An opportunity is offered to calculate likelihood profiles for specified parameters.

Likelihood profiling proceeds by sequentially fixing the parameter of interest at one of a vector of predefined values for that parameter, then maximizing the likelihood of that value with respect to all the other parameters.

A vector of likelihood values is created corresponding to each fixed parameter value.

The vector of negative ln-likelihoods is transformed and scaled to a vector of likelihoods from which a probability distribution of parameter values can be created and plotted.

This so-called likelihood profile can be used to draw statistical inferences concerning the parameter’s values.

In particular, likelihood profiles facilitate a likelihood ratio test which permits an analyst to make a probabilistic statement of confidence concerning a parameter’s value with respect to a postulated value.

A likelihood ratio test is based on the premise that twice the difference in negative ln-likelihood between two competing nested models (let’s say A and B) is chi-square distributed with the degrees of freedom determined by the difference in the number of free parameters between the two competing models.

In the case of likelihood profiling, model A is any accepted fit to the model with an associated value for the parameter of interest and model likelihood, model B is any one of the values in the vector of profiled model likelihoods corresponding to a fixed value of the parameter of interest, and there is one (1) degree of freedom.

By the way, comparison of a negative ln-likelihood profile with its corresponding probability distribution nicely illustrates the relationship between the shape of the negative ln-likelihood profile and the corresponding probability distribution.

A Gaussian (normal) probability distribution has an associated quadratic negative ln-likelihood profile, and the second partial derivative of the likelihood function with respect to the parameter is constant, as expected for a quadratic function.

An asymmetric negative ln-likelihood profile results in a skewed associated probability distribution, which when calculating a covariance matrix numerically will result in inconsistent estimates of parameter covariances across the three grids of numerically calculated second partial derivatives.

Smartstats© will guide you through the procedure for calculating likelihood profiles.

The numerical output of the procedure to calculate the likelihood profiles can be saved in a text file.