profile likelihood [ was Re: R-beta: R0.62.3 problems]

[Douglas Bates]

>>>>> "Bill" == Bill Simpson <wsimpson at uwinnipeg.ca> writes:

  >> 4. Is there no way to convince core that the default option for
  >> signif stars should be FALSE??? At a minimum, modern software
  >> should be printing out profile likelihood intervals (even SAS
  >> does for logistic regression now) and not standard errors for
  >> non-normal models. The intervals, let alone tests, based on s.e.s
  >> can be meaningless for glms. Are there no likelihoodlums (except
  >> me) or Bayesians out there?
  Bill> Hi Jim,

  Bill> I agree with you 100%. I am likelihood freak too and would
  Bill> like to have profile likelihood conf intervals.

  Bill> Also I would love to know how to compute profile likelihood
  Bill> confidence intervals when doing MLE with nlm. I have never
  Bill> seen this described anywhere in enough detail to do it
  Bill> myself. (I would incorporate that into the little nonlinear
  Bill> model fitting section I wrote for Notes on R.)

The splines library contains constructors and methods for
"backSplines".  These are piecewise polynomials that provide inverse
interpolation of a monotone function.  They would be used to construct
profile likelihood intervals by
 - determine the MLE and approximate standard errors
 - for each parameter {
    while (statistic < critical value) {
      create a step size as some fraction of the standard error
      step from the MLE of that parameter
      re-optimize on the other parameters
      convert the change in likelihood (or residual sum-of-squares) to
        a z-like or t-like statistic
      adjust step size if needed
    }  # I skipped the part about do this in both directions
   fit an interpolation spline to the values of the statistic versus
     the parameter being profiled
   form a backSpline and evaluate it at +/- critical value

Of course, the tricky part is setting up the optimization for all but
the parameter being profiled.  The backSplines just give you a
convenient way of evaluating the interval once you have done all the
real work.
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