Reference no: EM132273896

Exercise 1: This question concerns the power of the two tests. Use the same assumptions as exercise 2, except we will now assume that the population does have a reduction in viral load when treated with this therapy.

a. Write down the interpretation of the power of an hypothesis test, in the context of conducting repeated studies.

b. Repeat the simulations carried out in Exercise 2, except now assume that the population mean reduction is 0.2, 0.4, 0.6, 0.8 and 1.0 (i.e. carry out 5 additional sets of simulations). For each study decide whether the null hypothesis would be rejected or not at the 5% significance level, based on the results of each of the two types of hypothesis tests. For each value of the mean reduction, obtain a simulation-based estimate of the power for each test.

c. For the mean reduction ranging from 0 to 1, plot a simulation-based estimate of the power function for each test (plot both functions on the same graph).

d. What advantage does the t-test have? Based on your simulation results, is it an important advantage and why do you think it arises?

Exercise 2:

Aim: We want to compare the performance of the 2 sample t-test and the Wilcoxon test in this exercise. We fix the (nominal) level of these tests at α = 5% .

This will be achieved through simulations using the following scenario. We want to assess the effect of a blood-pressure lowering medication in a specific population. Imagine a study where 200 patients are assigned to either a placebo or the test drug. Each arm of the study has 100 patients. We focus on the diastolic blood pressure and assume that its distribution is normal with mean μ₀ = 80 and a standard deviation of σ = 10 in the placebo group. We will also assume that the distribution of diastolic blood pressure in the treated group is normal with mean μ = 80 - δ where δ represents the reduction in diastolic blood pressure or the treatment effect. The same standard deviation as in the placebo group, σ = 10 , can be used.

a. Carry out simulations similar to those done in Exercise 3b, with δ = 0, 2, 4, 6, 8 and 10. The main difference is that we now have tests based on 2 different samples. Tabulate and plot the same simulation-based estimates of the power functions (as functions of δ ) for each test, as was carried out in Exercise 3c.

b. Which test seems to have an advantage when the distribution is indeed normal in both groups?

We now consider that the distribution of diastolic blood pressure in the treated group is a "contaminated" normal. Approximately 97% of observations come from the same normal distribution as before and the remainder are sampled from a distribution with a larger mean and variance N(130 , γ² ) where γ = 2σ = 20 . Outliers are likely to arise in the treated group since about 3% of the observations are sampled from a different population.

c. Repeat part (a) using the data.
d. Does the t-test maintain its nominal significance level of 5%? Which test seems to have an advantage now?

Attachment:- exercises.rar