which test seems to have an advantage when the distribution is indeed Normal in both groups

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Exercise 1:

This question concerns the significance level (or type I error) of the two tests (i.e., one-sample t-test and the signed rank test). Consider 1000 studies repeated on the same population in which there is no average reduction in viral load from the treatment (i.e., 𝑦̅ = 0). Suppose that the variance of the reduction between individuals is 1.0 (on the scale used to measure viral load). This question involves simulating the results from these studies based on normally distributed reductions, using a sample size of 50.

a. Write down the interpretation of the significance level of a hypothesis test, in the context of conducting repeated studies.

b. Using statistical software, simulate 1000 repeated studies based on the assumptions above (Stata and R code is available on Canvas). 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. Hence, obtain a simulation-based estimate of the significance level for each of the two tests.

c. Based on the results in part (b), do the two tests seem to have the significance level that you would expect? Biostatistics Collaboration of Australia 3.3

Exercise 2:

This question concerns the power of the two tests (i.e., one-sample t-test and the signed rank test). Use the same assumptions as in Exercise 2, except we will now assume that the population does indeed have a reduction in viral load when treated with combination therapy.

a. Write down the interpretation of the power of a 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 of these possible mean reductions by appropriately modifying the simulation parameters). 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 (i.e., 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 3:

Suppose we have a population in which there is no reduction in viral load on average (as in Exercise 2). Suppose we conduct 𝑘 studies on this population and test whether there is a statistically significant reduction in each case. The test statistic that we use is unimportant for this exercise. a. Write down an expression for the probability that at least one of these studies yields a result that is statistically significant, using a significance level of 𝛼 for each test? (Hint: the probability that at least one is significant is 1 minus the probability that none are significant). b. Calculate this for 𝛼 = 0.05 and 𝑘 = 1, 2, 3, 5, 10, 15, 20. c. By referring to your results in part (b), summarise the problem with conducting multiple tests and concluding statistical significance if one of them rejects the null hypothesis. d. Supposes we let our significance level depend on the number of tests we are conducting. In particular, if we are conducting 𝑘 tests, suppose we use a significance level of 𝛼/𝑘 for each test, where 𝛼 = 0.05. Based on this, repeat the calculation in part (b), to determine the probability that at least one of the studies will yield a statistically significant result. e. Using the results of part (d), explain how this correction to the significance level rectifies the problem identified in part (c). (This is called Bonferroni correction for multiple comparisons, which is one of several different approaches to deal with the problem.)

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