[R-sig-ME] REML=TRUE with non-Gaussian responses

John Maindonald john@m@|ndon@|d @end|ng |rom @nu@edu@@u
Wed Mar 25 07:29:05 CET 2020

There are comments that Maira might find helpful at

The usual (unbiased) variance estimate in an lm() type model
is effectively a REML estimate, not a maximum likelihood
estimate.   Maximum likelihood estimates of the residual
variance are highly biased in small samples.  The older (but
still, where it can be used, insightful) analysis of variance
table for suitably balanced designs extended this idea.
Expected values of stratum mean squares are, except for the
Residual stratum, linear combinations of variance components,
(well, I have simplified somewhat) and give equations that can
be successively solved to obtain what are basically the variance

See Searle’s article at https://doi.org/10.3168/jds.s0022-0302(91)78599-8

“[This] ANOVA method of estimating variance components [is] ...
  based on equating ex- pected mean squares to their values
computed from data."

REML estimates extend these ideas.

John Maindonald             email: john.maindonald using anu.edu.au<mailto:john.maindonald using anu.edu.au>

On 25/03/2020, at 14:02, Ben Bolker <bbolker using gmail.com<mailto:bbolker using gmail.com>> wrote:

This is a good question, but we (I) need to know a little bit more about
the level of explanation you're looking for.

In a book chapter of mine (Chapter 13 in Fox et al., Ecological
statistics: contemporary theory and application) I wrote

A broader way of thinking about REML is that it  describes any
statistical method where we integrate over the fixed effects when
estimating the variances.

 That's clear, but very short.

 Or you might prefer:

Bellio and Brazzale Stat Comput (2011) 21: 173–183DOI

The restricted likelihood function was originally defined as  the
marginal  likelihood  of  a  set  of  residual  contrasts (Patterson
and  Thompson1971).  Alternatively, it may  be computed as an integrated
likelihood following a Bayesian argument (Stiratelli et al.1984), or as
a modified profile likelihood function (Severini2000, Chap. 9).

Millar's book says (in section 9.3):

For normal linear models (including mixed-effects models), parameters ψ
(the variance parameters) and λ ≡ β (the regression coefficients) are
orthogonal, and the Laplace approximation in (9.18) is exact since the
log-likelihood is quadratic in λ. It follows that REML is equivalent to
integrated likelihood in these models. This equivalence was first
reported by Harville (1974) ...

Harville, D. A. (1974) Bayesian inference for variance components using
only error contrasts, Biometrika 61: 383–385.

 So; can you say a little more about what you mean by "I would like to
understand what this exactly does"?

On 2020-03-24 1:59 p.m., Maira Fatoretto wrote:

I have a question about glmmTMB, when I using REML=TRUE for binomial

They suggest It may also be useful in some
cases with non-Gaussian responses (Millar 2011). However, I would like to
understand what this exactly does because in Millar (2011) there is not a
clear explication about this.

Thank you.

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