148.
Loughin, T. M., and Noble,W. (1997), A permutation test for effects in an unreplicate factorial design. Technometrics, 39, 180-190.
Miller, A. (2005), The analysis of unreplicated factorial experiments using all possible comparisons. Technometrics, 47(1), 51-63.
Montgomery, D. C. (2001), Design and analysis of experiments. Fifth edition. JohnWiley & Sons, Inc.1996), A hypothesis-testing approach toward identifying active contrasts. Technometrics, 38(2), 161-169.
Voss, D. T. (1988), Generalized modulus-ratio test for analysis of factorial designs with zero degrees of freedom for error. Communications in Statistic-Theory and Methods, 17,
3345-3359.
Voss, D. T. (1999), Analysis of orthogonal saturated designs. Journal of Statistical Planning and Inference, 78, 111-130.
Voss, D. T., and Wang,W. (2006), On adaptive testing in orthogonal saturated designs. Statistica Sinica, 16, 227-234.
Westfall, P. H., and Young, S. S. (1993), Resampling-based multiple testing, examples and methods for p-value adjustment. John Wiley & Sons, INC.
Wu, S. S., and Wang, W. (2007), Step-up simultaneous tests for identifying active effects in orthogonal saturated designs. Annals of Statistics, 35, 449-463.
Ye, K. Q., Hamada, M., and Wu, C. F. J. (2001), Astep-down Lenth method for analyzing unreplicated factorial designs. Journal of Quality Technology, 33(2), 140-152.
Zahn, D. A. (1975a), Modifications of and revised critical values for the half-normal plot. Technometrics, 17, 189-200.
Zahn, D. A. (1975b), An empirical study of the half-normal plot. Technometrics, 17, 201-211.
 
Journal of the Iranian Statistical Society
Volume 22, Issue 1
June 2023
Pages 161-174

  • Receive Date 03 July 2023
  • Revise Date 12 October 2023
  • Accept Date 23 December 2023