The Chi-Square Distribution and Test for Independence

1. The Chi-Square Distribution and Test for Independence Hypothesis testing between two or more categorical variables

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

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

4. Chi-square Degrees of freedom • 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

6. Using the Chi-Square Test • 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. • H0: The 2 variables are independent.

7. Requirements for Chi-Square test • 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 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.

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