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Chapter 10 Analyzing the Association Between Categorical Variables

Chapter 10 Analyzing the Association Between Categorical Variables

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Chapter 10 Analyzing the Association Between Categorical Variables

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  1. Chapter 10Analyzing the Association Between Categorical Variables • Learn …. How to detect and describe associations between categorical variables

  2. Section 10.1 What Is Independence and What is Association?

  3. Example: Is There an Association Between Happiness and Family Income?

  4. Example: Is There an Association Between Happiness and Family Income?

  5. Example: Is There an Association Between Happiness and Family Income? • The percentages in a particular row of a table are calledconditional percentages • They form theconditional distributionfor happiness, given a particular income level

  6. Example: Is There an Association Between Happiness and Family Income?

  7. Example: Is There an Association Between Happiness and Family Income? • 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

  8. Independence vs Dependence • For two variables to beindependent, the population percentage in any category of one variable is the same for all categories of the other variable • For two variables to bedependent (or associated),the population percentages in the categories are not all the same

  9. Example: Happiness and Gender

  10. Example: Happiness and Gender

  11. Example: Belief in Life After Death

  12. Example: Belief in Life After Death • 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

  13. Example: Belief in Life After Death • 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

  14. Independence vs Dependence • Even if variables are independent, we would not expect the sample conditional distributions to be identical • Because of sampling variability, eachsamplepercentage typically differs somewhat from thetrue populationpercentage

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

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

  17. 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

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

  19. Example: Happiness by Family Income

  20. 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

  21. Example: Happiness and Family Income

  22. 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)

  23. 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

  24. 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

  25. 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

  26. The Chi-Squared Distribution

  27. 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)

  28. 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

  29. The Chi-Squared Distribution

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

  31. 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)

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

  33. The Five Steps of the Chi-Squared Test of Independence 4. P-value: Right-tail probability above the observedvalue, 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

  34. 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

  35. Example: Aspirin and Heart Attacks Revisited

  36. 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

  37. Example: Aspirin and Heart Attacks Revisited • Report the test statistic and P-value for the chi-squared test: • The test statistic is 25.01 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

  38. 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

  39. 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

  40. 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

  41. Section 10.3 How Strong is the Association?

  42. In a study of the two variables (Gender and Happiness), which one is the response variable? • Gender • Happiness

  43. What is the Expected Cell Count for ‘Females’ who are ‘Pretty Happy’? • 898 • 801.5 • 902 • 521

  44. Calculate the • 1.75 • 0.27 • 0.98 • 10.34

  45. 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’

  46. 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

  47. 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

  48. 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

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

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