Question: Re: lm.series, design matrix
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16.2 years ago by
James Wettenhall1000 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
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