David Rindt

Nonparametric measures of dependence

PhD student under Dino Sejdinovic and David Steinsaltz working on Nonparametric measures of dependence and variable selection. In particular I’m focussing on the Hilbert Schmidt independence criterion. We try to extend existing techniques to new types of data, such as censored data and rankings. I started my PhD in October 2017.

Publications

2021

• T. Fernandez , A. Gretton , D. Rindt , D. Sejdinovic , A Kernel Log-Rank Test of Independence for Right-Censored Data, Journal of the American Statistical Association, 2021.
  
  With the incorporation of new data gathering methods in clinical research, it becomes fundamental for survival analysis techniques to deal with high-dimensional or/and non-standard covariates. In this paper we introduce a general non-parametric independence test between right-censored survival times and covariates taking values on a general (not necessarily Euclidean) space X. We show that our test statistic has a dual interpretation, first in terms of the supremum of a potentially infinite collection of weight-indexed log-rank tests, with weight functions belonging to a reproducing kernel Hilbert space (RKHS) of functions; and second, as the norm of the difference of embeddings of certain finite measures into the RKHS, similar to the Hilbert-Schmidt Independence Criterion (HSIC) test-statistic. We study the asymptotic properties of the test, finding sufficient conditions to ensure that our test is omnibus. The test statistic can be computed straightforwardly, and the rejection threshold is obtained via an asymptotically consistent Wild-Bootstrap procedure. We perform extensive simulations demonstrating that our testing procedure generally performs better than competing approaches in detecting complex nonlinear dependence.

  @article{FerGreRinSej2021,
    author = {Fernandez, Tamara and Gretton, Arthur and Rindt, David and Sejdinovic, Dino},
    title = {{{A Kernel Log-Rank Test of Independence for Right-Censored Data}}},
    journal = {Journal of the American Statistical Association},
    year = {2021},
    doi = {10.1080/01621459.2021.1961784}
  }
  

• D. Rindt , D. Sejdinovic , D. Steinsaltz , Consistency of permutation tests of independence using distance covariance, HSIC and dHSIC, Stat, vol. 10, no. 1, e364, 2021.
  
  The Hilbert–Schmidt independence criterion (HSIC) and its d-variable extension dHSIC are measures of (joint) dependence between random variables. While combining these statistics with a permutation test has become a popular method of testing the null hypothesis of (joint) independence, it had thus far not been proved that this results in a consistent test. In this work, we provide a simple proof that the permutation test with the test statistic HSIC or dHSIC is indeed consistent when using characteristic kernels. That is, we prove that under each alternative hypothesis, the power of these permutation tests indeed converges to 1 as the sample size converges to infinity. Since the test is consistent for each number of permutations, we further give a brief discussion of how the number of permutations relates to the power of the test and how the number of permutations may be selected in practice.

  @article{RinSejSte2021,
    author = {Rindt, David and Sejdinovic, Dino and Steinsaltz, David},
    title = {{{Consistency of permutation tests of independence using distance covariance, HSIC and dHSIC}}},
    journal = {Stat},
    doi = {10.1002/sta4.364},
    volume = {10},
    number = {1},
    pages = {e364},
    year = {2021}
  }
  

2020

• D. Rindt , D. Sejdinovic , D. Steinsaltz , A kernel and optimal transport based test of independence between covariates and right-censored lifetimes, International Journal of Biostatistics, 2020.
  
  We propose a nonparametric test of independence, termed optHSIC, between a covariate and a right-censored lifetime. Because the presence of censoring creates a challenge in applying the standard permutation-based testing approaches, we use optimal transport to transform the censored dataset into an uncensored one, while preserving the relevant dependencies. We then apply a permutation test using the kernel-based dependence measure as a statistic to the transformed dataset. The type 1 error is proven to be correct in the case where censoring is independent of the covariate. Experiments indicate that optHSIC has power against a much wider class of alternatives than Cox proportional hazards regression and that it has the correct type 1 control even in the challenging cases where censoring strongly depends on the covariate.

  @article{RinSejSte2020,
    author = {Rindt, David and Sejdinovic, Dino and Steinsaltz, David},
    title = {A kernel and optimal transport based test of independence between covariates and right-censored lifetimes},
    journal = {International Journal of Biostatistics},
    year = {2020},
    doi = {10.1515/ijb-2020-0022}
  }
  

2019

• D. Rindt , D. Sejdinovic , D. Steinsaltz , Nonparametric Independence Testing for Right-Censored Data using Optimal Transport, ArXiv e-prints:1906.03866, 2019.
  
  We propose a nonparametric test of independence, termed OPT-HSIC, between a covariate and a right-censored lifetime. Because the presence of censoring creates a challenge in applying the standard permutation-based testing approaches, we use optimal transport to transform the censored dataset into an uncensored one, while preserving the relevant dependencies. We then apply a permutation test using the kernel-based dependence measure as a statistic to the transformed dataset. The type 1 error is proven to be correct in the case where censoring is independent of the covariate. Experiments indicate that OPT-HSIC has power against a much wider class of alternatives than Cox proportional hazards regression and that it has the correct type 1 control even in the challenging cases where censoring strongly depends on the covariate.

  @unpublished{RinSejSte2019,
    author = {Rindt, David and Sejdinovic, Dino and Steinsaltz, David},
    title = {{{Nonparametric Independence Testing for Right-Censored Data using Optimal Transport}}},
    journal = {ArXiv e-prints:1906.03866},
    year = {2019}
  }