Source: Internat. Statist. Rev. Volume 74, Number 3
(2006), 391-402.
We consider exact F tests for the hypothesis of null random factor effect in the presence of interaction under the two factor mixed models involved in the mixed models controversy. We show that under the constrained parameter (CP) model, even in unbalanced data situations, MSB/MSE (in the usual ANOVA notation) follows an exact F distribution when the null hypothesis holds. We also obtain an exact F test for what is generally (and erroneously) assumed to be an equivalent hypothesis under the unconstrained parameter (UP) model. For unbalanced data, such a corresponding test statistic does not coincide with MSB/MSAB (the test statistic advocated for balanced data cases). We compute the power of the exact test under different imbalance patterns and show that although the loss of power increases with the degree of imbalance, it still remains reasonable from a practical point of view.
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription.
Read more about accessing full-text
References
[1] Crainiceanu, C.M. & Ruppert, D. (2004). Likelihood ratio tests in linear mixed models with one variance component. Journal of the Royal Statistical Society, B, 66, 165-185.
[2] Hinkelmann, K. (2000). Resolving the Mixed Models Controversy, Comments. The American Statistician, 54, 228.
[3] Hinkelmann, K. & Kempthorne, O. (1994). Design and Analysis of Experiments: Introduction to Experimental Design, Vol 1. New York: Wiley.
[4] Hocking, R.R. (1973). A discussion of the two-way mixed model. The American Statistician, 27, 148-152.
[5] Khuri, A.I. (1987). Measures of imbalance for unbalanced models. Biometrical Journal, 29, 383-396.
[6] Khuri, A.I., Mathew, T. & Sinha, B.K. (1998). Statistical Test for Mixed Linear Models. New York: Wiley.
[7] Kshirsagar, A.M. (1983). A Course in Linear Models. New York: Dekker.
[8] Lencina, V.B., Singer, J.M & Stanek, E.J. (2005). Much ado about nothing: the mixed models controversy revisited. International Statistical Review, 73, 9-20.
[9] Milliken, G.A. & Johnson, D.E. (1984). Analysis of Messy Data: Vol. 1. Designed Experiments. Belmont, CA: Lifetime Learning Publications.
[10] Montgomery, D.C. (1997). Design and Analysis of Experiments, 4th edition. New York: Wiley.
[11] Neter, J., Wasserman, W., Kutner, M.H. & Nachtsheim, C.J. (1996). Applied Linear Statistical Models (4th ed.). Homewood, Ill: Irwin.
[12] Öfversten, J. (1993). Exact Test for Variance Components in Unbalanced Mixed Linear Model. Biometrics, 49, 45-57.
[13] Rao, C.R. & Mitra, S.K. (1971). Generalized Inverse of Matrices and its Applications. New York: Wiley.
[14] SAS Institute. Inc. (1990). SAS Users Guide: Statistics (Version 6). Cary, NC.
[15] Searle, S.R. (1971). Linear Models. New York: Wiley.
[16] Searle, S.R. (1982). Matrix Algebra Useful for Statistics. New York: Wiley.
[17] Stram, D.O. & Lee, J.W. (1994). Variance Components Testing in the Longitudinal Mixed Effects Model. Biometrics, 50, 1171-1177.
[18] Voss, D.T. (1999). Resolving the Mixed Models Controversy. The American Statistician, 53, 352-356.
[19] Wolfinger, R. & Stroup, W.W. (2000). Resolving the Mixed Models Controversy, Comments. The American Statistician, 54, 228.
