Re: st: Bootstrapping question

[Nick Cox <njcoxstata@gmail.com>]

I agree, but note the bottom line, namely that Henry's problem is
about an ordinal variable with several proportions.

As a minute point, Jeffreys' procedure has a frequentist
interpretation as a continuity-corrected version of the so-called
exact (Clopper-Pearson) confidence interval.

See -ssc type cij.hlp- and the embedded references. (-cij-, -ciji- and
-ciw-, -ciwi- on SSC were superseded when StataCorp added those
procedures to official -ci-, -cii- but the help files include some
details that never got into -ci- documentation.)

Nick

On Fri, Feb 8, 2013 at 2:00 PM, JVerkuilen (Gmail)
<jvverkuilen@gmail.com> wrote:

> I think you might get a lot of benefit out of either a full likelihood
> or Bayesian analysis, but these aren't necessarily easy.
>
> Here's an example if you had binomial data with N = 10, with y = 0 successes.
>
>      L(p) = (1 - p)^10
>
>
> It turns out that L(p) is maximized at p = 0.0 (as one would expect),
> but this is of course a silly estimate for a probability, and the Wald
> interval is even sillier. Try
>
> cii 10 0, wald
>
> To determine 95% confidence interval from the likelihood function,
> find the p such that L(p) = 0.15, which is equivalent to -2 LL(p) =
> 3.84, for the .05 cutoff for a chi square(1). This can be done
> analytically with the likelihood above, but in general you need to
> solve it numerically. If you read off the table of values you find
> that the upper limit of the likelihood based confidence interval is
> 0.175.
>
> (One intermediate step is to normalize the likelihood such that the
> maximum value is scaled to equal 1. It's not needed here because the
> maximum value is 1.) In Stata the easiest way to do this is simply to
> generate p and the relevant likelihood (or equivalently -2
> log-likelihood) and plot.
>
> Example:
>
> set obs 101
> range temp 1 101
> gen p = (temp - 1)/100
> drop temp
> gen L = (1 - p)^10
> quietly sum L
> gen L0 = L/r(max)
> twoway (connected L0 p, sort)
>
>
> (Try all the other cii options such as wilson, agresti, jeffreys, too.
> Jeffreys is a Bayesian method that's very close to what I show.) Your
> problem is more complicated because it's multinomial, but this general
> approach might help.
>
> See Yudi Pawitan. (2001), In All Likelihood, Oxford.
> http://ukcatalogue.oup.com/product/9780198507659.do#.URR6e1qLzQA
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