Nonparametric Conditional Local Independence Testing

TL;DR

A model-free framework for testing the hypothesis that a counting process is conditionally locally independent of another process is developed, and a new functional parameter called the Local Covariance Measure (LCM), which quantifies deviations from the hypothesis is introduced.

Abstract

Conditional local independence is an asymmetric independence relation among continuous time stochastic processes. It describes whether the evolution of one process is directly influenced by another process given the histories of additional processes, and it is important for the description and learning of causal relations among processes. We develop a model-free framework for testing the hypothesis that a counting process is conditionally locally independent of another process. To this end, we introduce a new functional parameter called the Local Covariance Measure (LCM), which quantifies deviations from the hypothesis. Following the principles of double machine learning, we propose an estimator of the LCM and a test of the hypothesis using nonparametric estimators and sample splitting or cross-fitting. We call this test the (cross-fitted) Local Covariance Test ((X)-LCT), and we show that its level and power can be controlled uniformly, provided that the nonparametric estimators are consistent with modest rates. We illustrate the theory by an example based on a marginalized Cox model with time-dependent covariates, and we show in simulations that when double machine learning is used in combination with cross-fitting, then the test works well without restrictive parametric assumptions.

Figures (10)

• Figure 1: Local independence graph illustrating a dependence structure among the three processes $X$, $Z$ and $N$. Here $N$ is the indicator of death for an individual, $X$ is their cumulative pension savings and $Z$ is a covariate process. All nodes in this graph have implicit self-loops. There is no edge from $X$ to $N$, which indicates that death is not directly influenced by pension savings. This can be formalized as $N$ being conditionally locally independent of $X$, which is the hypothesis we aim to test.
• Figure 2: Local independence graphs illustrating how the three processes $X$, $Y$, and $Z$ could affect each other and time of death in the Cox example. There is no direct influence of $X$ (pension savings) on time of death in either of the two graphs, but in the left graph the death indicator is furthermore conditionally locally independent of $X$ given the history of $Z$ and $N$. In the right graph, $Z$ and $N$ does not block all paths from $X$ to $N$, thus conditioning on the history of $Z$ and $N$ only would not render $N$ conditionally locally independent of $X$.
• Figure 3: Histograms of the distributions of three different estimators of $\gamma_1$. Each histogram contains 1000 estimates fitted to samples of size $n=500$. The samples were sampled from a model that satisfies the hypothesis of conditional local independence and hence the ground truth is $\gamma_1=0$. See Section \ref{['sec:SamplingScheme']} for further details of the data generating process.
• Figure 4: A time dependent extension of \ref{['fig:endpoint_example']} showing the distribution of the sample paths $t \mapsto \hat{\gamma}_{t, \mathrm{plug-in}}^{(500)}$ and $t \mapsto \hat{\gamma}_{t, \mathrm{double}}^{(500)}$, the latter with and without using cross-fitting. The data were simulated under $H_0$ where $t\mapsto \gamma_t$ is the zero function. See Section \ref{['sec:SamplingScheme']} for further details of the data generating process.
• Figure 5: Empirical cumulative distribution functions of simulated $p$-values for the cross-fitted local covariance test and the hazard ratio test. The simulated data satisfies the hypothesis of conditional local independence, so the $p$-values are supposed to be uniformly distributed, and the CDF should fall on the diagonal dotted line.

Theorems & Definitions (83)

• Definition 2.1: Conditional local independence
• Definition 2.2: Residual Process
• Definition 2.3: Local Covariance Measure
• Proposition 2.4
• Proposition 2.5
• Proposition 4.1
• Definition 4.2
• Proposition 4.3
• Proposition 4.4
• Proposition 4.5