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Unformatted text preview: PHC 6020 – Homework Assignment 3 Due Thursday November 3, 2011 (Please show all your work, i.e. the steps that are necessary to reach your results.) 1. (20 pts) In class we derived the sample size formula for comparing two treatments which produce binary responses using variance stabilizing arcsin transformation. It turns out that the following transformation for sample proportion gives us what is called a symmetry transformation: ݃ሺሻ ൌ න 1 ሼݔሺ1 െ ݔሻሽ ଵ/ଷ ݀ݔ , where is the sample proportion. That is, this transformation makes the distribution of ݃ሺሻ look closest to symmetric and consequently gives the best approximation to a normal distribution. Let ܺ~ܤ݅݊ሺ݊, ߨሻ and ൌ ܺ/݊ . a. Using delta method, find the approximate variance of ݃ሺሻ . b. Consider the two sample problem where we compare the response rates of two treatments (1 and 2) in a randomized clinical trial. Suppose ݊ ଵ ൌ ݊ ଶ ൌ ݊/2 patients are to be put on each treatment. We are interested in testing the null hypothesis ܪ : ߨ ଵ ൌ ߨ ଶ or equivalently ܪ : ݃ሺߨ ଵ ሻ ൌ ݃ሺߨ ଶ ሻ (since ݃ሺሻ is a strictly increasing function of ). Construct a test statistic using this transformation such that the test statistic will approximately have a standard normal distribution under ܪ . c. What is the approximated distribution of the test statistic under the alternative ܪ : ߨ ଵ ൌ ߨ ଵ , ߨ ଶ ൌ ߨ ଶ where ߨ ଵ ് ߨ ଵ ? d. Based on the test statistic derived in part a., what is the necessary sample size if we want to test ܪ : ߨ ଵ ൌ ߨ ଶ at 0.05 level of significance (twosided) and detect the alternative ܪ : ߨ ଵ ൌ 0.45, ߨ ଶ ൌ 0.35 with power 90%? How does this compare to the sample size derived from the test statistic based on arcsin transformation? transformation?...
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 Fall '11
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 Normal Distribution, Standard Deviation, Variance, Probability theory

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