Re-Post: Combining Factors in model.matrix

You will need to explain to us why the object you list is `the design
matrix': have *you* a reference for that?  R is doing the conventional
thing, and I at least have no idea where your example comes from.

Perhaps I have used the wrong terminology?  My understanding of a design
matrix is that it identifies the factors are present for a
given experiment.

The design matrix is X in the regression usually represented by

y = Xb + e

and is called a model matrix in S/R.

Here, I have a two factor experiment, where each factor has two levels.
In the case I gave:
   t1 t2
1   1  0
2   1  1
3   0  0
4   0  1

I had expected this to represent four distinct experiments where
factor one is present in the first two and absent in the second two.

You can't have factors that are present/absent.  (You can have levels of
treatments which are present/absent.)  We understand the rows to represent
the individuals runs of a single experiment, but what do the columns
represent?

You seem to have coded variables t1 and t2 the opposite ways (the
reference level is 2 for t1 and 1 for t2), and your model has

the fit at

levels t1=2,t1=1 constrained to pass through the origin.  I

don't think R

has a simple syntax for that (although you can fake

anything), and I find

it hard to believe that is actually what you want.

That wasn't my intention, I want to retain the intercept term and
not constrain it to pass through the origin.

So why did you use ~ -1 + (t1+t2) ?  That explicitly removes the
intercept.

t1 <- factor(c(1,1,2,2));
t2 <- factor(c(1,2,1,2));
design <- model.matrix(~ t1+t2);
design;