# [BioC] Limma contrasts question

James W. MacDonald jmacdon at med.umich.edu
Wed Oct 22 15:07:16 CEST 2008

```Daniel Brewer wrote:

> Hi Jim,
>
> Could you go into the maths of the contrast formulas a bit?  I would
> like to get a really solid understanding of what it is doing for future
> analyses.

Once you understand what the coefficients are, the contrasts are just
simple algebra. In your case, all of the coefficients are estimating the
difference between the sample and PC3M (e.g., Knockdown - PC3M).

So the algebra is something like this:

2(Knockdown - PC3M) - (Scramble - PC3M)
=
2Knockdown - 2PC3M - Scramble + PC3M
=
2Knockdown - Scramble - PC3M
=
Knockdown - (Scramble + PC3M)/2

Which is knockdown minus the mean of the controls.

Note that this will be the numerator of the resulting t-statistic. The
denominator will be sort of an average of the variability within each of
the three groups being compared. So the question being answered is 'What
genes are different in Knockdown as compared to the average of the
controls?'. However, there is nothing here to test if the two controls
are similar at all (and you might not care).

So for instance, you might have a gene with average expression like this:

Knockdown = 10
PC3M = 4
Scramble = 7

If the intra-group variability is small for each sample type, then you
will likely get a significant t-statistic even though the two controls
are probably significantly different as well. Which is why I mentioned
earlier that you might want to test the Scramble - PC3M contrast as well.

Best,

Jim

>
> Many thanks
>
> Dan
>

--
James W. MacDonald, M.S.
Biostatistician
Hildebrandt Lab
8220D MSRB III
1150 W. Medical Center Drive
Ann Arbor MI 48109-0646
734-936-8662

```