{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

{[ snackBarMessage ]}