o need to tranform due to unequal variances
  - var(Y) = g(m)
   then h(Y) = antiderivative (1/sqrt g(Y))
  - for us:  log(tan(y/2) - .3)
  - results (note: some revised finger lengths and a 
    data reordering) -- view htime
SIZE FINGER FING TIME  HTIME  
		  CAT		
 1    64      1  100 -0.049
 1    72      2  140  0.388
 1    76      3   93 -0.122
 2    70      1   85 -0.210
 2    71      2   45 -0.942
 2    79      3  105  0.001
 3    70      1   40 -1.194
 3    75      2   37 -1.460
 3    77      3   35 -1.815
  

  - note: 1 = tiny .... <71; 2 = average...[71, 75]; 3 = big......>75
  - now rerun residual analysis.....all ok
  - run ANOVA and still a significant difference
  - form C.I. but lose interpretability; gain truth; lose 
   some confidence

o Now consider 2 factors 
  - our beads example has each with 3 levels:
    size: s,m,l
    finger: t,a,b
o what effects in 2-way? which estimable?
  - model
   Y(ijt) = m + t(ij) + e(ijt)
  where e(ijt) are N(0,sigma-squared)
     (Dean calls this the cell-means model)
  - like 1-way so LS estimates via normal
   equations, know from Gauss-Markov got BLUE
  - so long as stick to contrasts like then ok
  t(1.) - t(2.)  (effect of small and medium same)
  t(.1) - t(.2) (effect of tiny and average same)
  - alternately pose:
  Y(ijt) = m + a(i) + b(j) + (ab)(ij) + e(ijt)
where a(i) is main effect of size i
          b(j) is main effect of finger length j
        (ab)(ij) is an interaction effect
  2-way complete model; has observations
in each cell (treatment-level combo)
   * note: t(.j)/3 = m + b(j) if
    average effect of beads and of
    interaction is zero and thus
    represents the mean speed for
    j-type fingers.
o what is interaction?
  - for simplicity just view plot of
          1=t    3=b
s=1     .05    -.12
m=2  -.21      .00
  - (ab)(ij) preserves the effect which allows
this NOT to be true: the effect of switching
from small to medium beads is the same
(an improvement) across all finger lengths.
  - analysis easier if no interaction. view plot of
             1=t     2=a
2=m     -.21     -.94
3=L   -1.19     -1.81
  - if KNOW interaction is zero then use
  2-way main effects (2-way additive) model
   still check assumptions etc.
o more on the comparisons via contrasts
  - interaction
   * example (geometrically)
     [t(22) - t(12)] - [t(23) - t(13)]
    compares slopes
   * example (algebraically)
   [t(22) - t(12)] - [t(23) - t(13)] =
  a function of (ab)(ij) s only
   * any contrast has sum of c(ij)s =0
   additionally with interaction have
  sum over i of c(ij) =0 AND
  sum of j of c(ij) = 0
  - simple contrasts
   * have sum over c(j) for every i =0
    like t(11)-t(13) and/or t(21)-t(23)
  and/or t(31) - t(33) which compare
  small to large for each finger length
  -  main effects (note: often show
  contrasts as columns)
   * sum c(i)*t(i.) where sum of c(i) =0
   * like t(3.)-t(1.)
cells  size         finger   interaction
         main       main
         effect     effect
11   -1   	  -1	1
12  -1     	0	0
13  -1     	1	-1
21  0     	-1	0
22  0     	0	0
23  0     	1	0
31  1     	-1	-1
32  1     	0	0
33  1     	1	1
 note: interaction derived by
multiplying main effects (check
to see slope)
  - trend contrasts
   * can use if levels are equally spaced
   and equal sample sizes
   * can find these in A.2
   * mean?
   * formed?
      fit a line y = ax + b
      fit a parabola (quadratic) 
      y = a*x^2 + b x + c
   * for us:

SIZE FINGCAT A1 A2 B1 B2 HTIME  A1B1 A1B2 A2B1 A2B2

1    1      -1  1 -1  1 -0.049   1   -1   -1    1
1    2      -1  1  0 -2  0.388   0    2    0   -2
1    3      -1  1  1  1 -0.122  -1   -1    1    1
2    1       0 -2 -1  1 -0.210   0    0    2   -2
2    2       0 -2  0 -2 -0.942   0    0    0    4
2    3       0 -2  1  1  0.001   0    0   -2   -2
3    1       1  1 -1  1 -1.194  -1    1   -1    1
3    2       1  1  0 -2 -1.460   0   -2    0   -2
3    3       1  1  1  1 -1.815   1    1    1    1