How does lmer (from the R package lme4) compute log likelihood?

Question

I'm trying to understand the function lmer. I've found plenty of information about how to use the command, but not much about what it's actually doing (save for some cryptic comments here: http://www.bioconductor.org/help/course-materials/2008/PHSIntro/lme4Intro-handout-6.pdf). I'm playing with the following simple example:

library(data.table) library(lme4) options(digits=15)  n<-1000 m<-100 data<-data.table(id=sample(1:m,n,replace=T),key="id") b<-rnorm(m) data$y<-rand[data$id]+rnorm(n)*0.1 fitted<-lmer(b~(1|id),data=data,verbose=T) fitted 

I understand that lmer is fitting a model of the form Y_{ij} = beta + B_i + epsilon_{ij}, where epsilon_{ij} and B_i are independent normals with variances sigma^2 and tau^2 respectively. If theta = tau/sigma is fixed, I computed the estimate for beta with the correct mean and minimum variance to be

c = sum_{i,j} alpha_i y_{ij} 

where

alpha_i = lambda/(1 + theta^2 n_i) lambda = 1/[\sum_i n_i/(1+theta^2 n_i)] n_i = number of observations from group i 

I also computed the following unbiased estimate for sigma^2:

s^2 = \sum_{i,j} alpha_i (y_{ij} - c)^2 / (1 + theta^2 - lambda)

These estimates seem to agree with what lmer produces. However, I can't figure out how log likelihood is defined in this context. I calculated the probability density to be

pd(Y_{ij}=y_{ij}) = \prod_{i,j}[f_sigma(y_{ij}-ybar_i)]     * prod_i[f_{sqrt(sigma^2/n_i+tau^2)}(ybar_i-beta) sigma sqrt(2 pi/n_i)] 

where

ybar_i = \sum_j y_{ij}/n_i (the mean of observations in group i) f_sigma(x) = 1/(sqrt{2 pi}sigma) exp(-x^2/(2 sigma)) (normal density with sd sigma) 

But log of the above is not what lmer produces. How is log likelihood computed in this case (and for bonus marks, why)?

Edit: Changed notation for consistency, striked out incorrect formula for standard deviation estimate.

Answer 1

The links in the comments contained the answer. Below I've put what the formulae simplify to in this simple example, since the results are somewhat intuitive.

lmer fits a model of the form

, where

and

are independent normals with variances

and

respectively. The joint probability distribution of

and

is therefore

where

.

The likelihood is obtained by integrating this with respect to

(which isn't observed) to give

where

is the number of observations from group

, and

is the mean of observations from group

. This is somewhat intuitive since the first term captures spread within each group, which should have variance

, and the second captures the spread between groups. Note that

is the variance of

.

However, by default (REML=T) lmer maximises not the likelihood but the "REML criterion", obtained by additionally integrating this with respect to

to give

where

is given below.

Maximising likelihood (REML=F)

If

is fixed, we can explicitly find the

and

which maximise likelihood. They turn out to be

Note

has two terms for variation within and between groups, and

is somewhere between the mean of

and the mean of

depending on the value of

.

Substituting these into likelihood, we can express the log likelihood

in terms of

only:

lmer iterates to find the value of

which minimises this. In the output,

and

are shown in the fields "deviance" and "logLik" (if REML=F) respectively.

Maximising restricted likelihood (REML=T)

Since the REML criterion doesn't depend on

, we use the same estimate for

as above. We estimate

to maximise the REML criterion:

The restricted log likelihood

is given by

In the output of lmer,

and

are shown in the fields "REMLdev" and "logLik" (if REML=T) respectively.