o chapter 15 -- 15.1 &2
  recall ch 13 about
2**p factorial designs
 with single replicate
and blocking
- concept: run out of df so
confound higher order
interactions with blocks
(assume negligle)
- how?
2**p=(2**s)(2**(p-s))
n=vr=bk
EX1:
2**3=(2**1)(2**2)
b-1=1 so confound abc
EX2:
2**4=(2**2)(2**2)
b-1=3 so 
confound
abcd, abd..other?d
which block does
1001 go to?
o chapter 15 has
similar techniques but
objective differs
- objective:
2**p experiment, r=1
(single replicate),
no blocks, but
if want full factorial
need 2**p so wish
to confound (now called
alias) effects with
each other
(still hope/assume
high order ints neg)
- i cover technique
which uses ch.13
- EX:
2**3 on top of handout
A:size    S,L
B:color      B,O
C:music   N,Y
(did block on day
by confounding abc )
  * do this on handout..
  each obs to which block?
  * what is true about
   abc column in 0 vs. 1
   block?
  * if assume Mon=Tues
   and wish to do only 4
   obs an discuss
a,b,c,ab,ac,bc,abc  7 df
on n=4, but only n-1=3 df!
-DEFN: want a 1/2 - fraction or
1/2 - replicate of our 2**3
called a 2 ** (3-1) design
o Building fractional designs
- what if i omit 0 block?
  do it: cross them out
  * in the 4 obs left
   what is contrast
   coefficient of A*B*C (abc)?
-DEFN: the effect(s) which has all
 ones “contrast” is aliased to
 the mean and we write 
 I=effect(s) and refer to this as
the defining relation for the 
fractional design (the contrast/
effects are called words).
- Here: I=ABC
 note: 1,1..1 is not a contrast
A*B*C can not be tested
- what other effects are
 aliased (confounded)?
 do it: look at remaining 
compare
- see a=bc
meaning?
note: contrast is like looking
at SS, use it and normal
plot to find big effects
-find:
.5A+.5BC
so when testing may be
asking one of several questions:
1) if sig --> A or BC or both
2) if non --> A and BC neglible
         or opposite and cancel!
- for this reason fractionals often
  just used to screen upfront
o must we rely on contrasts
to find alias?
 one we have defining relation
I=ABC, just do letters multiply trick
(see p. 460 table 13.28)
o what will our experiment look like?
001,010,100,111
the 3 contrasts values are:
too few to plot, no MSE
- what does the other 1/2 look
like?
 * I=-ABC
-DEFN: the resolution of a
fractional factorial design
is teh number of letters in the
shortest word of the defining
relation.
 * EX: I=ABC we write
(2**(3-1)) III

o Another example
add D=light level L,H
(see bottom of handout)
- how many effects?
- if 2**(4-2)
  how many observations?
- recall blocking in ch 13  
  b= ?  k=?
  block how? confound
  abcd and abc bad
 * try abc and abd
  what else?
- assign obs to blocks
- pick a block
- what is the defining
  relation?
- what is the resolution?
- what are the 
  contrast values? 
- what is your decision?