1 Chapter 11: Analyzing the Association Between Categorical Variables Section 11.1: What is Independence and What is Association?

2 Learning Objectives 1. Comparing Percentages 2. Independence vs. Dependence

3 Learning Objective 1: Example: Is There an Association Between Happiness and Family Income?

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 Learning Objective 1: Example: Is There an Association Between Happiness and Family Income?

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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 Learning Objective 1: Example: Is There an Association Between Happiness and Family Income?

7 Learning Objective 2: 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 Learning Objective 2: Independence vs. Dependence 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

9 Learning Objective 2: Independence vs. Dependence 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

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 Learning Objective 2: Independence vs. Dependence

11 Chapter 11: Analyzing the Association Between Categorical Variables Section 11.2: How Can We Test Whether Categorical Variables Are Independent?

12 Learning Objectives 1. A Significance Test for Categorical Variables 2. What Do We Expect for Cell Counts if the Variables Are Independent? 3. How Do We Find the Expected Cell Counts? 4. The Chi-Squared Test Statistic 5. The Chi-Squared Distribution 6. The Five Steps of the Chi-Squared Test of Independence

13 Learning Objectives 7. Chi-Squared is Also Used as a “Test of Homogeneity” 8. Chi-Squared and the Test Comparing Proportions in 2x2 Tables 9. Limitations of the Chi-Squared Test

14 Learning Objective 1: A Significance Test for Categorical Variables Create a table of frequencies divided into the categories of the two variables The hypotheses for the test are: H 0 : The two variables are independent H a : 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)

15 Learning Objective 2: 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 count values The mean of its distribution is called an expected cell count This is found under the presumption that H 0 is true

16 Learning Objective 3: 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).

17 Learning Objective 3: How Do We Find the Expected Cell Counts? Example

18 Learning Objective 4: 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

19 State the null and alternative hypotheses for this test H 0 : Happiness and family income are independent H a : Happiness and family income are dependent (associated) Learning Objective 4: Example: Happiness and Family Income

20 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 Learning Objective 4: Example: Happiness and Family Income

21 Learning Objective 4: Example: Happiness and Family Income

22 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 Learning Objective 4: The Chi-Squared Test Statistic

23 Learning Objective 5: 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

24 Learning Objective 5: The Chi-Squared Distribution

25 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) Learning Objective 5: The Chi-Squared Distribution

26 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 H 0 : independence Learning Objective 5: The Chi-Squared Distribution

27 Learning Objective 5: The Chi-Squared Distribution

28 Learning Objective 6: The Five Steps of the Chi-Squared Test of Independence 1. Assumptions: Two categorical variables Randomization Expected counts ≥ 5 in all cells

29 Learning Objective 6: The Five Steps of the Chi-Squared Test of Independence 2. Hypotheses: H 0 : The two variables are independent H a : The two variables are dependent (associated)

30 Learning Objective 6: The Five Steps of the Chi-Squared Test of Independence 3. Test Statistic:

31 Learning Objective 6: 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 H 0 when P-value ≤ significance level

32 Learning Objective 7: 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

33 Learning Objective 8: 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 p 1 in group 1 and p 2 in group 2 If the response variable is independent of the group, p 1 =p 2, so the conditional distributions are equal H 0 : p 1 =p 2 is equivalent to H 0 : independence

34 Learning Objective 8: Example: Aspirin and Heart Attacks Revisited

35 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 Learning Objective 8: Example: Aspirin and Heart Attacks Revisited

36 Report the test statistic and P-value for the chi- squared test: The test statistic is with a P-value of This is very strong evidence that the population proportion of heart attacks differed for those taking aspirin and for those taking placebo Learning Objective 8: Example: Aspirin and Heart Attacks Revisited

37 The sample proportions indicate that the aspirin group had a lower rate of heart attacks than the placebo group Learning Objective 8: Example: Aspirin and Heart Attacks Revisited

38 Learning Objective 9: 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

39 Learning Objective 9: 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

40 Learning Objective 9: Limitations of the Chi-Squared Test 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

41 Learning Objective 10: “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.

42 Chapter 11: Analyzing the Association Between Categorical Variables Section 11.3: How Strong is the Association?

43 Learning Objectives 1. Analyzing Contingency Tables 2. Measures of Association 3. Difference of Proportions 4. The Ratio of Proportions: Relative Risk 5. Properties of the Relative Risk 6. Large Chi-square Does Not Mean There’s a Strong Association

44 Learning Objective 1: 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

45 Learning Objective 1: 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

46 Learning Objective 1: 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

47 Learning Objective 2: 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

48 Learning Objective 3: Difference of Proportions An easily interpretable measure of association is the difference between the proportions making a particular response Case (a) 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

49 Learning Objective 3: 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 larger the absolute value of the difference of proportions

50 Learning Objective 3: Difference of Proportions Example: Do Student Stress and Depression Depend on 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

51 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) Learning Objective 3: Difference of Proportions Example: Do Student Stress and Depression Depend on Gender?

52 Learning Objective 4: The Ratio of Proportions: Relative Risk Another measure of association, is the ratio of two proportions: p 1 /p 2 In medical applications in which the proportion refers to an adverse outcome, it is called the relative risk

53 Learning Objective 4: 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

54 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 = For those who did not wear a seat belt, the proportion who died equaled 1601/164,128 = Learning Objective 4: Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents

55 The relative risk is the ratio: / = The proportion of subjects wearing a seat belt who died was times the proportion of subjects not wearing a seat belt who died Learning Objective 4: Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents

56 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 Learning Objective 4: Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents

57 Reversing the order of the rows, we calculate the ratio: / = 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 Learning Objective 4: Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents

58 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 Learning Objective 4: Example: Relative Risk for Seat Belt Use and Outcome of Auto Accidents

59 Learning Objective 5: Properties of the Relative Risk The relative risk can equal any nonnegative number When p 1 = p 2, the variables are independent and relative risk = 1.0 Values farther from 1.0 (in either direction) represent stronger associations

60 Learning Objective 6: 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

61 Chapter 11: Analyzing the Association Between Categorical Variables Section 11.4: How Can Residuals Reveal The Pattern of Association?

62 Learning Objectives 1. Association Between Categorical Variables 2. Residual Analysis

63 Learning Objective 1: Association Between Categorical Variables The chi-squared test and measures of association such as (p 1 – p 2 ) and p 1 /p 2 are fundamental methods for analyzing contingency tables The P-value for summarized the strength of evidence against H 0 : independence

64 Learning Objective 1: 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

65 Learning Objective 2: Residual Analysis A cell-by-cell comparison of the observed counts with the counts that are expected when H 0 is true reveals the nature of the evidence against H 0 The difference between an observed and expected count in a particular cell is called a residual

66 Learning Objective 2: Residual Analysis The residual is negative when fewer subjects are in the cell than expected under H 0 The residual is positive when more subjects are in the cell than expected under H 0

67 Learning Objective 2: 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

68 Learning Objective 2: Residual Analysis 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

69 Learning Objective 2: Example: Standardized Residuals for Religiosity and Gender “To what extent do you consider yourself a religious person?”

70 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 Learning Objective 2: Example: Standardized Residuals for Religiosity and Gender

71 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 Learning Objective 2: Example: Standardized Residuals for Religiosity and Gender

72 Chapter 11: Analyzing the Association Between Categorical Variables Section 11.5: What if the Sample Size is Small? Fisher’s Exact Test

73 Learning Objectives 1. Fisher’s Exact Test 2. Example using Fisher’s Exact Test 3. Summary of Fisher’s Exact Test of Independence for 2x2 Tables

74 Learning Objective 1: 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

75 Learning Objective 1: 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 the evidence that the variables are associated

76 Learning Objective 2: Fisher’s Exact Test 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

77 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 Learning Objective 2: Fisher’s Exact Test Example: Tea Tastes Better with Milk Poured First?

78 Results: Learning Objective 2: Fisher’s Exact Test Example: Tea Tastes Better with Milk Poured First?

79 Analysis: Learning Objective 2: Fisher’s Exact Test Example: Tea Tastes Better with Milk Poured First?

80 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? Learning Objective 2: Fisher’s Exact Test Example: Tea Tastes Better with Milk Poured First?

81 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 Learning Objective 2: Fisher’s Exact Test Example: Tea Tastes Better with Milk Poured First?

82 Learning Objective 3: Summary of Fisher’s Exact Test of Independence for 2x2 Tables Assumptions: Two binary categorical variables Data are random Hypotheses: H 0 : the two variables are independent (p 1 =p 2 ) H a : the two variables are associated (p 1 ≠p 2 or p 1 >p 2 or p 1 <p 2 )

83 Learning Objective 3: Summary of Fisher’s Exact Test of Independence for 2x2 Tables Test Statistic: First cell count (this determines the others given the margin totals) P-value: Probability that the first cell count equals the observed value or a value even more extreme as predicted by H a Conclusion: Report the P-value and interpret in context. If a decision is required, reject H 0 when P-value ≤ significance level