andrey.core.ci.NormalScores

class andrey.core.ci.NormalScores(data)

Fisher-Z test on the normal scores of each column: a rank-based test for monotone effects.

Each column is replaced by its normal scores Phi^-1(rank / (n + 1)) (van der Waerden), with tied values sharing their average rank, and the FisherZ test runs on the scores. The scores depend on a column only through its ranks, so the p-value of every query is unchanged by a strictly increasing transformation of any column. For data whose columns are monotone transformations of jointly Gaussian variables, a Gaussian copula or nonparanormal model (Liu, Lafferty and Wasserman 2009), the scores correlate as the latent Gaussian variables do, and the partial correlation of the scores tests the latent conditional independence. Where every column is Gaussian the scores lose little: the test has asymptotic relative efficiency 1 against Fisher-Z there.

A query reads the ranks of the data this test was built over. Under test-wise deletion (andrey.core.ci_missing), the test of a query is built over the rows its columns observe, so the ranks are taken within them. See FisherZ for the p-value and PartialCorrelationTest for the query paths.

Notes

The scores estimate the latent correlation without the sine transformation of Spearman’s rho that Harris and Drton (2013) use for the same model. On the grid measured for this test, the sine-transformed Spearman estimate rejected a true null more often than the nominal level (0.06 to 0.09 at 0.05 for n = 200 and 500, 2,000 runs per cell), which the scores did not.

The scores keep the level where Fisher-Z loses it. Each cell below is the fraction of 1,000 simulated data sets in which a true conditional independence is rejected at level 0.05. The columns are jointly Gaussian variables, passed through the identity (gauss), exp, x**3, or a heavy-tailed monotone map (heavy); the pair is independent given |S| common causes:

n     |S|   fisherz (gauss, exp, cubic, heavy)   normal_scores (gauss, exp, cubic, heavy)
200   0     0.051 0.042 0.045 0.064              0.048 0.045 0.054 0.058
200   2     0.042 0.238 0.285 0.312              0.046 0.054 0.058 0.044
500   0     0.042 0.041 0.055 0.056              0.045 0.052 0.048 0.053
500   2     0.050 0.325 0.437 0.460              0.048 0.050 0.042 0.056
2000  0     0.048 0.057 0.048 0.053              0.046 0.040 0.037 0.056
2000  2     0.041 0.517 0.558 0.535              0.041 0.036 0.051 0.057

On pc at its default level, over 10 random DAGs per row with expected degree 3, the paired difference in skeleton errors (normal_scores minus fisherz; negative favors normal_scores), with a 95% interval, is:

d    n      Gaussian        exp             mixed monotone
10   500    -0.1 +- 0.2     -4.2 +- 1.9     -1.9 +- 1.3
10   2000   +0.0 +- 0.0     -3.2 +- 2.3     -2.9 +- 1.9
20   500    +0.1 +- 0.2     -13.2 +- 3.0    -8.9 +- 2.8
20   2000   +0.2 +- 0.5     -16.7 +- 4.6    -11.0 +- 2.5

The test is as fast as Fisher-Z: a fit on 10 to 20 columns and 500 to 2,000 rows takes between 4 and 20 milliseconds for both. Its p-values are the same whatever the marginals, as the ranks do not see them.

Limitations. Ranks measure monotone dependence: a relation that is not monotone, such as y = x**2 for a symmetric x, has zero rank correlation, and the test finds nothing where "kci", "rcot", "cmiknn", and "gpdc" do. Conditioning removes only the monotone part of a dependence. The p-value is the Fisher-Z one, which holds asymptotically.