Statistics
The Chi-square Test
April 23, 2009
Outline
1. The null hypothesis
H0 : There is no relationship between the two categorical variables
2. The expected count of a cell (if the null hypothesis is true)
expected count =
row total × column total
table total
3. We want a test statistic that is large if the expected counts are very different than the
actual counts and small if the expected counts are similar to the actual counts.
4. The chi-square statistic: (the sum is over all squares in the table)
2
X =
X (observed − expected)2
expected
Notice that X 2 is positive and tends to be close to 0 if the null hypothesis is true and
farther away from 0 if the null hypothesis is false.
5. If H0 is true, then X 2 has a chi-square distribution with (r − 1)(c − 1) degrees of freedom.
6. Miscellaneous issues concerning chi-square
(a) Conditions for inference (all expected values ≥ 1 and 80% of the cell counts ≥ 5).
(b) After H0 is rejected, then what?