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 logodds scale of the logit link makes changes in x have decreasing eﬀect
on π . For the inverseCauchy 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,
hwk35.1.xlsx, for
these
computations. It is possible to get statistical software to give all
of these tests. See hwk35.1.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, hwk35.1.MPJ (note
that the test stat given by Minitab is a ztest,
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
 Hall
 Derivative, logit link

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