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(a) A trial is to be conducted to compare the efficacy of verum acupuncture to conventional therapy (a combination of drugs, physical therapy and exercise) for the treatment of chronic lower back pain. The primary outcome is response after 6 months defined as 33% improvement or better on 3 pain-related items on the Von Korff Chronic Pain Grade Scale questionnaire or a 12% improvement or better on the back-specific Hanover Functional Ability Questionnaire. In what follows, let pe be the probability of response for a patient receiving acupuncture and let pc be the probability of response for a patient receiving the conventional treatment. If you are using Stata then compute the required sample size with and without continuity correct, provide code too. Suppose it is required to produce a confidence interval for pc of half width no greater than 0.05. Find the minimum sample size, ne, required: (2 marks] I. If it is anticipated that pe = 0.6; ii. With no prior assumption for pe (b) Consider the null hypothesis Ho: Pc=pe. If it is anticipated that p = 0.5, find the minimum sample size required for power 1-B = 0.7 against the alternative p. - Pe=0.10 if a two- tailed test with significance level a = 0.05 is used. [2 marks] (c) Consider the null hypothesis Ho: pes p. If it is anticipated that pe= 0.5, find the minimum sample size required for power 1-B = 0.7 against the alternative po- p=0.10 if the one- tailed test with significance level a = 0.05 is used. [2 marks] (d) A placebo group that received a sham acupuncture treatment (superficial needling at non- acupuncture points) was also included. Let n be the number of subjects and p, be the response probability for this group. Consider the null hypothesis, Ho: p. = p; and suppose it is required to have power 1 - B = 0.7 against the alternative pe- p = 0.2, when a two- sided test with significance level a = 0.05 is to be used. If it is anticipated that pe=0.5 and taking the n, to be fixed at the value obtained in (b), then find the minimum number n:for the placebo group. [4 marks] [Hint: Use the quantity Pe - P.(1 - Pa) a +Px(1 - p.), to derive an expression for the required sample size.) (a) A trial is to be conducted to compare the efficacy of verum acupuncture to conventional therapy (a combination of drugs, physical therapy and exercise) for the treatment of chronic lower back pain. The primary outcome is response after 6 months defined as 33% improvement or better on 3 pain-related items on the Von Korff Chronic Pain Grade Scale questionnaire or a 12% improvement or better on the back-specific Hanover Functional Ability Questionnaire. In what follows, let pe be the probability of response for a patient receiving acupuncture and let pc be the probability of response for a patient receiving the conventional treatment. If you are using Stata then compute the required sample size with and without continuity correct, provide code too. Suppose it is required to produce a confidence interval for pc of half width no greater than 0.05. Find the minimum sample size, ne, required: (2 marks] I. If it is anticipated that pe = 0.6; ii. With no prior assumption for pe (b) Consider the null hypothesis Ho: Pc=pe. If it is anticipated that p = 0.5, find the minimum sample size required for power 1-B = 0.7 against the alternative p. - Pe=0.10 if a two- tailed test with significance level a = 0.05 is used. [2 marks] (c) Consider the null hypothesis Ho: pes p. If it is anticipated that pe= 0.5, find the minimum sample size required for power 1-B = 0.7 against the alternative po- p=0.10 if the one- tailed test with significance level a = 0.05 is used. [2 marks] (d) A placebo group that received a sham acupuncture treatment (superficial needling at non- acupuncture points) was also included. Let n be the number of subjects and p, be the response probability for this group. Consider the null hypothesis, Ho: p. = p; and suppose it is required to have power 1 - B = 0.7 against the alternative pe- p = 0.2, when a two- sided test with significance level a = 0.05 is to be used. If it is anticipated that pe=0.5 and taking the n, to be fixed at the value obtained in (b), then find the minimum number n:for the placebo group. [4 marks] [Hint: Use the quantity Pe - P.(1 - Pa) a +Px(1 - p.), to derive an expression for the required sample size.)