Testing Independence with Chi-Square: Examples & Analysis

1. Set 7 Bivariate distribution Chi-square test of independence

2. Bivariate Categorical & Discrete Data • Example 1 • X = Smoking habit • Y = On-the-job-accident • Are smoking and accident occurrence independent? • Example 2 • X = Number of rooms in a house • Y = Number of bathrooms in a house • Are the two variables independent? • Example 3 • X = Number of children in a household • Y = Attitude toward a local proposition • Are the two variables independent?

3. Contingency Table • Cross-tabulation of individuals according to two characteristics • Example: Smoking and On-the-Job accident study • Data: Table of observed frequencies (Observed counts) Accident Smoking

4. Table of proportions Joint distribution f(x,y) • Observed probabilities Marginal distribution of Smoking f(x) Marginal distribution accident f(y) • Are smoking and accident occurrence independent?

5. Conditional distributions of smoking • Distribution of smoking among accident (or no accident) • Column percentages

6. Conditional distributions of accident • Distribution of accident in each level of smoking • Row percentages

7. Are the Two Variables Independent? • Example 1 • X = Smoking habit • Y = On-the-job-accident • Are smoking and accident occurrence independent? • X and Y independent when: • f(y|x) = f(y) for all values of x and y • f(x|y) = f(x) for all values of x and y • f(x,y) = f(x) f(y) for all values of x and y

8. Independent model • Values of one variable do not give any information about the probability of other variable • Conditional and marginal distributions all are equal • Example: Column percentages

9. Independent model • Conditional and marginal distributions all are equal • Example: Row percentages

10. Independent model • Product rule: f(x,y) = f(x)f(y), for all pairs (x,y) .24 x .52=.1248 Note that for all pairs (x,y) f(y|x) = f(y) f(x|y) = f(x)

11. Independent model: Expected counts • Counts given by the independent model • Expected count (x,y) = nf(x)f(y), for all (x,y) 66 x .24 x .52 = 66 x .1248

12. Deviation of data from independent model • For each pair (x,y), compute the deviation between the observed and expected counts given by the independent model • Chi-square deviation

13. Chi-square statistic • Chi-square statistic for testing independent is the total chi-square deviations • Is the discrepancy statistically significant? • When the counts are large, the distribution of the X2 statistic is approximately c2 distribution • df=(r-1)(c-1) • r = number of rows • c = number of columns • Approximation works well when frequencies > 5

14. Test of independence • Is the discrepancy statistically significant? • Is above a threshold? • Select an upper tail probability threshold a • a =.10, .05, .01 • a is called the significance level • Find the chi-square threshold from the table • Reject the independent model at a level when X2 > the threshold

15. Example of a chi-square density • Upper 5 percentile c2 • P(c2 > c) = 0.05 • df From table 0.05

16. MINITAB computation • Frequency table in worksheet • Stat >> Tables >> Chi-square Test • Individual data in worksheet (coded) • Stat >> Tables >> Cross tabulation and Chi-square • Row • Column • Chi-square • Chi-square analysis • Expected cell counts • Each cell’s contribution to the Chi-square statistic