Download class 15 anova 1

T-Tests and ANOVA I
Class 15
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 design
Non-experimental designs
IF TIME PERMITS
Writing up research
Quiz 2: Nov. 12 -- up to and including one-way ANOVA
Quiz 3: Dec. 3 – What we’ve covered by Dec. 3
Class Assignment: Assigned Dec. 1, Due Dec. 10
SD: The Standard Error of Differences Between Means
TARANT.
Sub. 1
Sub. 2
Sub. 3
Sub. 4
6
5
4
5
PICT.
3
3
2
3
X Tarant. = 5.00 X Pic = 2.75
D
D-D
(D-D)2
3
2
2
2
-. 75
.25
.25
.25
.56
.07
.07
.07
D = 2.25
Note: D is the
average diff. btwn
Tarant mean and
Pict. mean.
Σ (D-D)2 = .77
SD2 = Sum (D -D)2 / N - 1; = .77 / 3 = .26 [VARIANCE OF DIFFS]
SD = √SD2 = √.26 = .51
[STD. DEV. OF DIFFS]
SE of D = σD = SD / √N = .51 / √4 = .51 / 2 = .255 [STD. ERROR OF DIFFS]
t = D / SE of D
= 2.25 / .255 = 8.823 = Diffs Between means is
8.23 times greater than error.
Understanding SD and Experiment Power
Power of Experiment: Ability of expt. to detect actual differences.
Small SD indicates that average difference between
pairs of variable means should be large or small, if
null hyp true?
Small SD will therefore increase or decrease our
chance of confirming experimental prediction?
Small
Increase it.
Assumptions of Dependent T-Test
1. Samples are normally distributed
2. Data measured at interval level
(not ordinal or categorical)
Conceptual Formula for Dependent Samples
(Within Subjects) T-Test
t=
D − μD
Experimental Effect
=
sD / √N
Random Variation
D = Average difference between mean Var. 1 – mean Var. 2.
It represents systematic variation, aka experimental effect.
μD = Expected difference in true population = 0
It represents random variation, aka the the null effect.
sD / √N = Standard Error of Mean Differences.
Estimated standard
error of differences between all potential sample means.
It represents the likely random variation between means.
Within-Subjects T Test Tests Difference
between Obtained Difference Between Means
and Null Difference Between Means:
It Tests a Difference between Differences!
Mean, SD of Obtained Diff between Picture vs. Real Tarantula
Mean, SD of Actual Diff of Null Effect
Diff btwn Means (Effect) less than
shared variance (Error)
Diff btwn Means (Effect) more than
shared variance (Error)
Dependent (w/n subs) T-Test SPSS Output
t = expt. effect / error
t = X / SE
t = -7 / 2.83 = -2.473
Note:
Mean = mean diff
pic anx - real anx.
= 40 - 47 = - 7
SE = SD / √n
2.83 = 9.807 / √12
Independent (between-subjects) t-test
1. Subjects see either spider pictures OR real tarantula
2. Counter-balancing less critical (but still important). Why?
3. DV: Anxiety after picture OR after real tarantula
Data (from spiderBG.sav)
Subject
Condition
Anxiety
1
1
30
2
2
35
3
1
45
22
2
50
23
1
60
24
2
39
Assumptions of Independent T-Test
DEPENDENT T-TEST
1. Samples are normally distributed
2. Data measured at least at interval level (not
ordinal or categorical)
INDEPENDENT T-TESTS ALSO ASSUME
3. Homogeneity of variance
4. Scores are independent (b/c come from diff. people).
Logic of Independent Samples T-Test
(Same as Dependent T-Test)
t =
observed difference
between sample means
−
expected difference
between population means
(if null hyp. is true)
SE of difference between sample means
Note: SE of difference of sample means in independent
t test differs from SE in dependent samples t-test
Conceptual Formula for
Independent Samples T-Test
t=
(X1 − X2) − (μ1 − μ2)
Est. of SE
Experimental Effect
=
Random Variation
(X1 − X2) = Diffs. btwn. samples
It represents systematic variation, aka experimental effect.
(μ1 − μ2) = Expected difference in true populations = 0
It represents random variation, aka the the null effect.
Estimated standard error of differences between all potential sample
means.
It represents the likely random variation between means.
Computational Formulas for
Independent Samples T-Tests
t=
√(
X1 − X2
2
s1
N1
2
s2
+
N2
)
√
When N1 = N2
sp2 =
X1 − X2
t=
sp
2
2
sp
n1 + n2
When N1 ≠ N2
(n1 -1)s12 + (n2 -1)s22
n1 + n2 − 2
=
Weighted
average of
each groups SE
Independent (between subjects)
T-Test SPSS Output
t = expt. effect / error
t = (X1 − X2) / SE
t = -7 / 4.16 = - 1.68
Note: CI
crosses “0”
Dependent (within subjects)
T-Test SPSS Output
t = expt. effect / error
t = X / SE
t = -7 / 2.83 = -2.473
Note: SE = SD / √n
2.83 = 9.807 / √12
Mean = mean diff
pic anx - real anx.
= 40 - 47 = - 7
Note: CI
does not
cross “0”
Dependent T-Test is Significant;
Independent T-Test Not Significant.
A Tale of Two Variances
Independent T -Test
60
60
55
55
50
50
45
45
Anxiety
Anxiety
Dependent T-Test
40
35
40
35
30
30
25
25
20
20
Picture
Real T
SE = 2.83
Picture
Real T
SE = 4.16
Dependent (within subjects)
T-Test SPSS Output
t = expt. effect / error
t = X / SE
t = -7 / 2.83 = -2.473
Note:
Mean = mean diff
pic anx - real anx.
= 40 - 47 = - 7
SE = SD / √n
2.83 = 9.807 / √12
Independent (between subjects)
T-Test SPSS Output
t = expt. effect / error
t = (X1 − X2) / SE
t = -7 / 4.16 = - 1.68
Dependent T-Test is Significant;
Independent T-Test Not Significant.
A Tale of Two Variances
Independent T -Test
60
60
55
55
50
50
45
45
Anxiety
Anxiety
Dependent T-Test
40
35
40
35
30
30
25
25
20
20
Picture
Real T
Picture
Real T
SE = 2.83
SE = 4.16
T = 7 / 2.83 = 2.47, p = .031
T = 7 / 4.16 = 1.68, p = .107
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.
ANOVA “MOUNTAINS”
ANOVA TASK:
Random Samples or Meaningfully Different Samples
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
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
Activity Ratings of Those Who Disclose
As a Function of Birth Order
6
5
4
3
2
1
0
Youngest
Oldest
ANOVA Compares Between Group Differences to Within
Group Differences
Within Group Differences: Comprised of random error only.
Between Group Differences: Comprised of treatment effects and random error
(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 = true diff. + error
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 sub-samples are random.
“First Born and Last Born rate disclosers equally”
Alternative Hypothesis
Different sub-samples do not represent the same overall
population. Instead each represent distinct populations.
Differences between them are systematic, not random.
“First Born rate disclosers differently than do Last Born”
Logic of F Ratio
F =
Differences Among Treatment Groups
Differences Among Subjects Treated Alike
F =
(Treatment Effects) + (Experimental Error)
Experimental Error
F =
Between-group Differences
Within Group Differences
Partitioning Deviations with Actual Data
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 = 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 = 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!)