In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, quasi-likelihood methods are used to estimate parameters in a statistical model when exact likelihood methods, for example
maximum likelihood estimation, are computationally infeasible. Due to the wrong likelihood being used, quasi-likelihood estimators lose asymptotic efficiency compared to, e.g., maximum likelihood estimators. Under broadly applicable conditions, quasi-likelihood estimators are consistent and asymptotically normal. The asymptotic covariance matrix can be obtained using the so-called
sandwich estimator
A sandwich is a food typically consisting of vegetables, sliced cheese or meat, placed on or between slices of bread, or more generally any dish wherein bread serves as a container or wrapper for another food type. The sandwich began as a po ...
. Examples of quasi-likelihood methods are the
generalized estimating equations
In statistics, a generalized estimating equation (GEE) is used to estimate the parameters of a generalized linear model with a possible unmeasured correlation between observations from different timepoints. Although some believe that Generalized es ...
and pairwise likelihood approaches.
History
The term quasi-likelihood function was introduced by
Robert Wedderburn in 1974 to describe a function that has similar properties to the log-
likelihood function but is not the log-likelihood corresponding to any actual
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
.
[
] He proposed to fit certain quasi-likelihood models using a straightforward extension of the algorithms used to fit
generalized linear models.
Application to overdispersion modelling
Quasi-likelihood estimation is one way of allowing for
overdispersion, that is, greater variability in the data than would be expected from the
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of Sample (statistics), sample data (and similar data from a larger Statistical population, population). A statistical model repres ...
used. It is most often used with models for
count data or grouped binary data, i.e. data that would otherwise be modelled using the
Poisson or
binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
.
Instead of specifying a probability distribution for the data, only a relationship between the mean and the variance is specified in the form of a
variance function
In statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function is a measure of heteroscedasticity and plays a large role in many settings of statistica ...
giving the variance as a function of the mean. Generally, this function is allowed to include a multiplicative factor known as the overdispersion parameter or scale parameter that is estimated from the data. Most commonly, the variance function is of a form such that fixing the overdispersion parameter at unity results in the variance-mean relationship of an actual probability distribution such as the binomial or Poisson. (For formulae, see the
binomial data example and
count data example under
generalized linear models.)
Comparison to alternatives
Random-effects models
In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are dra ...
, and more generally
mixed models (
hierarchical models
A hierarchical database model is a data model in which the data are organized into a tree-like structure. The data are stored as records which are connected to one another through links. A record is a collection of fields, with each field containi ...
) provide an alternative method of fitting data exhibiting overdispersion using fully specified probability models. However, these methods often become complex and computationally intensive to fit to binary or count data. Quasi-likelihood methods have the advantage of relative computational simplicity, speed and robustness, as they can make use of the more straightforward algorithms developed to fit
generalized linear models.
See also
*
Quasi-maximum likelihood estimate
*
Extremum estimator
Notes
References
*
*
{{DEFAULTSORT:Quasi-Likelihood
Likelihood
Maximum likelihood estimation