Factorial ANOVA with unbalanced data: A fresh look at the types of sums of squares

In this paper we endeavour to provide a largely non-technical description of the issues surrounding unbalanced factorial ANOVA and review the arguments made for and against the use of Type I, Type II and Type III sums of squares. Though the issue of which is the 'best' approach has been debated in the literature for decades, to date confusion remains around how the procedures differ and which is most appropriate. We ultimately recommend use of the Type II sums of squares for analysis of main effects because when no interaction is present it tests meaningful hypotheses and is the most statistically powerful alternative.

[1]  P. Lachenbruch.  Statistical Power Analysis for the Behavioral Sciences (2nd ed.) , 1989 .

[2]  David C. Howell,et al.  Nonorthogonal Analysis of Variance: Putting the Question before the Answer , 1982 .

[3]  S. von Felten,et al.  Analysis of variance with unbalanced data: an update for ecology & evolution. , 2010, The Journal of animal ecology.

[4]  J. Lewsey,et al.  A STUDY OF TYPE II AND TYPE III POWER FOR TESTING HYPOTHESES FROM UNBALANCED FACTORIAL DESIGNS , 2001 .

[5]  John A. Nelder,et al.  The Computer Analysis of Factorial Experiments: In Memoriam—Frank Yates , 1995 .

[6]  Mark I. Appelbaum,et al.  Nonorthogonal analysis of variance--once again. , 1980 .

[7]  Frank Yates,et al.  The Analysis of Multiple Classifications with Unequal Numbers in the Different Classes , 1934 .

[8]  Øyvind Langsrud,et al.  ANOVA for unbalanced data: Use Type II instead of Type III sums of squares , 2003, Stat. Comput..

[9]  R. R. Hocking,et al.  Methods of Analysis of Linear Models with Unbalanced Data , 1978 .

[10]  Jacob Cohen.  Statistical Power Analysis for the Behavioral Sciences , 1969, The SAGE Encyclopedia of Research Design.

[11]  P. Gallo.  CENTER-WEIGHTING ISSUES IN MULTICENTER CLINICAL TRIALS , 2000, Journal of biopharmaceutical statistics.

[12]  Sanford Weisberg,et al.  An R Companion to Applied Regression , 2010 .

[13]  Ruth G. Shaw,et al.  Anova for Unbalanced Data: An Overview , 1993 .

[14]  S. Weiss,et al.  Gambling behaviors of former athletes: the delayed competitive effect. , 2008 .

[15]  Allan Stewart-Oaten,et al.  Rules and judgments in statistics : three examples , 1995 .

[16]  J. J. Higgins,et al.  Tests of Hypotheses for Unbalanced Factorial Designs Under Various Regression/Coding Method Combinations , 1978 .

[17]  S. R. Searle.  Comments on J. A. Nelder. ‘The statistics of linear models: back to basics’ , 1995 .

[18]  M. Kutner.  Hypothesis Testing in Linear Models (Eisenhart Model I) , 1974 .

[19]  James E. Carlson,et al.  Analysis of nonorthogonal fixed-effects designs. , 1974 .

[20]  J. Overall,et al.  Concerning least squares analysis of experimental data. , 1969 .

[21]  Mark I. Appelbaum,et al.  Some problems in the nonorthogonal analysis of variance. , 1974 .

[22]  John A. Nelder,et al.  The statistics of linear models: back to basics , 1995 .

[23]  Deborah A. Prentice,et al.  Contrast tests of interaction hypothesis. , 1997 .