×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.