Permutation procedure when Z is continuous

Usage

perm_cont(X, Z)

Arguments

X: a numeric or factor vector of length n containing the variable to be tested (the condition to be tested).
Z: a numeric vector of length n containing the covariate. Multiple variables are not allowed.

Value

X_star a vector of permuted X.

Details

X_star is a conditional permutation of X: draws are without replacement, so X_star is a bijective rearrangement of X and its marginal distribution is preserved exactly. The permut for observation i is drawn with a weight that decreases with the distance between fitted values of X given Z, so they are matched on Z.

For binary X the fitted value is a linear-probability approximation of the propensity score (matching on the propensity score balances Z, [Rosenbaum and Rubin 1983] and allows to match on that single number rather than on Z). Of note, this approximation can fall outside [0, 1] when the association between X and Z is strong.

This is the conditional permutation test (CPT) of Berrett et al. (2020), which draws a permutation with probability proportional to the likelihood of the permuted assignment under the conditional law of X given Z. The sequential draw used here approximates that distribution rather than sampling from it exactly. Permuting rather than resampling X, as the conditional randomisation test (CRT) of Candes et al. (2018) does, preserves the empirical marginal distribution of X exactly regardless of the error in the fitted conditional law model (and never produces degenerate single-level designs).

Permutation weights are Gaussian in the distance between fitted values, with bandwidth sd(fit) * n^(-1/3) (polynomial weights leave too much mass on distant candidates, so the neighborhood was not local and the X-Z relationship was not preserved). Of note, the bandwidth exponent is larger than Silverman's 1/5 because his rate is optimized for density estimation but leaves a first-order bias that invalidates inference built on it (Hall, 1992; Armstrong and Kolesar, 2020); o(n^(-1/4)) is the condition Kim et al. (2022) prove for the analogous local permutation test.

References

Berrett TB, Wang Y, Barber RF, Samworth RJ (2020). The conditional permutation test for independence while controlling for confounders. Journal of the Royal Statistical Society Series B, 82(1), 175-197. doi:10.1111/rssb.12340

Candes E, Fan Y, Janson L, Lv J (2018). Panning for gold: 'model-X' knockoffs for high dimensional controlled variable selection. Journal of the Royal Statistical Society Series B, 80(3), 551-577. doi:10.1111/rssb.12265

Hemerik J, Goeman JJ (2018). Exact testing with random permutations. TEST, 27(4), 811-825. doi:10.1007/s11749-017-0571-1

Rosenbaum PR, Rubin DB (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41-55. doi:10.1093/biomet/70.1.41

Kim I, Neykov M, Balakrishnan S, Wasserman L (2022). Local permutation tests for conditional independence. The Annals of Statistics, 50(6), 3388-3414. doi:10.1214/22-AOS2233

Hall P (1992). Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density. The Annals of Statistics, 20(2), 675-694. doi:10.1214/aos/1176348651

Armstrong TB, Kolesar M (2020). Simple and honest confidence intervals in nonparametric regression. Quantitative Economics, 11(1), 1-39. doi:10.3982/QE1199

Examples


set.seed(123)
X <- rbinom(n = 100, size = 1, prob = 0.5)
Z <- rnorm(100, 0, 1)
X_star <- perm_cont(X, Z)
table(X, X_star)
#>    X_star
#> X    0  1
#>   0 30 23
#>   1 23 24