Chapter 13 - ANOVA

1
Chapter 13 - ANOVA
2
ANOVA

• Be able to explain in general terms and using an
  example what a one-way ANOVA is (370). Know the
  purpose of the one-way ANOVA (371, 2). Carefully
  read page 372 in which the logic of the one-way
  ANOVA is described. You should be able to explain
  this back to me on the exam (372).
• One way refers to only 1 independent variable,
  with 2 or more independent levels (usually three
  or more!).
• Used to test whether difference among means of
  multiple groups is due to chance only, or to
  treatment as well.

3
Logic of F test

• Logic of the F test
• F(observed)
• Variance Betwn Groups var of scores w/in
  groups treatment
• Variance Within Groups var of scores w/in
  groups
• If treatment 0? Smaller F
• If treatment 10? Bigger F
• If F hovers around 1 no treatment
  effect.
• (The higher the number generated from the F,
  the greater the likelihood of a treatment effect.)

4
F Distribution

• 13-2.Be able to describe the F distribution and
  how it is created (e.g. many samples in which the
  F is calculated and put into a frequency
  distribution (373). Know the five characteristics
  of the F distribution. You should be able to
  explain the difference between dfB and dfW (374).
• The F Distribution
• A. F(observed) Variance Between Groups
• Variance within groups
• B. Calculate F(observed) a bunch of times create
  a frequency distribution of Fs
• C. Use this distribution to determine the
  probability of obtaining a particular F by chance
  alone.
• If likely no treatment effect
• If unlikely treatment effect statistically
  significant

5
Five characteristics of the F distribution

• 1. The mean of the distribution approaches 1 as
  the sample size increases.
• 2. The F distribution is unimodal.
• 3. The sampling distribution is positively
  skewed.
• 4. A different distribution exists for each DF
  (thus, a family of distributions)
• 5. When comparing only two groups will get the
  same result as the t-test (two tailed only)

6
F-test degrees of freedom

• Degrees of freedom for the F
• Two different numbers of degrees of freedom.
• A. One set of degrees of freedom depends on the
  number of sample group means being compared, and
• B. The other set depends on the number of
  subjects in each group.
• dfB number of groups-1 k-1
• dfW group 1 (n-1) group 2 (n-1) b group 3.

7
F Distribution
8
Shavelson Chapter 13

• S13-4A. Know the following regarding the one way
  ANOVAA. The design requirements (377) as well
  as the assumptions (378).B. Be able to create
  and recognize correct examples of hypotheses (Ho
  and H1) (378)C. Know the decision rules for
  rejecting or not the null hypothesis. Given
  data, you should be able to generate both the
  Fobserved and Fcritical and determine whether or
  not to reject the null. I will not ask you to
  calculate the sum of squares. Given a particular
  study, you should be able to say what rejecting
  (or not) the null means. (384)
• Assumptions of one-way ANOVA
• Scores are independent of each other
• The scores of the population from which the
  sample was drawn should be normally distributed
  (but little problem with this when levels of IV
  are fixed)
• The variances of the populations from which the
  samples were drawn should be equal homogeneity
  of variance (little problem with this when groups
  are all the same size (equal Ns)

9
Shavelson Chapter 13

• S13-4A continued
• Design requirements for the one-way ANOVA
• One IV with 2 or more levels
• The levels of the Iv can differ quantitatively or
  qualitatively
• Participant may only appear in one group (that is
  one level of IV) and was randomly selected from
  the population

10
Shavelson Chapter 13

• S13-4B
• Null Hypotheses used
• Ho ?1 ?2 ?3
• (All means are equal)
• H1 ?i ? ?i'
• (At least one of the pairs of means differ from
  one another)
• Only two-tailed tests!
• e.g. directional would be goofy ?1gt ?2lt ?3gt ?4
  etc.

11
Shavelson Chapter 13
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
Strength of Association

• S13-6. Be able to explain how one would obtain
  information about the size of a treatment effect
  (in general terms). Given the value of the
  omega-square (e.g. .71) explain what that number
  means. (387-388)
• Strength of Association (size of treatment
  effects)
• Omega-square indicates the amount of variability
  in the DV accounted for by the IV
• Thus, a larger number (e.g. .71) indicates a
  larger effect by the IV (more DV variability is
  accounted for by knowing the IV)
• Said another way, with an omega-square of .71,
  there is a strong relationship between the IV and
  DV.

16
Post hocs

• S13-7. Why would one conduct post hoc
  comparisons? (389). Be able to describe the
  general steps taken in conducting a post hoc
  comparison. This includes creating the null and
  alternative hypotheses writing the comparisons
  as a set of weighted means and finally when this
  comparison would be run. (390-394)
• Post hoc comparisons
• When a significant F is found, used to determine
  which means caused the significant F (or said
  another way, determine which pair, or combination
  of pairs of means have a significant difference
  between them)

17
Post hocs

• S13-7. Why would one conduct post hoc
  comparisons? (389). Be able to describe the
  general steps taken in conducting a post hoc
  comparison. This includes creating the null and
  alternative hypotheses writing the comparisons
  as a set of weighted means and finally when this
  comparison would be run. (390-394)
• Post hoc comparisons
• When a significant F is found, used to determine
  which means caused the significant F (or said
  another way, determine which pair, or combination
  of pairs of means have a significant difference
  between them)
• Scheffe Post hoc comparison
• Used to detect significant differences between
  pairs of means and combinations of means.
• Tukeys HSD test
• Used to test all pairs (only pairs, no complex
  combinations)
• More powerful than the Scheffe for this

18
Post hocs
19
Post hocs