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A spectral form of the dispersion model in designs with arbitrarily unequal block sizes. (English) Zbl 0758.62033
Summary: Under a usual additive mixed model, a spectral decomposition is given to each observed vector $$Y$$ from a class of block designs with arbitrarily unequal block sizes. As a result, an analysis of variance, in the sense defined by F. A. Graybill and R. A. Hultquist [Ann. Math. Stat. 32, 261-269 (1961; Zbl 0109.375)] and T. P. Speed [Ann. Stat. 15, 885-941 (1987; Zbl 0637.62070)], exists for such a design. The intrablock anova is discussed. Furthermore based on the dispersion structure, it is seen that the unbalancedness forces some useful information beyond the intrablock analysis, and even interblock comparisons. Let a group be a set of observations from the blocks of same size, a further decomposition of the spectral form provides an intergroup comparison which completes the anova of such designs.
##### MSC:
 62K10 Statistical block designs 62J10 Analysis of variance and covariance (ANOVA)
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##### References:
 [1] Brown, K.G., On analysis of variance in the mixed model, Ann. statist., 12, 4, 1488-1499, (1984) · Zbl 0558.62062 [2] Brown, L.D.; Cohen, A., Point and confidence estimation of a common Mean and recovery of inter-block information, Ann. statist., 2, 963-976, (1974) · Zbl 0305.62019 [3] Cohen, A.; Sackrowitz, H.B., Hypothesis testing for the common Mean and for balanced incomplete block designs, Ann. statist., 5, 195-211, (1977) · Zbl 0381.62024 [4] Cunningham, E.P.; Henderson, C.R., An iterative procedure for estimating fixed effects and variance components in mixed model situations, Biometrics, 24, 13-25, (1968) [5] Fisher, R.A., Discussion in Wishart, J., J. roy. statist. soc. suppl., 1, 26-61, (1934) [6] Graybill, F.A.; Hultquist, R.A., Theorems concerning Eisenhart’s model II, Ann. math. statist., 32, 261-269, (1961) · Zbl 0109.37501 [7] Hartley, H.O.; Rao, J.N.K., Maximum likelihood estimation for the mixed analysis of variance model, Biometrika, 54, 93-108, (1967) · Zbl 0178.22001 [8] Houtman, A.M.; Speed, T.P., Balance in designed experiments with orthogonal block structure, Ann. statist., 4, 1069-1085, (1983) · Zbl 0566.62065 [9] Nelder, J.A., The combination of information in generally balanced designs, J. roy. statist. soc. ser. B., 30, 303-311, (1968) · Zbl 0164.49101 [10] Patterson, H.D.; Thompson, R., Recovery of interblock information when block sizes are unequal, Biometrika, 58, 545-554, (1971) · Zbl 0228.62046 [11] Rao, C.R., Estimation of variance and covariance components in linear models, J. amer. statist. assoc., 67, 337, 112-116, (1972) · Zbl 0231.62082 [12] Speed, T.P., What is an analysis of variance?, Ann. statist., 15, 3, 885-911, (1987) · Zbl 0637.62070
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