Question: Re: lm.series, design matrix

0

James Wettenhall •

**1000**wrote:Dear Chunrong,
[I've included a few maths formulae in this email, which will
only line up correctly in a fixed-width font like Courier.]
The linear model says that for each gene, the Expected Value of
the "observed" log-ratios vector is equal to the design matrix
multipled by the vector of log-ratio(s) (parameters) to be
estimated.
Sometimes estimating the log-ratio(s) of interest is
simply a matter of averaging the "observed" log-ratios from the
raw data. (But it's also nice to estimate confidence statistics
based on correlation between replicates.)
For the Swirl Zebrafish dye-swap example (in the Limma manual),
for each gene, there is only one log-ratio to be estimated
(Swirl Mutant versus Wild Type) so there is only one column
in the design matrix. Estimating the log-ratio for each gene
can be done by taking a weighted average using -1's for dye
swaps.
For the ApoAI KO data set, the design matrix would be easier
to construct if we chose to estimate parameters (log-ratios) for
(Wild Type versus Reference) and (ApoAI Knockout versus Reference),
because this is the way the arrays were hybridized. Then the
design matrix would look like this: (16 rows in total)
1 0
: :
1 0
0 1
: :
0 1
but because we are really interested in estimating log-ratios
for (Wild Type vs Reference) and (ApoAI Knockout vs Wild Type),
we have to post-multiply our design matrix above by the
matrix
1 0
1 1
[ PLEASE READ THIS IN COURIER FONT ]
because
M (WT vs Ref) 1 0 M (WT vs Ref)
[ ] = [ ] [ ]
M (KO vs Ref) 1 1 M (KO vs WT)
where M is a log ratio being estimated.
With the simpler parameterization, we had :
E{M_observed1(WT vs Ref)} 1 0
: : : M (WT vs Ref)
E{M_observed8(WT vs Ref)} = [1 0] [
E{M_observed1(KO vs Ref)} 0 1 M (KO vs Ref)
: : :
E{M_observed8(KO vs Ref)} 0 1
where E{*} is the Expected Value of *
So now we have:
E{M_observed1(WT vs Ref)} 1 0
: : : 1 0 M (WT vs Ref)
E{M_observed8(WT vs Ref)} = [1 0] [ ] [ ]
E{M_observed1(KO vs Ref)} 0 1 1 1 M (KO vs WT)
: : :
E{M_observed8(KO vs Ref)} 0 1
so :
E{M_observed1(WT vs Ref)} 1 0
: : : M (WT vs Ref)
E{M_observed8(WT vs Ref)} = [1 0] [ ]
E{M_observed1(KO vs Ref)} 1 1 M (KO vs WT)
: : :
E{M_observed8(KO vs Ref)} 1 1
Regards,
James