Reference distributions that are exact at every sample size, obtained by conditioning on a statistic sufficient for the nuisance parameter rather than by appealing to an asymptotic approximation. Two forms of conditioning are supplied. For a two-way table the row and column totals are sufficient for the marginal parameters, so conditioning on both margins leaves the Fisher-Yates law, sampled here exactly by random permutation of the column labels; statistics include Goodman's linear-by-linear association and, for nominal rows against an ordinal column, the dispersion of the row mean scores, whose components are the group contrasts themselves. A two-table form conditions further on the total score to cancel a common association parameter, giving a nuisance-free test that two tables share it. For a linear model under a null in which the tested structure is absent, a statistic invariant to the coefficients and the error scale depends on the data only through the residual direction, which is uniform on the sphere of the residual space; the reference stays exact when the statistic is maximised over candidate structures, and so remains valid after a spatial weight matrix has been chosen by fit. Every replicate draws from its own stream, so results do not depend on the thread count. The same C sources back the 'Python' package 'exactcond'.
Exact conditional reference distributions for tables and for linear models whose structure was selected from the same data. Pure C back-end, shared byte for byte with the Python package of the same name.
A nuisance parameter is removed by conditioning on a statistic sufficient for it. What survives carries no nuisance, so a reference drawn from it is exact at every sample size, with no appeal to an asymptotic approximation and no bootstrap.
Both margins fixed. In an r by c table the row and column totals are sufficient for the marginal parameters, so conditioning on both leaves the Fisher-Yates law. It is sampled exactly, and by an argument needing no appeal to Patefield's algorithm: the conditional law given both margins is precisely the law of the table formed by pairing a fixed vector of row labels with a uniformly random permutation of the column labels.
Direction only. Under a null in which the tested structure does not enter the data-generating process, a statistic invariant to the regression coefficients and to the error scale depends on the data only through the residual direction, which is uniform on the sphere of the residual space whatever those nuisance parameters are. The reference stays exact when the statistic is maximised over candidates, and so remains valid after a spatial weight matrix has been selected by fit.
For nominal rows against an ordinal column, stat = "dispersion" is the between-row dispersion of
the mean score. Its components are the row mean scores themselves, so where the column score is a
signed difference the statistic is assembled out of the very contrasts being reported, and a reader
can follow the arithmetic from the table to the test. A rank test on such a variable answers a
different question, and answers it on data that are mostly ties.
library(exactcond)
tab <- matrix(c(3, 228, 72, 10, 548, 67, 7, 276, 131), nrow = 3, byrow = TRUE)
ec_rc_test(tab, v = c(-1, 0, 1), B = 200000, seed = 4207)
ec_rc_test() reports the largest null draw alongside the p-value, and says so when the observed
statistic exceeded every draw: that is a resolution floor, not a p-value, and the two should not be
written the same way. Every p-value comes with a Wilson interval for the Monte Carlo error in the
number of draws actually taken.
ec_rc_two() samples the paired law directly rather than convolving two separately estimated null
distributions. The convolution needs the observed score to be reachable under no association; for a
strongly associated table it is not, at any number of draws, and the failure arrives precisely when
both tables are strongly associated, which is the case in which the correct answer is a large
p-value.
ec_multi_test() answers "does ANY of these items differ across the groups" with ONE exact test:
the sum or the maximum over items of a per-item statistic, referred to the law obtained by permuting
the group labels across respondents or, when a cluster is given, across clusters that keep their
members. This is Pesarin's nonparametric combination with the direct and Tippett combining functions: one
p-value for the whole battery, so nothing to correct for multiplicity, and the per-item statistics
referred to the same permutations for a closed follow-up in the manner of Westfall and Young. The sum has power against
many items moving a little, the maximum against one moving a lot, and the cluster permutation is the
exact treatment of an interviewer or fieldworker effect, at negligible cost when there is none.
ec_diffnet_test() carries the same idea to a network. A differential network normally rests on a
Gaussian null; here the edge statistic is unchanged and only its reference is, from a chi-square to
a permutation of the group labels, which is exact whatever the marginals are and distribution-free
on ranks. One p-value for the whole network, per-edge p-values from the same permutations, and a
cluster permutation for interviewer effects. The route is the established one (NetCoMi's diffnet,
the Network Comparison Test, exchangeability blocks); what is here is the implementation.
Every replicate draws from its own stream, seeded from the base seed and the replicate index, so results do not depend on the thread count and are identical across the R and Python fronts. A byte-parity harness checks that agreement to fifteen significant figures over every entry point; it lives alongside the Python front rather than in this repository, because it needs both trees present to run.