LOCO CRT: Conditional Randomization Test

• LOCO–CRT is a method for assessing conditional independence in the model-X framework by evaluating the impact of omitting individual covariates.
• It computes predictive loss differences between full and leave-one-out models to yield robust, finite-sample valid p-values for each variable.
• The approach reduces computational cost through model reuse and supports analytic solutions in Gaussian covariate cases, enhancing its practical applicability.

The leave-one-covariate-out conditional randomization test (LOCO–CRT) is a computationally efficient methodology for assessing conditional independence within the model-X framework. It tests the null hypothesis that a response variable $Y$ is independent of a given covariate $X_j$ conditional on all other covariates $X_{-j}$, under the assumption that the marginal distribution of the covariate vector $X$ is known or can be accurately sampled. LOCO–CRT yields valid $p$-values useful for error-rate control in variable selection, minimizing algorithmic randomness and enabling practical application even in high-dimensional settings (Katsevich et al., 2020).

1. Statistical Formulation and Model-X Framework

The setup assumes i.i.d. samples $(X_i, Y_i)$, $i=1,\ldots,n$, with $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$ and arbitrary $Y_i$. The model-X assumption requires that the joint distribution $P_X$ of $X_j$0 is fully known or directly sampleable, while $X_j$1 remains unrestricted. For each variable $X_j$2, the test targets

$X_j$3

where $X_j$4 denotes all features except $X_j$5. This hypothesis asserts that, conditional on $X_j$6, $X_j$7 carries no information regarding $X_j$8.

2. LOCO Test Statistic

Given a loss function $X_j$9, e.g., squared loss for regression or logistic loss for classification, define:

• $X_{-j}$0: any fitted predictor trained on $X_{-j}$1,
• $X_{-j}$2: the same model class fitted using $X_{-j}$3.

The observed LOCO statistic for coordinate $X_{-j}$4 is: $X_{-j}$5 $X_{-j}$6 quantifies the change in predictive loss incurred by omitting $X_{-j}$7. Under $X_{-j}$8, $X_{-j}$9 is expected to be small; under the alternative, it should be large.

3. Algorithmic Procedure and P-value Calculation

The LOCO–CRT algorithm uses null randomization to obtain a valid $X$0-value for each variable. For each $X$1:

1. Fit both $X$2 (on $X$3) and $X$4 (on $X$5).
2. Compute $X$6 as the average loss difference.
3. For $X$7:
   • For each $X$8, sample $X$9.
   • Construct $p$0.
   • Compute $p$1, the analogous test statistic using the null-resampled $p$2.
4. Calculate

$p$3

Because $p$4 are exchangeable under $p$5, $p$6 is valid in finite samples. For simultaneous inference across coordinates, conventional multiplicity corrections (Bonferroni, Holm, Benjamini–Hochberg) can be applied.

4. Theoretical Guarantees

The principal theoretical result is finite-sample validity of LOCO–CRT $p$7-values: $p$8 given correct randomization from $p$9 and i.i.d. sampling. Under the null, $(X_i, Y_i)$0 is a super-uniform $(X_i, Y_i)$1-value. Familywise error rate (FWER) can be controlled at level $(X_i, Y_i)$2 by rejecting all $(X_i, Y_i)$3 with $(X_i, Y_i)$4. The computational efficiency is achieved by fitting $(X_i, Y_i)$5 and $(X_i, Y_i)$6 only once per variable; null sampling changes only $(X_i, Y_i)$7, not the fitted models.

5. L1ME–CRT Variant for L1-regularized M-Estimators

For L1-regularized estimators (e.g., Lasso, elastic net), refitting after each variable exclusion is computationally intensive. The L1ME–CRT modification capitalizes on the empirical observation that, for coordinates $(X_i, Y_i)$8 with $(X_i, Y_i)$9 in the full-data fit, the cross-validated penalty parameter $i=1,\ldots,n$0 is typically stable after exclusion, under restricted eigenvalue and Lipschitz loss conditions: $i=1,\ldots,n$1 Hence, for the “inactive” set $i=1,\ldots,n$2, the same $i=1,\ldots,n$3 can be reused for $i=1,\ldots,n$4, obviating additional cross-validations. This reduces computational overhead to near the number of “active” variables, $i=1,\ldots,n$5 in sparse regimes.

6. Closed-form Solution in the Multivariate Gaussian Covariate Case

Assuming $i=1,\ldots,n$6, the conditional law

$i=1,\ldots,n$7

with $i=1,\ldots,n$8, enables analytic computation. With squared error loss and ordinary least squares,

$i=1,\ldots,n$9

where $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$0 and $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$1 are fitted values excluding and including $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$2. Under $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$3, $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$4 follows a $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$5 distribution, and the $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$6-value is given by

$X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$7

recovering the classical partial $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$8-test or $X_i = (X_{i1}, \dots, X_{ip}) \in \mathbb{R}^p$9-test for a single coefficient in normal linear regression, with no need for Monte Carlo.

7. Computational and Practical Considerations

Let $Y_i$0 denote the cost of fitting a model and $Y_i$1 the cost of scoring $Y_i$2 samples:

• A single LOCO–CRT test requires $Y_i$3.
• Testing all $Y_i$4 features costs $Y_i$5, but with L1ME–CRT, the actual model refits may be far fewer than $Y_i$6 due to reuse for inactive features.

Typically, $Y_i$7 is $Y_i$8 for OLS or $Y_i$9 for Lasso; $P_X$0 is $P_X$1. Monte Carlo sample sizes $P_X$2–2000 are common, balancing $P_X$3-value granularity and computational burden.

Implementation suggestions include precomputing random seeds for reproducibility, vectorizing null-feature sampling when feasible, and using persistent model objects in languages like R or Python to avoid redundant refitting. These practicalities further enhance runtime efficiency.

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For foundational and related methodologies, see "Panning for gold: 'model-X' knockoffs for high-dimensional controlled variable selection” (Candès, Fan, Janson & Lv), “Gene hunting with hidden Markov model knockoffs” (Sesia, Candès & Sabatti), and “Multiple testing with the conditional randomization test” (Li & Barber) in addition to the primary development of LOCO–CRT (Katsevich et al., 2020).