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On choosing between fixed and random block effects in some no-interaction models. (English) Zbl 0958.62069
Summary: The purpose of this article is to strengthen the understanding of the relationship between a fixed-blocks and random-blocks analysis in models that do not include interactions between treatments and blocks. Treating the block effects as random has been recommended in the literature for balanced incomplete block designs (BIBD) because it results in smaller variances of treatment contrasts. This reduction in variance is large if the block-to-block variation relative to the total variation is small. However, this analysis is also more complicated because it results in a subjective interpretation of results if the block variance component is non-positive. The probability of a non-positive variance component is large precisely in those situations where a random-blocks analysis is useful – that is, when the block-to-block variation, relative to the total variation, is small.
In contrast, the analysis in which the block effects are fixed is computationally simpler and less subjective. The loss in power for some BIBD with a fixed effects analysis is trivial. In such cases, we recommend treating the block effects as fixed. For response surface experiments designed in blocks, however, an opposite recommendation is made. When block effects are fixed, the variance of the estimated response surface is not uniquely estimated, and in practice this variance is obtained by ignoring the block effects. It is argued that a more reasonable approach is to treat the block effects to be random than to ignore it.

##### MSC:
 62K10 Statistical block designs 62K20 Response surface designs
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##### References:
 [1] Box, G.E.P; Draper, N.R, Empirical model building and response surfaces, (1987), Wiley New York · Zbl 0482.62065 [2] Burns, J.C; Giesbrecht, F.G; Harvey, R.W; Linnerud, A.C, Central Appalachian Hill land pasture evaluation using cows and calves. I. ordinary and generalized least squares analysis for an unbalanced grazing experiment, Agron. J., 75, 865-871, (1983) [3] Feingold, M, A test statistic for combined intra- and inter-block estimates, J. statist. plann. inference, 12, 103-114, (1985) [4] Ganju, J; Lucas, J.M, Analysis of unbalanced data from an experiment with random block effects and unequally spaced factor levels, Amer. statist., 54, 5-11, (2000) [5] Giesbrecht, F.G., Burns, J.C., 1985. Two-stage analysis based on a mixed model: large-sample asymptotic theory and small-sample simulation results. Biometrics 41, 477-486. · Zbl 0653.62057 [6] Gill, J.L, Design and analysis of experiments in the animal and medical sciences, (1979), Iowa State University Press Ames [7] Graybill, F.A, An introduction to linear statistical models, (1961), McGraw-Hill New York · Zbl 0121.35605 [8] Khuri, A.I, Response surface models with block effects, Technometrics, 31, 159-171, (1992) · Zbl 0850.62618 [9] Khuri, A.I, Effect of blocking on the estimation of a response surface, J. appl. statist., 21, 305-316, (1994) [10] SAS Institute, 1992. SAS Technical Report P-229. [11] Searle, S.R, Linear models, (1971), Wiley New York [12] Searle, S.R; Casella, G; McCullogh, C.E, Variance components, (1992), Wiley New York [13] Swallow, W.H; Monahan, J.F, Monte Carlo comparison of ANOVA, MIVQUE, REML and ML estimators of variance components, Technometrics, 26, 47-57, (1984) · Zbl 0548.62051 [14] Yates, F, The recovery of inter-block information in balanced incomplete block designs, Ann. eugenics, 10, 317-325, (1940)
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