1. Bivariate Analyses

2. Bivariate Procedures I Overview
Chi-square test
T-test
Correlation

3. Chi-Square Test Relationships between nominal variables Types:
2x2 chi-square
Gender by Political Party
2x3 chi-square
Gender by Dosage (Hi vs. Med. Vs. Low)

4. Starting Point: The Crosstab Table
Example:
Gender (IV)
Males Females
Democrat
Party (DV)
Republican
Total

5. Column Percentages Gender (IV) Males Females Democrat 9% 91%
Party (DV)
Republican 91% 9%
Total 100% 100%

6. Row Percentages Gender (IV) Males Females Total Democrat 5% 95% 100%
Party (DV)
Republican 83% 17% 100%

7. Full Crosstab Table Males Females Total Democrat 1 20 21 5% 95%
5% 95%
9% 91% 64%
Republican
83% 17%
91% 9% 36%
Total
33% 67% 100%

8. Research Question and Hypothesis
Is gender related to party affiliation?
Hypothesis:
Men are more likely than women to be Republicans
Null hypothesis:
There is no relation between gender and party

9. Testing the Hypothesis
Eyeballing the table:
Seems to be a relationship
Is it significant?
Or, could it be just a chance finding?
Logic:
Is the finding different enough from the null?
Chi-square answers this question
What factors would it take into account?

10. Factors Taken into Consideration
1. Magnitude of the difference
2. Sample size
Biased coin example
Magnitude of difference:
60% heads vs. 99% heads
Sample size:
10 flips vs. 100 flips vs. 1 million flips

11. Chi-square Chi-Square starts with the frequencies:
Compare observed frequencies with frequencies we expect under the null hypothesis

12. What would the Frequencies be if there was No Relationship?
Males Females Total
Democrat
Republican
Total

13. Expected Frequencies (Null)
Males Females Total
Democrat
Republican
Total

14. Comparing the Observed and Expected Cell Frequencies
Formula:

15. Calculating the Expected Frequency
Simple formula for expected cell frequencies
Row total x column total / Total N
21 x 11 / 33 = 7
21 x 22 / 33 = 14
12 x 11 / 33 = 4
12 x 22 / 33 = 8

16. Observed and Expected Cell Frequencies
Males Females Total
Democrat
Republican
Total

17. Plugging into the Formula
O - E Square Square/E
Cell A = 1-7 = /7 = 5.1
Cell B = = /14 = 2.6
Cell C = 10-4 = /4 = 9
Cell D = 2-8 = /8 = 4.5
Sum = 21.2
Chi-square = 21.2

18. Is the chi-square significant?
Significance of the chi-square:
Great differences between observed and expected lead to bigger chi-square
How big does it have to be for significance?
Depends on the “degrees of freedom”
Formula for degrees of freedom:
(Rows – 1) x (Columns – 1)

19. Chi-square Degrees of Freedom
2 x 2 chi-square = 1
3 x 3 = ?
4 x 3 = ?

20. Chi-square Critical Valuesdf
P = 0.05
P = 0.01
P = 0.001 1
3.84
6.64
10.83 2
5.99
9.21
13.82 3
7.82
11.35
16.27 4
9.49
13.28
18.47 5
11.07
15.09
20.52 6
12.59
16.81
22.46 7
14.07
18.48
24.32 8
15.51
20.09
26.13 9
16.92
21.67
27.88 10
18.31
23.21
29.59
* If chi-square is > than critical value, relationship is significant

21. Chi-Square Computer Printout

22. Chi-Square Computer Printout

23. Multiple Chi-square Exact same procedure as 2 variable X2
Used for more than 2 variables
E.g., 2 x 2 x 2 X2
Gender x Hair color x eye color

24. Multiple chi-square example

25. Multiple chi-square example

26. The T-test Groups T-test Pairs T-test
Comparing the means of two nominal groups
E.g., Gender and IQ
E.g., Experimental vs. Control group
Pairs T-test
Comparing the means of two variables
Comparing the mean of a variable at two points in time

27. Logic of the T-test A T-test considers three things:
1. The group means
2. The dispersion of individual scores around the mean for each group (sd)
3. The size of the groups

28. Difference in the Means
The farther apart the means are:
The more confident we are that the two group means are different
Distance between the means goes in the numerator of the t-test formula

29. Why Dispersion Matters
Small variances
Large variances

30. Size of the Groups
Larger groups mean that we are more confident in the group means
IQ example:
Women: mean = 103
Men: mean = 97
If our sample was 5 men and 5 women, we are not that confident
If our sample was 5 million men and 5 million women, we are much more confident

31. The four t-test formulae
1. Matched samples with unequal variances
2. Matched samples with equal variances
3. Independent samples with unequal variances
4. Independent samples with equal variances

32. All four formulae have the same
Numerator
X1 - X2 (group one mean - group two mean)
What differentiates the four formulae is their denominator
denominator is “standard error of the difference of the means”
each formula has a different standard error

33. Independent sample with unequal variances formula
Standard error formula (denominator):

34. T-test Value Look up the T-value in a T-table (use absolute value )
First determine the degrees of freedom
ex. df = (N1 - 1) + (N2 - 1)
= 70
For 70 df at the .05 level =1.67
ex > 1.67: Reject the null
(means are different)

35. Groups t-test printout example

36. Pairs t-test example

37. Pearson Correlation Coefficient (r )
Characteristics of correlational relationships:
1. Strength
2. Significance
3. Directionality
4. Curvilinearity

38. Strength of Correlation:
Strong, weak and non-relationships
Nature of such relations can be observed in scatter diagrams
Scatter diagram
One variable on x axis and the other on the y-axis of a graph
Plot each case according to its x and y values

39. Scatterplot: Strong relationshipB O K R E A D I N G
Years of Education

40. Scatterplot: Weak relationshipM E
Years of Education

41. Scatterplot: No relationship
Years of Education

42. Strength increases…
As the points more closely conform to a straight line
Drawing the best fitting line between the points:
“the regression line”
Minimizes the distance of the points from the line:
“least squares”
Minimizing the deviations from the line

43. Significance of the relationship
Whether we are confident that an observed relationship is “real” or due to chance
What is the likelihood of getting results like this if the null hypothesis were true?
Compare observed results to expected under the null
If less than 5% chance, reject the null hypothesis

44. Directionality of the relationship
Correlational relationship can be positive or negative
Positive relationship
High scores on variable X are associated with high scores on variable Y
Negative relationship
High scores on variable X are associated with low scores on variable Y

45. Positive relationship exampleB O K R E A D I N G
Years of Education

46. Negative relationship exampleC I L P E J U D
Years of Education

47. Curvilinear relationships
Positive and negative relationships are “straight-line” or “linear” relationships
Relationships can also be strong and curvilinear too
Points conform to a curved line

48. Curvilinear relationship exampleF A M I L Y S Z E
SES

49. Curvilinear relationships
Linear statistics (e.g. correlation coefficient, regression) can mask a significant curvilinear relationship
Correlation coefficient would indicate no relationship

50. Pearson Correlation Coefficient
Numerical expression of:
Strength and Direction of straight-line relationship
Varies between –1 and 1

51. Correlation coefficient outcomes
-1 is a perfect negative relationship
-.7 is a strong negative relationship
-.4 is a moderate negative relationship
-.1 is a weak negative relationship
0 is no relationship
.1 is a weak positive relationship
.4 is a moderate positive relationship
.7 is a strong positive relationship
1 is a perfect positive relationship

52. Pearson’s r (correlation coefficient)
Used for interval or ratio variables
Reflects the extent to which cases have similar z-scores on variables X and Y
Positive relationship—z-scores have the same sign
Negative relationship—z-scores have the opposite sign

53. Positive relationship z-scores
Person Xz Yz A B C D E

54. Negative relationship z-scores
Person Xz Yz A B C D E

55. Conceptual formula for Pearson’s r
Multiply each cases z-score
Sum the products
Divide by N

56. Significance of Pearson’s r
Pearson’s r tells us the strength and direction
Significance is determined by converting the r to a t ratio and looking it up in a t table
Null: r = .00
How different is what we observe from null?
Less than .05?

57. Computer Printout