Profile likelihood
A profile likelihood shows how the likelihood changes when one parameter is fixed at a range of values and the others are re-estimated. A bounded profile means the data determine the parameter; a flat one means they do not.
Definition
The profile likelihood of a parameter is the maximum of the likelihood over all the other parameters, plotted against the value of the parameter. It shows how much worse the fit becomes as the parameter moves away from its estimate.
Confidence intervals
A confidence interval can be read from the profile as the range where twice the drop of the log-likelihood stays below a threshold (3.84 for 95% with one parameter, from the chi-square distribution). Unlike a Wald interval it can be asymmetric.
Bounded and flat profiles
- Bounded: the profile rises on both sides, so the data determine the parameter.
- Flat or one-sided: the profile stays near the maximum on one or both sides, so the data cannot tell values apart. This is a practical identifiability problem, or, if the model structure causes it, a structural identifiability one.
On this site
The R package pkident classifies the likelihood profiles of nlmixr2 fits as bounded or flat, to tell whether a flat profile is a property of the model or of the data. The nimotuzumab analysis asks how complex a model the data can support.
Frequently asked questions
What does a flat likelihood profile mean?
The data do not distinguish between a wide range of values of that parameter, so it is not practically identifiable from them.
How is a confidence interval taken from a profile?
It is the range of parameter values where twice the difference in log-likelihood from the maximum is below the chi-square threshold, 3.84 for 95% with one parameter.