Follow-up tests
o Single query vs. multiple queries
  - Bad weather = cloudy and freezing
o single
           ____
  - since Y(i .)   is N( m + t(i), sigma-squared/r(i)) then
    ____
    {     Y(i.) - (m+t(i))}/sqrt(sigma-squared/r(i))
  ----------------------------------------------------------
   sqrt {  [(n-v)MSE/sigma-squared]/(n-v)}

         is independent N(0,1) divided by sqrt of chi-squared divided
                  by its df       =     t with df = n-v
  - Example
  - turn into a test for equality to zero by squaring and recalling that
  t squared is an F..........ssc
  - Example

o multiple
  - a prior vs. a posteriori = “data snooping”  L vs M  and L vs. S  ... 
        peek at M vs. S?
  - 5 methods:
1. Bonferroni       any contrast     prior
2. Tukey         all pairwise          prior
3. Dunnett    treatment vs. control      prior
4. Hsu     best vs. rest        prior
5. Scheffe     any contrast    data-snoop
  - general form:
 sum of c(i)*t(i)-estimates  +-   w* sqrt (var of c(i)*t(i)-estimates)

1. Bonferroni
w = t df = n-v , alpha/2m  where m is # of comparisons
SE = sqrt (MSE* sum of c(i)-squared/r(i))
Example
2. Tukey
w = q v(1), v(2), alpha / sqrt (2)    table A.8
SE = sart (MSE * (1/r(i)  + 1/r(s) ))
Example
3. Dunnett
w = D1 (one-sided) v-1, n-v, alpha        table A.9
       D2 (two-sided) v-1, n-v, alpha        table A.10
SE = sqrt (MSE * (2/r))           note: equal sample sizes
Example
4. Hsu
w = table A.9
SE = sqrt (MSE * (2/r))
Example
  -  max if big is good; min if small is good
  - sample mean is candidate for best under max if lwbnd is > 0     
  - sample mean is candidate for best under min if upbnd is < 0 
Scheffe
w = sqrt ((v-1)* F (dfn=v-1, dfd=n-v, alpha))  
  if do means, too, change v-1 to v
SE = sqrt ( MSE *    sum of c(i)-squared/r(i))
  if do means then use sqrt (MSE/r(i))
  - if accept H0 in ANOVA do not do follow-up with Scheffe
  - if only interested in a few a prior s just do Bonferroni etc.
Example