Question

3) Suppose we are interested in comparing the effectiveness of two different antibiotics, A and B, in treating gonorrhea. Each person receiving antibiotic A is matched with an equivalent person (age within 5 years, same sex), to whom antibiotic B is given. These people are asked to return to the clinic within 1 week to see if the gonorrhea has been eliminated. Suppose the results are as follows:

            1) For 40 pairs of people, both antibiotics are successful.

            2) For 20 pairs of people, antibiotic A is effective whereas antibiotic B is not.

            3) For 16 pairs of people, antibiotic B is effective whereas antibiotic A is not.

            4) For 3 pairs of people, neither antibiotic is effective.

1. a. What is your null and alternative hypothesis?
2. b. Test for the relative effectiveness of the two antibiotics. Which test can you use?
3. c. Calculate the percent agreement.

Answer 1

a) The null and alternative hypotheses are:

H0: Two antibiotics are independent

Ha: Two antibiotics are dependent.

-----------------------------------------------------------------------------------------------

b) We can use chi square test.

Observed	B successful	B is not effective	Total
A is effective	40	20	60
A is not effective	16	3	19
Total	56	23	79

For expected value, use formula:

Expected values= (row total x column total)/grand total

For example, for first cell, Expected value = 60 * 56/79 = 42.53

Expected	B successful	B is not effective	Total
A is effective	42.53	17.47	60
A is not effective	13.47	5.53	19
Total	56	23	79

For Chi square contribution, use formula:

Chi square contribution = (O-E)²/E

For example, for first cell, Chi square contribution = (40-42.53)²/42.53 = 0.15

Chi square	B successful	B is not effective	Total
A is effective	0.15	0.37	0.52
A is not effective	0.48	1.16	1.63
Total	0.63	1.53	2.15

The test statistic is:

There are r=2 rows and c=2 columns. Therefore, degrees of freedom is:

df = (r-1)(c-1) = (2-1)(2-1) = 1

At α = 0.05, the critical value of chi square is 3.841.

Since test statistic = 2.15 is less than critical value, fail to reject null hypothesis. We can conclude that two antibiotics are independent.

-----------------------------------------------------------------------------------------------

c) The percent agreement expected by chance is: