The structural model for a nested design with fixed factor A and random factor B is given by the formula
where μ and the αi are constants, the βj(i) and εijk are variables and αi is the effect of the ith level in Factor A and βj(i) is the effect of the jth level in Factor B nested within the ith level of Factor A.
We assume that for all i, j and k, the εijk are pairwise independent and the βj(i) are independent of the error terms. In addition, we assume that
We can conclude that for all i, j and k, the xijk are independent and normally distributed with mean
and variance
The estimators of μ, αi, βj(i) and εijk are as for fixed factor nested models, as are the values for SSA, SSB(A), etc. This time we have
The null and alternative hypotheses are as follows. There is no interaction between factors A and B.
| Test Desired | Null H0 | Alt Hyp H1 | Statistical Test |
| Effect of factor A | αi = 0 for all i | αi ≠ 0 for some i | MSA/MSB(A) ∼ F(dfB,dfB(A)) |
| Effect of B(A) | βj = 0 for all j | βj ≠ 0 for some j | MSB(A)/MSE ∼ F(dfB(A),dfE) |
An unbiased estimator of 
Dear Charles,
could in another mixed nested design the fixed factor A also be nested within the random factor B?
Would it be possible to compare levels of A within one level of B?
Regards,
Carl
Carl,
I only know about the three combinations described on the website. See also https://online.stat.psu.edu/stat503/lesson/14/14.1
I haven’t considered a design with a fixed factor nested within a random factor (or even whether this makes sense). See also
https://stats.stackexchange.com/questions/232109/does-it-make-sense-for-a-fixed-effect-to-be-nested-within-a-random-one-or-how-t
Charles
Charles,
For testing the effect of the random factor B(A), should the null hypothesis be the variance of that random factor equals zero instead? And the alternative should be the variance does not equal zero. Please confirm.
-Sun
Hello Sun,
We can test either the random factor or the fixed factor. For Example 1 the random factor consists of the offices and the fixed factor consists of the training programs. Although we can test either factor, we aren’t really interested in the random factor, only the fixed factor.
Charles
Dear Charles,
Let me tell you, this site is wonderful and really good.
I was thinking, whether it is possible to do four way nested anova? If yes, can you please share the same.
Regards,
It is probably possible, but Real Statistics doesn’t support such a design.
It is pretty difficult to interpret a 3-way design. I would guess that I would get a big headache if I were to try to interpret a 4-way design.
Charles
Dear Sir
my heartfelt thanks for your wonderful support in statistical analyses. I need your guidance on how to carry out Mixed model analysis with nested variable. I am asking this pertaining to analysis of multiple choice assay. Reviewers asked me to carry out this on my data. Kindly guide me.
thanks in advance
regds
Jayanthi
Do you have a specific question_
Charles