interest focuses on the estimation of only one, or at most a few of them, with the others being considered as nuisance parameters. Several alternative approaches have been developed to eliminate such nuisance parameters, so that a likelihood can be written as a function of only the parameter (or parameters) of interest: the main approaches are profile, conditional, and marginal likelihoods. These approaches are also useful when a high-dimensional likelihood surface needs to be reduced to one or two parameters of interest in order to allow a graph. Profile likelihood It is possible to reduce the dimensions by concentrating the likelihood function for a subset of parameters by expressing the nuisance parameters as functions of the parameters of interest and replacing them in the likelihood function. In general, for a likelihood function depending on the parameter vector that can be partitioned into , and where a correspondence can be determined explicitly, concentration reduces computational burden of the original maximization problem. For instance, in a linear regression with normally distributed errors, , the coefficient vector could be partitioned into (and consequently the design matrix ). Maximizing with respect to yields an optimal value function . Using this result, the maximum likelihood estimator for can then be derived as where is the projection matrix of . This result is known as the Frisch–Waugh–Lovell theorem. Since graphically the procedure of concentration is equivalent to slicing the likelihood surface along the ridge of values of the nuisance parameter that maximizes the likelihood function, creating an isometric profile of the likelihood function for a given , the result of this procedure is also known as profile likelihood. In addition to being graphed, the profile likelihood can also be used to compute confidence intervals that often have better small-sample properties than those based on asymptotic standard errors calculated from the full likelihood. Conditional likelihood Sometimes it is possible to find a sufficient statistic for the nuisance parameters, and conditioning on this statistic results in a likelihood which does not depend on the nuisance parameters. One example occurs in 2×2 tables, where conditioning on all four marginal totals leads to a conditional likelihood based on the non-central hypergeometric distribution. This form of conditioning is also the basis for Fisher's exact test. Marginal likelihood Sometimes we can remove the nuisance parameters by considering a likelihood based on only part of the information in the data, for example by