Re: Next step in covariance matrix

[Jason Stover]

On Tue, Oct 27, 2009 at 06:38:19AM +0000, John Darrington wrote:
> Just to make sure I understand things correctly, consider the following 
> example, 
> where x and y are numeric variables and A and B are categorical ones:
> 
> x y A B
> =======
> 3 4 x v
> 5 6 y v
> 7 8 z w
> 
> We replace the categorical variables with bit_vectors:
> 
> x y A_0 A_1 A_2  B_0 B_1
> ========================
> 3 4  1   0   0    1   0
> 5 6  0   1   0    1   0
> 7 8  0   0   1    0   1
> 
> and arbitrarily drop the (say zeroth) subscript:
> 
> x y  A_1 A_2   B_1
> ==================
> 3 4   0   0     0
> 5 6   1   0     0
> 7 8   0   1     1
> 
> That will produce a 5x5 matrix. 5 is calculated from n + m - p,  where 
> n is the number of numeric  variables, m is the total number of  categories,
> and p is the number of categorical variables.  

This is correct. 

> However I don't see how such a matrix can be very useful. A better one would 
> involve 
> the  products of the categorical and numeric variables:
> 
> x y  x*A_1 x*A_2  y*A_1 y*A_2   x*B_1 y*B_1
> ===========================================
> 3 4     0   0        0     0       0     0
> 5 6     5   0        6     0       0     0
> 7 8     0   7        0     8       7     8
> 
> This makes an 8x8 matrix, where 8 is calculated from n + n * (m - p) , 
> which happens to be identical to n * (1 + m - p).  But this involves
> a whole lot more calculations.

This second choice would give you the covariance of x and y, and the
covariances of the *interactions* between x and A, x and B, y and A,
and y and B, but not the covariance between (say) x and A. The
covariance between x and A would be stored in the first matrix you
mentioned, in elements (0,2), (0,3), (2,0) and (3,0) assuming we kept
both upper and lower triangles.

You mention that matrix not being very useful, and in a sense it
isn't: No human would care about the covariance between x and the
column corresponding to the first bit vector of A. But in another
sense, that matrix is absolutely necessary: It's used to solve the
least squares problem, whose solution we use to tell us if A and our
dependent variable are related. That relation is shown via analysis of
variance, whose p-value is many computations away from the covariance
matrix, but depends on it nevertheless.

This matrix is unnecessary for a one-way ANOVA, whose computations from
the matrix above can be simplified into the simple sums used in
oneway.q.  But for a bigger model, with many factors and interactions
and covariates, we need that first matrix because we can't reduce the
problem to a few easy-to-read summations.