Variance components and REML "within" calculation

[#1]

How is the within variation calculated? even when there is only 1 data point available for a combination of factors?

I have a set of data with only one data point available for each factor combination and there is a pretty large amount of variation attributed to "within" and I'm curious how it is determining the within variation.

For example:

Day	Equipment	Analyst	Data
1	Z	A	##
1	Z	B	##
1	Y	A	##
1	Y	B	##
2	Z	A	##
2	Z	B	##
2	Y	A	##
2	Y	B	##

Thanks!

[#2]

You have 8 degrees of freedom in the data. You estimate the intercept (fixed effect) and three variance components (random effect), so you have sufficient degrees of freedom.

Your design with simulated response (null effects):