Cross-Tabulations

1. Cross-Tabulations

2. Cross-Tabs The level of measurement used for cross-tabulations are mostly nominal. Even when continuous variables are used (such as age and income), they are converted to categorical variables. When continuous variables are converted to categorical variables, important information (variation) is lost.

3. Data Types Prentice-Hall

4. Categorical Data • Categorical random variables yield responses that classify • Example: Gender (female, male) • Measurement reflects number in category • Nominal or ordinal scale • Examples • Did you attend a community college? • Do you live on-campus or off-campus? Prentice-Hall

5. Why Concerned about Categorical Random Variables? • Survey data tends to be categorical … hot/comfortable/cold, sunny/cloudy/fog/rain, yes/no… • Know limitations • nature of relationship • causality • Widely used in marketing for decision-making

6. Cross-Tabs The Chi-square, 2, statistic is used to test the null hypothesis. [Unfortunately, Chi-square, like many other statistics that indicate statistical significance, tells us nothing about the magnitude of the relation.] Prentice-Hall

7. c2 Test of Independence • Shows whether a relationship exists between two categorical variables • One sample is drawn • Does not show nature of relationship • Does not show causality • Used widely in marketing • Uses contingency table Prentice-Hall

8. Critical Value What is the criticalc2 value iftable has 2 rows and 3 columns, a =.05? If fo = fe, c2 = 0. Do not reject H0 a = .05 df = (2 - 1)(3 - 1) = 2 c2 Table (Portion) Prentice-Hall

9. c2 Test of Independence Hypotheses & Statistic • Hypotheses • H0: Variables are not dependent • H1: Variables are dependent (related) • Test statistic • Degrees of freedom: (r - 1)(c - 1) Observed frequency Expected frequency Prentice-Hall

10. c2 Test of Independence Expected Frequencies • Statistical independence means joint probability equals product of marginal probabilities • P(A and B) = P(A)·P(B) • Compute marginal probabilities • Multiply for joint probability • Expected frequency is sample size times joint probability Prentice-Hall

11. c2 Test of Independence An Example You’re a marketing research analyst. You ask a random sample of 286 consumers if they purchase Diet Pepsi or Diet Coke. At the 0.05 level of significance, is there evidence of a relationship? Prentice-Hall

12. Expected Frequencies Prentice-Hall

13. Expected Frequencies fe³ 1 in all cells 132·116286 132·154286 132·170286 154·170286 Prentice-Hall

14. c2 Test of Independence Prentice-Hall

15. c2 Test of Independence Test Statistic: Decision: Conclusion: H0: Not Dependent H1: Dependent a = .05 df = (2 - 1)(2 - 1) = 1 Critical Value(s): Reject at a = .05 a = .05 There is evidence of a relationship Prentice-Hall

16. Cross-Tabs Please provide the requested information by checking (once) in each category.   What is your: • age ____ < 18 ___ 18 - 26 ____ > 26 • gender ____ male ____ female • course load __ < 6 units __ 6 – 12 units __ > 12 units • gpa __ < 2.0 __ 2.0 - 2.5 __ 2.6 - 3.0 __ 3.1 - 3.5 __ > 3.5 • annual income __ < $15k __ $15k - $40k ___ > $40k

17. Cross-Tabs The information is coded and entered in the file student.sf by letting the first response be recorded as a 1, the second as a 2, etc.

18. Cross-Tabs The hypothesis test generally referred to as a test of dependence. The researcher wishes to determine whether the variables are dependent, or, exhibit a relationship.

19. Cross-Tabs Let’s investigate whether a relationship between a student’s gpa and units attempted exists. H0: GPA and UNITS are not dependent H1: GPA and UNITS are dependent.

20. Cross-Tabs Chi-Square Test ------------------------------------------ Chi-Square Df P-Value ------------------------------------------ 3.67 8 0.8853 ------------------------------------------

21. Cross-Tabs p-value = 0.8853, RetainH0 thus, GPA and UNITS are not dependent [Based on our data, there is no evidence to support the concept that a relationship exists between gpa and units attempted.]

22. Cross-Tabs Let’s investigate whether a relationship between a student’s age and units attempted exist. H0: AGE and UNITS are not dependent H1: AGE and UNITS are dependent.

23. Cross-Tabs Chi-Square Test ------------------------------------------ Chi-Square Df P-Value ------------------------------------------ 9.89 4 0.0423 ------------------------------------------

24. Cross-Tabs p-value = 0.0423, Reject H0 thus, AGE and UNITS are dependent [Based on our data, there is sufficient evidence to support the concept that a relationship exists between age and units attempted.]

25. Cross-Tabs Frequency Table for age by units Units<6 6-12 >12 AGE Total -------------------------------------------------------- <18 | 10 | 19 | 17 | 46 | 17.24% | 20.88% | 33.33% | 23.00% -------------------------------------------------------- Age18-26 | 24 | 22 | 16 | 62 | 41.38% | 24.18% | 31.37% | 31.00% -------------------------------------------------------- >26 | 24 | 50 | 18 | 92 | 41.38% | 54.95% | 35.29% | 46.00% -------------------------------------------------------- UNITS Total 58 91 51 200 29.00% 45.50% 25.50% 100.00%

26. Questions?

27. ANOVA