What you have described is, as Jos Jansen pointed out, a three-way
ANOVA, not a two-way ANOVA. Each specimen yields 8 sample values from
points that were determined as the corners of a cube, that is as points
that might be labelled A, B, C, ..., H thus:
Front Back
Left Right Left Right
Top A B E F
Bottom C D G H
The dimensions of the cube may reasonably be called Height (Top,
Bottom),
Breadth (Left, Right) and Depth (Front, Back), represented in a data
file by codes (1,2) respectively. At each of the 8 points you have
measured one or more variables that I shall call collectively Y.
Your data file has the following structure (although the variables
need not appear in this order):
..........Specimen ID (1, 2, ..., 21)
: ..........Position (A, B, C, D, E, F, G, H)
: : ..........Measured variable(s) Y (or the rank thereof?)
: : : .........Height (1 = Top, 2 = Bottom)
: : : : ........Breadth (1 = Left, 2 = Right)
: : : : : .......Depth (1 = Front, 2 = Back)
: : : : : :
1 A Y 1 1 1
1 B Y 1 2 1
1 C Y 2 1 1
1 D Y 2 2 1
1 E Y 1 1 2
1 F Y 1 2 2
1 G Y 2 1 2
1 H Y 2 2 2
2 A Y 1 1 1
.
:
21 H Y 2 2 2
A three-way ANOVA using Height, Breadth, Depth as factors would yield
(1) a main effect due to Height, representing the difference between
the averages of ABEF and CDGH; (2) a main effect of Breadth, being the
difference between the averages of ACEG and BDFH; (3) a main effect of
depth: ABCD vs. EFGH, equivalent to your mu_*F vs. mu_*B. The
differences between these averages of four points are more precise than
the individual differences (e.g., A-C, B-D, E-G, F-H, for Height) that
you first proposed to examine.
In addition to the three main effects, the ANOVA would yield three
two-way interactions: (4) between Height and Breadth, (5) between
Height and Depth, (6) between Breadth and Depth; and (7) a three-way
interaction among all three factors. These can be thought of as asking
whether there is (4) a systematic difference in the Height effect
between Left and Right (or equivalently a systematic difference in the
Breadth effect between Top and Bottom), AEDH vs. BFCG; etc. If these
effects turn out to be negligible, one has only main effects to
interpret.
You later wrote, "I ranked all my data within each biological specimen."
I'm not sure what you meant by that. I think Jos interpreted that to
mean that you substituted ranks for the measured values themselves, in
such a way that the data for A to H, in each specimen, are now the
integers 1, 2, 3, 4, 5, 6, 7, 8 in some order, the order depending on
what the original measures were. Is that in fact what you meant?
If so, you are correct in observing that your conclusions can only be
of the form "top values are less than (or greater than) bottom values";
you cannot say how much less (or more) in units of the measured
variable. And if so, there will be no differences at all among the 21
specimens (for each specimen, the average is 4.5; no variation).
On Thu, 29 Jan 2004, Marlies wrote:
> I have a set of data, obtained from 21 biological specimen. In each
> specimen we obtained eight samples, which are positioned like the
> corners of a cube, meaning every sample has one unique position within
> the specimen based on three 'directional' parameters: Front-Back;
> Top_Down and Left-Right.
>
> I am interested in 'checking' whether or not the eight samples taken
> from one and the same specimen can be considered equal. I started with
> comparing every sample positions with all sample positions (one-way
> ANOVA) but this gives too much information (27 pairs of positions) to
> conclude anything comprehensive.
>
> So now we try to compare the data according to the three axes of
> symmetry.
> Front-Back differences
> Top-Bottom differences
> Left-Right differences
>
> I'll use the Front-Back differences as an example.
>
> ---------------- Example of proposed analysis
>
> Data can be written down as follows: (A to H inserted -- DFB.)
>
> | Front | Back | Total
> ------------------------------------------------------------
> Top_Left | mu_TLF A | mu_TLB E | mu_TL*
> Top_Right | mu_TRF B | mu_TRB F | mu_TR*
> Down_Left | mu_DLF C | mu_DLB G | mu_DL*
> Down_Right | mu_DRF D | mu_DRB H | mu_DR*
> -------------------------------------------------------------
> Total | mu_*F | mu_*B | mu_**
-----------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
.
.
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