Mixed model runs well in R whereas a random effect has only one level

Question

How do you explain that ? There's only one operator but the mixed model returns an estimate for the operator random effect. Furthermore the sample effect is confounded with the interaction sample:operator. Below is the R code.

> dd
   sample operator         y
9      10      SCF 0.9153188
10     10      SCF 0.9884982
19    100      SCF 2.0798781
20    100      SCF 2.0464027
29   1000      SCF 3.0401590
30   1000      SCF 3.0114448
39  10000      SCF 4.1348324
40  10000      SCF 4.0840063
49  1e+05      SCF 5.1235795
50  1e+05      SCF 5.1106381
59  1e+06      SCF 6.0803404
60  1e+06      SCF 6.2353263
> str(dd)
'data.frame':   12 obs. of  3 variables:
 $ sample  : Factor w/ 6 levels "10","100","1000",..: 1 1 2 2 3 3 4 4 5 5 ...
     $ operator: Factor w/ 1 level "SCF": 1 1 1 1 1 1 1 1 1 1 ...
 $ y       : num  0.915 0.988 2.08 2.046 3.04 ...
> lmer(y ~ (1|sample)+(1|operator)+(1|sample:operator), data=dd) 
Linear mixed model fit by REML 
Formula: y ~ (1 | sample) + (1 | operator) + (1 | sample:operator) 
   Data: dd 
  AIC   BIC logLik deviance REMLdev
 18.6 21.03 -4.302    9.932   8.605
Random effects:
 Groups          Name        Variance   Std.Dev.
 sample:operator (Intercept) 1.87954740 1.370966
 sample          (Intercept) 1.87954925 1.370967
 operator        (Intercept) 0.00063096 0.025119
 Residual                    0.00283931 0.053285
Number of obs: 12, groups: sample:operator, 6; sample, 6; operator, 1

Fixed effects:
            Estimate Std. Error t value
(Intercept)   3.5709     0.7921   4.508

For those who are more familiar with SAS the corresponding code is:

PROC MIXED DATA=dd;
CLASS sample operator;
MODEL y=;
RANDOM sample operator sample*operator;
RUN;

This is nothing but the crossed 2-way ANOVA with random effects.

Answer 1

I have asked this question on the r-sig-mixed-models mailing list and below is the answer from Douglas Bates:

Stéphane,

You are correct that there should not be an attempt to fit such a model. In theory the deviance function is degenerate (a singular Hessian matrix at every parameter value). In practice it will not be exactly degenerate and the optimizer will attempt to locate a minimum determined by round-off error.

Again, it is probably good to throw an error in this case and to warn when the number of levels in a grouping factor for random effects term(s) is small (say, < 5). I'll add an issue on the github repository, https://github.com/lme4/lme4/issues, regarding this.

By the way, the reason for printing out the number of levels in the factors as part of the random-effects summary is so that the user will notice when these are small, which usually means a mistake in the model formula. Of course, it requires that the user look at that part of the summary and anyone who has used SAS is accustomed to ignoring large parts of the output.

Anyway, the issue is now recorded as https://github.com/lme4/lme4/issues/24