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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
Comments: To: Jeffrey Berman <jberman@memphis.edu>
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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