Hi everybody:
This is just a small statistics curio: how to analyze a 2x2x2
confounding design (considered by Ching Chun Li "one of most important
achievements in the art of experimental design and statistical
analysis") with SPSS.
When I was a student, I was faced with this same example, and, while I
failed miserably to grasp its meaning, I was able to do the analysis by
hand (squaring and summing a lot of numbers...). Now I'm older&wiser,
and I finally understand its beauty, and I want to save everybody the
pain of computing it by hand.
It is a 2x2x2 randomized block design, but with the problem that every
block allows only for 4 treatments, not 8. Therefore, the8 treatments
formed by all the combination of A, B and C at tow levels) are assigned
according to the following layout:
abc a b c
I 30 16 24 19
II 23 16 28 16
III 25 19 20 16
ab ac bc 1
IV 28 16 27 8
V 18 25 16 10
VI 23 22 17 18
Those who are used to orthogonal contrasts as a tool to decompose ANOVA
sum of squares (*), will see that this distribution of treatments has
confounded absolutely the 3 way interaction with the blocks (hence the
name). Therefore, the statistical analysis will not be analyze that effect.
(*) this is the matrix:
Effect A: 1 1 -1 -1 1 1 -1 -1
Effect B: 1 -1 1 -1 1 -1 1 -1
Effect C: 1 -1 -1 1 -1 1 1 -1
Int. A*B:1 -1 -1 1 1 -1 -1 1
Int .A*C: 1 -1 1 -1 -1 1 -1 1
Int. B*C: 1 1 -1 -1 -1 -1 1 1
A*B*C: -1 -1 -1 -1 1 1 1 1
Funny (grim smile), last time, as a student, I had to analyze this data,
the university had suffered a terrorist attack (another car bomb), like
now. As the French saying goes: "Plus ça change, plus c'est la même
chose" (the more that changes, the more it's the same thing).
Now, the analysis with SPSS:
DATA LIST FREE/block A B C outcome treatment (6 F8).
BEGIN DATA
1 1 1 1 30 1 1 1 0 0 16 2 1 0 1 0 24 3 1 0 0 1 19 4
2 1 1 1 23 1 2 1 0 0 16 2 2 0 1 0 28 3 2 0 0 1 16 4
3 1 1 1 25 1 3 1 0 0 19 2 3 0 1 0 20 3 3 0 0 1 16 4
4 1 1 0 28 5 4 1 0 1 16 6 4 0 1 1 27 7 4 0 0 0 8 8
5 1 1 0 18 5 5 1 0 1 25 6 5 0 1 1 16 7 5 0 0 0 10 8
6 1 1 0 23 5 6 1 0 1 22 6 6 0 1 1 17 7 6 0 0 0 18 8
END DATA.
VAL LABEL A B C 0'No' 1'Yes'.
VAR LEVEL BLOCK treatment (NOMINAL).
VAL LABEL treatment 1'abc' 2'a' 3'b' 4'c' 5'ab' 6'ac' 7'bc' 8'1'.
* Table 24.1 of Ching Chun Li book (+ matrix of orthogonal contrasts as
in table 21.5) *.
UNIANOVA
outcome BY block treatment
/RANDOM = block
/METHOD = SSTYPE(1)
/CONTRAST (treatment)=SPECIAL( 1 1 -1 -1 1 1 -1 -1
1 -1 1 -1 1 -1 1 -1
1 -1 -1 1 -1 1 1 -1
1 -1 -1 1 1 -1 -1 1
1 -1 1 -1 -1 1 -1 1
1 1 -1 -1 -1 -1 1 1)
/DESIGN = block treatment .
* Analysis not using orthogonal contrasts *.
UNIANOVA
outcome BY block A B C
/RANDOM = block
/METHOD = SSTYPE(1)
/DESIGN = block A B C A*B A*C B*C .
Best regards,
Marta García-Granero
--
For miscellaneous statistical stuff, visit:
http://gjyp.nl/marta/=====================
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