LAB wednesday
o finish ch 7 n-way factorial
  - like 2-way but more complex
   computations
  - how to work with 1 obs per 
     cell via orthogonal trick
   & normal plot
  - Taguchi approach
  - rules 9-16 about C.I.
  will discuss via example on 
  3-way for beads
  - also more on interaction plots
o only one (could do with 2) obs
  - too expensive?
  - have size (a=3), finger (b=3)
    and color (c=2) 
   * note: old reps given last
   time, say were with black 
   beads
  - ANOVA table
Source       SS    df
S			a-1=2


F			b-1=2
C			c-1=1
S*F			(a-1)(b-1)=4
S*C			(a-1)(c-1)=2
F*C			(b-1)(c-1)=2
S*F*C		(a-1)(b-1)(c-1)=4
Error
------------------------------
total			n-1=17

  - do orthogonal trend 
  contrast trick A2(equal ss)
  - see main effects set up
  and interaction by multiplication
  - before collapse a few
  interactions for error, now...
  - theory:
sum of (c(i)*Y(~.))
is N(sum of (c(i)*m), ?)
........
so 
sum(c(i)*Y(~.) divided by
             2                          2
sqrt(sum c(i) /r) is N(0,sigma )
under no effects at all
  - 1. find normalized contrasts
(see first handout)
  2. plot them on normal prob paper
  a) lorder low to high
  b) find (rank-.375)/(n+.25)
   c) find Pr(z<above value)
       call this ?
   d) x-axis is ?; y is normalized
     contrast
  3) off line=>significant effect
  4) assumes normality, independence
      and equal variance
note: slope is sigma
Example: quad on size (est1,?)

o Taguchi approach
  - controversial since abit ad hoc
  - objective: choose a best 
  mean performer where variability
  is small, particularly for effect
  which could be noise
  - mostly visual
  - Example S, F, C; F is noise
  plot size*color against finger
  choice?

o 3-way
  - get 18 more people to string
s,m,l; t,a,b; black and orange
(see 2nd handout)
  - 3-way interaction?  look
at plots over color.
Note: // ism one way SSABC=0
  - test
H0: (abc)(ijk)-(abc)(i.k)
-(abc)(.jk)-(abc)(ij.)+
(ab)(ij.)+(ac)(i.k)+
(bc)(.jk)-(abc)(...)=0 
for all i,j,k
how? think rules and Y-bars
  - check SAS printout
   - note: most interactions
resoundingly negligible
  what of S*F?  and S*C?
  p-value=.05  p-value=.98
what is plot difference?
  - need means
   st  sa sb  mt  ma  mb  lt  la   lb
o -.0 .4 -.1 -.2 -.9  .0 -1.2 -1.5 -1.8 
o  .0 .2 -.1 -.1 -.7  .3 -1.0 -1.0 -1.5 
b  .1 .2 -.4 -.8  .3 -.2  1.0 -1.8 - .7   
b  .0 .1 -.2 -.5 -.6 -.1 -1.2 -1.3 -2.2

look at plot S*F average over color
(down columns: x-axis: sml, symbol=f )
look at plot S*C average over finger 
 (top 2x3, next 2x3 etc.: x-axis:sml,
  symbol=c)  

  - just accept all interactions
  are negligible then test main
  effects
  - “use small beads”
  - scheffe follow-up
what if beads shipped mixed:
30% small & 70% large
versus medium