Statistics for the Social Sciences

1
Statistics for the Social Sciences

• Psychology 340
• Spring 2005

Analysis of Variance (ANOVA)
2
Outline

• Brief review of last time ANOVA
• Post-hoc and planned comparisons
• Effect sizes in ANOVA
• ANOVA in SPSS

3
Example

• Effect of knowledge of prior behavior on jury
  decisions
• Dependent variable rate how innocent/guilty
• Independent variable 3 levels
• Criminal record
• Clean record
• No information (no mention of a record)

4
Analysis of Variance
Test statistic

• More than two groups

Criminal record Clean record No information
10 5 4
7 1 6
5 3 9
10 7 3
8 4 3

• Need a measure that describes several difference
  scores
• Variance
• Variance is essentially an average squared
  difference

5
Testing Hypotheses with ANOVA

• Hypothesis testing a five step program

• Step 1 State your hypotheses

• Step 2 Set your decision criteria
• Step 3 Collect your data
• Step 4 Compute your test statistics
• Compute your estimated variances
• Compute your F-ratio
• Compute your degrees of freedom (there are
  several)
• Step 5 Make a decision about your null hypothesis

• Additional tests Planned comparisons Post hoc
  tests
• Reconciling our multiple alternative hypotheses

6
Testing Hypotheses with ANOVA

• Hypothesis testing a five step program

• Step 1 State your hypotheses

• Null hypothesis
• H0 all the groups are equal

• Alternative hypotheses (HA)
• Not all of the populations all have same mean

Choosing between these requires additional test
7
1 factor ANOVA

• Planned contrasts and Post-hoc tests
• Further tests used to rule out the different
  alternative hypotheses

• reject

• reject

• fail to reject

• Alternative hypotheses (HA)
• Not all of the populations all have same mean

8
Why do the ANOVA?

• Whats the big deal? Why not just run a bunch of
  t-tests instead of doing an ANOVA?
• Experiment-wise error
• The type I error rate of the family (the entire
  set) of comparisons
• ?EW 1 - (1 - ?)c where c of
  comparisons
• e.g., If you conduct two t-tests, each with an
  alpha level of 0.05, the combined chance of
  making a type I error is nearly 10 in 100 (rather
  than 5 in 100)
• Planned comparisons and post hoc tests are
  procedures designed to reduce experiment-wise
  error

9
Which follow-up test?

• Planned comparisons
• A set of specific comparisons that you planned
  to do in advance of conducting the overall ANOVA
• General rule of thumb, dont exceed the number of
  conditions that you have (or even stick with one
  fewer)
• Post-hoc tests
• A set of comparisons that you decided to examine
  only after you find a significant (reject H0)
  ANOVA

10
Planned Comparisons

• Different types
• Simple comparisons - testing two groups
• Complex comparisons - testing combined groups
• Bonferroni procedure
• Use more stringent significance level for each
  comparison
• Basic procedure
• Within-groups population variance estimate
  (denominator)
• Between-groups population variance estimate of
  the two groups of interest (numerator)
• Figure F in usual way

11
Post-hoc tests

• Generally, you are testing all of the possible
  comparisons (rather than just a specific few)
• Different types
• Tukeys HSD test
• Scheffe test
• Others (Fishers LSD, Neuman-Keuls test, Duncan
  test)
• Generally they differ with respect to how
  conservative they are.

12
Effect sizes in ANOVA

• The effect size for ANOVA is r2
• Sometimes called ?2 (eta squared)
• The percent of the variance in the dependent
  variable that is accounted for by the independent
  variable

• Size of effect depends, in part, on degrees of
  freedom
• See table 12-7 in textbook for what is considered
  small medium and large

13
ANOVA in SPSS

• Lets see how to do a between groups 1-factor
  ANOVA in SPSS (and the other tests too)