Question

Suppose the manufacturer of a certain drug claims the adverse event rate of the drug is 20% (ie. 20% of people who take the drug have an adverse event), but you think the adverse event rate is higher. (In fact, you think the adverse event rate is 30%.) So, you want to do a study to show the adverse event rate is higher than 20%. If the adverse event rate is really 30% and you obtain a sample of size 10 patients, what is the power of your study for testing Ho: p=0.2 vs Ha: p > 0.2 with a significance level of 0.05? To address this question, answer the following: a) State in words what "power" means in the context of this problem. b) Determine the minimum number of adverse events among 10 patients that would need to happen to reject your null hypothesis. In other words, determine the minimum number of adverse events so that the one-sided p-value is less than 0.05. c) Now, calculate the probability of observing the number of events from part b or more events under the assumption that the true rate is 30%. In other words, calculate the power (the probability of rejecting the null hypothesis if the adverse event rate is really 0.3). d) Now do steps b) and c) to determine the power if the rate were really 70%. e) (*1 point) Compare the power in c) and d). This comparison illustrates (choose the best answer): i) power is higher with alternatives closer to the null hypothesis. ii) power is higher with alternatives farther from the null hypothesis. iii) power is higher with a smaller Type I error rate. iv) power is higher with a larger Type I error rate.

Answer 1

Solution

Part (a)

Concept Base

α = P(Type I Error) = probability of rejecting a null hypothesis when it is true

β = P(Type II Error) = probability of accepting a null hypothesis when it is not true, i.e., Alternative is true.

Significance level α of a test = maximum of P(Type I Error)

Power of a test = 1 – β = probability of rejecting a null hypothesis when it is not true, i.e., Alternative is true. Or equivalently, power is the probability of accepting a true alternative.

So, in the context of this problem, "power" means the probability of concluding that the adverse event rate is more than 20% when actually it is more than 20%. Answer

Part (b)

Let X = number of adverse events among 10 patients. Then, X ~ B(10, 02)........................................................................... (1)

Then, we want to find a value x₀ for X such that P(X > x₀) < 0.05 or P(X ≤ x₀) > 0.95

Using Excel Function: Statistical BINOMDIST, P(X ≤ x) for various values of x is tabulated below:

n	P	x	P(X = x)	P(X ≤ x)
10	0.2	0	0.10737418	0.10737418
		1	0.26843546	0.3758
		2	0.30198989	0.6778
		3	0.20132659	0.8791
		4	0.08808038	0.9672

From the above table we find P(X ≤ 3) < 0.95 < P(X ≤ 4). So, x₀ = 4

Thus, the minimum number of adverse events among 10 patients that would need to happen to reject your null hypothesis is 4 Answer 2

Part (c)

Now, if the true rate is 30%, X ~ B(10, 03).

So, probability probability of rejecting the null hypothesis if the adverse event rate is really 0.3 = P(X > 4)

= 0.3504 [Using Excel Function: Statistical BINOMDIST]

Thus, the power is 0.3504. Answer 4

Part (d)

Following the steps of (c) and (d) identically,

X ~ B(10, 03).

So, probability probability of rejecting the null hypothesis if the adverse event rate is really 0.7 = P(X > 4)

= 0.89844 [Using Excel Function: Statistical BINOMDIST]

Thus, the power is 0.8984. Answer 5

Part (e)

Clearly, power for 0.7 > power for 0.3 and 07 is farther than 0.3 from 0.2.

So,

Option ii, i.e., power is higher with alternatives farther from the null hypothesis. Answer 6

DONE