Quasi-Likelihood Methods

Abstract

In this chapter, quasi-likelihood methods are shown. If the random component of a GLM is specified then the likelihood function can be used and the role of maximum likelihood method for estimating parameters of a model is well established. In GLM, the response or outcome variable follows a specific probability distribution under the family of exponential distributions. However, in many instances for non-normal errors with GLMs, such as for Poisson and binomial errors as fixed functions of the mean, the dispersion parameter cannot vary independently which restricts the use of GLM to some extent. In this situation, there is a need for models based on quasi-likelihood where exact likelihood is not necessary. The quasi-likelihood method depends on the first two moments, where the second moment is expressed as a function of the first moment. It may be noted that if there exists true likelihood of a distribution but does not belong to the exponential family of distributions then a quasi-likelihood can also be used alternatively. Hence, in the absence of a specified random component for GLMs where the distributions belong to exponential family of distributions or in some cases if the probability distributions may not belong to the family of exponential distributions, in both situations an alternative may be the use of quasi-likelihood method of estimation. In this chapter, quasi-likelihood functions are introduced and the estimation of parameters is illustrated for various data types.