Psychology 9

1
Psychology 9

• Quantitative Methods in Psychology
• Jack Wright
• Brown University
• Section 26 (old 28)

Note. These lecture materials are intended
solely for the private use of Brown University
students enrolled in Psychology 9, Spring
Semester, 2002-03. All other uses, including
duplication and redistribution, are unauthorized.

2
AGENDA

• Pairwise comparisons
• Post hoc
• Fishers LSD
• Tukeys HSD
• Planned or a priori
• Bonferroni

3
Beyond the omnibus F test

• An investigator wants to know if psychology
  students have different levels of anxiety.
• She measures the anxiety in each group
• PY47 PY1
  PY9
• Mean anxiety 2 4
  6
• She then performs an ANOVA, and finds a
  significant difference between the groups.
• Now what?

4
Pairwise comparisons

• Test for specific differences between means while
  protecting against Type I error
• Post hoc
• Require significant F in order to use
• More conservative than a priori comparisons
• Planned or a priori
• Decided in advance and generally based on prior
  research
• Can be done regardless of results from an
  ANOVA

5
Example

• PY47 PY1
  PY9
• Mean anxiety 2 4
  6
• Source SS df MS F
  p
• Between 24 2 12 12
  
• Within 6 6 1
• Total 30 8

n per group 3
6
A post hoc test Fishers LSD

• Fishers Least Significant Difference (LSD) also
  called a protected t-test
• Use MSwithin to estimate standard error of diff.
  between means for the entire experiment
• ?2MSwithin / n (n per condition)
• sqrt (2 (1) / 3)
• .82

Use this as the denominator in a t-test
comparing the means
7
LSD continued

• Find tobserved for means of PY9 and PY1
• tobserved (6 - 4) / .82
• 2.44
• Find tcritical in table for df6, p .05
• tcritical 2.45
• 2.44

8
Tukeys HSD

• By todays standards, LSD is not conservative
  enough
• main protection is significant F
• Tukeys Honestly Significant Difference (HSD).
• Like LSD, must have significant ANOVA result
• However, more conservative than LSD, with alpha
  based on number of comparisons possible.

9
Example of HSD

• Using ANOVA summary table, obtain
• HSD q ?(MSwithin/n)
• q is penalty for number of comparisons
• From table of Studentized range statisitic.
• For alpha .05, 3 means, and df 6
• q 4.34
• HSD 4.34 ?(1/3)
• 4.34 .58
• 2.52
• Note that only one comparison meets this threshold

10
An a priori comparison Bonferroni

• Bonferroni corrects alpha based on of
  comparisons you decided to perform prior to
  seeing the results.
• EG placebo, low dose treatment, high dose
  treatment
• compare placebo to low dose group, placebo to
  high dose group
• Therefore, two comparisons, not three

11
Example using Bonferroni

• 1. Use estimate of standard error of difference
  from LSD procedure
• Based on MSwithin .82 from earlier work
• 2. Based on number of planned comparisons (k),
  obtain corrected alpha
• corrected alpha alpha/k
• for k 2, alpha .05, corrected .05/2 .025
• for df 6, critical t 3.14
• 3. Using this t critical, proceed as in LSD

12
Bonferroni example

• Perform comparisons
• PY9 vs. PY1 4 2 / .82
• 2.44 2.44 fail to reject
• PY9 vs. PY47 6 2 / .82
• 4.88 4.88 3.14,
  reject null

13
SUMMARY

• Post hoc
• LSD requires significant F relatively liberal
• HSD requires significant F more conserative
  penalizes according to the number of comparisons
  possible.
• A priori
• Bonferroni penalizes according to of
  comparisons we want to do.
• comparisons must be determined in advance