Unformatted text preview: Fall 2011 p.5 2. Problem 4.32 in Agresti.
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
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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- Fall '11
- Derivative, logit link