1. Chi-square Basics

2. The Chi-square distribution
Positively skewed but becomes symmetrical with increasing degrees of freedom
Mean = k where k = degrees of freedom
Variance = 2k
Assuming a normally distributed dataset and sampling a single z2 value at a time
2(1) = z2
If more than one… 2(N) =

3. Why used?
Chi-square analysis is primarily used to deal with categorical (frequency) data
We measure the “goodness of fit” between our observed outcome and the expected outcome for some variable
With two variables, we test in particular whether they are independent of one another using the same basic approach.

4. One-dimensional
Suppose we want to know how people in a particular area will vote in general and go around asking them.
How will we go about seeing what’s really going on?
Republican
Democrat
Other 20 30 10

5. Hypothesis: Dems should win district
Solution: chi-square analysis to determine if our outcome is different from what would be expected if there was no preference

6. Plug in to formula 20 30 10 Observed Expected Republican Democrat
Other
Observed 20 30 10
Expected

7. Reject H0
The district will probably vote democratic
However…

8. Conclusion
Note that all we really can conclude is that our data is different from the expected outcome given a situation
Although it would appear that the district will vote democratic, really we can only conclude they were not responding by chance
Regardless of the position of the frequencies we’d have come up with the same result
In other words, it is a non-directional test regardless of the prediction

9. More complex What do stats kids do with their free time? 30 40 20 10TV
Nap
Worry
Stare at Ceiling
Males 30 40 20 10
Females

10. Example for males TV: (100*50)/200 = 25
Is there a relationship between gender and what the stats kids do with their free time?
Expected = (Ri*Cj)/N
Example for males TV: (100*50)/200 = 25 TV
Nap
Worry
Stare at Ceiling
Total
Males 30 40 20 10
100
Females 50 70 60
200

11. df = (R-1)(C-1) R = number of rows C = number of columns 30 (25)TV
Nap
Worry
Stare at Ceiling
Total
Males (E)
30 (25)
40 (35)
20 (30)
10 (10)
100
Females (E)
20 (25)
30 (35)
40 (30) 50 70 60 20
200

12. Interpretation
Reject H0, there is some relationship between gender and how stats students spend their free time

13. Other
Important point about the non-directional nature of the test, the chi-square test by itself cannot speak to specific hypotheses about the way the results would come out
Not useful for ordinal data because of this

14. Assumptions Normality Inclusion of non-occurences Independence
Rule of thumb is that we need at least 5 for our expected frequencies value
Inclusion of non-occurences
Must include all responses, not just those positive ones
Independence
Not that the variables are independent or related (that’s what the test can be used for), but rather as with our t-tests, the observations (data points) don’t have any bearing on one another.
To help with the last two, make sure that your N equals the total number of people who responded

15. Measures of Association
Contingency coefficient
Phi
Cramer’s Phi
Odds Ratios
Kappa
These were discussed in 5700