Exercise 1 Suppose that you are helping to design a study that seeks to show that a new medication is superior by a margin to a placebo for improving cognitive abilities among individuals with mild dementia. For the outcome, Cognitive Abilities Scale (CAS), a higher score corresponds to better cognitive abilities. The mean CAS in the two conditions will be compared using a two-sample \(t\) test. Participants will be randomized 3:1 to the new medication:placebo conditions. The margin of superiority is 2 and the common standard deviation is assumed to be 10.

1. State the null and alternative hypotheses.

2. If the means are expected to be 30 and 36 in the placebo and medication groups, respectively, compute the sample sizes required to achieve \(80\%\) power, when using a one-sided test with \(\alpha\) of 0.025.

Exercise 2 You are helping to design a study that aims to demonstrate that an app-based intervention (Intervention 1) is noninferior to an in-person intervention (Intervention 2) for controlling hypertension in individuals with high blood pressure. The primary outcome variable is systolic blood pressure in mmHg; for this outcome, a lower mean is considered a better outcome. We will compare the means in the two groups using a two-sample \(t\) test assuming equal variance and using equal allocation. We will use one-sided \(\alpha\) of 0.025. The common standard deviation is assumed to be \(\sigma = 15\) mmHg. The noninferiority margin is 3 mmHg.

1. State the null and alternative hypotheses.

2. Although noninferiority studies are based on one-sided tests, inference for noninferiority studies often uses two-sided confidence intervals. What confidence coefficient should be used for a two-sided CI such that the inference will correspond to one-sided \(\alpha\) of 0.025 (meaning, for a \(X\%\) CI, what should be the value of \(X\))?

3. Assuming that \(\mu_1=\mu_2\), compute the sample size required for \(80\%\) power.

4. Compute the sample size requirement again but assuming that outcomes for Intervention 1 are slightly worse (higher) than for Intervention 2, with \(\mu_1=\mu_2+1\). Is the required \(N\) higher or lower? Why?

Exercise 3 A study is planned to test whether an internet-delivered psychological treatment is equivalence to a treatment as usual (TAU) face-to-face treatment in terms of reducing anxiety and depressive symptoms, as measured by change in the Hospital Anxiety and Depression Scale (HADS). A pilot study found that the mean change score for TAU was 3.9 with a standard deviation of 6.0. The equivalence limit is specified as 2. The study plans to use a parallel group design with equal allocation.

1. State the null and alternative hypotheses.

2. Assuming no true difference in mean change scores in the two groups, what sample size is needed for 80% power, when specifying a significance level of 0.05? Compute the sample size using both the normal approximation formula and using the “exact” Owen’s Q method (in R).

3. What is the confidence coefficient for a two-sided confidence interval that will correspond to the significance level of 0.05, that is, for an \(X\%\) confidence interval, what is \(X\)?

4. Suppose that the internet-delivered treatment is actually less effective than TAU, resulting in a lower mean change. Conduct a sensitivity analysis showing how the sample size requirement changes as the true difference in mean change increases. :::{.content-hidden} Solution

Exercise 4 Zegels et al (2013) http://dx.doi.org/10.1016/j.joca.2012.09.017 reports a three-arm randomized trial in patients with knee osteoarthritis that compared single dose chondroitin 4&6 sulfate (CS), 3 daily dose CS and placebo. You are helping to design a new three-arm trial that will compare single-dose CS, a new experimental drug BS and placebo. The primary hypothesis is that CS and BS are equivalent; secondary hypotheses are that CS and BS are each superior by a margin to placebo. Using the information reported in the article to help provide parameter estimates, compute the following. You can assume equal variances in the three groups. The outcome measure is change in the total score of algo-functional Lequesne index (LI). For this exercise, you can rely mainly on the information in Table II of the paper.

1. What sample size per group is needed to achieve 80% power to conclude that CS and BS are equivalent in an intent to treat analysis, using a significance level of 0.05? Include a sensitivity analysis. Use an equivalence margin of 1.0.

2. The plan calls for doing a per protocol analysis to compare CS and BS. Assuming 15% of patient do not adhere to their assigned treatment, what sample size per group would you need for a per protocol analysis of the equivalence hypothesis?

3. What sample size per group is needed to achieve 80% power to conclude that CS and BS are each superior to placebo, using a one-sided test of superiority with \(\alpha = 0.025\), using an intent to treat analysis? Use a superiority margin of 1.0.

4. Will the sample sizes found for (b) be adequate for (c)? What sample size per group is needed to provide adequate power for all three analyses?

5. Write a succinct paragraph describing your sample size analysis, such as you would for it to be included in a trial protocol.

Exercise 5 In a pharmacokinetic study, investigators want to compare two formulations of a drug with respect to an outcome variable called the area under the curve (AUC). They plan to use a parallel group design with equal allocation. AUC is highly skewed and the data will be log-transformed prior to analysis. (Note: It is more common to use a crossover design for pharmacokinetic studies. Crossover designs are discussed later in the course.)

1. State the null and alternative hypotheses.

2. What is the required sample size when specifying a CV of 30%, true geometric mean ratio of 1, 90% power, significance level of 0.05 and equivalence bounds of (80%, 125%)?