Profile Likelihood Confidence Intervals
Compute profile likelihood confidence intervals for ‘beta’ or ‘real’ parameters of a spatially explicit capture-recapture model,
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numeric or character vector of parameters
confidence level (1 – alpha)
optional dataframe of values at which to evaluate model
integer for level of detail in reporting (0,1,2)
absolute tolerance (passed to uniroot)
numeric vector of outer starting values – optional
other arguments (not used)
parm is numeric its elements are interpreted as the indices of
‘beta’ parameters; character values are interpreted as ‘real’
parameters. Different methods are used for beta parameters and real
parameters. Limits for the j-th beta parameter are found by a
numerical search for the value satisfying -2(l_j(beta_j) - l) = q, where l is the maximized log
likelihood, l_j(beta_j) is the maximized profile log
likelihood with beta_j fixed, and q is the
100(1-alpha) quantile of the
chi-square distribution with one degree of freedom. Limits
for real parameters use the method of Lagrange multipliers (Fletcher and
Faddy 2007), except that limits for constant real parameters are
backtransformed from the limits for the relevant beta parameter.
bounds is provided it should be a 2-vector or matrix of 2
columns and length(parm) rows.
A matrix with one row for each parameter in
parm, and columns
giving the lower (lcl) and upper (ucl) 100*
Calculation may take a long time, so probably you will do it only after selecting a final model.
The R function
uniroot is used to search for the roots of
-2(l_j(beta_j) - l) = q within a
suitable interval. The interval is anchored at one end by the MLE, and
at the other end by the MLE inflated by a small multiple of the
asymptotic standard error (1, 2, 4 or 8 SE are tried in turn, using the
smallest for which the interval includes a valid solution).
A more efficient algorithm was proposed by Venzon and Moolgavkar (1988);
it has yet to be implemented in secr, but see
the package Bhat for another R implementation.
Evans, M. A., Kim, H.-M. and O'Brien, T. E. (1996) An application of profile-likelihood based confidence interval to capture–recapture estimators. Journal of Agricultural, Biological and Experimental Statistics 1, 131–140.
Fletcher, D. and Faddy, M. (2007) Confidence intervals for expected abundance of rare species. Journal of Agricultural, Biological and Experimental Statistics 12, 315–324.
Venzon, D. J. and Moolgavkar, S. H. (1988) A method for computing profile-likelihood-based confidence intervals. Applied Statistics 37, 87–94.
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