If we have 4 levels of first treatment and 4 levels of second treatment in first year and 4 levels of first treatment and 2 levels of second treatment in second year. Is it called unbalanced design, unbalanced data or unbalanced replications?
If we have 4 levels of first treatment and 4 levels of second treatment in first year and 4 levels of only first treatment in second year. Is it called unbalanced design, unbalanced data or unbalanced replications?
It's called an incomplete factorial design. "Unbalanced" generally means that sample sizes for different treatment combinations are unequal; "incomplete" means that sample sizes for some treatment combinations are zero.
There is a literature out there about estimation of effects and contrasts for incomplete designs, but I think much of that literature is based on a carefully planned (a priori) pattern of incompleteness. In that sense, it's like fractional factorial design in that the actual design must be carefully chosen so that you are able to estimate the effects that are of the most interest to you.
If you give SAS procedures like MIXED or GLIMMIX a factorial design for which the data are incomplete, it generally will return results for overall tests of A, B, Year and their interactions. You'll note that degrees of freedom will reflect the missing combinations. Certain lsmeans will be reported as non-estimable.
Although people have brought me incomplete designs, I can't say that anyone has brought me one that was planned. So typically I suggest analyzing "slices" through the design space so that each slice is a complete factorial. For example, I might analyze the complete 2x2x2 for AxBxYear in one analysis and the 4x4 for AxB for just the first year, and then pull results together in interpretation. Not very elegant, but functional.