science

Quasi-likelihood

Quasi-likelihood is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Quasi-likelihood rather than just read about it. In short: In statistics, 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.

Key takeaways

Quasi-likelihood belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Quasi-likelihood to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Quasi-likelihood from memory before moving on to harder problems.

Reference excerpt

In statistics, 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. Examples of quasi-likelihood methods include the generalized estimating equations 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. 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 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. 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 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, and more generally mixed models (hierarchical models) 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 McCullagh, Peter; Nelder, John (1989). Generalized Linear Models (second ed.). London: Chapman and Hall. ISBN 0-412-31760-5.{{cite book}}: CS1 maint: publisher location (link) Hardin, James; Hilbe, Joseph (2007). Generalized Linear Models and Extensions (second ed.). College Station: Stata Press.{{cite book}}: CS1 maint: publisher location (link)

Worked examples

Example 1 — a first encounter with Quasi-likelihood

Start with the simplest possible case. Write down what Quasi-likelihood claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quasi-likelihood before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quasi-likelihood ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quasi-likelihood

In research: Quasi-likelihood appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Quasi-likelihood in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Quasi-likelihood is common in secondary-school and first-year university syllabi. It links to neighbouring topics Likelihood, Maximum likelihood estimation, so understanding it makes those chapters shorter.
In everyday life: Look for Quasi-likelihood outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Quasi-likelihood in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Quasi-likelihood means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Quasi-likelihood out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quasi-likelihood in simple terms?

Why does Quasi-likelihood matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quasi-likelihood?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quasi-likelihood.