Question

A study aimed to determine if grapefruit juice has beneficial effects on the pharmacokinetics of oral digoxin, a drug often prescribed for heart ailments. Seven healthy non-smoking volunteers participated in the study. Subjects took digoxin with water for 2 weeks, no digoxin for 2 weeks, and then digoxin with grapefruit juice for 2 weeks. The peak plasma digoxin concentrations (Cmax; ng/mL) when subjects took digoxin under the two conditions are given in the following table:

Subject 1 2 3 4 5 6 7

Water 2.34 2.46 1.87 3.09 5.59 4.05 6.21

Grapefruit Juice 3.03 3.46 1.97 3.81 3.07 2.62 3.44

Decrease -0.69 -1.00 -0.10 -0.72 2.52 1.43 2.77

While small, note that the sample size was chosen carefully by the authors. In their paper they state that “assuming an � level of 0.05, a sample size of seven subjects has a power of 85% to detect a 25% change in digoxin Cmax”. Lower values of Cmax are better since they imply that digoxin is available in the body for longer. Is there any evidence that grapefruit juice increases the effectiveness of oral digoxin, by decreasing Cmax? (a) Identify one issue in the design of this experiment that undermines being able to use the data to answer this question. How could the design be improved? [2 marks]

(b) Suppose mew is the true mean decrease in digoxin Cmax with grapefruit compared to water. Define the null and alternative hypotheses for this study in symbols. [1 mark]

(c) We have two sets of Cmax measurements, one for water and one for grapefruit. Briefly explain why we work with the differences rather than carrying out a twosample t-test to compare the treatments. [1 mark]

(d) The seven differences in Cmax have mean 0.601 ng/mL with standard deviation 1.609 ng/mL. Use these values to test the hypotheses in (b). What do you conclude? [2 marks]

(e) How many of the seven subjects had a lower Cmax value with grapefruit juice? Use this to find the P-value for a sign test of whether grapefruit juice tends to lower Cmax. What do you conclude? [2 marks]

(f) For a sign test from seven subjects, what is the minimum number of reductions in Cmax needed to give evidence of an effect at the 5% level? [1 mark]

Answer 1

a) The one issue will be the choice of sample sample size. Since there are only 7 observations which is a reletively smaller sample size, this will impact the power of the test. Smaller the sample size smaller the power and chances of getting type II error is more. here in this problem they considerd 85% power that means there is 15% chance for type II error. This is the issue they could have considerd for. They could have considered larger sample size to reduce the type II error.

S.No.	Water	Grape fruit juice	Difference
1	2.34	3.03	-0.69
2	2.46	3.46	-1
3	1.87	1.97	-0.1
4	3.09	3.81	-0.72
5	5.59	3.07	2.52
6	4.05	2.62	1.43
7	6.21	3.44	2.77
Sum	25.61	21.4	4.21
Mean	3.66	3.06	0.60

b)Hypothesis

given is true mean decrease in digoxin Cmax with garpefruit compared to water. the hypothesis may be constructed as follows.

both water and grape fruit juice have same effectiveness

grape fruit juice has higher effective than water

c) If we use this paired comparision method any noise( variabilites among the persons) gets cancels out when we take the difference. And in another way standard deviation gets reduced greatly in this method when compared to pooled standard deviation.

d) test statistic for a given mean and standard deviation can be calculated from the following formulae

where S_d is the standard deviation

= 0.988

t_0.05,6 = 1.943

since t₀ < t_0.05,6 we fail to reject null hypothesis, that is there is no evidence that the two methods are different.