Summary
In this paper, we summarize some recent developments in the analysis of nonparametric models where the classical models of ANOVA are generalized in such a way that not only the assumption of normality is relaxed but also the structure of the designs is introduced in a broader framework and also the concept of treatment effects is redefined. The continuity of the distribution functions is not assumed so that not only data from continuous distributions but also data with ties are included in this general setup. In designs with independent observations as well as in repeated measures designs, the hypotheses are formulated by means of the distribution functions. The main results are given in a unified form. Some applications to special designs are considered, where in simple designs, some well known statistics (such as the Kruskal-Wallis statistic and the χ2-statistic for dichotomous data) come out as special cases. The general framework presented here enables the nonparametric analysis of data with continuous distribution functions as well as arbitrary discrete data such as count data, ordered categorical and dichotomous data.
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References
Akritas, M. G. (1990). The rank transform method in some two-factor designs. J. Amer. Statist. Assoc. 85, 73–78.
Akritas, M. G. (1991). Limitations on the rank transform procedure: A study of repeated measures designs, Part I. J. Amer. Statist Assoc. 86, 457–460.
Akritas, M. G., and Arnold, S. F. (1994). Fully nonparametric hypotheses for factorial designs I: Multivariate repeated measures designs. J. Amer. Statist. Assoc. 89, 336–343.
Akritas, M. G., Arnold, S. F. and Brunner, E. (1997). Nonparametric hypotheses and rank statistics for unbalanced factorial designs. J. Amer. Statist. Assoc. 92, 258–265.
Akritas, M. G. and Brunner, E. (1996). Rank Tests for Patterned Alternatives in Factorial Designs with Interactions. Festschrift on the Occasion of the 65 th birthday of Madan L Pari, VSP-International Science Publishers, Utrecht, The Netherlands, 277–288.
Akritas, M. G. and Brunner, E. (1997). A unified approach to ranks tests in mixed models. J. Statist. Plann. Inference 61, 249–277.
Blair, R. C, Sawilowski, S. S. and Higgens, J. J. (1987). Limitations of the rank transform statistic in tests for interactions. Comm. in Statist., Part B — Simulation and Computation 16, 1133–1145.
Boos, D. D. and Brownie, C. (1992). Rank based mixed model approach to multisite clinical trials. Biometrics 48, 61–72.
Box, G. E. P. (1954). Some theorems on quadratic forms applied in the study of analysis of variance problems, I. Effect of inequality of variance in the one-way classification. Ann. Math. Statist. 25, 290–302.
Brunner, E. and Denker, M. (1994). Rank statistics under dependent observations and applications to factorial designs. J. Statist. Plann. Inference 42, 353–378.
Brunner, E., Dette, H. and Munk, A. (1997). Box-type approximations in nonparametric factorial designs. J. Amer. Statist. Assoc. 92, 1494–1502.
Brunner, E. and Langer, F. (1999). Nichtparametrische Analyse longitudinaler Daten, Oldenbourg, München.
Brunner, E., Munzel, U. and Puri, M. L. (1999). Rank-Score Tests in Factorial Designs with Repeated Measures. J. Mult. Analysis 70, 286–317.
Brunner, E. and Neumann, N. (1982). Rank Tests for Correlated Random Variables. Biometr. J. 24, 373–389.
Brunner, E. and Neumann, N. (1986a). Rank tests in 2x2 designs. Statist. Neerlandica 40, 251–271.
Brunner, E. and Neumann, N. (19866). Two-sample rank tests in general models. Biometr. J. 28, 395–402.
Brunner, E. and Puri, M. L. (1996). Nonparametric methods in design and analysis of experiments. Handbook of Statistics 13, (S. Ghosh and C.R. Rao, Eds.), 631–703.
Brunner, E. and Puri, M. L. (2000). A class of rank-score tests in factorial designs. J. Statist. Plann. Inference (to appear).
Brunner, E., Puri, M. L. and Sun, S. (1995). Nonparametric methods for stratified two-sample designs with application to multi clinic trials. J. Amer. Statist. Assoc. 90, 1004–1014.
Conover, W. J. and Iman, R. L. (1976). On some alternative procedures using ranks for the analysis of experimental designs. Common. Statist. A5, 14, 1349–1368.
Conover, W. J. and Iman, R. L. (1981). Rank transformations as a bridge between parametric and nonparametric statistics (with discussion). Amer. Statist. 35 124–133.
Govindarajulu, Z. (1975). Robustness of Mann-Whitney-Wilcoxon test to dependence in the variables. Studia Scientiarum Mathematicarum Hungarica 10, 39–45.
Graybill, F. A. (1976). Theory and Applications of the Linear Model. Duxbury Press, North Scituate, Mass.
Groggle, D. J. and Skillings, J. H. (1986). Distribution-free tests for main effects in multifactor designs. Amer. Statist. 40, 99–102.
Hettmansperger, T. P. and Norton, R. M. (1987). Tests for patterned alternatives in k-sample problems. J. Amer. Statist. Assoc. 82, 292–299.
Hora, S. C. and Iman, R. L. (1988). Asymptotic relative efficiencies of the rank-transformation procedure in randomized complete block designs. J. Amer. Statist. Assoc. 83, 462–470.
Hollander, M., Pledger, G. and Lin, P. E. (1974). Robustness of the Wilcoxon test to a certain dependency between samples. Ann. Statist. 2, 177–181.
Kepner, J. L. and Robinson, D. H. (1988). Nonparametric Methods for Detecting Treatment Effects in Repeated Measures Designs. J. Amer. Statist. Assoc., 83, 456–461.
KOCH, G. G. (1969). Some aspects of the statistical analysis of ‘split-plot’ experiments in completely randomized layouts. J. Amer. Statist. Assoc. 64, 485–506.
Koch, G. G. (1970). The use of nonparametric methods in the statistical analysis of a complex split plot experiment. Biometrics 26, 105–128.
Koch, G. G. and Sen, P. K. (1968). Some aspects of the statistical analysis of the ‘mixed model’. Biometrics, 24, 27–48.
Kruskal, W. H. (1952). A nonparametric test for the several sample problem. Ann. Math. Statist. 23, 525–540.
Kruskal, W. H. and Wallis, W. A. (1952). The use of ranks in one-criterion variance analysis. J. Amer. Statist. Assoc. 47, 583–621.
Kruskal, W. H. and Wallis, W. A. (1953). Errata in: The use of ranks in one-criterion variance analysis. J. Amer. Statist. Assoc. 48, 907–911.
Kulle, B. (1999). Nichtparametrisches Behrens-Fisher-Problem im Mehr-Stichprobenfall. Diplomarbeit, Universitat Göttingen, Inst, für Mathematische Stochastik.
Langer, F. (1998). Berücksichtigung von Kovariablen im nichtparametrischen gemischten Modell. Dissertation, Universität Göttingen, Inst, für Mathematische Stochastik.
Lemmer, H. H. (1980). Some empirical results on the two-way analysis of variance by ranks. Comm. Statist. Theory and Methods 14, 1427–1438.
Levy, P. (1925). Calcul des Probabilities. Gauthiers-Villars, Editeurs, Paris.
Mack, G. A. and Skillings, J. H. (1980). A Friedman-type rank test for main effects in a two-factor ANOVA. J. Amer. Statist. Assoc. 75, 947–951.
Marden, J. I. and Muyot, M. E. (1995). Rank tests for main and interaction effects in analysis of variance. J. Amer. Statist. Assoc. 90, 1388–1398.
Mathai, A. M. and Provost, S. B. (1992). Quadratic Forms in Random Variables. Marcel Dekker, New York.
Mehra, K. L. and Puri, M. L. (1967). Multi-sample analogues of some one-sample tests. Ann. Math. Statist. 38, 523–549.
Munzel, U. (1996). Multivariate nichtparametrische Verfabren für feste Faktoren in mehrfak-toriellen Versuchsanlagen. Dissertation, Universität Göttingen, Inst, für Mathematische Stochastik.
Munzel, U. (1999a). Linear rank score statistics when ties are present. Statist. Probab. Lett. 41, 389–395.
Munzel, U. (19996). Nonparametric methods for paired samples. Statistica Neerlandica 53, 277–286.
Munzel, U. and Brunner, E. (2000). Nonparametric methods in multivariate factorial designs. J. Statist. Plann. Inference (to appear).
Page, E. B. (1963). Ordered hypotheses for multiple treatments: A significance test for linear ranks. J. Amer. Statist. Assoc. 58, 216–230.
Patel, K. M. and HOEL, D. G. (1973). A nonparametric test for interaction in factorial experiments. J. Amer. Statist. Assoc. 68, 615–620.
PURl, M. L. (1964). Asymptotic efficiency of a class of c-sample tests. Ann. Math. Statist. 35, 102–121.
Puri, M. L. and Sen, P. K. (1969). Analysis of covariance based on general rank scores. Ann. Math. Statist. 40, 610–618.
Quade, D. (1967). Rank analysis of covariance. J. Amer. Statist. Assoc. 62, 1187–1200.
Raviv, A. (1978). A Non-parametric Test for Comparing Two Non-independent Distributions. J. R. Statist. Soc. B 40, 253–261.
Rinaman, W. C., Jr. (1983). On distribution-free rank tests for two-way layouts. J. Amer. Statist. Assoc. 78, 655–659.
Ruymgaart, F.H. (1980). A unified approach to the asymptotic distribution theory of certain midrank statistics. In: Statistique non Parametrique Asymptotique, 1–18, J.P. Raoult (Ed.), Lecture Notes on Mathematics, No. 821, Springer, Berlin.
Sen, P. K. (1967). On some non-parametric generalizations of Wilk’s test for HM,Hvc and Hmvc. Ann. Inst. Statist. Math. 19, 541–571.
Siemer, A. (1999). Berücksichtigung von heterogen verteilten Kovariablen in einem nicht-parametrischen Modell. Diplomarbeit, Universität Göttingen, Inst, für Mathematische Stochastik.
Thompson, G. L. (1990). Asymptotic Distribution of Rank Statistics Under Dependencies with Multivariate Applications. J. Mult. Analysis 33, 183–211.
Thompson, G. L. (1991). A unified approach to rank tests for multivariate and repeated measures designs. J. Amer. Statist. Assoc. 86, 410–419.
Thompson, G. L. and Ammann, L. P. (1989). Efficacies of rank-transform statistics in two-way models with no interaction. J. Amer. Statist. Assoc. 84, 325–330.
Thompson, G. L. and Ammann, L. P. (1990). Efficiencies of interblock rank statistics for repeated measures designs. J. Amer. Statist. Assoc. 85, 519–528.
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Brunner, E., Puri, M.L. Nonparametric methods in factorial designs. Statistical Papers 42, 1–52 (2001). https://doi.org/10.1007/s003620000039
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DOI: https://doi.org/10.1007/s003620000039