o why stop at 2 factors
  - add: color, sex, timeof day...
  - messy to veiw; concepts same 
o need to understand (at least)
 interaction in 3-way; higher?
  - complete model
Y(ijkt)=m+a(i)+b(j)+c(k)+
 (ab)(ij)+(ac)(ik)+bc(jk)+
  (abc)(ijk)+e(ijkt)
usual assumptions
  - which terms are needed?
ask which factors interact.
* with 2-way we said:
the pattern or profile of means 
of A for vairous levels of B 
behaved the same (in a relative
manner) ,i.e. compared
parallelism of lines = no 2-way
interation if parallel, SSAB=0
* with 3-way we say:
the pattern or profile of means 
of A&B for vairous levels of C 
behave the same (in a relative 
manner), i.e. compare entire plot;
if same = no 3-way
interaction, SSABC=0
(or can have SSAB at every level
of C be zero and then SSABC=0 also
since sum (SSAB over c(k) =
SSAB + SSABC)
  - Examples a=2 b=3 c=2
o what does model look like if we
can logically deduce relationship
between factors?
  - discern with line graphs
  - have nC2 (n choose 2 combos)
connections => n order interaction
  - to decide whether to connect
A & B ask: if i view average responses
over all levels of A for each level of
B do I expect a different pattern? If
yes, then interaction.
  - Example
look at string tiems for S,M,L for
orange vs. black beads.  what if
really high time for small black
because they are hard to see?
*    A -- B
    /  | /  \
  E    |/    C
       D --
A=size, B=finger, C=color, 
D=sex E=time of day

so
Y(ijklmt)=m+a(i)+b(j)+c(k)+
 d(l)+e(m)+(ab)(ij)+(ad)(il)+
 (bd)(jk)+(abd)(ijl)+
 (acd)(ikl) +e(ijklmt)
o how to build contrasts in n-way?
  - as usual, make up cell means
comparisons, i.e. 1 -1 0 
  - divisor is # observations in 
each level of the effect
  - interactions gotten by 
multiplication of other contrasts
  - Example
  A=size=3 levels
  B=finger=3 levels
cells
labeled  	A1	B1	A1B1	
11		1	1	1
12		1	-1	-1
13		1	0	0
21		-1	1	-1
22		-1	-1	1
23		-1	0	0
31		0	1	0
32		0	-1	0	
33		0	0	0
divisor 	3	3	1
o rules for conducting hypothesis
test for n-way, equal sample
sizes, COMPLETE model
1. write down the factorial
effect of interest 
              size finger sex
			A  B  D
# levels		3  3  2
subscripts		i  j  l
2. v=df=(a-1)(b-1)(d-1)
3. multiply out df and replace
with subscripts
abd-ab-ad-bd+a+b+d-1
ijl-ij-il-jl+i+j+l-1
4. to test H0: (abd)(ijl) = 0
find SS(effect) =
rce*sum of squared
(sample means associated
with subscripts above)
5. SSTOT as usual
6. SSE=SSTOT-all other SS
df(error) = (n-1) - all other df
7. MS(effect) = SS(effect)/df(effect)
MSE=SSE/df(error)
8. F = MS(effect)/MSE compare to
F-dist with dfn=df(effect) 
& dfd=df(error)