1. Testing for an Association between two Categorical Variables
STAT 250
Dr. Kari Lock Morgan
Testing for an Association between two Categorical Variables
SECTION 7.2
χ2 test for association (7.2)

2. Painkillers and Miscarriage
Is use of painkillers during pregnancy associated with miscarriage?
Scientists interviewed 1009 women soon after they got a positive pregnancy test about their use of painkillers around the time of conception or the early weeks of pregnancy
The researchers then kept track of which of the pregnancies ended in miscarriage
Li, D-K., et. al. (2003). “Exposure to non-steroidal anti-inflammatory drugs during pregnancy and risk of miscarriage: population based cohort study,” British Medical Journal, 327(7411): 1.

3. Painkillers and Miscarriage
No Miscarriage
TOTAL
No painkiller
103
659
762
Aspirin 5 17 22
Ibuprofen 13 40 53
Acetaminophen 24
148
172
145
864
1009
Does this data provide evidence that these two variables are associated?

4. Two Categorical Variables
The statistics behind a χ2 test easily extends to two categorical variables
A χ2 test for association (often called a χ2 test for independence) tests for an association between two categorical variables
Everything is the same as a chi-square goodness-of-fit test, except:
The hypotheses
The expected counts
Degrees of freedom for the χ2-distribution

5. Hypotheses General hypotheses:
H0: The two variables are not associated
Ha: The two variables are associated
Painkillers and miscarriage:
H0: Type of painkiller taken is not associated with whether or not pregnancy ends in miscarriage
Ha: Type of painkiller taken is associated with whether or not pregnancy ends in miscarriage

6. Expected Counts Miscarriage No Miscarriage TOTAL No painkiller 762
Aspirin 22
Ibuprofen 53
Acetaminophen
172
145
864
1009

7. Expected Count Give the expected count for Aspirin, Miscarriage. 2.1
No Miscarriage
TOTAL
No painkiller
762
Aspirin 22
Ibuprofen 53
Acetaminophen
172
145
864
1009
Give the expected count for Aspirin, Miscarriage.
2.1
3.16
4.72
5.65

8. Chi-Square Statistic Observed (expected) Miscarriage No Miscarriage
TOTAL
No painkiller
103 (109.5)
659 (652.5)
762
Aspirin
5 ( )
17 (18.8) 22
Ibuprofen
13 (7.6)
40 (45.4) 53
Acetaminophen
24 (24.7)
148 (147.3)
172
145
864
1009

9. Chi-Square Statistic
Miscarriage
No Miscarriage
No painkiller
103 (109.5)
659 (652.5)
Aspirin
5 (3.16)
17 (18.8)
Ibuprofen
13 (7.6)
40 (45.4)
Acetaminophen
24 (24.7)
148 (147.3)
Give the contribution to the χ2 statistic for the Aspirin, Miscarriage category.
0.7
1.07
1.7
2.07

10. StatKey
χ2 = 6.168

11. What Next?
χ2 = 6.168
What next?

12. Randomization Distribution

13. Conclusion
Can we conclude that type of painkiller taken is associated with having a miscarriage?
Yes No

14. Conclusion
Can we conclude that type of painkiller taken is not associated with having a miscarriage?
Yes No

15. Chi-Square (χ2) Distribution
If each of the expected counts are at least 5, AND if the null hypothesis is true, then the χ2 statistic follows a χ2 –distribution, with degrees of freedom equal to
df = (number of rows – 1)(number of columns – 1)
Painkillers and Miscarriage:
df = (4 – 1)(2 – 1) = 3

16. Theoretical Distribution
Miscarriage
No Miscarriage
No painkiller
103 (109.5)
659 (652.5)
Aspirin
5 (3.16)
17 (18.8)
Ibuprofen
13 (7.6)
40 (45.4)
Acetaminophen
24 (24.7)
148 (147.3)
Can we also use the theoretical χ2 distribution to get the p-value?
Yes No

17. NSAIDs?
Headline coming out of this paper:Use of NSAIDs in pregnancy increases risk of miscarriage
NSAIDs (Nonsteroidal anti-inflammatory drugs) are a special class of painkillers that include aspirin and ibuprofen (but not acetaminophen)
Is taking NSAIDs or not associated with miscarriage?

18. NSAIDs and Miscarriage
No Miscarriage
TOTAL
No painkiller
103
659
762
Aspirin 5 17 22
Ibuprofen 13 40 53
Acetaminophen 24
148
172
145
864
1009
Miscarriage
No Miscarriage
TOTAL
No NSAIDs
127
807
934
NSAIDs 18 57 75
145
864
1009

19. NSAIDs and Miscarriage
No Miscarriage
TOTAL
No NSAIDs
127
807
934
NSAIDs 18 57 75
145
864
1009
How should we analyze this data?
Test for difference in proportions using a randomization test
Test for a difference in proportions using the z-statistic and normal distribution
Chi-Square Test for Association
Any of the above
None of the above

20. Two Categorical Variables with Two Categories
If you are testing for an association between two categorical variables each with two categories, test for a difference in proportions and chi-square test for association will give you identical p-values

21. Hypotheses
H0: taking NSAIDs around the time of conception or early in pregnancy is not associated with having a miscarriage
Ha: taking NSAIDs around the time of conception or early in pregnancy is associated with having a miscarriage

22. Expected Counts
Miscarriage
No Miscarriage
TOTAL
No NSAIDs
127
807
934
NSAIDs 18 57 75
145
864
1009
What is the expected count for the NSAIDs, Miscarriage cell?
10.8
12.5
15.6
16.4

23. Expected Counts
Miscarriage
No Miscarriage
TOTAL
No NSAIDs
127
807
934
NSAIDs
18 (10.8) 57 75
145
864
1009
What is the contribution to the chi-square statistic for the NSAIDs, Miscarriage cell?
3.21
4.13
4.84
5.4

24. StatKey

25. Conclusion
Can we conclude that taking NSAIDs around the time of conception or in early pregnancy is associated with having a miscarriage?
Yes No

26. Conclusion
Can we conclude that taking NSAIDs around the time of conception or in early pregnancy causes increased risk of miscarriage?
Yes No

27. That’s Not All!
A much more recent study (March 2014) reexamined this issue.
Daniel, S. et. al. (2014). Fetal Exposure to nonsteroidal anti-inflammatory drugs and spontaneous abortions, Canadian Medical Association Journal, 186(5).

28. NSAIDs and Miscarriage

29. NSAIDs and Miscarriage

30. Results

31. Results

33. ???
The first study found a significant association between NSAIDs and miscarriage, with those taking NSAIDS having significantly higher risk of miscarriage
The second study found a significant association between NSAIDs and miscarriage, with those taking NSAIDS having a significantly lower risk of miscarriage
WHAT’S GOING ON????

34. To Do
Read Section 7.2
Do HW 7.2 (due Friday, 4/17)