Abstract
Two-way nested design with mixed effects model arises in many practical situations. In the classical analysis of variance set-up, a test for the absence of the random effects is obtained under the assumption that the random effects and the errors are normally distributed. The present paper avoids this assumption and provides an asymptotically distribution-free test procedure for the above problem. The asymptotic null distribution of the test statistic is obtained. Actual implementation of the test is straight forward given the prior information on quantiles of the intra-block differences of observations. In the absence of such information, working test procedures are proposed. The performances of these tests are compared with the classical analysis of variance test through simulations. The tests are then illustrated by some real data sets.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Banerjee T. (1984). Utilization of additional information in non-parametric tests. Calcutta Statistical Association Bulletin 33:69–84
Brunner E., Denker M. (1994). Rank statistics under dependent observations and applications to factorial designs. Journal of Statistical planning and Inference 42:353–378
Brunner E., Neumann N. (1982). Rank tests for correlated random variables. Biometrical Journal 24:373–389
Brunner E., Puri M.L. (1996). Nonparametric methods in design and analysis of experiments. In: Ghosh S., Rao C.R. (eds) Handbook of statistics (Vol 13). Elsevier Science Publishers, Amsterdam, pp. 631–703
Chatterjee S.K., Banerjee T. (1986). Combining alternative rank tests for the multiple regression problem. Calcutta Statistical Association Bulletin 35:169–188
Chatterjee S.K., Banerjee T. (1991). Combination of multiple scores for nonparametric testing in multivariate linear regression set-up. Communications in Statistics Theory and Methods 19:2967–2999
Dean A.M., Wolfe D.A. (1996). Nonparametric analysis of experiments. In: Ghosh S., Rao C.R. (eds) Handbook of statistics (Vol 13). Elsevier Science Publishers, Amsterdam, pp. 705–758
Dean A.M., Voss D. (1999). Design and analysis of experiments. Springer, New York
Friedmann M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. Journal of the American Statistical Association 32:675–699
Hajek J. (1969). Non-parametric statistics. Holden Day, San Francisco
Hajek J. (1970). Miscellaneous problems of rank test theory. In: Puri M.L. (eds) Non-parametric techniques in statistical inference. Cambridege university Press, Cambridge
Hajek J., Sidak Z. (1967). Theory of rank tests. Academic, New York
Hogg R.V., Fisher D.M., Randles R.H.(1975). A two-sample adaptive distribution-free test. Journal of the American Statistical Association 70:656–661
Kruskal W.H., Wallis W.A. (1952). The use of ranks in one-criterion variance analysis. Journal of the American Statistical Association 47:583–621
Lemmer H.H., Stoker D.J. (1967). A distribution-free analysis of variance for the two-way classification. South African Statistical Journal 1:67–74
Montgomery D.C. (1984). Design and analysis of experiments (2nd ed). Wiley, New York
Morgan J.P. (1996). Nested designs. In: Ghosh S., Rao C.R. (eds) Handbook of statistics (Vol 13). Elsevier Science Publishers, Amsterdam, pp. 939–976
Scheffe H. (1959). The analysis of variance. John Wiley, New York
Sidak Z., Sen P.K., Hajek J. (1999). Theory of rank tests (2nd ed). Academic, San Diego
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Banerjee, T., Biswas, A. Testing for the Absence of Random Effects in a Two-way Nested Design with Mixed Effects Model: A Nonparametric Approach. AISM 59, 197–210 (2007). https://doi.org/10.1007/s10463-006-0049-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-006-0049-5