Chapter 13: Categorical Data Analysis Statistics

McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 2 Where We’ve Been Presented methods for making inferences about the population proportion associated with a two-level qualitative variable (i.e., a binomial variable) Presented methods for making inferences about the difference between two binomial proportions

Where We’re Going McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 3 Discuss qualitative (categorical) data with more than two outcomes Present a chi-square hypothesis test for comparing the category proportions associated with a single qualitative variable – called a one-way analysis Present a chi-square hypothesis test relating two qualitative variables – called a two-way analysis

13.1: Categorical Data and the Multinomial Experiment Properties of the Multinomial Experiment 1. The experiment consists of n identical trials. 2. There are k possible outcomes (called classes, categories or cells) to each trial. 3. The probabilities of the k outcomes, denoted by p 1, p 2, …, p k, where p 1 + p 2 + … + p k = 1, remain the same from trial to trial. 4. The trials are independent. 5. The random variables of interest are the cell counts n 1, n 2, …, n k of the number of observations that fall into each of the k categories. McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 4

13.2: Testing Categorical Probabilities: One-Way Table Suppose three candidates are running for office, and 150 voters are asked their preferences.  Candidate 1 is the choice of 61 voters.  Candidate 2 is the choice of 53 voters.  Candidate 3 is the choice of 36 voters. Do these data suggest the population may prefer one candidate over the others? McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 5

13.2: Testing Categorical Probabilities: One-Way Table Candidate 1 is the choice of 61 voters. Candidate 2 is the choice of 53 voters. Candidate 3 is the choice of 36 voters. n =150 McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 6

13.2: Testing Categorical Probabilities: One-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 7 Reject the null hypothesis

Test of a Hypothesis about Multinomial Probabilities: One-Way Table H 0 : p 1 = p 1,0, p 2 = p 2,0, …, p k = p k,0 where p 1,0, p 2,0, …, p k,0 represent the hypothesized values of the multinomial probabilities H a : At least one of the multinomial probabilities does not equal its hypothesized value where E i = np 1,0, is the expected cell count given the null hypothesis. 13.2: Testing Categorical Probabilities: One-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 8

Conditions Required for a Valid  2 Test: One-Way Table 1. A multinomial experiment has been conducted. 2. The sample size n will be large enough so that, for every cell, the expected cell count E ( n i ) will be equal to 5 or more. 13.2: Testing Categorical Probabilities: One-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 9

LegalizationDecriminalizationExisting LawNo Opinion 7%18%65%10% 13.2: Testing Categorical Probabilities: One-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 10 Example 13.2: Distribution of Opinions About Marijuana Possession Before Television Series has Aired Table 13.2: Distribution of Opinions About Marijuana Possession After Television Series has Aired LegalizationDecriminalizationExisting LawNo Opinion

13.2: Testing Categorical Probabilities: One-Way Table 11McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.2: Testing Categorical Probabilities: One-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 12 Expected Distribution of 500 Opinions About Marijuana Possession After Television Series has Aired LegalizationDecriminalizationExisting LawNo Opinion 500(.07)=35500(.18)=90500(.65)=325500(.10)=50

13.2: Testing Categorical Probabilities: One-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 13 Expected Distribution of 500 Opinions About Marijuana Possession After Television Series has Aired LegalizationDecriminalizationExisting LawNo Opinion 500(.07)=35500(.18)=90500(.65)=325500(.10)=50

13.2: Testing Categorical Probabilities: One-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 14 Expected Distribution of 500 Opinions About Marijuana Possession After Television Series has Aired LegalizationDecriminalizationExisting LawNo Opinion 500(.07)=35500(.18)=90500(.65)=325500(.10)=50 Reject the null hypothesis

13.2: Testing Categorical Probabilities: One-Way Table Inferences can be made on any single proportion as well:  95% confidence interval on the proportion of citizens in the viewing area with no opinion is McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 15

13.3: Testing Categorical Probabilities: Two-Way Table Chi-square analysis can also be used to investigate studies based on qualitative factors.  Does having one characteristic make it more/less likely to exhibit another characteristic? McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 16

13.3: Testing Categorical Probabilities: Two-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 17 Column 12  cRow Totals 1n 11 n 12  n 1c R1R1 Row2n 21 n 22  n 2c R2R2  rn r1 n r2  n rc RrRr Column TotalsC1C1 C1C1 C1C1 n The columns are divided according to the subcategories for one qualitative variable and the rows for the other qualitative variable.

13.3: Testing Categorical Probabilities: Two-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 18

13.3: Testing Categorical Probabilities: Two-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 19 The results of a survey regarding marital status and religious affiliation are reported below (Example 13.3 in the text). ABCDNoneTotals Divorced Married, never divorced Totals Marital Status Religious Affiliation H 0 : Marital status and religious affiliation are independent H a : Marital status and religious affiliation are dependent

13.3: Testing Categorical Probabilities: Two-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 20 The expected frequencies (see Figure 13.4) are included below: ABCDNoneTotals Divorced39 (48.95) 19 (18.56) 12 (12.99) 28 (27.74) 18 (12.76) 116 Married, never divorced 172 (162.05) 61 (61.44) 44 (43.01) 70 (75.26) 37 (42.24) 384 Totals Marital Status Religious Affiliation The chi-square value computed with SAS is , with p-value = Even at the  =.10 level, we cannot reject the null hypothesis.

13.3: Testing Categorical Probabilities: Two-Way Table 21McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

13.4: A Word of Caution About Chi-Square Tests Relative ease of use Widespread applications Misuse and misinterpretation McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 22

13.4: A Word of Caution About Chi-Square Tests McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis 23 Sample is from the correct population Expected counts are ≥ 5 Avoid Type II errors by not accepting non-rejected null hypotheses Avoid mistaking dependence with causation To produce (possibly) valid  2 results Be sure