Download class 13 anova 1

ANOVA I
Class 13
Schedule for Remainder of Semester
1. ANOVA: One way, Two way
2. Planned contrasts
3. Moderated multiple regression
4. Data management
5. Survey design
6. Non-experimental designs
7. Writing up research
Class assignment: After completing data analyses series.
ANOVA
ANOVA = Analysis of Variance
Next 4-5 classes focus on ANOVA
One-Way ANOVA – tests differences between 2 or more
independent 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 XXX Variance
Amount Shared Variance
Same as
Amount Groups Differ
Amount Groups XXXX
Basic Principle of ANOVA
Amount Distributions Differ
Amount Distributions Overlap
Same as
Amount Distinct Variance
Amount Shared Variance
Same as
Amount Groups Differ
Amount Groups Same
Population Parameters
Mean: The average score, measure, or response
 =  (X - 
2
)2
N
S 2 =  (x - X )2
n -1
Standard Deviation: The positive square root of the variance.
1 SD = .34 of entire distribution
=
 (X - )2
N
POPULATION
SAMPLE
Population Parameters
Mean: The average score, measure, or response
 = X
X= X
n
N
Variance: The average amount that individual scores vary
around the mean
2 =  (x - X )2
2
S
2
 =  (X - )
n -1
N
Standard Deviation: The positive square root of the variance.
1 SD = .34 of entire distribution
=
 (X - 
)2
N
POPULATION
S=  (x - X)2
n -1
SAMPLE
PEOPLE WHO DISCLOSE THEIR EMOTIONS ARE:
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
Birth Order Means
Activity Ratings as a Function of Birth Order
6
5
4
3
2
1
0
Oldest
Youngest
ANOVA Compares Between Group Differences to
Within Group Differences
Within Group Differences: Comprised of ???
Between Group Differences: Comprised of ????
ANOVA:
Between Group Differences
?????
When Null Hyp. is true: Between group = _XXXX
Within group
= YYYY
When Alt Hyp. is true: Between group = XXXXX
Within group
= YYYY
ANOVA ≤ 1
ANOVA > 1
ANOVA Compares Between Group Differences to
Within Group Differences
Within Group Differences: Comprised of random error only.
Between Group Differences: Comprised of random error + treatment effects
(error + true differences)
ANOVA:
Between Group Differences
Within Group Differences
When Null Hyp. is true: Between group = error
Within group
= error
ANOVA ≤ 1
When Alt Hyp. is true: Between group = error + true diff.
Within group
=
error
ANOVA > 1
Logic of Inferential Statistics:
Is the null hypothesis supported?
Null Hypothesis:
Different sub-samples are equivalent representations of
same overall population.
Differences between them are random.
“First Born don’t differ from Last Born re. disclosers”
Alternative Hypothesis
Different sub-samples do not represent the same overall
population. Instead represent distinct populations.
Differences between them are systematic, not random.
“First Born DO differ from Last Born re. disclosers”
Logic of F Ratio
F =
Differences Among Treatment Means
Differences Among Subjects Treated Alike
F =
XXXXXX + (Experimental Error)
Experimental Error
F =
Between-group Differences
????? Differences
Logic of F Ratio
F =
Differences Among Treatment Means
Differences Among Subjects Treated Alike
F =
Treatment Effect + (Experimental Error)
Experimental Error
F =
Between-group Differences
Within-group Differences
Average Scores Around the Mean
“Oldest Child”
Average
AS1
(AS1 - A)
(AS1 -A)2
1.33
-1.80
3.24
2.00
-1.13
1.28
3.33
0.20
0.04
4.33
1.20
1.44
4.67
1.54
2.37
3.13
0.00
1.67
AS1 = individual scores in condition 1 (Oldest: 1.33, 2.00…)
A = Mean of all scores in a condition (e.g., 3.13)
(AS - A)2 = Squared deviation between individual score and condition mean
Sum of Squared Deviations
Total Sum of Squares = Sum of Squared between-group deviations
+ Sum of Squared within-group deviations
SSTotal = SS????? + SS?????
Sum of Squared Deviations
Total Sum of Squares = Sum of Squared between-group deviations
+ Sum of Squared within-group deviations
SSTotal = SSBetween + SSWithinb
Computing the Sums of Squares
Parameter
Total
Sum of Squares
Between Groups
Sum of Squares
Within Groups
Sum of Squares
Code
SST
SSB
SSS/A
Formula
 (AS -
s [(A -
T)2
T)2]
[(AS - A)2]
Steps
1. Subtract each individual score from total mean.
2. Square each deviation.
3. Sum up all deviations, across all factor levels.
1. Subtract each group mean from the total
mean.
2. Square this deviation.
3. Multiply squared deviation by number of scores
in the group.
4. Repeat for each group.
5. Sum each group's squared deviation to get
total.
1. Subtract group mean from each individual
score in group.
2. Square these deviations.
3. Sum all squared deviations, within each group.
4. Sum the sums of each group's squared
deviations.
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 = 3.13
Youngest = 5.47
Total = 4.30
(0)
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
Logic of F Test and Hypothesis Testing
Form of F Test:
Purpose:
Between Group Differences
Within Group Differences
Test null hypothesis: Between Group = XXXX = YYYY
Interpretation:
If null hypothesis is not supported (F > 1) then
Between Group diffs are not simply random error, but
instead reflect ??????.
Result:
Null hypothesis is rejected, alt. hypothesis is
therefore ___ PROVED ___ SUPPORTED
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!)