Designs Robust against Violation of Normality Assumption in the Standard F-test
18 Jul 2015-Communications in Statistics-theory and Methods (Taylor & Francis)-Vol. 44, Iss: 14, pp 2898-2910
TL;DR: In this article, the robustness of incomplete block designs against non-normality in the observations has been investigated and a new measure of robustness is introduced for further study, which is called robustness measure.
Abstract: In this article, designs are found for which the F-test of analysis of variance is insensitive to violation of normality assumption. Atiqullah (1962) proved that the F-test for treatments adjusting for blocks in the intra-block analysis of a balanced incomplete block design is robust against non-normality in the observations. Here an attempt has been made to identify other designs robust in this sense. In particular, it is observed that for testing relevant hypothesis, a partially balanced incomplete block design in block design setup, under certain conditions, is robust. Robustness of a balanced treatment incomplete block design and a partially balanced treatment incomplete block design (Biswas, 2012), in treatment-control design setup, is also studied. Moreover, a new measure of robustness is introduced for further study. The performance of the F-test in presence of non-normality in the observations for a quadratically balanced design is also examined.
Citations
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01 Jan 2016
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21 citations
TL;DR: The effect of normal model departure on the optimal operating condition estimates of the control factors obtained in the robust design framework is investigated and a methodology is constructed to deal with the effect ofnormal model departure.
17 citations
01 Dec 2020
TL;DR: In this article, an attempt has been made to find robust designs in a block design set-up with covariates, when there is one single wild observation in the study variable.
Abstract: Presence of one or more aberration in the observations affects inference procedure in statistical analysis. Block designs robust against presence of aberrations can be found in the literature for both regression and block design set-ups. In this paper, an attempt has been made to find robust designs in a block design set-up with covariates, when there is one single wild observation in the study variable. Specifically, effects on the estimation of a full set of orthonormal treatment contrasts and that on the estimation of covariate parameters have been considered.
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278 citations
"Designs Robust against Violation of..." refers background or methods in this paper
...Example 4.1 (Clatworthy, 1973, SR4) Let us consider the two-associate semi-regular group divisible PBIB design with parameters v = 4, r = 4, k = 2, b = 8,m = 2, n = 2, λ1 = 0, λ2 = 2 in the test treatments and augment each block of it with one control treatment....
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...In the tables of Clatworthy (1973) there are only two two-associate PBIB designs, viz....
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...Example 3.1 (Clatworthy, 1973: SR4) Let us consider the two-associate semi-regular group divisible PBIB design with parameters v = 4, r = 4, k = 2, b = 8,m = 2, n = 2, λ1 = 0, λ2 = 2....
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TL;DR: In this paper, the optimal augmented block and optimal augmented row-column designs for estimating certain contrasts of new treatments are presented, which may be used for estimating contrasts of check effects, new variety effects, of new variety versus checks, or of all check and new varieties simultaneously.
Abstract: When some treatments (checks) are replicated r times and other treatments (new treatments) are replicated less than r times, an augmented desigii may be used. These designs may be minimum variance designs for estimating contrasts of check effects, of new variety effects, of new variety versus checks, or of all check and new varieties simultaneously. In this paper optimal augmented block and optimal augmented row-column designs for estimating certain contrasts of new treatments are presented.
256 citations
"Designs Robust against Violation of..." refers methods in this paper
...…incomplete block (PBIB) designs, (ii) balanced treatment incomplete block (BTIB) designs; and (iii) two-associate partially balanced treatment incomplete block (PBTIB) designs (Biswas, 2012), a sub-class of supplemented designs (Pearce, 1983) or augmented designs (Federer and Raghavarao, 1975)....
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TL;DR: In this paper, it was shown that the extent of non-normality in the regression variables (the x's) is not the only variable that determines sensitivity to nonnormality.
Abstract: 1. SUMMARY A number of statistical procedures involve the comparison of a 'regression' mean square with a 'residual' mean square using the normal-theory F distribution for reference. The use of the procedure for the analysis of actual data implies that the distribution of the meansquare ratio is insensitive to moderate non-normality. Many investigators, in particular Pearson (1931), Geary (1947), Gayen (1950), have considered the sensitivity of this distribution to parent non-normality for important special cases and a very general investigation was carried out by David & Johnson (1951a, b). The principal object of this paper is to demonstrate the overriding influence which the numerical values of the regression variables have in deciding sensitivity to non-normality and to demonstrate the essential nature of this dependency. We first obtain a simple approximation to the distribution of the regression F statistic in the non-normal case. This shows that it is 'the extent of non-normality' in the regression variables (the x's), which determines sensitivity to non-normality in the observations (the y's). Our results are illustrated for certain familiar special cases. In particular the well-known robustness of the analysis of variance test to compare means of equal-sized groups and the notorious lack of robustness of the test to compare two estimates of variance from independent samples are discussed in this context. We finally show that it is possible to choose the regression variables so that, to the order of approximation we employ, non-normality in the y's is without effect on the distribution of the test statistic. Our results demonstrate the effect which the choice of experimental design has in deciding robustness to non-normality.
133 citations
TL;DR: In this article, a theory of optimal incomplete block designs for comparing several treatments with a control is developed, and a procedure for making exact joint confidence statements for this multiple comparisons problem is described.
Abstract: In this paper we develop a theory of optimal incomplete block designs for comparing several treatments with a control. This class of designs is appropriate for comparing simultaneously p ≥ 2 test treatments with a control treatment (the so-called multiple comparisons with a control (MCC) problem) when the observations are taken in incomplete blocks of common size k < p + 1. For this problem we propose a new general class of incomplete block designs that are balanced with respect to (wrt) test treatments. We shall use the abbreviation BTIB to refer to such designs. We study their structure and give some methods of construction. A procedure for making exact joint confidence statements for this multiple comparisons problem is described. By using a new concept of admissibility of designs, it is shown how “inferior” designs can be eliminated from consideration, and attention limited to a small class of BTIB designs that can be constructed from so-called generator designs in the minimal complete class of such d...
99 citations
"Designs Robust against Violation of..." refers methods in this paper
...In the class of incomplete treatment-control block designs, Bechhofer and Tamhane (1981) introduced balanced treatment incomplete block designs (BTIB designs)....
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TL;DR: In this article, existing knowledge on optimal designs for comparing test treatments with controls under 0-, 1-, and 2-way elimination of heterogeneity models is presented. But the results are motivated through numerical examples.
Abstract: This article outlines existing knowledge on optimal designs for comparing test treatments with controls under 0-, 1- and 2-way elimination of heterogeneity models. The results are motivated through numerical examples.
98 citations