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 and SAS give the same results.
> 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