Modeling through group invariance: an interesting example with potential applications. (English) Zbl 1101.62341

Summary: A particular linear group symmetry model, called the dyadic symmetry model, is studied in some detail. Statistical procedures analogous to (multivariate) analysis of variance are introduced. This model may be suitable for various kinds of data collected on pairs of sampling units. Examples include (complete) diallel cross experiments in genetics and social relations analysis in psychology, for which ad hoc methods of analysis have been developed independently in those disciplines.
Our approach is based entirely on formal data structure following the principle of group symmetry, and hence its applicability is not restricted to any specific substantive areas. This paper illustrates the benefits that can be derived from the exploration of mathematical meanings in the development of statistical methods.


62H05 Characterization and structure theory for multivariate probability distributions; copulas
62P15 Applications of statistics to psychology
62A01 Foundations and philosophical topics in statistics
62J10 Analysis of variance and covariance (ANOVA)
62P10 Applications of statistics to biology and medical sciences; meta analysis
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[1] ANDERSSON, S. A. (1975). Invariant normal models. Ann. Statist. 3 132-154. · Zbl 0373.62029 · doi:10.1214/aos/1176343004
[2] ANDERSSON, S. A. (1990). The lattice structure of orthogonal linear-models and orthogonal variance component models. Scand. J. Statist. 17 287-319. · Zbl 0717.62057
[3] ANDERSSON, S. A. and MADSEN, J. (1998). Sy mmetry and lattice conditional independence in a multivariate normal distribution. Ann. Statist. 26 525-572. · Zbl 0943.62047 · doi:10.1214/aos/1028144848
[4] BECHTEL, G. G. (1967). The analysis of variance and pairwise scaling. Psy chometrika 32 47-65.
[5] BECHTEL, G. G. (1971). A dual scaling analysis for paired compositions. Psy chometrika 36 135- 154. · doi:10.1007/BF02291394
[6] BOX, G. E. P. and TIAO, G. C. (1973). Bayesian Inference in Statistical Analy sis. Addison-Wesley, Reading, MA. · Zbl 0271.62044
[7] CAPPÉ, O. and ROBERT, C. P. (2000). Markov chain Monte Carlo: 10 years and still running! J. Amer. Statist. Assoc. 95 1282-1286. JSTOR: · Zbl 1072.60506 · doi:10.2307/2669770
[8] COCKERHAM, C. C. and WEIR, B. S. (1977). Quadratic analyses of reciprocal crosses. Biometrics 33 187-204. · Zbl 0351.62053 · doi:10.2307/2529312
[9] DAVID, H. A. (1988). The Method of Paired Comparisons, 2nd ed. Griffin, London. · Zbl 0665.62075
[10] DAWID, A. P. (1988). Sy mmetry models and hy potheses for structured data lay outs (with discussion). J. Roy. Statist. Soc. Ser. B 50 1-34. JSTOR:
[11] DEMPSTER, A. P., LAIRD, N. M. and RUBIN, D. B. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1-38. JSTOR: · Zbl 0364.62022
[12] KENNY, D. A. (1994). Interpersonal Perception. Guilford, New York.
[13] LEV, J. and KINDER, E. (1957). New analysis of variance formulas for treating data from mutually paired subjects. Psy chometrika 22 1-15. · Zbl 0085.35404 · doi:10.1007/BF02289205
[14] LI, H. (2000). Comment on ”Invariance and factorial models,” by P. McCullagh. J. Roy. Statist. Soc. Ser. B 62 250-251. JSTOR: · doi:10.1111/1467-9868.00229
[15] LIU, C. and RUBIN, D. B. (1994). The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika 81 633-648. JSTOR: · Zbl 0812.62028 · doi:10.1093/biomet/81.4.633
[16] LIU, C., RUBIN, D. B. and WU, Y. N. (1998). Parameter expansion to accelerate EM: The PX-EM algorithm. Biometrika 85 755-770. JSTOR: · Zbl 0921.62071 · doi:10.1093/biomet/85.4.755
[17] MCCULLAGH, P. (2000). Invariance and factorial models (with discussion). J. Roy. Statist. Soc. Ser. B 62 209-256. JSTOR: · doi:10.1111/1467-9868.00229
[18] MENG, X.-L. and RUBIN, D. B. (1991). Using the EM algorithm to obtain asy mptotic variancecovariance matrices: The SEM algorithm. J. Amer. Statist. Assoc. 86 899-909.
[19] MENG, X.-L. and RUBIN, D. B. (1993). Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika 80 267-278. JSTOR: · Zbl 0778.62022 · doi:10.1093/biomet/80.2.267
[20] MENG, X.-L. and VAN Dy K, D. A. (1997). The EM algorithm-an old folk song sung to a fast new tune (with discussion). J. Roy. Statist. Soc. Ser. B 59 511-567. JSTOR: · Zbl 1090.62518 · doi:10.1111/1467-9868.00082
[21] MENG, X.-L. and VAN Dy K, D. A. (1998). Fast EM-ty pe implementations for mixed-effects models. J. Roy. Statist. Soc. Ser. B 59 559-578. · Zbl 0909.62073 · doi:10.1111/1467-9868.00140
[22] OAKES, D. (1999). Direct calculation of the information matrix via the EM algorithm. J. Roy. Statist. Soc. Ser. B 61 479-482. JSTOR: · Zbl 0913.62036 · doi:10.1111/1467-9868.00188
[23] PERLMAN, M. D. (1987). Group sy mmetry models. Comment on ”A review of multivariate analysis,” by M. J. Schervish. Statist. Sci. 2 421-425.
[24] RAO, P. S. R. S. (1997). Variance Components Estimation. Chapman and Hall, London. · Zbl 0996.62501
[25] SEARLE, S. R., CASELLA, G. and MCCULLOCH, C. E. (1992). Variance Components. Wiley, New York. · Zbl 0850.62007
[26] VAN Dy K, D. A., MENG X.-L. and RUBIN, D. B. (1995). Maximum likelihood estimation via the ECM algorithm: computing the asy mptotic variance. Statist. Sinica 5 55-75. · Zbl 0824.62021
[27] WARNER, R. M., KENNY, D. A. and STOTO, M. (1979). A new round robin analysis of variance for social interaction data. J. Personality Soc. Psy ch. 37 1742-1757.
[28] ROCHESTER, NEW YORK 14642 E-MAIL: liheng@bst.rochester.edu
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