Class TwoIndicatorTest

java.lang.Object
edu.cmu.tetrad.search.rlcd.TwoIndicatorTest
public final class TwoIndicatorTest extends Object
Tells a direct edge between two indicators from a latent common cause, using third-order cumulants. Rank constraints cannot: if L --> X1, L --> X2 and X1 --> X2, then {X1, X2} against anything outside has rank 1, just as if a second latent sat over the pair, and RLCD reports the second latent.

Setting. X1 and X2 share a factor L with reference variables R1..Rm (m at least 2), whose only connection to X1 and X2 is through L and which share no other source among themselves. Errors are independent; the model is linear.

The part of the third cumulant k(Xi, Xj, Xk), i, j, k in {1, 2}, due to L is k(Xi, Ra, Rb) cov(Xj, Ra) cov(Xk, Rb) / cov(Ra, Rb)^2 for any two distinct references (averaged here over the ordered pairs). Subtracting it leaves residual cumulants R_ijk involving only the sources X1 and X2 share beyond L and their own errors. If X1 --> X2, X1's only source beyond L is the one it passes to X2, so T12 = R111 R122 - R112^2 = 0; if X2 --> X1, T21 = R222 R112 - R122^2 = 0; if a further latent (or correlated errors) is shared, both are nonzero provided the variables' own errors are skewed. Each is put on the scale n = T / (|first product| + second term squared), which lies in [-1, 1] and is unchanged by rescaling any variable.

The decision is conservative: an edge has to be shown, not merely not refuted. X1 --> X2 is returned when the 90% percentile bootstrap interval for n12 lies inside (-delta, delta) (an equivalence test at 5%) and the 95% interval for n21 excludes 0; X2 --> X1 likewise with the roles swapped. LATENT is returned when both 95% intervals exclude 0 and neither n is shown equivalent to 0. Anything else is UNDECIDED, which is the only answer possible when the errors are Gaussian (every n is then 0 or a ratio of noise).

In simulation (linear, three clean references, N = 5000), with delta = 0.1 no latent was read as an edge, and about half of the true edges were found; the rest came back undecided, or occasionally latent: n is a ratio of sample cumulants and has a small bias under a true edge at N in the thousands (about -0.016 at N = 5000 in one design, vanishing by N = 20000), which the percentile intervals carry. That errs toward keeping a latent. Basic (bias-corrected) intervals recover some of those edges but behave erratically under Gaussian errors, so they are not used. Weakly skewed or ceiling-truncated data, too few rows, or references that are not pure indicators of the same factor will all degrade it; the last can make it wrong rather than undecided. The simulation is impurity_cumulant_sim.py (kept outside the repository); edu.cmu.tetrad.search.harness.RlcdTwoIndicatorStudy measures the step inside RLCD.

Author:
josephramsey

  • Method Details

    • test

      public static TwoIndicatorTest.Result test(double[] x1, double[] x2, double[][] refs, double delta, int numBoot, long seed)
      Runs the test.
      Parameters:
      x1 - the first variable's values.
      x2 - the second variable's values.
      refs - the references' values, refs[r][row]; at least two.
      delta - the equivalence margin for n, in (0, 1).
      numBoot - the number of bootstrap resamples (a few hundred).
      seed - the bootstrap seed.
      Returns:
      the result; UNDECIDED with NaN statistics if there are too few complete rows.