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  "Title": "Exact Conditional Reference Distributions for Tables and\nSelected Linear Models",
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  "Description": "Reference distributions that are exact at every sample\nsize, obtained by conditioning on a statistic sufficient for\nthe nuisance parameter rather than by appealing to an\nasymptotic approximation. Two forms of conditioning are\nsupplied. For a two-way table the row and column totals are\nsufficient for the marginal parameters, so conditioning on both\nmargins leaves the Fisher-Yates law, sampled here exactly by\nrandom permutation of the column labels; statistics include the\nlinear-by-linear association of Goodman (1979)\n<doi:10.1080/01621459.1979.10481650> and, for nominal rows\nagainst an ordinal column, the dispersion of the row mean\nscores, whose components are the group contrasts themselves. A\ntwo-table form conditions further on the total score to cancel\na common association parameter, giving a nuisance-free test\nthat two tables share it. For a linear model under a null in\nwhich the tested structure is absent, a statistic invariant to\nthe coefficients and the error scale depends on the data only\nthrough the residual direction, which is uniform on the sphere\nof the residual space; the reference stays exact when the\nstatistic is maximised over candidate structures, and so\nremains valid after a spatial weight matrix has been chosen by\nfit. Permutation tests for a battery of categorical items and\nfor differences between correlation networks follow the\nnonparametric combination of Pesarin and Salmaso (2010)\n<doi:10.1002/9780470689516>, with exchangeability blocks as in\nWinkler et al. (2014) <doi:10.1016/j.neuroimage.2014.01.060>.\nEvery replicate draws from its own stream, so results do not\ndepend on the thread count. The same C sources back the\n'Python' package 'exactcond'.",
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