Transcript Psychology of Emotions

One-Way ANOVA
Class 16
Schedule for Remainder of Semester
1.
2.
3.
4.
5.
6.
7.
ANOVA: One way, Two way
Planned contrasts
Correlation and Regression
Moderated Multiple Regression
Survey Question Wording
Non-experimental designs
Writing up research
Quiz 2: Nov. 12 -- up to and including one-way ANOVA I
Quiz 3: Nov. 26 -- up to and including simple Regression
Class Assignment Supplied:
Class Assignment Due:
Nov. 26
Dec. 10
ANOVA
ANOVA = Analysis of Variance
Next 4-5 classes focus on ANOVA and Planned Contrasts
One-Way ANOVA – tests differences between 2 or more
independent groups. (t-test only 2 groups)
Goals for ANOVA series:
1. What is ANOVA, tasks it can do, how it works.
2. Provide intro to SPSS for Windows ANOVA
3. Objective: you will be able to run ANOVA on SPSS,
and be able to interpret results.
Notes on Keppel reading:
1. Clearest exposition on ANOVA
2. Assumes no math background, very intuitive
3. Language not gender neutral, more recent eds. are.
Basic Principle of ANOVA
Amount Distributions Differ
Amount Distributions Overlap
Same as
Amount Distinct Variance
Amount Shared Variance
Same as
Amount Treatment Groups Differ
Amount Treatment Groups the Same
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
Do Means Significantly Differ?
Oldest
3.13
Youngest
4.30
Oldest
3.13
5.47
Youngest
4.30
5.47
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 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
F Ratio
F = Between Group Difference
Within Group Differences
F = Treatment Effects + Error
Error
Ronald Fisher, 1890-1962
F Ratio if Null True, if Alt. True
Null Hyp true: F = (Treatment Effects = 0) + Error
Error
Null Hyp true: F =
Error
Error
=
1
Alt. Hyp true: F = (Treatment Effects > 0) + Error
Error
Alt. Hyp true: F = (Treatment Effects) + Error = >1
Error
ANOVA JOB: Estimate Magnitude of Variances
How much do systematic (meaningful) diffs. between
experimental conditions exceed random error?
NEED TWO MEASURES OF VARIABILTY TO ANSWER
THIS QUESTION
1.
Treatment effects (Between Group Var.)
2.
Random diffs between subjects (Within Group Var.)
Thus, ANOVA = Analysis of Variances
Key Point: Each score contains both
group effect and random 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)
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
(-1.14)
(-0.47)
(-0.14)
( 0.20)
( 1.53)
Youngest (a2) = 5.47
Why are these
"0" sums a
problem?
How do we
fix this?
(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
Hey, Can We Compute F Now?
Why the F Not?
F=
Estimate Between Group Diffs
Estimate Within Group Diffs
SSA = Total Btwn Diffs = 13.61
SSW = Total Within Diffs = 12.27
Does F =
NO! Why not?
13.61
12.27
= 1.11
?
Need AVERAGE estimates of Btwn. Diffs.
variability and Within Diffs. variability.
How Do We Obtain AVERAGE
Variance Estimates?
SSA = Total Btwn Diffs = 13.61
SSW = Total Within Diffs = 12.27
Can we get Ave. Between by dividing
SSA by number of groups?
NO
Can we get Ave. Within by dividing SSW by
number of subjects within each group?
NO
Why not? Why must life be so hard and complicated?
Because we need est. of average of scores that can vary, not
average of all scores.
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
1 df?
Coin flip = __
5 df?
Dice = __
Japanese game that rivals cross-word puzzle?
Sudoku – The Exciting Degrees of Freedom Game
4
df for just
this section?
8
5
2
5
8
4
7
1
9
9-4-1
=4
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
General 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
Ave. Between Group Variance
=
MSS/A
F
=
13.61
1.51
Ave. 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
1
13.61
Within Groups
12.27
8
1.533
Total
25.88
9
Source of
Variation
F
8.877
Significance
of F
.018
Analysis of Variance Summary Table:
SPSS
F Distribution Notation
"F (1, 8)" means:
The F distribution with:
1 df in the numerator (1 df associated with treatment groups
(= between-group variation))
and
8 df 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 and F' Distributions (from Monte Carlo Experiments)
Which Distribution Do Data Support: 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)
Critical Values for F (1, 8)
What must our F be in order to reject null hypothesis?
≥ 5.32
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 or Accept null hypothesis?
Reject or Accept alternative hypothesis?
Have we proved alt. hypothesis?
No, we supported it. There's a chance (p < .05), that we are wrong.
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.
TYPE I AND TYPE II ERRORS
Errors in Hypothesis Testing
Reality
Decision
Reject Null
Accept Alt.
Accept Null
Reject Null
Null Hyp. True
Null Hyp. False
Alt. Hyp. False
Alt. Hyp. True
Incorrect:
Correct
TypeI IError
Error
Type
Correct
Incorrect:
Type
II Error
Type
II Error
Avoiding Type I and Type II Errors
Avoiding Type I error:
1. Reduce the size of the Type I rejection region (i.e., go
from p < .05 to p < .01 for example).
Avoiding Type II error
1. Reduce size of Type II rejection region, BUT
a. Not permitted by basic sci. community
b. But, OK in some rare applied contexts
2. Increase sample size
3. Reduce random error
a. Standardized instructions
b. Train experimenters
c. Pilot testing , etc.