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

2. Agenda
Wrapping up the t-test discussion: effect size, how big of a sample, and t-test.do program example
Chi-Square and Chi-Square test of independence

3. Discussion from Last Class: Statistical Significance vs
Discussion from Last Class: Statistical Significance vs. Practical Significance

4. Effect Size
Values of .8 or greater are usually large effects, .5 medium, .2 small.
Effect size is a measure of practical significance, not statistical significance.
Cohen’s d is a common effect size calculation.
So, if you found a very large effect size (.9) but the effect was not statistically significant…
1) how would you interpret this result?
2) what could you do about it?

5. How Big of A Sample?
We can use a power analysis to determine how big our sample needs to be in order to reject the null that there is no difference.
Requires an idea of the two means and their S.D.’s (usually through pre-testing or literature)
In STATA, use “sampsi” command.
A power analysis is based on our best guesses about our sample(s). It is not a guarantee of significance.

6. Chi-Square test of independence and the chi-square distribution

7. Application of the Chi-Square
Used with two categorical variables. This will produce a crosstabulation (crosstab table) of cells.
How do you know if the cell counts are different than what would be expected by chance alone?
Or, are the observed cell counts in the cells independent from one another?
“It’s like playing Russian roulette. If you keep on going, sooner or later you’re going to lose”

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

9. 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.

10. Chi-Square Test of Independence

11. 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.

12. 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

13. Example Crosstab: Gender x Student
Student
Not Student
Total
Males 46
(40.97) 71
(76.02)
117
Females 37
(42.03) 83
(77.97)
120
154
237
Observed
Expected

14. Special Cases Fisher’s Exact Test Strength of Association
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.
Cramer’s V is interpreted much like a correlation. Values range from 0-1, where under .2 is weak, .2 to .4 is strong, and anything higher is very strong.

15. Practical Examples:
chi2dist.do
chisquare.do
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