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.
Again, thank Michael, Jim, Donald, and Jay for your thoughts regarding my
confusion of factor effect and mean difference in N-way ANOVA.
First let me confess how my confusion started. I have been trying to write a
program that runs four-way ANOVA with a nested design. After I run a test on
the program, I was surprised to notice that, with same p value, the F test
for a factor effect does not necessarily lead to the same significance of t
test in the difference between the (only) two means within that factor. I
went back and forth trying to find anything I might have coded incorrectly,
but failed. As I could not make sense out of it, I went back to the simplest
case, unpaired t test versus one-way ANOVA. This was how my original
question came up.
Now I am convinced that the unpaired t test and the factor effect (F test)
in one-way ANOVA are basically the same thing. The question is: How about
the case in N-way ANOVA? I forgot to mention in my last message that for
N-way ANOVA, the term inside the square root in the denominator of the t
test for the mean difference of the factor (with only two levels) should be
the same -- either MSE or some interaction term -- as the one for the F test
for the corresponding factor effect except for a multiplier inside the
square root, right? Is this what Donald refers to as the "planned
comparison"?
So the two tests should still be the same in N-way ANOVA, right? My algebra
does say so. Then I must have done something wrong in the coding of my
program, and have to work harder to debug the inconsistence. Thank Donald
for your continuing help, and I did purchase Keppel's book right after you
recommended it a few month ago .
Thanks again,
Gang
----- Original Message -----
From: "Donald Burrill" <[EMAIL PROTECTED]>
Sent: Friday, June 11, 2004 5:11 AM
Subject: Re: Follow-up: Unpaired t test and one-way ANOVA
> On Thu, 10 Jun 2004, Gang Chen wrote in part:
>
> > Now I have another question: In N-way ANOVA, does the above conclusion
> > hold as well? Among the N factors there is one factor A that has two
> > levels: would the F test for the significance of factor A be
> > equivalent to the t test (two-sided) for the difference between the
> > two factor level means? I feel it should, but want to confirm it.
>
> Depends on what precisely you want to mean by "the t test (two sided)".
> Mike's and Jim's replies ("No, the tests aren't equivalent") appear to
> have been based on their assuming that you meant a simple-minded t-test
> that wholly ignored the other factors in the ANOVA. (Which would be a
> really silly thing to do, if factors other than A, and/or interactions,
> were significant in the ANOVA.) If what you actually meant was what
> some folks formally call the planned comparison (or planned contrast)
> between the two levels of Factor A, then yes: a planned comparison is
> in fact a t-test, using the proper error variance from the ANOVA.
>
> You should be able to verify that assertion algebraically, from the
> standard procedures for pursuing planned comparisons, in the special
> case of a 2-level factor.
>
> (For that matter, any contrast, planned or post hoc, is a t-test of
> sorts; but the critical value for the test is not necessarily taken
> from the F distribution with 1 and k degrees of freedom, except in the
> case of a 2-level effect. And then one hardly ever bothers to carry
> out a formal comparison, because the F value in the ANOVA tells you
> whether the levels differ "significantly" or not, and you need only
> inspect the two means to figure out the direction of the effect.)
>
> (It's hard to know how deeply to pursue the question. I hope you have
> access to, and are consulting, a thorough treatment of multiple
> comparisons in ANOVA designs in general, of the sort one would find,
> say, in Keppel. Of course, if you want the underlying mathematics of
> it all, you can't do better than Scheffe 1959.)
> -- DFB.
> ------------------------------------------------------------
> Donald F. Burrill [EMAIL PROTECTED]
> 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
