1, Proof the sample size formula for two sample proportion difference, with unequal sample sizes.
2, The sample size calculation for longitudinal study (case I), under m=2, formula 1* and 2* are the same. For understanding this, in slide 18 of lecture 10 notes, please show: when xij=(0, 2), m=2, two sample size formula are the same.
3, Consider independent and identically distributed observations from a Poisson distribution with rate parameter λ. The maximum likelihood estimator of λ is T/n, where T is the total number of events in n units, such as T is the epileptic seizures in n patient-years of exposure. Now consider two groups with parameters λA and λB with sample sizes Qn and (1-Q)n, respectively, where 0<Q<1.
(Part of the content of this question comes from Problem 2.4 of “randomization and clinical trial” book, page 32)
(i) Derive the basic expression for sample size calculation (based on two-sided test), with fixed power and minimum detected difference Δ.
(ii) Similarly, write down the formula for power calculation, as if Δ and power are known.
(iii) Consider a study to compare a drug vs. placebo in the treatment of epileptics. What parameter will have to be estimated from prior knowledge on similar studies, apart from the parameters that we could fixed in the beginning of this study design.
4, We plan to assess healing performance on a disease (binary outcome) with an investigational drug A (Z=1) and placebo (Z=0). (Part of the content of this question comes from Problem 2.5 of “randomization and clinical trial” book, page 32)
a. Prior studies suggest that the control healing rate is on the order of 20%. Investigators believe that a minimal, clinically-meaningful increase in healing on the experimental therapy is 5%. For 80% power, compute the number of patients needed, assuming equal sample sizes for both treatment arms.
b. From part a, investigate the changes in total sample size (two treatment arms) that occur with changing the allocation proportion Q, where nA= Qn, nB= (1-Q)n. (Hint, let Q vary within (0.1, 0.2, 0.3, …. 0.9), and use the sample size formula for two sample proportion difference, with unequal sample sizes, to calculate the results.)
5, For a clinical trial comparing two sample means (continuous outcomes), with unequal sample sizes, nA= Qn, nB= (1-Q)n. Suppose the standard deviation on treatment A is σA, and the standard deviation on treatment B is σB, where σA≠ σB . (Part of the content of this question comes from Problem 2.6 of “randomization and clinical trial” book, page 32)
a) Derive the sample size formula, in terms of fixed power and minimum detected difference Δ.
b) Show that, for fixed n, the value of Q that maximizes power is given by Q*= σA/( σA + σB ). (Allocating according to the ratio of the standard deviations is called Neyman allocation. This result implies that equal allocation does not maximize power when the two treatments have different standard deviations).
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