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For mixed model, when we estimate variance component by restricted maximum likelihood estimation procedure, an iterative algorithm is required to solve the estimating equations for variance component. Two main iterative algorithms are :

(1) Expectation-Maximization Algorithm ;

(2) Newton-Raphson Algorithm .

I am using lmer to estimate parameters of mixed model by restricted maximum likelihood estimation method. But I don't know which iterative algorithm it uses for solving the estimating equations for variance components. Does it use (1) or (2) or other iterative algorithm ?

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Neither of the above. The paper describing the algorithms used in lmer [Bates et al. J. Statistical Software (2015) 1-48, available via vignette("lmer",package="lme4")] specifies that the likelihood conditional on top-level variance-covariance parameters is computed by penalized least squares (section 3.6); the algorithm then uses a derivative-free nonlinear optimizer (often Powell's BOBYQA algorithm) to minimize the negative log-likelihood over the space of variance-covariance parameters.

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