ANOVA I Part 2

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ANOVA I (Part 2)
Class 14
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How Do You Regard Those Who Disclose?
EVALUATIVE DIMENSION Good Bad Beautiful
Ugly Sweet Sour POTENCY DIMENSION Strong
Weak Large Small Heavy Light ACTIVITY
DIMENSION Active Passive Fast Slow Hot
Cold
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Birth Order Means
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Logic of F Test and Hypothesis Testing     Form
of F Test Between Group Differences
Within Group Differences   Purpose Test
null hypothesis Between Group
Within Group Random Error   Interpretation If
null hypothesis is not supported (F gt 1) then
Between Group diffs are not simply random
error, but instead reflect effect of the
independent variable.   Result Null
hypothesis is rejected, alt. hypothesis is
supported
(BUT NOT PROVED!)
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F Ratio F Between Group Difference
Within Group Differences
F Error Treatment Effects
Error
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Birth Order and Ratings of Activity Deviation
Scores
AS Total Between Within
(AS T) (A
T) (AS A)
Level a1 Oldest Child
1.33 (-2.97) (-1.17) (-1.80)
2.00 (-2.30) (-1.17) (-1.13)
3.33 (-0.97) (-1.17) ( 0.20)
4.33 (0.03) (-1.17) ( 1.20)
4.67 (0.37) (-1.17) ( 1.54)
Level a2 Youngest Child
4.33 (0.03) (1.17) (-1.14)
5.00 (0.07) (1.17) (-0.47)
5.33 (1.03) (1.17) (-0.14)
5.67 (1.37) (1.17) ( 0.20)
7.00 (2.70) (1.17) ( 1.53)
Sum (0) (0)
(0) Mean scores Oldest (a1) 3.13
Youngest (a2) 5.47 Total (T) 4.30
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    Sum of Squared Deviations     Total Sum of
Squares Sum of Squared between-group deviations
Sum of
Squared within-group deviations     SSTotal
SSBetween SSWithin
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Computing Sums of Squares from Deviation
Scores Birth Order and Activity Ratings
(continued)
SS Sum of squared diffs, AKA sum of
squares SST Sum of squares., total (all
subjects) SSA Sum of squares, between
groups (treatment) SSs/A Sum of squares,
within groups (error)
SST (2.97)2 (2.30)2 (1.37)2
(2.70)2 25.88 SSA (-1.17)2
(-1.17)2 (1.17)2 (1.17)2 13.61 SSs/A
(-1.80)2 (-1.13)2 (0.20)2 (1.53)2
12.27
Total (SSA SSs/A) 25.88
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Birth Order and Activity Ratings Deviation Scores
Level a1 Oldest
Level a2 Youngest
Mean Scores Oldest 3.13 Youngest
5.47 Total 4.30
SST (2.97)2 (2.30)2 ... (1.37)2
(2.70)2 25.88 SSA (-1.17)2 (-1.17)2
... (1.17)2 (1.17)2 13.61 SSs/A
(-1.80)2 (-1.13)2 ... (0.20)2 (1.53)2
12.27 Total 25.88
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Degrees of Freedom   df Number of
observations free to vary.
5 6 4 5 4 24 Number of
observations n 5 Number of estimates 1
(i.e. sum, which 24) df n - estimates 5
-1 4   5 6 4 5 4 24 5 6
X 5 4 24 20 X 24
X 4
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Degrees of Freedom for Fun and Fortune
Coin flip __ df? Dice __ df? Japanese
game that rivals cross-word puzzle?
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Sudoku The Exciting Degrees of Freedom Game
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Degrees of Freedom Formulas for the Single Factor
(One Way) ANOVA
Source Type Formula Meaning
. Groups dfA a 1 df for Tx groups
Between-groups df Scores dfs/A a(s
1) df for individual scores
Within-groups df Total dfT as
1 Total df (note dfT dfA dfs/A)
Source Type Formula Semantic
Differential Study Groups dfA a 1 2 1
1 Scores dfs/A a(s 1) 2 (5 1 ) 8
Total dfT as 1 (2 5) - 1 9

(note dfT dfA dfs/A)
Note a levels in factor A s subjects
per condition
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Mean Squares Calculations
Note What happens to MS/W as n increases?
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F Ratio Computation
F MSA Between Group Variance
MSS/A Within Group Variance
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Analysis of Variance Summary Table   One Factor
(One Way) ANOVA
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Analysis of Variance Summary Table   One Factor
(One Way) ANOVA
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F Distribution Notation   "F (1, 8)"
means   The F distribution with   one df in
the numerator (1 df associated with treatment
groups ( between-group variation))   and   8
degrees of freedom in the denominator (8 df
associated with the overall sample (
within-group variation))
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F Distribution for (2, 42) df
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Criterion F and p Value For F (2, 42) 3.48
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F or F'?     If F is correct, then Ho
supported u1 u2 (First born Last
born)   If F' is correct, then H1 supported
u1 ? u2 (First born ? Last born)
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F Distribution
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F Distribution Notation   "F (1, 8)"
means   The F distribution with   one df in
the numerator (1 df associated with treatment
groups/between-group variation)   and   8
degrees of freedom in the denominator (8 df
associated with the overall sample/within-group
variation)
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Decision Rule Regarding F   Reject null
hypothesis when F observed gt ? (m,n) Reject
null hypothesis when F observed gt 5.32 (1,
8).   F (1,8) 8.88 gt ? 5.32   Decision
Reject null hypothesis Accept alternative
hypothesis   Note We haven't proved alt.
hypothesis, only supported it. Format for
reporting our result   F (1,8) 8.88, p lt
.05   F (1,8) 8.88, p lt .02 also OK, based on
our results.   Conclusion First Borns regard
help-seekers as less "active" than do Last
Borns.
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• Summary of One Way ANOVA
•  
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• 1. Specify null and alt. hypotheses
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• 2. Conduct experiment
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• Calculate F ratio Between Group Diffs
• Within Group
  Diffs
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• 4. Does F support the null hypothesis? i.e., is
  
• Observed F gt Criterion F, at p lt .05?
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• ___ p gt .05, accept null hyp.
• ___ p lt .05, accept alt. hyp.