First, a correction about two-way ANOVA with factor B nested within
factor A, that is B(A) in the design notation with which I am familiar.
It is not correct to describe it as reducible "to a series of one-way
ANOVAs", because the one-way ANOVAs will not (in general) have removed
the other factor from the error SS. It may reasonably be said to be
reducible FROM a two-way ANOVA with crossed factors, since (as you
correctly indicate) the SS for B(A) is equal to the sum of the SS for B
and the SS for AB in the crossed design. (The corresponding degrees of
freedom behave similarly.) Thus if you run a 2-way ANOVA you will get
the following sources of variation:
A
B
AB
Error (within cells) (formally equivalent to R(AB) where R is the
"replication" factor nested within the smalles cells of the design:
what one writer has called "the ubuquitous nested factor", but which
software often labels merely "Error"
and you merely need to add together the SS (and DF) for B and AB to
yield the SS and DF for the nested design:
A
B(A)
Error
For a four-way design AxBxC(D) the corresponding result is that the
nested effect C(D) is equivalent to (C + CD) everywhere in the design;
so you run a standard 4-way crossed design, just as though AxBxCxD, and
produce the summary table thus:
D
C(D) = C + CD
A
AD
AC(D) = AC + ACD
B
BD
BC(D) = BC + BCD
BA
BAD
BAC(D) = BAC + BACD
Error [formally, R(ABC(D)).]
If (as is often the case) the nested factor C is a random factor, the
rest being fixed factors, the mean squares for C(D), AC(D), BC(D), and
BAC(D) serve as the denominator mean squares (aka "error mean squares")
for the immediately preceding terms in the table as displayed above.
What MINITAB labels "Error" is the error term for C(D) and its
interactions (presuming that A, B, and D are fixed factors; if one or
more of them are random, things get a bit more complicated).
You may find it convenient to consult any good text on design (e.g.,
Keppel, "Design and Analysis"; or Glass & Stanley, "Statistical Methods
in Education and Psychology" (Chapter 18, particularly sections 18.4 et
seq., "Rules of thumb for writing the ANOVA table")) on how to set up
the ANOVA table (and carry out computations) for any complete balanced
design.
Nested designs may be analyzed fairly easily in MINITAB, if you have
access to it: one merely specifies the design in a conventional kind of
way (MINITAB conventions, of course, but they're fairly clear). For the
design above, a suitable MINITAB command would be:
ANOVA Y = A ! B ! C(D) ! D ;
random C.
In the ANOVA design statement, "!" means "construct all possible
interactions involving these factors"; and MINITAB will refrain from
trying to construct impossible interactions like DC(D) .
If you want to verify the logic you think you need, ask MINITAB to
produce a table of the expected mean squares by including the subcommand
"ems" (for "expected mean square"), thus:
ANOVA Y = A ! B ! C(D) ! D ;
random C;
ems.
HTH. -- DFB.
On Wed, 22 Oct 2003, Gang Chen wrote:
> I need to run a four-way ANOVA with one factor (i.e., subject) nested
> within another factor. I did a little research on the funtion 'anovan'
> in Matlab, and it seems that 'anovan' would not allow me to do nesting
> directly. My understanding about two-way balanced ANOVA with nested
> design is that it could be reduced to a series of one-way ANOVA's since
>
> nested SSB(A) = crossed (SSB + SSAB)
>
> Does such a reduction still hold for four-way ANOVA with nested
> design? I mean, can I reduce a four-way balanced ANOVA to a few
> three-way ANOVA's? If true, how? And, does anybody happen to have
> such a Matlab program available for N-Way ANONA with nested design?
-----------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
.
.
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