1. Statistics for the Social Sciences
Psychology 340
Fall 2013
Thursday, November 7, 2013
Correlation and Regression

2. Homework #11 due Thursday 11/14
Ch 15 #1, 2, 5, 6, 8, 9, 14

3. Exam III Results
Will be available on Tuesday, 11/12

4. Last Few Weeks of the Semester
Review of Correlation (CH 15)
Review of Simple Linear Regression (CH 16)
Multiple Regression (CH 16)
Hypothesis testing with Correlation and Regression (CH 15, 16)
Chi-Squared Test (CH 17)

5. Statistical analysis follows design
We have finished the top part of the chart!
Focus on this section for rest of semester

6. Correlation Write down what (you think) a correlation is.
Write down an example of a correlation
Association between scores on two variables
age and coordination skills in children, as kids get older their motor coordination tends to improve
price and quality, generally the more expensive something is the higher in quality it is

7. Correlation and Causality
Correlational research design
Correlation as a statistical procedure
Correlation as a kind of research design (observational designs)

8. Another thing to consider about correlation
A correlation describes a relationship between two variables, but DOES NOT explain why the variables are related
Suppose that Dr. Steward finds that rates of spilled coffee and severity of plane turbulence are strongly positively correlated.
One might argue that turbulence cause coffee spills
One might argue that spilling coffee causes turbulence

9. Another thing to consider about correlation
A correlation describes a relationship between two variables, but DOES NOT explain why the variables are related
Suppose that Dr. Cranium finds a positive correlation between head size and digit span (roughly the number of digits you can remember).
One might argue that bigger your head, the larger your digit span 1 21 24 15 37
One might argue that head size and digit span both increase with age (but head size and digit span aren’t directly related)
AGE

10. Relationships between variables
Properties of a correlation
Form (linear or non-linear)
Direction (positive or negative)
Strength (none, weak, strong, perfect)
To examine this relationship you should:
Make a scatterplot - a picture of the relationship
Compute the Correlation Coefficient - a numerical description of the relationship

11. Graphing Correlations
Steps for making a scatterplot (scatter diagram)
Draw axes and assign variables to them
Determine range of values for each variable and mark on axes
Mark a dot for each person’s pair of scores

12. Scatterplot X Y Plots one variable against the other
Each point corresponds to a different individual Y X 1 2 3 4 5 6 X Y A

13. Scatterplot X Y Plots one variable against the other
Each point corresponds to a different individual Y X Y 6 A 5 B 4 3 2 1 X 1 2 3 4 5 6

14. Scatterplot X Y Plots one variable against the other
Each point corresponds to a different individual Y X Y 6 A 5 B 4 C 3 2 1 X 1 2 3 4 5 6

15. Scatterplot X Y Plots one variable against the other
Each point corresponds to a different individual Y X Y 6 A 5 B 4 C 3 D 2 1 X 1 2 3 4 5 6

16. Scatterplot X Y Plots one variable against the other
Each point corresponds to a different individual Y X Y 6 A 5 B 4 C 3 D 2 E 1 X 1 2 3 4 5 6

17. Scatterplot X Y Plots one variable against the other
Each point corresponds to a different individual Y X Y 6
Imagine a line through the data points A 5 B 4 C 3
Useful for “seeing” the relationship
Form, Direction, and Strength D 2 E 1 X 1 2 3 4 5 6

18. Form
Linear
Non-linear

19. Direction Positive Negative Y Y X X X & Y vary in the same direction
As X goes up, Y goes up
positive Pearson’s r
X & Y vary in opposite directions
As X goes up, Y goes down
negative Pearson’s r

20. Strength The strength of the relationship
Spread around the line (note the axis scales)
Correlation coefficient will range from -1 to +1
Zero means “no relationship”.
The farther the r is from zero, the stronger the relationship

21. Strength
r = -1.0
“perfect negative corr.”
r2 = 100%
r = 0.0
“no relationship”
r2 = 0.0
r = 1.0
“perfect positive corr.”
r2 = 100%
-1.0
0.0
+1.0
The farther from zero, the stronger the relationship

22. The Correlation Coefficient
Formulas for the correlation coefficient:
You may have used this one in PSY138
Common alternative

23. The Correlation Coefficient
Formulas for the correlation coefficient:
You may have used this one in PSY138
Common alternative

24. Computing Pearson’s r (using SP)
Step 1: SP (Sum of the Products)
X Y
mean
3.6
4.0

25. Computing Pearson’s r (using SP)
Step 1: SP (Sum of the Products)
X Y
2.4 =
-2.6 =
1.4 =
-0.6 =
-0.6 =
mean
3.6
4.0
0.0
Quick check

26. Computing Pearson’s r (using SP)
Step 1: SP (Sum of the Products)
X Y
2.4
2.0 =
-2.6
-2.0 =
1.4
2.0 =
-0.6
0.0 =
-0.6
-2.0 =
mean
3.6
4.0
0.0
0.0
Quick check

27. Computing Pearson’s r (using SP)
Step 1: SP (Sum of the Products)
X Y
2.4
4.8 * =
2.0
-2.6
5.2 * =
-2.0
1.4
2.8 * =
2.0
-0.6
0.0 * =
0.0
-0.6
1.2 * =
-2.0
mean
3.6
4.0
0.0
0.0
14.0 SP

28. Computing Pearson’s r (using SP)
Step 2: SSX & SSY

29. Computing Pearson’s r (using SP)
Step 2: SSX & SSY
X Y 2 =
2.4
5.76
2.0
4.8
6.76 2 =
-2.6
-2.0
5.2
1.4
1.96 2 =
2.0
2.8
-0.6
0.36 2 =
0.0
0.0
-0.6
0.36 2 =
-2.0
1.2
mean
3.6
4.0
0.0
15.20
0.0
14.0
SSX

30. Computing Pearson’s r (using SP)
Step 2: SSX & SSY
X Y
2.4
5.76
2.0 2 =
4.0
4.8
-2.6
6.76
-2.0 2 =
4.0
5.2
1.4
1.96
2.0 2 =
4.0
2.8
-0.6
0.36
0.0 2 =
0.0
0.0 2 =
4.0
-0.6
0.36
-2.0
1.2
mean
3.6
4.0
0.0
15.20
0.0
16.0
14.0
SSY

31. Computing Pearson’s r (using SP)
Step 3: compute r

32. Computing Pearson’s r (using SP)
Step 3: compute r
X Y
2.4
5.76
2.0
4.0
4.8
-2.6
6.76
-2.0
4.0
5.2
1.4
1.96
2.0
4.0
2.8
-0.6
0.36
0.0
0.0
0.0
-0.6
0.36
-2.0
4.0
1.2
mean
3.6
4.0
0.0
15.20
0.0
16.0
14.0 SP
SSX
SSY

33. Computing Pearson’s r
Step 3: compute r
15.20
16.0
14.0 SP
SSX
SSY

34. Computing Pearson’s r
Step 3: compute r
15.20
16.0
SSX
SSY

35. Computing Pearson’s r
Step 3: compute r
15.20
SSX

36. Computing Pearson’s r
Step 3: compute r

37. Computing Pearson’s r Step 3: compute r Y Appears linearX 1 2 3 4 5 6
Appears linear
Positive relationship
Fairly strong relationship
.89 is far from 0, near +1

38. The Correlation Coefficient
Formulas for the correlation coefficient:
You may have used this one in PSY138
Common alternative

39. Computing Pearson’s r (using z-scores)
Step 1: compute standard deviation for X and Y (note: keep track of sample or population)
For this example we will assume the data is from a population
X Y

40. Computing Pearson’s r (using z-scores)
Step 1: compute standard deviation for X and Y (note: keep track of sample or population)
For this example we will assume the data is from a population
X Y
2.4
-2.6
1.4
-0.6
0.0
5.76
6.76
1.96
0.36
Mean
3.6
15.20
SSX
1.74
Std dev

41. Computing Pearson’s r (using z-scores)
Step 1: compute standard deviation for X and Y (note: keep track of sample or population)
For this example we will assume the data is from a population
X Y
2.4
5.76
2.0
-2.0
0.0
4.0
-2.6
6.76
4.0
0.0
1.4
1.96
-0.6
0.36
-0.6
0.36
Mean
3.6
4.0
15.20
16.0
SSY
1.74
1.79
Std dev

42. Computing Pearson’s r (using z-scores)
Step 2: compute z-scores
X Y
2.4
5.76
2.0
4.0
1.38
-2.6
6.76
-2.0
4.0
1.4
1.96
2.0
4.0
-0.6
0.36
0.0
0.0
-0.6
0.36
-2.0
4.0
Mean
3.6
4.0
15.20
16.0
1.74
1.79
Std dev

43. Computing Pearson’s r (using z-scores)
Step 2: compute z-scores
X Y
2.4
5.76
2.0
4.0
1.38
-2.6
6.76
-2.0
4.0
-1.49
1.4
1.96
2.0
4.0
0.8
-0.6
0.36
0.0
0.0
- 0.34
-0.6
0.36
-2.0
4.0
- 0.34
Mean
3.6
4.0
15.20
16.0
0.0
Quick check
1.74
1.79
Std dev

44. Computing Pearson’s r (using z-scores)
Step 2: compute z-scores
X Y
2.4
5.76
2.0
4.0
1.38
1.1
-2.6
6.76
-2.0
4.0
-1.49
1.4
1.96
2.0
4.0
0.8
-0.6
0.36
0.0
0.0
- 0.34
-0.6
0.36
-2.0
4.0
- 0.34
Mean
3.6
4.0
15.20
16.0
1.74
1.79
Std dev

45. Computing Pearson’s r (using z-scores)
Step 2: compute z-scores
X Y
2.4
5.76
2.0
4.0
1.38
1.1
-2.6
6.76
-2.0
4.0
-1.49
-1.1
1.4
1.96
2.0
4.0
0.8
1.1
-0.6
0.36
0.0
0.0
- 0.34
0.0
-0.6
0.36
-2.0
4.0
- 0.34
-1.1
Mean
3.6
4.0
15.20
16.0
0.0
1.74
1.79
Quick check
Std dev

46. Computing Pearson’s r (using z-scores)
Step 3: compute r
X Y
2.4
5.76
2.0
4.0
1.38 * =
1.1
1.52
-2.6
6.76
-2.0
4.0
-1.49
-1.1
1.4
1.96
2.0
4.0
0.8
1.1
-0.6
0.36
0.0
0.0
- 0.34
0.0
-0.6
0.36
-2.0
4.0
- 0.34
-1.1
Mean
3.6
4.0
0.0
15.20
0.0
16.0
0.0
0.0
1.74
1.79
Std dev

47. Computing Pearson’s r (using z-scores)
Step 3: compute r
X Y
2.4
5.76
2.0
4.0
1.38
1.1
1.52
-2.6
6.76
-2.0
4.0
-1.49
-1.1
1.64
1.4
1.96
2.0
4.0
0.8
1.1
0.88
-0.6
0.36
0.0
0.0
- 0.34
0.0
0.0
-0.6
0.36
-2.0
4.0
- 0.34
-1.1
0.37
Mean
3.6
4.0
0.0
15.20
0.0
16.0
0.0
0.0
4.41
1.74
1.79
Std dev

48. Computing Pearson’s r (using z-scores)
Step 3: compute r Y X 1 2 3 4 5 6
Appears linear
Positive relationship
Fairly strong relationship
.88 is far from 0, near +1

49. A few more things to consider about correlation
Correlations are greatly affected by the range of scores in the data
Consider height and age relationship
Extreme scores can have dramatic effects on correlations
A single extreme score can radically change r
When considering "how good" a relationship is, we really should consider r2 (coefficient of determination), not just r.

50. Correlation in Research Articles
Correlation matrix
A display of the correlations between more than two variables
Why have a “-”?
Acculturation
Why only half the table filled with numbers?

51. Hypothesis testing with correlation
H0: No population Correlation
HA: There is a population correlation
(Can also have directional hypothesis)
Use table B6 in appendix B (gives critical values for r, at different sample sizes, based on t/F statistic that can be calculated)

52. Correlation in SPSS
Enter the data in two columns (one for X and one for Y), with scores for each individual in the same row.
Analyze -->Correlate-->Bivariate Move labels from the two variables into the Variables box Check Pearson box Click OK Output includes correlation (r) and p-value (significance level for the correlation)

53. Next time Predicting a variable based on other variables
Read Chapter 16