In chronic diseases such as multiple sclerosis, medications are approved for sale after the US Food and Drug Administration evaluates both the effectiveness of the medication in delaying the progression of the disease and the extent to which the medication provides symptom relief. An earlier study showed that two treatments (labeled A and B) were equally effective in delaying the progression of disease. This problem examines simulated data from a study comparing how well the treatments relieve disease symptoms. A total of 410 participants were randomly assigned to one of the two treatments. The effect of disease symptoms on daily living was assessed by responses to the EuroQol 5D-5L (EQ-5D-5L), a questionnaire widely used to study health-related quality of life; higher scores are indicative of better quality of life. The EQ-5D-5L was administered to each participant before the start of treatment and again after 6 months of treatment. Investigators planned to compare the mean change in the EQ-5D-5L score between the two groups in order to assess whether the extent of symptom relief differs by treatment. The data from the study are contained in the file quality_of_life.Rdata. The dataset quality.of.life contains the following variables: - treatment.group: The treatment to which the participant was assigned, coded A or B. - pre.treatment.score: The EQ-5D-5L score just before treatment started. - post.treatment.score: The EQ-5D-5L score after 6 months of treatment. Do the data from this study provide definitive evidence that the two treatments are different in their ability to offer symptom relief?

a) Outline a plan for analyzing the data. Specify how the variables in the dataset will be used, in addition to any variables that might be created. Clearly indicate which methods will be used in the analysis. If a hypothesis test will be conducted, state the null and alternative hypotheses and use significance level = 0.05.

b) Conduct the analysis and summarize the numerical results. Be sure to provide an answer to the original question in the context of the data.

c) The sponsors of treatment A decide to perform a second study comparing A and B, but to save money, they decide to administer the EQ-5D-5L only once, 6 months after the start of the therapy. Earlier studies have shown that the population average post-treatment score on A is 64.3, with standard deviation 8.5. The sponsors believe that treatment B will have a population average post-treatment score of 63 and also have standard deviation 8.5. How large should this second study be to have power 0.80 of rejecting the null hypothesis of no treatment difference between A and B, when = 0.05?

d) When the investigators behind the first study learn about the design of the second study, they send a message to the study team: "You may be saving some money by not administering the questionnaire before treatment starts, but your study will be much larger than if you were to use our design." What justifies that message? Provide a specific explanation.