A Priori and Post Hoc Comparisons

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Chapter 12

• A Priori and Post Hoc Comparisons
• Multiple t-tests
• Linear Contrasts
• Orthogonal Contrasts
• Trend Analysis
• Bonferroni t
• Fisher Least Significance Difference
• Studentized Range Statistic
• Dunnetts Test

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Trend Analysis

• The logic of trend analysis is exactly the same
  logic we just talked about with contrasts!

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Example

• You collect axon firing rate scores from rats in
  one of four conditions.
• Condition 1 10 mm of Zeta inhibitor
• Condition 2 20 mm of Zeta inhibitor
• Condition 3 30 mm of Zeta inhibitor
• Condition 4 40 mm of Zeta inhibitor
• Condition 5 50 mm of Zeta inhibitor
• You think Zeta inhibitor reduces the number of
  times an axon fires are you right?

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What does this tell you ?
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Trend Analysis
Contrast Codes!
-2 -1 0
1 2
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Trend Analysis
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a1 -2, a2 -1, a3 0, a4 1, a5 2
L 7.2
F crit (1, 20) 4.35
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Note
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Example

• You place subjects into one of five different
  conditions of anxiety.
• 1) Low anxiety
• 2) Low-Moderate anxiety
• 3) Moderate anxiety
• 4) High-Moderate anxiety
• 5) High anxiety
• You think subjects will perform best on a test at
  level 3 (and will do worse at both lower and
  higher levels of anxiety)

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What does this tell you ?
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-2 1 2
1 -2
Contrast Codes!
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Trend Analysis

• Create contrast codes that will examine a
  quadratic trend.
• -2, 1, 2, 1, -2

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a1 -2, a2 1, a3 2, a4 1, a5 -2
L 10
F crit (1, 20) 4.35
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Trend Analysis

• How do you know which numbers to use?
• Page 742

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Linear
(NO BENDS)
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Quadratic
(ONE BEND)
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Cubic
(TWO BENDS)
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Practice

• You believe a balance between school and ones
  social life is the key to happiness. Therefore
  you hypothesize that people who focus too much on
  school (i.e., people who get good grades) and
  people who focus too much on their social life
  (i.e., people who get bad grades) will be more
  depressed.
• You collect data from 25 subjects
• 5 subjects F
• 5 subjects D
• 5 subjects C
• 5 subjects B
• 5 subjects A
• You measured their depression

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Practice

• Below are your findings interpret!

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Trend Analysis

• Create contrast codes that will examine a
  quadratic trend.
• -2, 1, 2, 1, -2

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a1 -2, a2 1, a3 2, a4 1, a5 -2
L -12.8
F crit (1, 20) 4.35
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Remember

• Freshman, Sophomore, Junior, Senior
• Measure Happiness (1-100)

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ANOVA

• Traditional F test just tells you not all the
  means are equal
• Does not tell you which means are different from
  other means

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Why not

• Do t-tests for all pairs
• Fresh vs. Sophomore
• Fresh vs. Junior
• Fresh vs. Senior
• Sophomore vs. Junior
• Sophomore vs. Senior
• Junior vs. Senior

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Problem

• What if there were more than four groups?
• Probability of a Type 1 error increases.
• Maximum value comparisons (.05)
• 6 (.05) .30

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Chapter 12

• A Priori and Post Hoc Comparisons
• Multiple t-tests
• Linear Contrasts
• Orthogonal Contrasts
• Trend Analysis
• Bonferroni t
• Fisher Least Significance Difference
• Studentized Range Statistic
• Dunnetts Test

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Bonferoni t

• Controls for Type I error by using a more
  conservative alpha

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• Do t-tests for all pairs
• Fresh vs. Sophomore
• Fresh vs. Junior
• Fresh vs. Senior
• Sophomore vs. Junior
• Sophomore vs. Senior
• Junior vs. Senior

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• Maximum probability of a Type I error
• 6 (.05) .30
• But what if we use
• Alpha .05/C
• .00833 .05 / 6
• 6 (.00855) .05

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t-table

• Compute the t-value the exact same way
• Problem normal t table does not have these p
  values
• Test for significance using the Bonferroni t
  table (page 751)

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Practice
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Practice
Fresh vs. Sophomore t .69 Fresh vs. Junior t
2.41 Fresh vs. Senior t -1.55 Sophomore vs.
Junior t 1.72 Sophomore vs. Senior t
-2.24 Junior vs. Senior t -3.97 Critical t
6 comp/ df 20 2.93
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Bonferoni t

• Problem
• Silly
• What should you use as the value in C?
• Increases the chances of the Type II error!

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Fisher Least Significance Difference

• Simple
• 1) Do a normal omnibus ANOVA
• 2) If there it is significant you know that there
  is a difference somewhere!
• 3) Do individual t-test to determine where
  significance is located

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Fisher Least Significance Difference

• Problem
• You may have an ANOVA that is not significant and
  still have results that occur in a manner that
  you predict!
• If you used this method you would not have
  permission to look for these effects.

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Remember
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Remember
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Chapter 12

• A Priori and Post Hoc Comparisons
• Multiple t-tests
• Linear Contrasts
• Orthogonal Contrasts
• Trend Analysis
• Bonferroni t
• Fisher Least Significance Difference
• Studentized Range Statistic
• Dunnetts Test

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Studentized Range Statistic
Which groups would you likely select to determine
if they are different?
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Studentized Range Statistic
Which groups would you likely select to determine
if they are different?
This statistics controls for Type I error if
(after looking at the data) you select the two
means that are most different.
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Studentized Range Statistic

• Easy!
• 1) Do a normal t-test

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Studentized Range Statistic

• Easy!
• 2) Convert the t to a q

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Studentized Range Statistic

• 3) Critical value of q (note this is a
  two-tailed test)
• Figure out df (same as t)
• Example 20
• Figure out r
• r the number of groups

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Studentized Range Statistic

• 3) Critical value of q note this is a two-tailed
  test)
• Figure out df (same as t)
• Example 20
• Figure out r
• r the number of groups
• Example 4

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Studentized Range Statistic

• 3) Critical value of q
• Page 744
• Example
• q critical /- 3.96

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Studentized Range Statistic

• 4) Compare q obs and q critical same way as t
  values
• q -5.61
• q critical / 3.96

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Practice

• You collect axon firing rate scores from rates in
  one of four conditions.
• Condition 1 10 mm of Zeta inhibitor
• Condition 2 20 mm of Zeta inhibitor
• Condition 3 30 mm of Zeta inhibitor
• Condition 4 40 mm of Zeta inhibitor
• Condition 5 50 mm of Zeta inhibitor
• You are simply interested in determining if any
  two groups are different from each other use
  the Studentized Range Statistic

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Studentized Range Statistic

• Easy!
• 1) Do a normal t-test

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Studentized Range Statistic

• Easy!
• 2) Convert the t to a q

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Studentized Range Statistic

• 3) Critical value of qnote this is a two-tailed
  test)
• Figure out df (same as t)
• Example 20
• Figure out r
• r the number of groups
• Example 5

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Studentized Range Statistic

• 3) Critical value of q
• Page 744
• Example
• q critical /- 4.23

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Studentized Range Statistic

• 4) Compare q obs and q critical same way as t
  values
• q -4.34
• q critical / 4.23

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Dunnetts Test

• Used when there are several experimental groups
  and one control group (or one reference group)
• Example
• Effect of psychotherapy on happiness
• Group 1) Psychoanalytic
• Group 2) Humanistic
• Group 3) Behaviorism
• Group 4) Control (no therapy)

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Psyana vs. Control Human vs. Control Behavior vs.
Control
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Psyana vs. Control 47.8 51.4 -3.6 Human vs.
Control 50.8 51. 4 -0.6 Behavior vs.
Control 59 51.4 7.6
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Psyana vs. Control 47.8 51.4 -3.6 Human vs.
Control 50.8 51. 4 -0.6 Behavior vs.
Control 59 51.4 7.6
How different do these means need to be in order
to reach significance?
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Dunnetts t is on page 753 df Within groups df
/ k number of groups
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Dunnetts t is on page 753 df 16 / k 4
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Dunnetts t is on page 753 df 16 / k 4
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Psyana vs. Control 47.8 51.4 -3.6 Human vs.
Control 50.8 51. 4 -0.6 Behavior vs.
Control 59 51.4 7.6
How different do these means need to be in order
to reach significance?
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Practice

• As a graduate student you wonder what
  undergraduate students (freshman, sophomore,
  etc.) have different levels of happiness then you.

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Dunnetts t is on page 753 df 25 / k 5
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Fresh vs. Grad -17.5 Soph vs. Grad
-21.5 Jun vs. Grad -31.5 Senior vs. Grad
-8.5
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