Problem 1 A study to determine whether a new drug to treat a

Problem 1. A study to determine whether a new drug to treat a specific brain tumor was administered to 100 patients. The old treatment had a standard deviation of 18 months, and the new treatment was thought to have similar variability from patient to patient. The investigators calculated that their study would have approximately 80% power if the new drug would lengthen the average survival by 5 months and the individual observations came from a Normal distribution. What would happen to the power of the test, would it go up, down, or stay the same (briefly explain), if:

(a) The study had 200 subjects (instead of the 100 in the original design)?

(b) The new drug improved survival 3 months (instead of the 5 months originally thought).

(c) The standard deviation of survival time for individual patients was actually 24 months (instead of the original 18 months)?

(d) Determine how many observations are needed in order for the study in problem #1 to truly have 80% power (your result may not exactly match the number presented here).

Problem 2. In reality the study in problem #1 would need to have two different treatment groups in order to prove the new drug increases survival time (with random assignment to groups): one group of patients receiving the new treatment, and the other group, the controls, receiving the old treatment (often labeled as “standard of care”). For this two-sample situation (which we will use a z-based approach since we are going to assume the true is known), we can assume the following properties: individual survival times in each group are being sampled from Normal distributions with the same standard deviation ( months in both groups) and a mean difference of months (you can assume the old treatment had an average survival of µ years). That is the observed survival times can be thought of as being i.i.d. r.v.s from the following distributions: Yi,trt N(µ + , 2 ) and independently Yj,ctrl N(µ, 2 ). for i = 1, 2, ..., ntrt and j = 1, 2, ..., nctrl.

(a) What are the hypotheses? What will be the sampling distribution of the statistic (Y¯ trt Y¯ ctrl) under the null hypothesis? What will it be under the alternative hypothesis? Determine the z-based test statistic for this setting. Note: the government (FDA) mandates that all clinical trials are to be performed as two-sided tests at the = 0.05 level.

(b) Determine in terms of ntrt, nctrl, , z1/2 , and z given a desired value of power = 1 . Hint: think about how far the sampling distribution of (Y¯ trt Y¯ ctrl) under HA needs to be from the 1 sampling distribution of (Y¯ trt Y¯ ctrl) under H0 (like in the plot on slide 13 in Unit 7 lecture notes).

(c) Let ntrt = nctl = n. Using your expression in part (b), solve for n (so a total of 2n patients are needed for this study).

(d) Using the same conditions as in problem #1 part (d), determine the total number of patients needed for this 2-treatment study where the number of patients is equal in the two treatment groups.

(e) Compare the total number of patients needed in part 1(d) and 2(d). Why does this result make sense?

Note from student. I have already solved prob 1, but it is provided for context for prob 2.

Solution