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.