PSYC 225 -- Dr. King

Chapter 13 - The Lowdown on One-way ANOVA (Between Subjects/Independent Groups)

What you don't need to know from Chap. 13
   * Scheffe Test (p. 399) - it's used incorrectly here anyway
   * F-max Test (a test for homogeneity of variance, but not a good one)

What's ANOVA all about?
   * it's like a grown-up t-test for more than two groups (two or more groups)
   * if there are two groups, then F = t-squared

The idea...
   * can no longer test a difference between means because there are too many
   * so now we test the variance of the group means

                  variance between groups
              F = -----------------------
                  variance within groups

Partitioning the variance...

          |-- 10 --|
          |-- 12 --|- within groups variance    M = 12.333 --|
          |-- 15 --|      (unexplained)                      |
          |  ----                                            |
          |--  9                                             |  between
  total  -|--  8                                M = 7.333  --|- groups
 variance |--  5                                             |  variance
          |  ----                                            |  (explained)
          |-- 16                                             |
          |-- 18                                M = 19.333 --|
          |-- 24

   total variance = between groups variance + within groups variance

What do we ultimately want to know?
   * are those means really different?
   * or are the differences in the sample means just random noise?
   * there are no directional tests in ANOVA (we just look for differences)

What we need to calculate...
   * total variance
   * within groups variance
   * between groups variance = total - within

   * memorize me; I will be your friend! SS = (look it up if you don't remember)

Let's see it happen!  (PRACTICE THIS!!!)

          grp 1      grp 2      grp 3
         -----------------------------
           10          9         16
           12          8         18
           15          5         24
         -----------------------------           grand totals
      sum  37         22         58  -----sum----->  117  -|
    sumsq 469        170       1156  -----sum-----> 1795  -|
        n   3          3          3  -----sum----->    9  -| calculate
                                                           |
       SS  12.6667     8.6667    34.6667   |-NO!-|   274 <-|

What we are trying to get...
   * SS - from above
   * df (degrees of freedom)
      * df(betw) = no. of groups - 1
      * df(within) = total no. of subjects - no. of groups
      * df(total) = total no. of subjects - 1
   * MS (the variance) = SS/df

        Source                 SS      df   MS       F
        -------------------------------------------------
        Between treatments    218.000   2  109.000  11.68
        Within treatments      56.000   6    9.3333
        Total                 274.000   8
        -------------------------------------------------
        eta-squared = 218 / 274 = 0.80

Critical value of F comes from a table: F(alpha=.05, df=2,6) = 5.14

Post hoc tests - to find which groups are actually different.
   * the Tukey HSD test (HSD for honestly significant difference)

          HSD = q * sqrt(MSwithin / n per grp)
              = 4.34 * sqrt(9.333/3)
              = 7.66

   * there are many, many, many other such tests!

Assumptions for ANOVA...
   * random sampling or assignment
   * independent scores
   * normal distributions
   * homogeneity of variance

A problem from the book - problem no. 21 on page 412.

Recommended problems
   * 7th ed.: 7, 11, 14, 15, 16, 17, 18, 21, 22
   * 6th ed.: 7, 11, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25