Why is it necessary to use ML estimation instead of REML to compare multilevel linear models?

Question

In Discovering Statistics Using SPSS 4e, Andy Field writes on p835 that:

SPSS gives you the choice of two methods for estimating the parameters in the analysis: maximum likelihood (ML), which we have encountered before, and restricted maximum likelihood (REML). The conventional wisdom seems to be that ML produces more accurate estimates of fixed regression parameters, whereas REML produces more accurate estimates of random variances (Twisk, 2006). As such, the choice of estimation procedure depends on whether your hypotheses are focused on the fixed regression parameters or on estimating variances of the random effects. However, in many situations the choice of ML or REML will make only a small difference to parameter estimates. Also, if you want to compare models you must use ML.

I'm most interested in the last sentence. Why is it that to compare models you must use ML and not REML?

Answer 1

(RE)ML estimation is an iterative process. ML estimate the variances as if the fixed parameters are known, so doesn't account for the degrees of freedom lost in their estimation. REML adjusts for the uncertainty about the fixed parameters. So you generally cannot use REML to compare models, because whatever difference in the fixed part (parameters and constrasts) invalidates the comparison.
However you can use REML to compare models if their fixed part are exactly equal.