Date: Thu, 17 Feb 2005 13:04:07 -0800
Reply-To: LUCINDA M TEAR <lucindatear@msn.com>
Sender: "SPSSX(r) Discussion" <SPSSX-L@LISTSERV.UGA.EDU>
From: LUCINDA M TEAR <lucindatear@msn.com>
Subject: Re: nested anovas and reading expected mean squares
Content-Type: text/plain; charset="iso-8859-1"
Thank you so much! I had no idea this controversy existed and really
appreciate your response! Thank you!
----- Original Message -----
From: "Jeffrey Berman" <jberman@memphis.edu>
To: "SPSS Discussion List" <SPSSX-L@LISTSERV.UGA.EDU>
Cc: "Jeffrey Berman" <jberman@memphis.edu>; "LUCINDA M TEAR"
<lucindatear@msn.com>
Sent: Wednesday, February 16, 2005 9:57 PM
Subject: Re: nested anovas and reading expected mean squares
> On 2/16/05 12:20 PM, LUCINDA M TEAR <lucindatear@msn.com> wrote:
>
> > Hello - I submitted this last week but no one responded. Thought I'd
> > try
> > again. Thank you for any help!
> >
> >
> >> Hello:
> >>
> >> I am running a nested ANOVA with the following design:
> >> Factor A - fixed - 5 levels
> >> Factor B - Fixed - 3 levels
> >> Factor C(B) - Random, nested within Factor B - 3 levels per level of
> >> factor B
> >> AB interaction
> >> AC(B) interaction
> >> Error (unequal sample sizes in each cell).
> >>
> >> The way I calculate the EMS,
> >> A would be a pseudo F test and I should test
> >> B against C(B)
> >> AC(B) against Error
> >> AB against AC(B) and
> >> C(B) against Error
> >>
> >> If I run the model within Univariate GLM as
> >> /DESIGN = A B C(B) A*B A*C(B)
> >> Or
> >> /DESIGN = A B C*B A*B A*C*B
> >>
> >> From the error terms specified at the end of the output, I think that
> >> SPSS
> >> tested C(B) against AC(B) and A against AC(B):
> >>
> >> Error for Intercept: .984 MS(C(B)) + .016 MS(Error)
> >> Error for A: .973 MS(A * C(B)) + .027 MS(Error)
> >> Error for B: .990 MS(C(B)) + .010 MS(Error)
> >> Error for C(B): .977 MS(A * C(B)) + .023 MS(Error)
> >> Error for AB: .981 MS(A * C(B)) + .019 MS(Error)
> >> Error for AC(B): MS(Error)
> >> I'm not sure how to read the EMS table or some of my problem with my
> >> EMS
> >> is that I didn't completely account for the different sample sizes
> >> within
> >> each cell (I just called them all "n" while calculating my EMS).
> >>
> >> Can anyone give me any leads here? Are my EMS calculations wrong or am
> >> I
> >> not understanding the output? I'm happy to provide my EMS calculations
> >> or
> >> the entire data set.
>
> Lucinda:
>
> I suspect you're running into the same issue I encountered when developing
> examples with SPSS for my graduate statistics course. SPSS (and SAS) use
> a
> different model for deriving expected mean squares than that found in many
> analysis of variance textbooks. SPSS uses an "unconstrained parameters"
> model, whereas the derivation in the textbooks is a "constrained
> parameters"
> model. Differences between the models arise in situations in which some
> factors are random and others are fixed.
>
> There is a spirited debate among statisticians about which model is more
> appropriate or general. For competing perspectives, see:
>
> McLean, R. A., Sanders, W. L., & Stroup, W. W. (1991). A unified approach
> to mixed linear models. American Statistician, 45, 54-64.
>
> Voss, D. T. (1999). Resolving the mixed models controversy. American
> Statistician, 53, 352 - 356.
>
> Wolfinger, R., & Stroup, W. (2000). [Letter to the editor]. American
> Statistician, 54, 228-229.
>
> David Nichols of SPSS posted about this issue some time ago and I have
> attached his message below.
>
> Personally, I tend to be persuaded by Voss's arguments in favor of the
> constrained parameters model, at least in cases in which there are no
> empty
> cells in the design. However, this is not the model used when relying on
> the RANDOM subcommand of GLM (or UNIANOVA).
>
> -Jeffrey Berman
> University of Memphis
>
>
> ----- Forwarded message follows -----
>
> 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)
> 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
> ----------------------------------------------------------------------------
> -
>
> ----- End of forwarded message -----
>
>
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