Abstract
<title>Abstract</title> <p> <bold>Background.</bold> The test of independence in a two-way contingency table is among the most common procedures in medicine and the social sciences, yet the software defaults, Pearson's chi-square and Fisher's exact test, fail in two directions: they misbehave when the table is sparse and collapse when the margins are heterogeneous. No existing statistic holds both calibration and power across the whole plane of table shape, sparsity, and marginal balance. <bold>Methods.</bold> We derive T_root, a closed-form, parameter-free statistic for independence. It variance-stabilizes each cell with the Anscombe root, centers it at its own mean under the exact margin-conditional null, and refers the sum of squared residuals to a three-moment chi-square from that null. Conditioning on the margins makes the reference mean an identity rather than an approximation, so no fitted constant enters the construction. We validate it by Monte Carlo over an exhaustive grid of shapes from 2x2 to 100x100 and eleven levels of marginal heterogeneity (about 1.4 million cells), benchmarked against an exact permutation reference. <bold>Results.</bold> Above a stated expected-count screen, T_root holds nominal size at every shape. Its mean absolute size error over 378,415 well-estimated cells is 0.0099, against 0.0238 for the Cressie-Read 2/3 statistic and 0.0412 for Pearson's. Its size stays flat along the heterogeneity axis while Pearson's rises to 0.171, and it holds nominal to an expected-count CVe of 4.5, where even a three-moment reference on the exact moments of Pearson's X^2 reaches six times nominal. Its power matches Pearson's where both control size, and it holds up in the heterogeneous collapse zone where the 2/3 member fails. Every result is a single deterministic p-value, with no permutation or bootstrap. <bold>Conclusions.</bold> T_root is a comprehensive closed-form statistic for contingency-table independence, sparse or heterogeneous. It subsumes the Cressie-Read 2/3 statistic and collapses a fragmented toolkit of exact, asymptotic, corrected, and routed tests into a single closed-form choice. The one remaining corner is small count, where the expected-count screen routes the table to the exact margin-conditional test that guarantees size. </p>