  We present a symbolic decomposition of the Pearson chi-square statistic with
unequal cell probabilities, by presenting Hadamard-type matrices whose columns
are eigenvectors of the variance-covariance matrix of the cell counts. All of
the eigenvectors have non-zero values so each component test uses all cell
probabilities in a way that makes it intuitively interpretable. When all cell
probabilities are distinct and unrelated we establish that such decomposition
is only possible when the number of multinomial cells is a small power of 2.
For higher powers of 2, we show, using the theory of orthogonal designs, that
the targeted decomposition is possible when appropriate relations are imposed
on the cell probabilities, the simplest of which is when the probabilities are
equal and the decomposition is reduced to the one obtained by Hadamard
matrices. Simulations are given to illustrate the sensitivity of various
components to changes in location, scale skewness and tail probability, as well
as to illustrate the potential improvement in power when the cell probabilities
are changed.
