Chapter 10 Analyzing the Association Between Categorical Variables

Chapter 10 Analyzing the Association Between Categorical Variables l Learn …. How to detect and describe associations between categorical variables Agresti/Franklin Statistics, 1 of 90

Ø Section 10. 1 What Is Independence and What is Association? Agresti/Franklin Statistics, 2 of 90

Example: Is There an Association Between Happiness and Family Income? Agresti/Franklin Statistics, 3 of 90

Example: Is There an Association Between Happiness and Family Income? Agresti/Franklin Statistics, 4 of 90

Example: Is There an Association Between Happiness and Family Income? l The percentages in a particular row of a table are called conditional percentages l They form the conditional distribution for happiness, given a particular income level Agresti/Franklin Statistics, 5 of 90

Example: Is There an Association Between Happiness and Family Income? Agresti/Franklin Statistics, 6 of 90

Example: Is There an Association Between Happiness and Family Income? l 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 Agresti/Franklin Statistics, 7 of 90

Independence vs Dependence l l 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 Agresti/Franklin Statistics, 8 of 90

Example: Happiness and Gender Agresti/Franklin Statistics, 9 of 90

Example: Happiness and Gender Agresti/Franklin Statistics, 10 of 90

Example: Belief in Life After Death Agresti/Franklin Statistics, 11 of 90

Example: Belief in Life After Death l 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 Agresti/Franklin Statistics, 12 of 90

Example: Belief in Life After Death l 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 Agresti/Franklin Statistics, 13 of 90

Independence vs Dependence l l 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 Agresti/Franklin Statistics, 14 of 90

Ø Section 10. 2 How Can We Test whether Categorical Variables are Independent? Agresti/Franklin Statistics, 15 of 90

A Significance Test for Categorical Variables l The hypotheses for the test are: H 0: The two variables are independent Ha: The two variables are dependent (associated) • The test assumes random sampling and a large sample size Agresti/Franklin Statistics, 16 of 90

What Do We Expect for Cell Counts if the Variables Are Independent? l The count in any particular cell is a random variable • Different samples have different values for the count l The mean of its distribution is called an expected cell count • This is found under the presumption that H 0 is true Agresti/Franklin Statistics, 17 of 90

How Do We Find the Expected Cell Counts? l Expected Cell Count: • For a particular cell, the expected cell count equals: Agresti/Franklin Statistics, 18 of 90

Example: Happiness by Family Income Agresti/Franklin Statistics, 19 of 90

The Chi-Squared Test Statistic l 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 Agresti/Franklin Statistics, 20 of 90

Example: Happiness and Family Income Agresti/Franklin Statistics, 21 of 90

Example: Happiness and Family Income l State the null and alternative hypotheses for this test l H 0: Happiness and family income are independent l Ha: Happiness and family income are dependent (associated) Agresti/Franklin Statistics, 22 of 90

Example: Happiness and Family Income l l 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 Agresti/Franklin Statistics, 23 of 90

Example: Happiness and Family Income l 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 Agresti/Franklin Statistics, 24 of 90

The Chi-Squared Distribution l To convert the test statistic to a Pvalue, we use the sampling distribution of the statistic l For large sample sizes, this sampling distribution is well approximated by the chi-squared probability distribution Agresti/Franklin Statistics, 25 of 90

The Chi-Squared Distribution Agresti/Franklin Statistics, 26 of 90

The Chi-Squared Distribution l 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) Agresti/Franklin Statistics, 27 of 90

The Chi-Squared Distribution l 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 Agresti/Franklin Statistics, 28 of 90

The Chi-Squared Distribution Agresti/Franklin Statistics, 29 of 90

The Five Steps of the Chi. Squared Test of Independence 1. Assumptions: • Two categorical variables • Randomization • Expected counts ≥ 5 in all cells Agresti/Franklin Statistics, 30 of 90

The Five Steps of the Chi. Squared Test of Independence 2. Hypotheses: l H 0: The two variables are independent l Ha: The two variables are dependent (associated) Agresti/Franklin Statistics, 31 of 90

The Five Steps of the Chi. Squared Test of Independence 3. Test Statistic: Agresti/Franklin Statistics, 32 of 90

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 Agresti/Franklin Statistics, 33 of 90

Chi-Squared is Also Used as a “Test of Homogeneity” l The chi-squared test does not depend on which is the response variable and which is the explanatory variable l 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 Agresti/Franklin Statistics, 34 of 90

Example: Aspirin and Heart Attacks Revisited Agresti/Franklin Statistics, 35 of 90

Example: Aspirin and Heart Attacks Revisited l What are the hypotheses for the chisquared test for these data? l The null hypothesis is that whether a doctor has a heart attack is independent of whether he takes placebo or aspirin l The alternative hypothesis is that there’s an association Agresti/Franklin Statistics, 36 of 90

Example: Aspirin and Heart Attacks Revisited l 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 l This is very strong evidence that the population proportion of heart attacks differed for those taking aspirin and for those taking placebo Agresti/Franklin Statistics, 37 of 90

Example: Aspirin and Heart Attacks Revisited l The sample proportions indicate that the aspirin group had a lower rate of heart attacks than the placebo group Agresti/Franklin Statistics, 38 of 90

Limitations of the Chi-Squared Test If the P-value is very small, strong evidence exists against the null hypothesis of independence But… l The chi-squared statistic and the Pvalue tell us nothing about the nature of the strength of the association l Agresti/Franklin Statistics, 39 of 90

Limitations of the Chi-Squared Test l We know that there is statistical significance, but the test alone does not indicate whethere is practical significance as well Agresti/Franklin Statistics, 40 of 90