Question

In a pair-matched case-control study of risk factors for carcinoma, dietary history of 30 patients with the disease and 30 control subjects were collected. Below is a table that summarizes their history of corn and corn oil consumption.

Pairs	Cases - Carcinoma patients	Controls - healthy subjects
1	Yes	Yes
2	Yes	Yes
3	Yes	Yes
4	Yes	Yes
5	Yes	Yes
6	Yes	Yes
7	Yes	No
8	Yes	No
9	Yes	No
10	Yes	No
11	Yes	No
12	Yes	No
13	Yes	No
14	Yes	No
15	Yes	No
16	Yes	No
17	Yes	No
18	Yes	No
19	Yes	No
20	No	No
21	No	No
22	No	No
23	No	No
24	No	No
25	No	No
26	No	Yes
27	No	Yes
28	No	Yes
29	No	Yes
30	No	Yes

a. What is estimated odds ratio? what is the 95% confidence interval of population odds ratio?

b. Perform hypothesis test and see if there is any difference in efficacy between two treatments. level of sig - 0.05. state null and alternate hypothesis, test statistic, p value and your conclusion.

Answer 1

This is the question of case-control data where the counts in this table(table structured using the information above) represent the number of pairs and not the number of individuals.

Cases/control	Yes	No
Yes	6	13
No	5	6

No. of Yes-yes pairs are 6, No. of Yes-No pairs are 13, No. of No-Yes pairs are 5, No. of No-No pairs are 6 (from the information provided in the question)

a) Here Yes-Yes and No-No are known as the concordant pairs of the sample and Yes-No and No-Yes are known as discordant pairs. Although there are N = 30 pairs but we are only interested in the discordant pairs which are u = 13 and v = 5.

Odds Ratio =

The confidence interval for odds-ratio =

where z = 1.96 at 95% level of confidence.

Now first let us find

=

= 0.526

Now we will calculate the CI for odds ratio=

=

= -0.076,1.99

Hence the 95% confidence interval for case control data's odd ratio is [-0.076,1.99].

b) Also, When the number of discordant pairs (u + v) is 10 or greater, you can test H₀: OR = 1 with McNemar’s chi-square statistic.

Since u+v = 13+5 = 18

H₀: OR = 1

H₁: OR > 1

The regular and continuity-correct McNemar’s chi squares are shown below:

Because of the relation between the chi-square distributions and z distributions, the above formulas can be re-expressed: