exclusion_test {leakyIV}

Testing Exclusion

Description

Performs a Monte Carlo test against the null hypothesis that minimum leakage is zero, a necessary but insufficient condition for exclusion.

Usage

exclusion_test(
  dat,
  normalize = TRUE,
  method = "mle",
  approx = TRUE,
  n_sim = 1999L,
  parallel = TRUE,
  return_stats = FALSE,
  ...
)

Arguments

dat

Input data. Either (a) an n \times d data frame or matrix of observations with columns for treatment, outcome, and candidate instruments; or (b) a d \times d covariance matrix over such variables. Note that in either case, the order of variables is presumed to be treatment (X), outcome (Y), leaky instruments (Z). exclusion_test requires at least two candidate instruments Z.

normalize

Scale candidate instruments to unit variance?

method

Estimator for the covariance matrix. Options include (a) "mle", the default; (b) "shrink", an analytic empirical Bayes solution; or (c) "glasso", the graphical lasso. See details.

approx

Use nearest positive definite approximation if the estimated covariance matrix is singular? See details.

n_sim

Number of Monte Carlo replicates.

parallel

Run Monte Carlo simulations in parallel? Must register backend beforehand, e.g. via doParallel.

return_stats

Return observed statistic and simulated null distribution?

...

Extra arguments to be passed to graphical lasso estimator if method = "glasso". Note that the regularization parameter rho is required as input, with no default.

Details

The classic linear instrumental variable (IV) model relies on the exclusion criterion, which states that instruments Z have no direct effect on the outcome Y, but can only influence it through the treatment X. This implies a series of tetrad constraints that can be directly tested, given a model for sampling data from the covariance matrix of the observable variables (Watson et al., 2024).

We assume that data are multivariate normal and impose the null hypothesis by modifying the estimated covariance matrix to induce a linear dependence between the vectors for Cov(Z, X) and Cov(Z, Y). Our test statistic is the determinant of the cross product of these vectors, which equals zero if and only if the null hypothesis is true. We generate a null distribution by simulating from the null covariance matrix and compute a p-value by estimating the proportion of statistics that exceed the observed value. Future releases will provide support for a wider range of data generating processes.

Numerous methods exist for estimating covariance matrices. exclusion_test provides support for maximum likelihood estimation (the default), as well as empirical Bayes shrinkage via corpcor::cov.shrink (Schäfer & Strimmer, 2005) and the graphical lasso via glasso::glasso (Friedman et al., 2007). These latter methods are preferable in high-dimensional settings where sample covariance matrices may be unstable or singular. Alternatively, users can pass a pre-computed covariance matrix directly as dat.

Estimated covariance matrices may be singular for some datasets or Monte Carlo samples. Behavior in this case is determined by the approx argument. If TRUE, the test proceeds with the nearest positive definite approximation, computed via Higham's (2002) algorithm (with a warning). If FALSE, the sampler will attempt to use the singular covariance matrix (also with a warning), but results may be invalid.

Value

Either a scalar representing the Monte Carlo p-value of the exclusion test (default) or, if return_stats = TRUE, a named list with three entries: psi, the observed statistic; psi0, a vector of length n_sim with simulated null statistics; and p_value, the resulting p-value.

References

Watson, D., Penn, J., Gunderson, L., Bravo-Hermsdorff, G., Mastouri, A., and Silva, R. (2024). Bounding causal effects with leaky instruments. arXiv preprint, 2404.04446.

Spirtes, P. Calculation of entailed rank constraints in partially non-linear and cyclic models. In Proceedings of the 29th Conference on Uncertainty in Artificial Intelligence, 606–615, 2013.

Friedman, J., Hastie, T., and Tibshirani, R. (2007). Sparse inverse covariance estimation with the lasso. Biostatistics, 9:432-441.

Schäfer, J., and Strimmer, K. (2005). A shrinkage approach to large-scale covariance estimation and implications for functional genomics. Statist. Appl. Genet. Mol. Biol., 4:32.

Higham, N. (2002). Computing the nearest correlation matrix: A problem from finance. IMA J. Numer. Anal., 22:329–343.

Examples

 
set.seed(123)

# Hyperparameters
n <- 200
d_z <- 4
beta <- rep(1, d_z)
theta <- 2
rho <- 0.5

# Simulate correlated residuals
S_eps <- matrix(c(1, rho, rho, 1), ncol = 2)
eps <- matrix(rnorm(n * 2), ncol = 2)
eps <- eps %*% chol(S_eps)

# Simulate observables from the linear IV model
z <- matrix(rnorm(n * d_z), ncol = d_z)
x <- z %*% beta + eps[, 1]
y <- x * theta + eps[, 2]
obs <- cbind(x, y, z)

# Compute p-value of the test
exclusion_test(obs, parallel = FALSE)