2. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 11 Analyzing the Association Between Categorical Variables Section 11.1 Independence and Dependence (Association)

3. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 3 Example: Happiness Table 11.2 Conditional Percentages for Happiness Categories, Given Each Category of Family Income.

4. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 4 The percentages in a particular row of a table are called conditional percentages. They form the conditional distribution for happiness, given a particular income level. Table 11.2 Conditional Percentages for Happiness Categories, Given Each Category of Family Income. Example: Happiness

5. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 5 Figure 11.1 MINITAB Bar Graph of Conditional Distributions of Happiness, Given Income, from Table 11.2. For each happiness category, you can compare the three categories of income on the percent in that happiness category. Question: Based on this graph, how would you say that happiness depends on income? Example: Happiness

6. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 6 Guidelines when constructing tables with conditional distributions:  Make the response variable the column variable.  Compute conditional proportions for the response variable within each row.  Include the total sample sizes. Example: Happiness

7. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 7 Independence vs. Dependence For two variables to be independent, the population percentage in any category of one variable is the same for all categories of the other variable. For two variables to be dependent (or associated), the population percentages in the categories are not all the same.

8. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 8 Are race and belief in life after death independent or dependent?  The conditional distributions in the table are similar but not exactly identical.  It is tempting to conclude that the variables are dependent. Table 11.4 Sample Conditional Distributions for Belief in Life After Death, Given Race Independence vs. Dependence

9. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 9 Are race and belief in life after death independent or dependent?  The definition of independence between variables refers to a population.  The table is a sample, not a population. Independence vs. Dependence

10. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 10 Even if variables are independent, we would not expect the sample conditional distributions to be identical. Because of sampling variability, each sample percentage typically differs somewhat from the true population percentage. Independence vs. Dependence

11. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 11 Analyzing the Association Between Categorical Variables Section 11.2 Testing Categorical Variables for Independence

12. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 12 Testing Categorical Variables for Independence Create a table of frequencies divided into the categories of the two variables:  The hypotheses for the test are: : The two variables are independent. : The two variables are dependent (associated). The test assumes random sampling and a large sample size (cell counts in the frequency table of at least 5).

13. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 13 Expected Cell Counts If the Variables Are Independent The count in any particular cell is a random variable.  Different samples have different count values. The mean of its distribution is called an expected cell count.  This is found under the presumption that is true.

14. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 14 How Do We Find the Expected Cell Counts? Expected Cell Count: For a particular cell, The expected frequencies are values that have the same row and column totals as the observed counts, but for which the conditional distributions are identical (this is the assumption of the null hypothesis).

15. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 15 Table 11.5 Happiness by Family Income, Showing Observed and Expected Cell Counts. We use the highlighted totals to get the expected count of 66.86 = (315 * 423)/1993 in the first cell. How Do We Find the Expected Cell Counts?

16. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 16 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.

17. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 17 State the null and alternative hypotheses for this test.  : Happiness and family income are independent  : Happiness and family income are dependent (associated) Example: Happiness and Family Income

18. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 18 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 106.955. Example: Happiness and Family Income

19. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 19 Example: Happiness and Family Income

20. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 20 Insight: 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. The Chi-Squared Test Statistic

21. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 21 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.

22. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 22 Figure 11.3 The Chi-Squared Distribution. The curve has larger mean and standard deviation as the degrees of freedom increase. Question: Why can’t the chi-squared statistic be negative? The Chi-Squared Distribution

23. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 23 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: The Chi-Squared Distribution

24. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 24 Main properties of the chi-squared distribution (cont’d):  The mean of the distribution equals the df value.  It is skewed to the right.  The larger the value, the greater the evidence against : independence. The Chi-Squared Distribution

25. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 25 Table 11.7 Rows of Table C Displaying Chi-Squared Values. The values have right-tail probabilities between 0.250 and 0.001. For a table with r = 3 rows and c = 3 columns, df = (r - 1) x (c - 1) = 4, and 9.49 is the chi-squared value with a right-tail probability of 0.05. The Chi-Squared Distribution

26. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 26 The Five Steps of the Chi-Squared Test of Independence 1.Assumptions:  Two categorical variables  Randomization  Expected counts in all cells

27. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 27 2. Hypotheses:  The two variables are independent  The two variables are dependent (associated) The Five Steps of the Chi-Squared Test of Independence

28. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 28 3. Test Statistic: The Five Steps of the Chi-Squared Test of Independence

29. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 29 4. P-value:  Right-tail probability above the observed value, for the chi-squared distribution with. 5. Conclusion:  Report P-value and interpret in context. If a decision is needed, reject when P-value significance level. The Five Steps of the Chi-Squared Test of Independence

30. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 30 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.

31. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 31 Chi-Squared and the Test Comparing Proportions in 2x2 Tables In practice, contingency tables of size 2x2 are very common. They often occur in summarizing the responses of two groups on a binary response variable.  Denote the population proportion of success by in group 1 and in group 2.  If the response variable is independent of the group,, so the conditional distributions are equal.  is equivalent to independence

32. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 32 Example: Aspirin and Heart Attacks Revisited Table 11.9 Annotated MINITAB Output for Chi-Squared Test of Independence of Group (Placebo, Aspirin) and Whether or Not Subject Died of Cancer. The same P-value results as with a two-sided Z test comparing the two population proportions.

33. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 33 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. Example: Aspirin and Heart Attacks Revisited

34. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 34 Report the test statistic and P-value for the chi- squared test:  The test statistic is 11.35 with a P-value of 0.001. This is very strong evidence that the population proportion of heart attacks differed for those taking aspirin and for those taking placebo. The sample proportions indicate that the aspirin group had a lower rate of heart attacks than the placebo group. Example: Aspirin and Heart Attacks Revisited

35. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 35 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. We know that there is statistical significance, but the test alone does not indicate whether there is practical significance as well.

36. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 36 The chi-squared test is often misused. Some examples are:  When some of the expected frequencies are too small.  When separate rows or columns are dependent samples.  Data are not random.  Quantitative data are classified into categories - results in loss of information. Limitations of the Chi-Squared Test

37. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 37 “Goodness of Fit” Chi-Squared Tests The Chi-Squared test can also be used for testing particular proportion values for a categorical variable.  The null hypothesis is that the distribution of the variable follows a given probability distribution; the alternative is that it does not.  The test statistic is calculated in the same manner where the expected counts are what would be expected in a random sample from the hypothesized probability distribution.  For this particular case, the test statistic is referred to as a goodness-of-fit statistic.

38. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 11 Analyzing the Association Between Categorical Variables Section 11.3 Determining the Strength of the Association

39. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 39 Analyzing Contingency Tables 1.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.

40. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 40 2.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. Analyzing Contingency Tables

41. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 41 3.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. Analyzing Contingency Tables

42. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 42 Measures of Association A measure of association is a statistic or a parameter that summarizes the strength of the dependence between two variables.  A measure of association is useful for comparing associations to determine which is stronger.

43. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 43 Difference of Proportions An easily interpretable measure of association is the difference between the proportions making a particular response. Table 11.12 Decision About Accepting a Credit Card Cross-Tabulated by Income.

44. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 44 Case (a) in Table 11.12 exhibits the weakest possible association – no association. The difference of proportions is 0. Case (b) exhibits the strongest possible association: The difference of proportions is 1. Difference of Proportions

45. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 45 In practice, we don’t expect data to follow either extreme (0% difference or 100% difference), but the stronger the association, the larger the absolute value of the difference of proportions. Difference of Proportions

46. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 46 Example: Student Stress, Depression and Gender  Which response variable, stress or depression, has the stronger sample association with gender?  The difference of proportions between females and males was 0.35 – 0.16 = 0.19 for feeling stressed.  The difference of proportions between females and males was 0.08 – 0.06 = 0.02 for feeling depressed. Table 11.13 Conditional Distributions of Stress and Depression, by Gender

47. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 47 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). Example: Student Stress, Depression and Gender

48. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 48 The Ratio of Proportions: Relative Risk Another measure of association, is the ratio of two proportions:. In medical applications in which the proportion refers to an adverse outcome, it is called the relative risk.

49. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 49 Example: Seat Belt Use and Outcome of Auto Accidents Treating the auto accident outcome as the response variable, find and interpret the relative risk. Table 11.14 Outcome of Auto Accident by Whether or Not Subject Wore Seat Belt.

50. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 50  The adverse outcome is death.  The relative risk is formed for that outcome.  For those who wore a seat belt, the proportion who died equaled 510/412,878 = 0.00124.  For those who did not wear a seat belt, the proportion who died equaled 1601/164,128 = 0.00975. Example: Seat Belt Use and Outcome of Auto Accidents

51. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 51 The relative risk is the ratio:  0.00124/0.00975 = 0.127  The proportion of subjects wearing a seat belt who died was 0.127 times the proportion of subjects not wearing a seat belt who died. Example: Seat Belt Use and Outcome of Auto Accidents

52. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 52 Many find it easier to interpret the relative risk but reordering the rows of data so that the relative risk has value above 1.0. Reversing the order of the rows, we calculate the ratio:  0.00975/0.00124 = 7.9  The proportion of subjects not wearing a seat belt who died was 7.9 times the proportion of subjects wearing a seat belt who died. Example: Seat Belt Use and Outcome of Auto Accidents

53. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 53 A relative risk of 7.9 represents a strong association  This is far from the value of 1.0 that would occur if the proportion of deaths were the same for each group.  Wearing a set belt has a practically significant effect in enhancing the chance of surviving an auto accident. Example: Seat Belt Use and Outcome of Auto Accidents

54. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 54 Properties of the Relative Risk  The relative risk can equal any nonnegative number.  When, the variables are independent and relative risk = 1.0.  Values farther from 1.0 (in either direction) represent stronger associations.

55. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 55 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.  For a given strength of association, larger values occur for larger sample sizes. Large values can occur with weak associations, if the sample size is sufficiently large.

56. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 11 Analyzing the Association Between Categorical Variables Section 11.4 Using Residuals to Reveal the Pattern of Association

57. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 57  The chi-squared test and measures of association such as and are fundamental methods for analyzing contingency tables.  The P-value for summarized the strength of evidence against independence. Association Between Categorical Variables

58. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 58  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. Association Between Categorical Variables

59. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 59 Residual Analysis  A cell-by-cell comparison of the observed counts with the counts that are expected when is true reveals the nature of the evidence against.  The difference between an observed and expected count in a particular cell is called a residual.

60. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 60  The residual is negative when fewer subjects are in the cell than expected under.  The residual is positive when more subjects are in the cell than expected under. Residual Analysis

61. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 61 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. Residual Analysis

62. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 62  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.  The se describes how much the difference would tend to vary in repeated sampling if the variables were independent. Its formula is complex, software can be used to find its value.  A large standardized residual value provides evidence against independence in that cell. Residual Analysis

63. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 63 Example: Religiosity and Gender “To what extent do you consider yourself a religious person?” Table 11.17 Religiosity by Gender, With Expected Counts and Standardized Residuals.

64. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 64 Interpret the standardized residuals in the table.  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. Example: Religiosity and Gender

65. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 65  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. Example: Religiosity and Gender

66. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 11 Analyzing the Association Between Categorical Variables Section 11.5 Small Sample Sizes: Fisher’s Exact Test

67. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 67 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.

68. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 68  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 the evidence that the variables are associated. Fisher’s Exact Test

69. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 69 Example: A Tea-Tasting Experiment 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.

70. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 70 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. Example: A Tea-Tasting Experiment

71. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 71 Results: Table 11.18 Result of Tea-Tasting Experiment. The table cross-tabulates what was actually poured first (milk or tea) by what Dr. Bristol predicted was poured first. She had to indicate which four of the eight cups had the milk poured first. Example: A Tea-Tasting Experiment

72. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 72 Analysis: Table 11.19 Result of Fisher’s Exact Test for Tea-Tasting Experiment. The chi-squared P- value is listed under Asymp. Sig. and the Fisher’s exact test P-values are listed under Exact Sig. “Sig” is short for significance and “asymp.” is short for asymptotic. Example: A Tea-Tasting Experiment

73. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 73 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? Example: A Tea-Tasting Experiment

74. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 74 The P-value of 0.243 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. If she had predicted all four cups correctly, the one-sided P-value would have been 0.014. We might then believe her claim. Example: A Tea-Tasting Experiment

75. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 75 Summary of Fisher’s Exact Test of Independence for 2x2 Tables 1.Assumptions:  Two binary categorical variables  Data are random 2.Hypotheses:  the two variables are independent  the two variables are associated

76. Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 76 3.Test Statistic:  First cell count (this determines the others given the margin totals). 4.P-value:  Probability that the first cell count equals the observed value or a value even more extreme as predicted by. 5.Conclusion:  Report the P-value and interpret in context. If a decision is required, reject when P-value significance level. Summary of Fisher’s Exact Test of Independence for 2x2 Tables