[BioC] technical reps, limma - theory

[Naomi Altman]

Consider the simple one-way design, with biological and technical 
reps.  Generally we would consider the biological reps to be blocks.  (For 
simplicity, we can think of this as a one-channel analysis).  The usual 
ANOVA seems to be at odds with what we get from limma (ignoring the eBayes 
step, which clearly cannot be recovered from the classical treatment).

The usual ANOVA (t treatments, b biological reps, n technical reps within 
biological reps, giving ntb arrays)

source          df              MS            F

treatment    t-1            MS(T)          MS(T)/MS(B)
bio rep        b-1           MS(B)          MS(B)/MSE (although we don't 
really care about this)
error=T*B    ntb-t-b+1   MSE

However, the Limma manual suggests using duplicateCorrelation and block to 
handle block designs, and this gives a different ANOVA.  In particular, the 
error d.f. for this ANOVA is ntb-t.  Using this method, the within block 
correlation is used in computing the t-statistics for the treatment, so you 
do not get the simple 1-way ANOVA that would come from ignoring block, but 
you cannot recover the p-value from the usual ANOVA, either.

If you put in bio rep as a fixed factor, then Limma will use the MSE as the 
denominator for the contrast tests, so this also does not recover the ANOVA.

I have not tried this computation with replicate spots (only replicate 
arrays) but either:

1) the usual ANOVA is right and duplicateCorrelation is doing something odd
2) duplicate Correlation is right and I don't understand the usual ANOVA
3) both methods are correct for somewhat different models, and I don't 
understand the statistical implications of this

I am not discounting 3 - as the statisticians in the crowd know, there are 
2 versions, constrained and unconstrained, for the simplest case of 
balanced ANOVA with fixed and random effects which lead to different 
p-values for some effects and both are defensible for almost any data set.

Anyways, I would like to understand this better.  So, I would welcome comments.

Naomi S. Altman                                814-865-3791 (voice)
Associate Professor
Bioinformatics Consulting Center
Dept. of Statistics                              814-863-7114 (fax)
Penn State University                         814-865-1348 (Statistics)
University Park, PA 16802-2111