o ch 13 = r=1, factorial
 2 level per treatment
 should block
called 2**p
- ex: BEADS
A(size): S,L  (0,1) (-1,1) (  ,a)
B(color): B,O
C(music): N,Y
also know need to block AM,PM
- what do treatment combos look like:
000  OR -1 -1 -1  OR (1)
001  OR -1 -1  1  OR (c) ...
- if want full factorial 
  with block
Source  df
Block   blk - 1
A        a-1
B        b-1
C        c-1
A*B     (a-1)*(b-1)
A*C     (a-1)*(c-1)
B*C     (b-1)*(c-1)
A*B*C   (a-1)*(b-1)*(c-1)
Error
Total   n - 1

* orthogonal contrasts
 (like trend A2) won’t
 even work on this small
 sample
- solution to our troubles?
choose to confound higher order
 interaction with block
*DEFN 2 effects are confounded
  when their estimators are
  identical
* EX: H0: AM-PM no help
       H0: no A*B*C
- this confounding seems good
 say, plots look like:

- how to do this using
orthogonal contrasts
note:2 level factor main 
effects contrast easy.
rest by multiplication

0	0	0	-1	-1	-1	1	1	1	-1	70.000000
0	0	1	-1	-1 	1	1	-1	-1	1	65.000000
0	1	0	-1	1	-1	-1	1	-1	1	64.000000
0	1	1	-1	1	1	-1	-1	1	-1	62.000000
1	0	0	1	-1	-1	-1	-1	1	1	43.000000
1	0	1	1	-1	1	-1	1	-1	-1	41.000000
1	1	0	1	1	-1	1	-1	-1	-1	39.000000
1	1	1	1	1	1	1	1	1	1	35.000000
                                                     where
                                                    to confound?
                                                   block 1s and -1s
                                                   together

		000 001 010 011 100 101 110 111
BLOCK
     AM	x            x  x            x 		
     PM          x   x            x   x

truly an incomplete block design
like ch.11, for now
analyze:
- recall normalized orthogonal
contrasts (independent) with df=1
sum of c(i)*Y(ijk)
has variance (+-1)**2 sigma2 +
 ........(+-1)**2 sigma2 = v*sigma2
for all main and interaction
(no divisor needed)
Do usual
sumc(i)Y(ijk)  rnk  blom  cacee
-------------  ---  ----  ------

and plot
                                  SSC

   A                             1326.125      1      1326.125   424.360  .031
   B                               45.125      1        45.125    14.440  .164
   C                               21.125      1        21.125     6.760  .234
   AB                                .125      1          .125      .040  .874
   AC                                .125      1          .125      .040  .874
   BC                                .125      1          .125      .040  .874
   ABC                              3.125      1         3.125

Total                            1395.875      7       199.411