hwk3soln - Fall 2011 p.5 2. Problem 4.32 in Agresti....

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Unformatted text preview: Fall 2011 p.5 2. Problem 4.32 in Agresti. Answer: This is the CDF of the Cauchy distribution. When α = 0 and β = 1 this is the standard Cauchy distribution. To compare this choice of link with a logit link, I have plotted π (x) versus x for these two link functions below. As you can see, the two links are fairly similar for π ∈ (.2, .8), both behaving fairly linearly. That is, for both links, a unit increase change in x has a near constant effect on π for π values not too close to 0 or 1. However, the range of π within which the behavior is nearly linear is substantially greater for the logit link. As one makes x (or the linear predictor in a GLM) more extreme, the linearity on the log-odds scale of the logit link makes changes in x have decreasing effect on π . For the inverse-Cauchy c.d.f. link, these effects decrease more rapidly at it takes much more extreme x values to reach 0 or 1. So, the Cauchy link would be appropriate for situations in which π is (for the most part) bounded away from 0 or 1, or when we desire to minimize the effect of extreme covariates (robust regression). Inverse Logit Cauchy 0.0 0.2 0.4 pi1 0.6 0.8 1.0 Cauchy and inverse logit functions −5 0 x 1 5 p.6 p.7 p.8 p.9 p.10 p.11 p.12 See the Excel spreadsheet, hwk3-5.1.xlsx, for these computations. It is possible to get statistical software to give all of these tests. See hwk3-5.1.sas for an example of how to get the Wald and LR tests. You can get the score test from Minitab, for example. See the Minitab project, hwk3-5.1.MPJ (note that the test stat given by Minitab is a z-test, which is the square root of the score test stat. p.14 p.15 8 8. 8. p.16 p.17 8 ...
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This note was uploaded on 11/13/2011 for the course STAT 8620 taught by Professor Hall during the Fall '11 term at University of Florida.

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