1. HYPOTHESIS TESTING BETWEEN TWO OR MORE CATEGORICAL VARIABLES The Chi-Square Distribution and Test for Independence

2. Agenda Wrapping up the t-test program example Chi-Square and Chi-Square test of independence 2

3. Chi-Square Distribution 3 The chi-square distribution results when independent variables with standard normal distributions are squared and summed.

4. Chi-square Degrees of freedom 4 df = (r-1) (c-1)  Where r = # of rows, c = # of columns  Thus, in any 2x2 contingency table, the degrees of freedom = 1.  As the degrees of freedom increase, the distribution shifts to the right and the critical values of chi- square become larger.

5. Chi-Square Test of Independence 5

6. Using the Chi-Square Test 6 Often used with contingency tables (i.e., crosstabulations)  E.g., gender x student The chi-square test of independence tests whether the columns are contingent on the rows in the table.  In this case, the null hypothesis is that there is no relationship between row and column frequencies.  H 0 : The 2 variables are independent.

7. Requirements for Chi-Square test 7 Must be a random sample from population Data must be in raw frequencies Variables must be independent Categories for each I.V. must be mutually exclusive and exhaustive

8. Example Crosstab: Gender x Student 8 StudentNot StudentTotal Males 46 (40.97) 71 (76.02) 117 Females 37 (42.03) 83 (77.97) 120 Total83154237 Observed Expected

9. Special Cases Fisher’s Exact Test  When you have a 2 x 2 table with expected frequencies less than 5. Strength of Association  Some use Cramer’s V (for any two nominal variables) or Phi (for 2 x 2 tables) to give a value of association between the variables. 9

10. Practical Examples: 10 chi2dist.do chisquare.do Auto.dta