o difference in average bead stringing speed?
  - H0: t(1)=t(2)=t(3)=0       H1: not all zero
  - estimable....contrast
  - BLUE (Gauss-Markov theorem)
o How to proceed?
  - reduced model is H0 and full model is H1
  - ssE0 (also called ssTot) is sums of squares where global mean is used
  - E(ssE0) is (n-1) (sigma-squared)
  - ssE is sums of squares where group means used
  - E(ssE) is (n-v) (sigma-squared)
o How to compare?
  - ssE0 = ssE + ssT  (partitioning the sums of squares)       
         where ssT is called the treatment sums of squares
  - when all the means are equal then E(ssT) is (v-1)(sigma-squared) if not
     get added (v-1)Q(t(i)) term, see Dean&Voss bottom p. 47
  - estimate sigma-squared under H1 get ssT/(v-1)  called MST 
                             (mean square for treatment)
  - estimate sigma-squared under H0 get ssE/(n-v)  called MSE
                             (mean square error)
  - view {SST/(v-1)}/{SSE/(n-v)} and believe in independence,
     we have ratio of chi-squares divided by their df...F-distribution
  - put this in an ANOVA table.......SAS output
data beads; 
input size time; 
cards; 
1 100 
1 93 
1 140 
2 105 
2 45 
2 85 
3 35 
3 40 
3 37 
; 
proc glm;
class size; 
model time=size; 
means size; 
run; 
quit;

                                         The SAS System        22:32 Sunday, January 14,
2001   1

                                        The GLM Procedure

                                    Class Level Information

                                 Class         Levels    Values

                                 size               3    1 2 3


                                   Number of observations    9

                                         The SAS System        22:32 Sunday, January 14,
2001   2

                                        The GLM Procedure

Dependent Variable: time

                                               Sum of
       Source                      DF         Squares     Mean Square    F Value    Pr > F

       Model                        2      8174.88889      4087.44444       7.75    0.0217

       Error                        6      3165.33333       527.55556

       Corrected Total              8     11340.22222


                       R-Square     Coeff Var      Root MSE     time Mean

                       0.720876      30.39959      22.96858      75.55556


       Source                      DF       Type I SS     Mean Square    F Value    Pr > F

       size                         2     8174.888889     4087.444444       7.75    0.0217


       Source                      DF     Type III SS     Mean Square    F Value    Pr > F

       size                         2     8174.888889     4087.444444       7.75    0.0217

                                         The SAS System        22:32 Sunday, January 14,
2001   3

                                        The GLM Procedure

                        Level of           -------------time------------
                        size         N             Mean          Std Dev

                        1            3       111.000000       25.3574447
                        2            3        78.333333       30.5505046
                        3            3        37.333333        2.5166115

  - recall the meaning of p-values, see table A.6

o Was 3 people per bead size good?  what is the definition of good?
  - control alpha-error
  - control beta-error
  - under H0 we have the F dfn=v-1  dfd=n-v
  - under H1 we have a non-central F dfn=v-1  dfd=n-v with
     d2 = (v-a)Q(t(i))/(sigma-squared)  where with equal 
     sample sizes this is
     d2 = r* sum of each [{t(i) - mean of t(i)} squared]/(sigma-squared)
  - imposing condition that biggest mean and smallest differ by delta wiht
     others in the middle then
      d2=r*(delta-squared)/(2*sigma-squared)

  - table A.7 gives phi=sqrt((d2/v)) and area=prob=pi=1-beta
  - using this we can solve for
            r=2v*sigma-squared*phi-squared/(delta-squared)
  - so for alpha, v1=v-1, v2=n-v, and 1-beta=pi we get phi...
    but trouble since v2 depends on r;  so iterate
  - show you Russ Lenth’s calc for power next week in lab
o Follow up tests
  - recall from last lecture that
                      __
    sum of c(i)*  Y(i.)    were normal with 
    mean of     sum of c(i)*t(i)  and variance of
            sum of (c(i)-squared)/r(i) * sigma-squared
  -  note: a good estimate of sigma-squared is the MSE
  -  can use to do follow ups
  -  may view normalized
o I. Pairwise follow ups
o II.  Treatment vs. Control follow ups
o III. Differences of averages follow ups
o IV. Trend follow ups