## Difference between quasi-likelihood and pseudo-likelihood

Super Contributor
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# Difference between quasi-likelihood and pseudo-likelihood

May I request someone to explain the difference between quasi-likelihood and pseudo-likelihood in simple terms?

SAS Super FREQ
Posts: 3,839

## Re: Difference between quasi-likelihood and pseudo-likelihood

I recommend reading the SAS/STAT Users Guide section "Classical Estimation Principles."

Among the good information there are these statements:

• Quasi-likelihood methods do not require that the joint distribution of the data be specified. These methods derive estimators based on only the first two moments (mean and variance) of the joint distributions and play an important role in the analysis of correlated data
• The pseudo-likelihood concept is also applied when the likelihood function is intractable, but the likelihood of a related, simpler model is available.

• An important difference between quasi-likelihood and pseudo-likelihood techniques is that the latter make distributional assumptions to obtain a likelihood function in the pseudo-model. Quasi-likelihood methods do not specify the distributional family.

Super Contributor
Posts: 499

## Re: Difference between quasi-likelihood and pseudo-likelihood

Thanks!

To complete understanding, may I request you to give a real time examples where these likelihood estimates takes place?

SAS Super FREQ
Posts: 3,839

## Re: Difference between quasi-likelihood and pseudo-likelihood

Because the likelihood formulas involve models, distributional assumptions, and formulas, understanding an example that compares maximum likelihood to pseudo- or quasi-likelihood will involve dealing with some math.

For an example that compares fill likelihood to quasi-likleihood, see the example "Quasi-likelihood Estimation for Proportions with Unknown Distribution"  Notice that the quasi-likelihood method involves assumptions about the DATA.

The pseudo-likelihood method can arises when you use a Taylor-series expansion to replace a complicated function or optimization by a simpler function, usually quadratic or linear.  You then solve the simpler model. In the GLIMMIX procedure, you can use the METHOD= option to choose a pseudo-likelihood model. For the grusome details, see a reference on Generalized Mixed Models Theory.

Super Contributor
Posts: 499

## Re: Difference between quasi-likelihood and pseudo-likelihood

In the example below, could you please tell me how to interpret Output 45.4.9?

https://support.sas.com/documentation/cdl/en/statug/68162/HTML/default/viewer.htm#statug_glimmix_exa...

SAS Super FREQ
Posts: 3,839

## Re: Difference between quasi-likelihood and pseudo-likelihood

By itself, Output 45.4.9 isn't very useful.  It becomes useful when you compare the model to other models. This example models the variance as _mu_**2 * (1-_mu_)**2. If you run further analyses with other potential forms for the variance function, the FitStatistics tables can be compared to help you determine which quasi-likelihood model for variance describes the data best.

Posts: 2,655

## Re: Difference between quasi-likelihood and pseudo-likelihood

The example you picked out uses a pseudo-likelihood (PL) method (restricted subject specific pseudolikelihood to be precise) to fit a quasi-likelihood model.  Think of quasi-likelihoods as the log likelihood divided by the deviance.  In a PL method the deviance is partialed out of the optimization step, and then re-introduced to calculate a quasi-likelihood.  Hence, it as a way of introducing overdispersion.

PL = linearization of the likelihood function, generally using a Taylor series expansion

QL = scaled likelihood, where the likelihood is divided by the deviance

I realize that this overgeneralizes what is going on, but it is how I generally think about the difference between the two.

Steve Denham

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