It is unfortunate, as Jochen Wilhelm says that SPSS lists one-way ANOVA as it does rather than as model comparison as it truly is. It is even more unfortunate that 100 years after Fisher used regression in his first ANCOVA paper (Article The Influence of Rainfall on the Yield of Wheat at Rothamsted

I'll answer your question with another question: What is the null hypothesis for the F-test in a one-way ANOVA?

Without the mathematical point of view, F is the proportion between variability/within variability. https://en.wikipedia.org/wiki/F-test

The H0 is that we have a normal variation in each group. If the variation between the groups is not dominantly more than that normal variation the group variation that we found is not an strange finding. I.e. it might be occurred by chance.

I think that if we have Wald approach (rather than F) we can interpret as means comparison (Wald test is a t test).

Many thanks for your answer!

Edited.

The t-test is a special case of an F-test ( t(n)² = F(1,n) ). So the t-test is a special form of an ANOVA, testing exactly one restiction in a linear model. ANOVA allows testing the simultaneuous restriction of several different coefficients. In this case, there is no simple or unique way anymore to connect F to t. Notably, the restrictions do not need to be that all coefficients are zero. Each coefficient can be resticted to a different value! The null hypothesis of an ANOVA does therefornot need to be what Bruce seems to expect as the answer to his question ;)

I personally think it's unfortunate to list "ANOVA" under "comparing means". I would rather list "ANOVA" under "comparing models", besidespoints like "Analysis of Deviance", "AIC, BIC, DIC etc.".

On the other hand, "one-way ANOVA" implies that all coefficients regarding one categorical predictor are restricted to zero, so that the restricted model in fact assumes that there is no difference between any of the groups. This way, the one-way ANOVA is a comparison of a group of means alltogether (an ominbus test). It's only not possible to specify what precisely leads to the rejection of that omnibus hypothesis (and because of this, post-hoc tests of pair-wise group comparisons are eventually required-but this is (usually) not related to the ANOVA, so the question remains why "ANOVA" is named when the actual task is to do post-hoc tests (like HSD, Dunnett etc.). I would findit better to see a menu listing "post-hoc tests" with, possibly, a submenu listing "multiple-to one procedures", "all-pairwise comparisons" and "planned contrasts". Or just a list of different multiple comparison procedures (they could also list ANOVA-protected tests like Fisher's LSD for 3 groups).

Many thanks!

Aren't post hoc tests F based? Which one of them are? Other than this fact that they have correction coefficient for P value.

The original question was about one-way ANOVA (and the ONEWAY command in SPSS). For that specific model, the usual null hypothesis states that all of the population means are equal. That is why the good folks at SPSS included the menu item for ONEWAY under Analyze > Compare Means.

You can use the UNIANOVA (or GLM) command (under Analyze > General Linear Model) to estimate the same one-way ANOVA model. But those commands are not limited to one-way ANOVA (like ONEWAY is).

Thanks again Bruce Weaver

This action of SPSS team helps the software to be friendly with beginners and students.

My university major is not biostatistics. From my intermediate leveled viewpoint, I do not agree with " all of the population means are equal "; because hereby, the F ratio will become very lower than one (between variation ~=0) that may create a left sided P

In Stata

To have an interaction in an ANOVA model, you must have at least two categorical explanatory variables (sometimes called factor variables). One-way ANOVA has one categorical explanatory variable, so cannot include any interactions.

This paper believes that ANOVA is for comparison of mean like Bruce Weaver

Article Analysis of variance (ANOVA) comparing means of more than two groups

My personal conclusion from this discussion:

Basically --> No

Practically --> Yes

I agree with you there Sayyed, as an amateur scientist even I know that ANOVAs don't necessarily mean a comparison between two group means

It is unfortunate, as Jochen Wilhelm says that SPSS lists one-way ANOVA as it does rather than as model comparison as it truly is. It is even more unfortunate that 100 years after Fisher used regression in his first ANCOVA paper (Article The Influence of Rainfall on the Yield of Wheat at Rothamsted

I am not sure what the main focus of this discussion it. Is it about the topic if ANOVA should be considered as a comparison of means or if SPSS should have put it elsewhere?

From the Command Syntax Reference manual entry for the ONEWAY command:

ONEWAY produces a one-way analysis of variance for an interval-level dependent variable by one numeric independent variable that defines the groups for the analysis.

• https://www.ibm.com/support/knowledgecenter/en/SSLVMB_26.0.0/statistics_reference_project_ddita/spss/base/syn_oneway.html

Given that the ONEWAY command estimates only models with one categorical explanatory variable and one quantitative outcome variable, it makes sense (I think) to have it where it is in the GUI menu system. I see no implication that ANOVA models in general are restricted to comparisons of means. Nor do I see any implication that one-way ANOVA is not a special case of the general linear model. YMMV.

Finally, note that the GLM and UNIANOVA commands are NOT filed under Compare Means in the GUI. You can find them under Analyze > General Linear Model > Univariate.

• https://www.ibm.com/support/knowledgecenter/en/SSLVMB_26.0.0/statistics_reference_project_ddita/spss/advanced/syn_glm_univariate.html
• https://www.ibm.com/support/knowledgecenter/en/SSLVMB_26.0.0/statistics_reference_project_ddita/spss/base/syn_unianova.html

What I do see here is that SPSS, like many other stats packages that have been around for a long time, are products of evolution. If one was creating a new stats package from scratch today, one would very likely have just one GLM procedure rather than separate procedures for all of the special types of GLMs (e.g., unpaired t-test, ANOVA, OLS regression). Certainly the UNIANOVA & GLM commands are relatively recent additions to SPSS. IIRC, MANOVA was the original command that allowed one to estimate such models. It is still available, but via command syntax only. Anyway, the objections to ONEWAY being located under Compare Means in the GUI strikes me as an objection to SPSS not having removed older procedures that could not be handled by GLM or UNIANOVA. An obvious problem with doing that would be that much old code would suddenly be "broken" if those old commands were removed.

Finally, SPSS is not the only stats package that has multiple procedures for estimating the same model. ;-)

Bruce Weaver Just a quick comment MANOVA is a case of a matrix of DV's rather than a vector so the conclusion that MANOVA is a GLM is a bit more difficult for a student without linear algebra to see. For a brief discussion, see the link:. https://stats.stackexchange.com/questions/209696/how-is-manova-a-case-of-the-glm. From the point of view of teaching one might look at Seber's work here: https://b-ok.cc/book/2617885/1d7d3d :

I prefer not to work out the MANOVA inference procedures that correspond to the F test even in my PhD level courses but I do cover the topic and give them a data set project they can do with either SAS or R. Best to all. D. Booth.

Saja Radi Thank you for your comment. Yes, I believe that accounting ANOVA as a mean comparison test is just a practical approach for easy interpretation.

David Eugene Booth Thank you for your comment. I did not have historical information whether ANOVA (F test) was the first approach or regression was the first approach to general linear model. But I know this that comparison of means has this meaning that we compare a quantitative variable in different groups using their differences in pooled SE (t test). This, is not what we are looking for in one way ANOVA. I agree with you that regression (Wald) and ANOVA (F) approach should be used along with each other (as in R, -anova- is a post estimation command for -lm-, or as in SPSS, there is an option for beta coefficient estimation in -GLM- command. However, R has more scientific approach while SPSS has more friendly approach).

Bruce Weaver Thank you again for donating your time. From your answer, I found that during updating the versions in these 30 years, ONEWAY was older than GLM and UNIANOVA in SPSS. Adding these new commands was a very good action to solve this misinterpretation. I have a new idea! completely equal means not only is not a null hypothesis, but also is a left sided alternative hypothesis according to F distribution (i.e. over equality is not normal!!!).

Rainer Duesing Thank you for your reply. I wanted to know researchgete members idea whether they agree with this classification of SPSS and why. Because, by seeing this classification, many students think that ANOVA is a multiple group equivalent for t test!

Seyyed Amir Yasin Ahmadi wrote: "I found that during updating the versions in these 30 years, ONEWAY was older than GLM and UNIANOVA in SPSS. Adding these new commands was a very good action to solve this misinterpretation."

What misinterpretation? I think you missed the major point I tried to make in my last post: The ONEWAY command in SPSS estimates only the one-way, between-subjects (independent groups) ANOVA model. And for that specific ANOVA model, the null hypothesis makes a statement about the relationship among the population means. The most common form of that statement is that all of the population means are equal. It may be that you can come up with another way of stating the null hypothesis that does not directly refer to the population means. But it will of necessity reduce to the same thing.

If your point is that ANOVA is used in regression models with no categorical variables, and that in such cases, there are no comparisons of means, think about this: Even in such a model, each coefficient represents a comparison of two fitted values. And in OLS regression, each fitted value is a conditional mean. So even in such a model, comparisons of means are present. They're just not group means.

Seyyed Amir Yasin Ahmadi You may find my above link to Fisher's ANCOVA paper and His daughter"s (Mrs. Geo. Box) quite interesting publications in regard to the history of the methods.

https://booksc.xyz/book/35193814/2b8a6b

https://booksc.xyz/book/35193814/2b8a6b

https://booksc.xyz/g/Box,%20Joan%20Fisher

https://b-ok.cc/book/541145/b1fb22

The last discusses the method of least squares(Legendre,1802) and

The method of least absolute deviations (by Laplace, 1804)

These should give you the entire history of the methods and much more.

Best wishes, D. Booth