Download class 14 anova 1b

ANOVA I (Part 2)
Class 14
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
Birth Order Means
Logic of F Test and Hypothesis Testing
Form of F Test:
Purpose:
Between Group Differences
Within Group Differences
Test null hypothesis:
Between Group = Within Group = Random Error
Interpretation: If null hypothesis is not supported (F > 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!)
F Ratio
F = Between Group Difference
Within Group Differences
F = Error + Treatment Effects
Error
Birth Order and Ratings of “Activity” Deviation Scores
AS
Total
(AS – T)
=
Between
(A – T)
+
Within
(AS – A)
+
+
+
+
+
(-1.80)
(-1.13)
( 0.20)
( 1.20)
( 1.54)
(-1.14)
(-0.47)
(-0.14)
( 0.20)
( 1.53)
Level a1: Oldest Child
1.33
2.00
3.33
4.33
4.67
(-2.97)
(-2.30)
(-0.97)
(0.03)
(0.37)
=
=
=
=
=
(-1.17)
(-1.17)
(-1.17)
(-1.17)
(-1.17)
Level a2: Youngest Child
4.33
5.00
5.33
5.67
7.00
Sum:
(0.03)
(0.07)
(1.03)
(1.37)
(2.70)
=
=
=
=
=
(1.17)
(1.17)
(1.17)
(1.17)
(1.17)
+
+
+
+
+
(0)
=
(0)
+
Mean scores: Oldest (a1) = 3.13
Youngest (a2) = 5.47
(0)
Total (T) = 4.30
Sum of Squared Deviations
Total Sum of Squares = Sum of Squared between-group deviations
+ Sum of Squared within-group deviations
SSTotal = SSBetween + SSWithin
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
Birth Order and Activity Ratings: Deviation Scores
AS
Total__
(AS - T)
Between
(A - T)
=
Within
(AS - A)
+
Level a1: Oldest
1.33
2.00
3.33
4.33
4.67
(-2.97)
(-2.30)
(-0.97)
(0.03)
(0.37)
=
=
=
=
=
(-1.17)
(-1.17)
(-1.17)
(-1.17)
(-1.17)
4.33
5.00
5.33
5.67
7.00
(0.03)
(0.70)
(1.03)
(1.37)
(2.70)
=
=
=
=
=
(1.17)
(1.17)
(1.17)
(1.17)
(1.17)
+
+
+
+
+
(-1.14)
(-0.47)
(-0.14)
(0.20)
(1.53)
Sum:
(0)
=
(0)
+
(0)
Level a2: Youngest
Mean Scores: Oldest = 3.13
+
+
+
+
+
Youngest = 5.47
(-1.80)
(-1.13)
(0.20)
(1.20)
(1.54)
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
SSs/A =(-1.80)2 + (-1.13)2 + ... + (0.20)2 + (1.53)2
Total
= 13.61
= 12.27
= 25.88
Degrees of Freedom
df =
Number of observations free to ???
df =
Number of independent
Observations
-
Number of restraints
df =
Number of independent
Observations
-
Number of population
estimates
5 + 6 + 4 + 5 + 4 = 24
Number of observations = n = 5
Number of estimates = 1 (i.e. sum, which = 24)
df = n - # estimates = 5 -X = Z
5 + 6 + 4 + 5 + 4 = 24
Degrees of Freedom
df =
Number of observations free to vary.
df =
Number of independent
Observations
-
Number of restraints
df =
Number of independent
Observations
-
Number of population
estimates
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
Degrees of Freedom for Fun and Fortune
Coin flip = __ df?
Dice = __ df?
Japanese game that rivals cross-word puzzle?
Sudoku – The Exciting Degrees of Freedom Game
4
8
5
2
5
8
4
7
1
9
3
4
5
6
8
2
7
9
1
5
3
1
9
7
6
3
2
8
2
6
Degrees of Freedom Formulas
for the Single Factor (One Way) ANOVA
Source
Type
Formula
Meaning
Groups
dfA
a–X
df for Tx groups;
Between-groups df
Scores
dfs/A
X(s –1)
df for individual scores
Within-groups df
Total
dfT
XY – 1
Total df (note: dfT = dfA + dfs/A)
Note: a = # levels in factor A; s = # subjects per condition
.
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
Groups
dfA
a–1
2 –1 = 1
Scores
dfs/A
a(s –1)
2 (5 –1 ) = 8
Total
dfT
as – 1
Semantic Differential Study
(2 * 5) - 1 = 9
(note: dfT = dfA + dfs/A)
Note: a = # levels in factor A; s = # subjects per condition
.
Mean Squares Calculations
Variance
Code
Calculation
Meaning
Mean Square
Between Groups
MSA
SSA
dfA
Between groups
variance
Mean Square
Within Groups
MSS/A
SSS/A
dfS/A
Within groups
variance
Variance
Code
Calculation
Data
Result
Mean Square
Between Groups
MSA
SSA
dfA
13.61
1
13.61
Mean Square
Within Groups
MSS/A
SSS/A
dfS/A
12.27
8
1.53
Note: What happens to MS/W as n increases?
F Ratio Computation
F
=
MSA
XXX Variance
=
MSS/A
F=
13.61
1.51
YYYY Variance
=
8.78
F Ratio Computation
F
=
MSA
Between Group Variance
=
MSS/A
F=
13.61
1.51
Within Group Variance
=
8.78
Analysis of Variance Summary Table:
One Factor (One Way) ANOVA
Source of Variation
Sum of
Squares
(SS)
df
Mean Square
(MS)
A
SSA
a-1
SSA
dfA
S/A
SSS/A
a (s- 1)
SSS/A
dfS/A
Total
SST
as - 1
F Ratio
MSA
MSS/A
Analysis of Variance Summary Table:
One Factor (One Way) ANOVA
Sum of
Squares
df
Mean
Square
(MS)
Between Groups
13.61
?
13.61
Within Groups
????
8
1.533
Total
25.88
9
Source of
Variation
F
????
Significance
of F
.018
Analysis of Variance Summary Table:
One Factor (One Way) ANOVA
Sum of
Squares
df
Mean
Square
(MS)
Between Groups
13.61
1
13.61
Within Groups
12.27
8
1.533
Total
25.88
9
Source of
Variation
F
8.877
Significance
of F
.018
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))
F Distribution for (2, 42) df
Criterion F and p Value
For F (2, 42) = 3.48
F or F′?
If F is correct, then Ho supported:
(First born ??? Last born)
If F' is correct, then H1 supported:
(First born ??? Last born)
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)
F’ Distribution
F Distribution Notation
"F (1, 8)" means:
The F distribution with:
??? 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 ??????)
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)
Decision Rule Regarding F
Reject null hypothesis when F observed >  (m,n)
Reject null hypothesis when F observed > 5.32 (1, 8).
F (1,8) = 8.88 >  = 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 < .05
F (1,8) = 8.88, p < .02 also OK, based on our results.
Conclusion: First Borns regard help-seekers as
less "active" than do Last Borns.
Summary of One Way ANOVA
1. Specify null and alt. hypotheses
2. Conduct experiment
3. Calculate F ratio Between Group Diffs
Within Group Diffs
4. Does F support the null hypothesis? i.e., is
Observed F > Criterion F, at p < .05?
___ p > .05, accept null hyp.
___ p < .05, accept alt. hyp.