Consider a $3\times 3$ contingency table $N = (n_{ij})$ which can be regarded as a matrix in $\mathbb{N}^{3 \times 3}$, whose row-sums and column-sums are restricted to be $$ \sum_{j} n_{ij} = n_i, \quad \sum_{i}n_{ij} = n_j $$ for given $(n_1,n_2,n_3)$. Let $\text{vec}(N) \in \mathbb{R}^9$ be the vectorized version of $N$. Then, $\text{vec}(N)$ lies in a 4-dimensional subspace of $\mathbb{R}^9$. What is the best way to parametrize this subspace?
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Best always depends on context. But you can parameterize an $n\times n$ matrix with fixed row and column totals with the $(n-1)^2$ values of any $(n-1)\times (n-1)$ submatrix.
Be careful though - the values in a contingency table can only be non-negative integers, which gives them quite a different feel to vector space type objects.
