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Vanleeuwen, Dawn M.; Birkes, David S.; Seely, Justus F.

Balance and orthogonality in designs for mixed classification models. (English) Zbl 0963.62059

Ann. Stat. 27, No. 6, 1927-1947 (1999).
Summary: A classification model is easiest to analyze when it has a balanced design. Many of the nice features of balanced designs are retained by error-orthogonal designs, which were introduced by the authors in J. Stat. Plann. Inference 73, No. 1-2, 373-389 (1998; Zbl 0933.62069). The present paper defines a kind of “partially balanced” design and shows that this partial balance is sufficient to ensure the error-orthogonality of a mixed classification model. Results are provided that make the partial balance condition easy to check. It is shown that for a maximal-rank error-orthogonal design, the Type I sum of squares for a random effect coincides with the Type II sum of squares.

Cited in 18 Documents

MSC:

62J10 Analysis of variance and covariance (ANOVA)
62K99 Design of statistical experiments

Keywords:

ANOVA; mixed linear model; sums of squares; orthogonal block structure; orthogonal design

Citations:

Zbl 0933.62069
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Full Text: DOI

References:

[1] BROWN, K. G. 1983. Sub-balanced data and the mixed analysis of variance. J. Amer. Statist. Assoc. 78 162 167. JSTOR: · Zbl 0501.62068 · doi:10.2307/2287124
[2] HOUTMAN, A. M. and SPEED, T. P. 1983. Balance in designed experiments with orthogonal block structure. Ann. Math. Statist. 42 1069 1085. Z. · Zbl 0566.62065
[3] NELDER, J. A. 1965. The analysis of randomized experiments with orthogonal block structure I. Block structure and the null analysis of variance. Proc. Roy. Soc. London Ser. A 273 147 162. Z. · Zbl 0124.10703 · doi:10.1098/rspa.1965.0012
[4] PUKELSHEIM, F. 1981. On the existence of unbiased nonnegative estimates of variance covariance components. Ann. Statist. 9 293 299. Z. · Zbl 0486.62066 · doi:10.1214/aos/1176345395
[5] PUKELSHEIM, F. 1993. Optimal Design of Experiments. Wiley, New York. Z. · Zbl 0834.62068
[6] RAO, C. R. and KLEFFE, J. 1988. Estimation of Variance Components and Applications. NorthHolland, New York. Z. · Zbl 0645.62073
[7] SEELY, J. 1971. Quadratic subspaces and completeness. Ann. Math. Statist. 42 710 721. Z. · Zbl 0249.62067 · doi:10.1214/aoms/1177693420
[8] SEELY, J. 1972. Completeness for a family of multivariate normal distributions. Ann. Math. Statist. 43 1644 1647. Z. · Zbl 0257.62018 · doi:10.1214/aoms/1177692396
[9] SEELY, J. 1977. Minimal sufficient statistics and completeness for multivariate normal families. Sankhya Ser. A 39 170 185. Z. · Zbl 0409.62004
[10] SEIFERT, B. 1979. Optimal testing for fixed effects in general balanced mixed classification models. Math. Operationsforsch. Statist. Ser. Statist. 10 237 255. Z. · Zbl 0415.62052 · doi:10.1080/02331887908801481
[11] SZATROWSKI, T. H. 1980. Necessary and sufficient conditions for explicit solutions in the multivariate normal estimation problem for patterned means and covariances. Ann. Statist. 8 802 810. Z. · Zbl 0507.62050 · doi:10.1214/aos/1176345072
[12] VANLEEUWEN, D. M., BIRKES, D. S. and SEELY, J. F. 1997. Balance and orthogonality in designs for mixed classification models. Technical Report 168, Dept. Statistics, Oregon State Univ. Z. · Zbl 0963.62059
[13] VANLEEUWEN, D. M., SEELY, J. F. and BIRKES, D. S. 1998. Sufficient conditions for orthogonal designs in mixed linear models. J. Statist. Plann. Inference 73 373 389. · Zbl 0933.62069 · doi:10.1016/S0378-3758(98)00071-8
[14] LAS CRUCES, NEW MEXICO 88003 CORVALLIS, OREGON 97331
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