1. How Can We Test whether Categorical Variables are Independent?
Section 10.2
How Can We Test whether Categorical Variables are Independent?

2. A Significance Test for Categorical Variables
The hypotheses for the test are:
H0: The two variables are independent
Ha: The two variables are dependent (associated)
The test assumes random sampling and a large sample size

3. What Do We Expect for Cell Counts if the Variables Are Independent?
The count in any particular cell is a random variable
Different samples have different values for the count
The mean of its distribution is called an expected cell count
This is found under the presumption that H0 is true

4. How Do We Find the Expected Cell Counts?
For a particular cell, the expected cell count equals:

5. Example: Happiness by Family Income

6. The Chi-Squared Test Statistic
The chi-squared statistic summarizes how far the observed cell counts in a contingency table fall from the expected cell counts for a null hypothesis

7. Example: Happiness and Family Income

8. Example: Happiness and Family Income
State the null and alternative hypotheses for this test
H0: Happiness and family income are independent
Ha: Happiness and family income are dependent (associated)

9. Example: Happiness and Family Income
Report the statistic and explain how it was calculated:
To calculate the statistic, for each cell, calculate:
Sum the values for all the cells
The value is 73.4

10. Example: Happiness and Family Income
The larger the value, the greater the evidence against the null hypothesis of independence and in support of the alternative hypothesis that happiness and income are associated

11. The Chi-Squared Distribution
To convert the test statistic to a P-value, we use the sampling distribution of the statistic
For large sample sizes, this sampling distribution is well approximated by the chi-squared probability distribution

12. The Chi-Squared Distribution

13. The Chi-Squared Distribution
Main properties of the chi-squared distribution:
It falls on the positive part of the real number line
The precise shape of the distribution depends on the degrees of freedom:
df = (r-1)(c-1)

14. The Chi-Squared Distribution
Main properties of the chi-squared distribution:
The mean of the distribution equals the df value
It is skewed to the right
The larger the value, the greater the evidence against H0: independence

15. The Chi-Squared Distribution

16. The Five Steps of the Chi-Squared Test of Independence
1. Assumptions:
Two categorical variables
Randomization
Expected counts ≥ 5 in all cells

17. The Five Steps of the Chi-Squared Test of Independence
2. Hypotheses:
H0: The two variables are independent
Ha: The two variables are dependent (associated)

18. The Five Steps of the Chi-Squared Test of Independence
3. Test Statistic:

19. The Five Steps of the Chi-Squared Test of Independence
4. P-value: Right-tail probability above the observed value, for the chi-squared distribution with df = (r-1)(c-1)
5. Conclusion: Report P-value and interpret in context
If a decision is needed, reject H0 when P-value ≤ significance level

20. Chi-Squared is Also Used as a “Test of Homogeneity”
The chi-squared test does not depend on which is the response variable and which is the explanatory variable
When a response variable is identified and the population conditional distributions are identical, they are said to be homogeneous
The test is then referred to as a test of homogeneity

21. Example: Aspirin and Heart Attacks Revisited

22. Example: Aspirin and Heart Attacks Revisited
What are the hypotheses for the chi-squared test for these data?
The null hypothesis is that whether a doctor has a heart attack is independent of whether he takes placebo or aspirin
The alternative hypothesis is that there’s an association

23. Example: Aspirin and Heart Attacks Revisited
Report the test statistic and P-value for the chi-squared test:
The test statistic is with a P-value of 0.000
This is very strong evidence that the population proportion of heart attacks differed for those taking aspirin and for those taking placebo

24. Example: Aspirin and Heart Attacks Revisited
The sample proportions indicate that the aspirin group had a lower rate of heart attacks than the placebo group

25. Limitations of the Chi-Squared Test
If the P-value is very small, strong evidence exists against the null hypothesis of independence
But…
The chi-squared statistic and the P-value tell us nothing about the nature of the strength of the association

26. Limitations of the Chi-Squared Test
We know that there is statistical significance, but the test alone does not indicate whether there is practical significance as well

27. How Strong is the Association?
Section 10.3
How Strong is the Association?

28. The following is a table on Gender and Happiness:
Not
Pretty
Very
Females
163
898
502
Males
130
705
379
In a study of the two variables (Gender and Happiness), which one is the response variable?
Gender
Happiness

29. The following is a table on Gender and Happiness:
Not
Pretty
Very
Females
163
898
502
Males
130
705
379
What is the Expected Cell Count for ‘Females’ who are ‘Pretty Happy’?
898
801.5
902
521

30. The following is a table on Gender and Happiness:
Not
Pretty
Very
Females
163
898
502
Males
130
705
379
What is the Expected Cell Count for ‘Females’ who are ‘Pretty Happy’?
898
801.5
902 = N*( )/N*( )/N
521

31. The following is a table on Gender and Happiness:
Not
Pretty
Very
Females
163
898
502
Males
130
705
379
Calculate the
1.75
0.27
0.98
10.34

32. The following is a table on Gender and Happiness:
Not
Pretty
Very
Females
163
898
502
Males
130
705
379
At a significance level of 0.05, what is the correct decision?
‘Gender’ and ‘Happiness’ are independent
There is an association between ‘Gender’ and ‘Happiness’

33. Analyzing Contingency Tables
Is there an association?
The chi-squared test of independence addresses this
When the P-value is small, we infer that the variables are associated

34. Analyzing Contingency Tables
How do the cell counts differ from what independence predicts?
To answer this question, we compare each observed cell count to the corresponding expected cell count

35. Analyzing Contingency Tables
How strong is the association?
Analyzing the strength of the association reveals whether the association is an important one, or if it is statistically significant but weak and unimportant in practical terms

36. Measures of Association
A measure of association is a statistic or a parameter that summarizes the strength of the dependence between two variables

37. Difference of Proportions
An easily interpretable measure of association is the difference between the proportions making a particular response

38. Difference of Proportions

39. Difference of Proportions
Case (a) exhibits the weakest possible association – no association
Accept Credit Card
The difference of proportions is 0
Income No
Yes
High
60%
40%
Low

40. Difference of Proportions
Case (b) exhibits the strongest possible association:
Accept Credit Card
The difference of proportions is 100%
Income No
Yes
High 0%
100%
Low

41. Difference of Proportions
In practice, we don’t expect data to follow either extreme (0% difference or 100% difference), but the stronger the association, the large the absolute value of the difference of proportions

42. Example: Do Student Stress and Depression Depend on Gender?

43. Example: Do Student Stress and Depression Depend on Gender?
Which response variable, stress or depression, has the stronger sample association with gender?

44. Example: Do Student Stress and Depression Depend on Gender?
The difference of proportions between females and males was 0.35 – 0.16 = 0.19
Gender
Yes No
Female
35%
65%
Male
16%
84%

45. Example: Do Student Stress and Depression Depend on Gender?
The difference of proportions between females and males was 0.08 – 0.06 = 0.02
Gender
Yes No
Female 8%
92%
Male 6%
94%

46. Example: Do Student Stress and Depression Depend on Gender?
In the sample, stress (with a difference of proportions = 0.19) has a stronger association with gender than depression has (with a difference of proportions = 0.02)

47. Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents

48. Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents
Treating the auto accident outcome as the response variable, find and interpret the relative risk

49. Large Does Not Mean There’s a Strong Association
A large chi-squared value provides strong evidence that the variables are associated
It does not imply that the variables have a strong association
This statistic merely indicates (through its P-value) how certain we can be that the variables are associated, not how strong that association is

50. How Can Residuals Reveal the Pattern of Association?
Section 10.4
How Can Residuals Reveal the Pattern of Association?

51. Association Between Categorical Variables
The chi-squared test and measures of association such as (p1 – p2) and p1/p2 are fundamental methods for analyzing contingency tables
The P-value for summarized the strength of evidence against H0: independence

52. Association Between Categorical Variables
If the P-value is small, then we conclude that somewhere in the contingency table the population cell proportions differ from independence
The chi-squared test does not indicate whether all cells deviate greatly from independence or perhaps only some of them do so

53. Residual Analysis
A cell-by-cell comparison of the observed counts with the counts that are expected when H0 is true reveals the nature of the evidence against H0
The difference between an observed and expected count in a particular cell is called a residual

54. Residual Analysis
The residual is negative when fewer subjects are in the cell than expected under H0
The residual is positive when more subjects are in the cell than expected under H0

55. Residual Analysis
To determine whether a residual is large enough to indicate strong evidence of a deviation from independence in that cell we use a adjusted form of the residual: the standardized residual

56. Residual Analysis (observed count – expected count)/se
The standardized residual for a cell:
(observed count – expected count)/se
A standardized residual reports the number of standard errors that an observed count falls from its expected count
Its formula is complex
Software can be used to find its value
A large value provides evidence against independence in that cell

57. Example: Standardized Residuals for Religiosity and Gender
“To what extent do you consider yourself a religious person?”

58. Example: Standardized Residuals for Religiosity and Gender

59. Example: Standardized Residuals for Religiosity and Gender
Interpret the standardized residuals in the table

60. Example: Standardized Residuals for Religiosity and Gender
The table exhibits large positive residuals for the cells for females who are very religious and for males who are not at all religious.
In these cells, the observed count is much larger than the expected count
There is strong evidence that the population has more subjects in these cells than if the variables were independent

61. Example: Standardized Residuals for Religiosity and Gender
The table exhibits large negative residuals for the cells for females who are not at all religious and for males who are very religious
In these cells, the observed count is much smaller than the expected count
There is strong evidence that the population has fewer subjects in these cells than if the variables were independent

62. What if the Sample Size is Small? Fisher’s Exact Test
Section 10.5
What if the Sample Size is Small? Fisher’s Exact Test

63. Fisher’s Exact Test
The chi-squared test of independence is a large-sample test
When the expected frequencies are small, any of them being less than about 5, small-sample tests are more appropriate
Fisher’s exact test is a small-sample test of independence

64. Fisher’s Exact Test
The calculations for Fisher’s exact test are complex
Statistical software can be used to obtain the P-value for the test that the two variables are independent
The smaller the P-value, the stronger is the evidence that the variables are associated

65. Example: Tea Tastes Better with Milk Poured First?
This is an experiment conducted by Sir Ronald Fisher
His colleague, Dr. Muriel Bristol, claimed that when drinking tea she could tell whether the milk or the tea had been added to the cup first

66. Example: Tea Tastes Better with Milk Poured First?
Experiment:
Fisher asked her to taste eight cups of tea:
Four had the milk added first
Four had the tea added first
She was asked to indicate which four had the milk added first
The order of presenting the cups was randomized

67. Example: Tea Tastes Better with Milk Poured First?
Results:

68. Example: Tea Tastes Better with Milk Poured First?
Analysis:

69. Example: Tea Tastes Better with Milk Poured First?
The one-sided version of the test pertains to the alternative that her predictions are better than random guessing
Does the P-value suggest that she had the ability to predict better than random guessing?

70. Example: Tea Tastes Better with Milk Poured First?
The P-value of does not give much evidence against the null hypothesis
The data did not support Dr. Bristol’s claim that she could tell whether the milk or the tea had been added to the cup first