I'm confused about my design matrix for a loop design experiment. This is a
six-membered direct design that can be drawn as a hexagram inscribed in a
hexagon (where the corners are treatments, obviously).

Here is what my targets look like:

Cy3 Cy5
A   B
B   C
C   D
D   E
E  RC
RC   A
A   C
B   D
C   E
D  RC
E   A
RC   B

There are three ways I can conceive of constructing my design matrix for
this. The first is to use the modelMatrix function with the parameters
argument, but I don't really understand how this should be done, and this
might be my problem. The second is to declare an arbitrary reference and use
the modelMatrix function, as described in the user's manual:

designmatrix <- modelMatrix(targets, ref="RC")

When I do this I get the following design matrix

A  B  C  D  E
-1  1  0  0  0
0 -1  1  0  0
0  0 -1  1  0
0  0  0 -1  1
0  0  0  0 -1
1  0  0  0  0
-1  0  1  0  0
0 -1  0  1  0
0  0 -1  0  1
0  0  0 -1  0
1  0  0  0 -1
0  1  0  0  0

Which is fine, except when I use this to fit my data:

designfit <- lmFit(RG, design=designmatrix, method="ls")

I see the following annotation in the generated data:

Warning message:
design matrix is singular

>From reading threads on the listserv, I understand this to mean that some of
the coefficients are non-estimable, but I don't know why this is the case.
So as a third possibility, I thought I might create my own design matrix.  I
came up with the following matrix, which is apparently also wrong:

       A  B  C  D  E RC
 [1,]  1 -1  0  0  0  0
 [2,]  0  1 -1  0  0  0
 [3,]  0  0  1 -1  0  0
 [4,]  0  0  0  1 -1  0
 [5,]  0  0  0  0  1 -1
 [6,] -1  0  0  0  0  1
 [7,]  1  0 -1  0  0  0
 [8,]  0  1  0 -1  0  0
 [9,]  0  0  1  0 -1  0
[10,]  0  0  0  1  0 -1
[11,] -1  0  0  0  1  0
[12,]  0 -1  0  0  0  1

I say it's apparently wrong because when I fit the data, I get the
following:

> designfittest = lmFit(RGAn, design=design, method="ls")
Coefficients not estimable: RC

Is this due to the same singularity problem? I did note on another web site
(http://discover.nci.nih.gov/microarrayAnalysis/Experimental.Design.jsp)
that it is sometimes desirable to append the matrix with a row of 1's to
force a single solution to the fit and wondered if this was what I was
missing.  I know a few other people who are struggling with this same type
of issue, and we'd all love some clarification.

Thanks,

Matt

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