1. T-Tests and ANOVA I
Class 15

2. Schedule for Remainder of Semester  
1. ANOVA: One way, Two way
2. Planned contrasts 
3. Correlation and Regression
4. Moderated Multiple Regression
5. Survey design 
6. Non-experimental designs   IF TIME PERMITS
7. Writing up research
Quiz 2: Nov 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

3. SD: The Standard Error of Differences Between Means
TARANT PICT D D - D (D-D)2
Sub
Sub
Sub
Sub
X Tarant. = X Pic = D = Σ (D-D)2 =
SD2 = Sum (D -D)2 / N - 1; = .77 / 3 = [VARIANCE OF DIFFS]
SD = √SD2 = √.26 = [STD. DEV. OF DIFFS]
SE of D = σD = SD / √N = .51 / √4 = .51 / 2 = [STD. ERROR OF DIFFS]
t = D / SE of D = 2.25 / = = Diffs Between means is
8.23 times greater than error.
Note: D is the average diff. btwn Tarant mean and Pict. mean.

4. 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
Small SD will therefore increase or decrease our chance of confirming experimental prediction?
Increase it.

5. Assumptions of Dependent T-Test
1. Samples are normally distributed
2. Data measured at interval level (not ordinal or categorical)

6. Conceptual Formula for Dependent Samples (Within Subjects) T-Test
D − μD
Experimental Effect
t = =
Random Variation
sD / √N
D = Average difference between mean Var. 1 – mean Var It represents systematic variation, aka experimental effect.
μD = Expected difference in true population = 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.

7. It Tests a Difference between Differences!
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) more than shared variance (Error)
Diff btwn Means (Effect) less than shared variance (Error)

8. Dependent (w/n subs) T-Test SPSS Output
t = expt. effect / error
t = X / SE
t = -7 / 2.83 =
Note:
SE = SD / √n
2.83 = / √12
Mean = mean diff pic anx - real anx.
= = - 7

9. Data (from spiderBG.sav)
Independent (between-subjects) t-test
Subjects see either spider pictures OR real tarantula
Counter-balancing less critical (but still important). Why?
DV: Anxiety after picture OR after real tarantula
Data (from spiderBG.sav)
Subject Condition Anxiety

10. Assumptions of Independent 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).

11. Logic of Independent Samples T-Test (Same as Dependent T-Test)
expected difference between population means (if null hyp. is true)
observed difference between sample means −
t =
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

12. Conceptual Formula for Independent Samples T-Test
(X1 − X2) − (μ1 − μ2)
Experimental Effect
t = =
Est. of SE
Random Variation
(X1 − X2) = Diffs. btwn. samples
It represents systematic variation, aka experimental effect.
(μ1 − μ2) = Expected difference in true populations = 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.

13. Computational Formulas for Independent Samples T-Tests
X1 − X2
t =
X1 − X2 ( ) √ 2 2
s1 s2
sp sp 2 + √ 2 + N1 N2 n1 n2
When N1 = N2
When N1 ≠ N2
Weighted average of each groups SE sp 2
(n1 -1)s1 + (n2 -1)s2 2 2 = =
n1 + n2 − 2

14. Independent (between subjects) T-Test SPSS Output
t = expt. effect / error
t = (X1 − X2) / SE
t = -7 / 4.16 =
Note: CI crosses “0”

15. Dependent (within subjects) T-Test SPSS Output
t = expt. effect / error
t = X / SE
t = -7 / 2.83 =
Note:
SE = SD / √n
2.83 = / √12
Note: CI does not cross “0”
Mean = mean diff pic anx - real anx.
= = - 7

16. Dependent T-Test is Significant; Independent T-Test Not Significant
Dependent T-Test is Significant; Independent T-Test Not Significant. A Tale of Two Variances
Independent T -Test
Dependent T-Test
SE = 4.16
SE = 2.83

17. Dependent (within subjects) T-Test SPSS Output
t = expt. effect / error
t = X / SE
t = -7 / 2.83 =
Note:
SE = SD / √n
2.83 = / √12
Mean = mean diff pic anx - real anx.
= = - 7

18. Independent (between subjects) T-Test SPSS Output
t = expt. effect / error
t = (X1 − X2) / SE
t = -7 / 4.16 =

19. Dependent T-Test is Significant; Independent T-Test Not Significant
Dependent T-Test is Significant; Independent T-Test Not Significant. A Tale of Two Variances
Independent T -Test
Dependent T-Test
SE = 2.83
T = 7 / 2.83 = 2.47, p = .031
SE = 4.16
T = 7 / 4.16 = 1.68, p = .107

20. 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.

21. ANOVA “MOUNTAINS”

22. Random Samples or Meaningfully Different Samples
ANOVA TASK:
Random Samples or Meaningfully Different Samples

23. 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

24. Population Parameters
Mean: The average score, measure, or response
 =  X N
 X n
X =
Variance: The average amount that individual scores vary around the mean
S 2 =  (x - X )2 n -1
 2 =  (X - ) N
Standard Deviation: The positive square root of the variance. 1 SD = .34 of entire distribution
S=  (x - X)2 n -1
 =  (X - ) N
POPULATION
SAMPLE

25. 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

26. Birth Order Means

27. Activity Ratings of Those Who Disclose As a Function of Birth Order

30. 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
When Alt Hyp. is true: Between group = true diff. + error Within group = error
ANOVA ≤ 1
ANOVA > 1

31. Logic of Inferential Statistics: Is the null hypothesis supported?
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”

32. Logic of F Ratio F = Differences Among Treatment Groups
F = Differences Among Treatment Groups
Differences Among Subjects Treated Alike   
F = (Treatment Effects) + (Experimental Error)
Experimental Error
F = Between-group Differences
Within Group Differences

35. Partitioning Deviations with Actual Data

36. Average Scores Around the Mean
“Oldest Child”
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
Average
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

37. Sum of Squared Deviations
Sum of Squared Deviations
Total Sum of Squares = Sum of Squared between-group deviations
+ Sum of Squared within-group deviations
SSTotal = SSBetween + SSWithinb

38. Computing the Sums of Squares
Parameter
Code
Formula
Steps
Total
Sum of Squares
SST
 (AS - T)2
1. Subtract each individual score from total mean.
2. Square each deviation.
3. Sum up all deviations, across all factor levels.
Between Groups Sum of Squares
SSB
s [(A - T)2]
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.
Within Groups Sum of Squares
SSS/A
[(AS - A)2]
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.

39. Birth Order and Ratings of “Activity” Deviation Scores
AS Total Between Within (AS – T) = (A – T) (AS –A)
Level a1: Oldest Child
(-2.97) = (-1.17) (-1.80) (-2.30) = (-1.17) (-1.13) (-0.97) = (-1.17) ( 0.20) (0.03) = (-1.17) + ( 1.20) (0.37) = (-1.17) ( 1.54)
Level a2: Youngest Child
(0.03) = (1.17) (-1.14) (0.07) = (1.17) (-0.47) (1.03) = (1.17) (-0.14) (1.37) = (1.17) + ( 0.20) (2.70) = (1.17) ( 1.53)
Sum: (0) = (0) (0)
Mean scores: Oldest = 3.13 Youngest = Total = 4.30

40. 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

41. 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 > 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!)