Subject Code : MAST90100
Country : Australia
Assignment Task:

Task:

In this assignment we compare two approaches to defining a confidence interval for the binomial proportion (based on the simple model discussed extensively in the text and lectures in relation to prevalence estimation). You should use the same notation as in the lectures, i.e.
θθ = the unknown true population proportion (or prevalence) of an outcome of interest
θθ? = PP = TT/nn = the sample proportion, where TT is the number of cases with the outcome among the total of nn individuals in a random sample.

You also need to use the fact (which can be taken for granted, i.e. a proof is not required!) that the arcsine (inverse sine) transformation of PP is also approximately normal, with variance that does not depend on θθ, as follows: YY = arcsin?√PP? ~ N?arcsin?√θθ?, 1 4nn ? .

(This is called a “variance-stabilising transformation”, because of the property that the variance of the transformed random variable is—approximately—independent of the underlying parameter; such transformations were important historically in statistics although less so now.)

1. (a) [3 marks] Use the result above (and the appropriate quantile from the standard normal distribution) to write out a formula for an approximate 90% confidence interval for φφ = arcsin?√θθ?, and use this to obtain a formula for an approximate 90% confidence interval for θθ. [Hint: for the second part, you will need to derive an expression for θθ as a function of φφ.] Make sure you use clear notation and wording to present your answers.
(b) [2 marks] Calculate the 90% confidence interval for θθ in the following two hypothetical examples, using both the method derived in part (a) and the “standard” method described in Lecture 2 and section 2.7 of IPB:
(i) 121 cases in a sample size of 1000.
(ii) 2 cases in a sample size of 90.

Short_Asst_1.docx 2
Are the results from the two methods similar in both cases (i) and (ii)?
Explain why or why not. What seems particularly “wrong” about the result of the standard method for case (ii)?
2. Now we ask you to compare these two methods for obtaining a confidence interval for θθ using a simulation experiment.

(a) [2 marks] Modify the program defined in the file ‘prevsim.ado’ (provided in the zip file of Stata code on the LMS site) so that it also calculates the confidence interval based on the arcsine transformation. This can be done by inserting the five extra lines shown below at the bottom of the existing code (before the “end” statement), to calculate the arcsine-transformed value of the proportion [N.B. the arcsine is given in Stata by the function asin()], then the limits of the CI for φφ, and finally the “back-transformed” CI for the prevalence (proportion).

 

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  • Posted on : April 22nd, 2019
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