Wrong error term for test of random factor in two-way mixed-effects ANOVA in SPSS In SPSS, when I run a two-way mixed-effects ANOVA (with factor A fixed and factor B random), the between subjects effects table reported by SPSS uses mean square for the interaction (MS_AB) as the error term for testing the main effect of factor B, the random factor. According to Keppel and Wickens (2004) and Maxwell and Delaney (2004, p. 477) the error term for the main effect of factor B, the random factor, should be MS_within:

Am I missing something or is this an error in SPSS's reporting? 
 A: I had also posted this on the SPSSX listserv and got the following link from a response by Bruce Weaver:

From: nichols@spss.com (David Nichols)
Subject: Expected mean squares and error terms in GLM
Date: 1996/11/05
Message-ID: <55oa9t$1tj@netsrv2.spss.com>#1/1
organization: SPSS, Inc.
newsgroups: comp.soft-sys.stat.spss

I've had a few questions from users about expected mean squares and
  error terms in GLM. In particular, with a two way design with A fixed
  and B random, many people are expecting to see the A term tested
  against A*B and B tested against the within cells term. In the model
  used by GLM, the interaction term is automatically assumed to be
  random, expected mean squares are calculated using Hartley's method of
  synthesis, and the results are not as many people are used to seeing.
  In this case, both A and B are tested against A*B. Here's some
  information that people may find useful.
It would appear that there's something of a split among statisticians
  in how to handle models with random effects. Quoting from page 12 of
  the SYSTAT DESIGN module documentation (1987):

There are two sets of distributional assumptions used to analyze a two
    factor mixed model, differing in the way interactions are handled. The
    first, used by SAS (1985, p. 469-470), can be traced to Mood (1950).
    Interaction terms are assumed to be a set of i.i.d. normal random
    variables. The second, used by DESIGN, is due to Anderson and Bancroft
    (1952). They impose the constraint that the interactions sum to zero
    over the levels of fixed factor within each level of the random
    factor.
According to Miller (1986, p. 144): "The matter was more or less
    resolved by Cornfield and Tukey (1956)." Cornfield and Tukey derive
    expected mean squares under a finite population model and obtain
    results in agreement with Anderson and Bancroft.
On the other side, Searle (1971) states: "The model that leads to
    [Mood's results] is the one customarily used for unbalanced data."

Statisticians have divided themselves along the following lines:
 Mood (1950, p. 344)           |   Anderson and Bancroft (1952)
 Hartley and Searle (1969)     |   Cornfield and Tukey (1956)
 Hocking (1985, p. 330)        |   Graybill (1961, p. 398)
 Milliken and Johnson (1984)   |   Miller (1986, p. 144)
 Searle (1971, sec. 9.7)       |   Scheffe (1959, p. 269)
 SAS                           |   Snedecor and Cochran (1967, p. 367)
 SPSS GLM*                     |   DESIGN

The references are:
  
  
*
  
*Cornfield, J., & Tukey, J. W. (1956). Average values of mean squares    in factorials. Annals of Mathematical Statistics, 27,
  907-949.
  
*Graybill, F. A. (1961). An introduction to linear statistical models    (Vol. 1). New York: McGraw-Hill.
  
*Hartley, H. O., & Searle, S. R. (1969). On interaction variance    components in mixed models. Biometrics, 25, 573-576.
  
*Hocking, R. R. (1985). The analysis of linear models. Monterey, CA:    Brooks/Cole.
  
*Miller, R. G., Jr. (1986). Beyond ANOVA, basics of applied    statistics. New York: Wiley.
  
*Milliken, G. A., & Johnson, D. E. (1984). Analysis of Messy Data,    Volume 1: Designed Experiments. New York: Van Nostrand Reinhold.
  
*Mood, A. M. (1950). Introduction to the theory of statistics. New    York: McGraw-Hill. Scheffe, H. (1959). The analysis of variance. New York: Wiley.
  
*Searle, S. R. (1971). Linear models. New York: Wiley.
  
*Snedecor, G. W., & Cochran, W. G. (1967). Statistical methods (6th    ed.). Ames, IA: Iowa State University Press.
SPSS can be added to the left hand column. We're assuming i.i.d.
  normally normally distributed random variables for any interaction
  terms containing random factors.
----------------------------------------------------------------------
David Nichols Senior Support Statistician SPSS, Inc.
Phone: (312) 329-3684 Internet: nichols@spss.com Fax: (312) 329-3668
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