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Type: Package
Title: Nonlinear Conditional Independence Tests
Version: 0.1.5
Date: 2019-11-11
Author: Christina Heinze-Deml <heinzedeml@stat.math.ethz.ch>, Jonas Peters <jonas.peters@math.ku.dk>, Asbjoern Marco Sinius Munk <fgp998@alumni.ku.dk>
Depends: R (≥ 3.1.0)
Maintainer: Christina Heinze-Deml <heinzedeml@stat.math.ethz.ch>
Description: Code for a variety of nonlinear conditional independence tests: Kernel conditional independence test (Zhang et al., UAI 2011, <doi:10.48550/arXiv.1202.3775>), Residual Prediction test (based on Shah and Buehlmann, <doi:10.48550/arXiv.1511.03334>), Invariant environment prediction, Invariant target prediction, Invariant residual distribution test, Invariant conditional quantile prediction (all from Heinze-Deml et al., <doi:10.48550/arXiv.1706.08576>).
License: GPL-2 | GPL-3 [expanded from: GPL]
LazyData: TRUE
Imports: methods, randomForest, quantregForest, lawstat, RPtests, caTools, mgcv, MASS, kernlab, pracma, mize
URL: https://github.com/christinaheinze/nonlinearICP-and-CondIndTests
BugReports: https://github.com/christinaheinze/nonlinearICP-and-CondIndTests/issues
RoxygenNote: 6.1.1
Suggests: testthat
NeedsCompilation: no
Packaged: 2019-11-11 16:14:39 UTC; heinzec
Repository: CRAN
Date/Publication: 2019-11-12 06:50:21 UTC

Wrapper function for conditional independence tests.

Description

Tests the null hypothesis that Y and E are independent given X.

Usage

CondIndTest(Y, E, X, method = "KCI", alpha = 0.05,
  parsMethod = list(), verbose = FALSE)

Arguments

Y

An n-dimensional vector or a matrix or dataframe with n rows and p columns.

E

An n-dimensional vector or a matrix or dataframe with n rows and p columns.

X

An n-dimensional vector or a matrix or dataframe with n rows and p columns.

method

The conditional indepdence test to use, can be one of "KCI", "InvariantConditionalQuantilePrediction", "InvariantEnvironmentPrediction", "InvariantResidualDistributionTest", "InvariantTargetPrediction", "ResidualPredictionTest".

alpha

Significance level. Defaults to 0.05.

parsMethod

Named list to pass options to method.

verbose

If TRUE, intermediate output is provided. Defaults to FALSE.

Value

A list with the p-value of the test (pvalue) and possibly additional entries, depending on the output of the chosen conditional independence test in method.

References

Please cite C. Heinze-Deml, J. Peters and N. Meinshausen: "Invariant Causal Prediction for Nonlinear Models", arXiv:1706.08576 and the corresponding reference for the conditional independence test.

Examples


# Example 1
set.seed(1)
n <- 100
Z <- rnorm(n)
X <- 4 + 2 * Z + rnorm(n)
Y <- 3 * X^2 + Z + rnorm(n)
test1 <- CondIndTest(X,Y,Z, method = "KCI")
cat("These data come from a distribution, for which X and Y are NOT
cond. ind. given Z.")
cat(paste("The p-value of the test is: ", test1$pvalue))

# Example 2
set.seed(1)
Z <- rnorm(n)
X <- 4 + 2 * Z + rnorm(n)
Y <- 3 + Z + rnorm(n)
test2 <- CondIndTest(X,Y,Z, method = "KCI")
cat("The data come from a distribution, for which X and Y are cond.
ind. given Z.")
cat(paste("The p-value of the test is: ", test2$pvalue))

Invariant conditional quantile prediction.

Description

Tests the null hypothesis that Y and E are independent given X.

Usage

InvariantConditionalQuantilePrediction(Y, E, X, alpha = 0.05,
  verbose = FALSE, test = fishersTestExceedance,
  mtry = sqrt(NCOL(X)), ntree = 100, nodesize = 5, maxnodes = NULL,
  quantiles = c(0.1, 0.5, 0.9), returnModel = FALSE)

Arguments

Y

An n-dimensional vector.

E

An n-dimensional vector. If test = fishersTestExceedance, E needs to be a factor.

X

A matrix or dataframe with n rows and p columns.

alpha

Significance level. Defaults to 0.05.

verbose

If TRUE, intermediate output is provided. Defaults to FALSE.

test

Unconditional independence test that tests whether exceedence is independent of E. Defaults to fishersTestExceedance.

mtry

Random forest parameter: Number of variables randomly sampled as candidates at each split. Defaults to sqrt(NCOL(X)).

ntree

Random forest parameter: Number of trees to grow. Defaults to 100.

nodesize

Random forest parameter: Minimum size of terminal nodes. Defaults to 5.

maxnodes

Random forest parameter: Maximum number of terminal nodes trees in the forest can have. Defaults to NULL.

quantiles

Quantiles for which to test independence between exceedence and E. Defaults to c(0.1, 0.5, 0.9).

returnModel

If TRUE, the fitted quantile regression forest model will be returned. Defaults to FALSE.

Value

A list with the following entries:

Examples

# Example 1
n <- 1000
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * (X)^2 + rnorm(n)
InvariantConditionalQuantilePrediction(Y, as.factor(E), X)

# Example 2
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * E + rnorm(n)
InvariantConditionalQuantilePrediction(Y, as.factor(E), X)

Invariant environment prediction.

Description

Tests the null hypothesis that Y and E are independent given X.

Usage

InvariantEnvironmentPrediction(Y, E, X, alpha = 0.05, verbose = FALSE,
  trainTestSplitFunc = caTools::sample.split,
  argsTrainTestSplitFunc = list(Y = E, SplitRatio = 0.8),
  test = propTestTargetE, mtry = sqrt(NCOL(X)), ntree = 100,
  nodesize = 5, maxnodes = NULL, permute = TRUE,
  returnModel = FALSE)

Arguments

Y

An n-dimensional vector.

E

An n-dimensional vector. If test = propTestTargetE, E needs to be a factor.

X

A matrix or dataframe with n rows and p columns.

alpha

Significance level. Defaults to 0.05.

verbose

If TRUE, intermediate output is provided. Defaults to FALSE.

trainTestSplitFunc

Function to split sample. Defaults to stratified sampling using caTools::sample.split, assuming E is a factor.

argsTrainTestSplitFunc

Arguments for sampling splitting function.

test

Unconditional independence test that tests whether the out-of-sample prediction accuracy is the same when using X only vs. X and Y as predictors for E. Defaults to propTestTargetE.

mtry

Random forest parameter: Number of variables randomly sampled as candidates at each split. Defaults to sqrt(NCOL(X)).

ntree

Random forest parameter: Number of trees to grow. Defaults to 100.

nodesize

Random forest parameter: Minimum size of terminal nodes. Defaults to 5.

maxnodes

Random forest parameter: Maximum number of terminal nodes trees in the forest can have. Defaults to NULL.

permute

Random forest parameter: If TRUE, model that would use X only for predicting Y also includes a random permutation of E. Defaults to TRUE.

returnModel

If TRUE, the fitted quantile regression forest model will be returned. Defaults to FALSE.

Value

A list with the following entries:

Examples

# Example 1
n <- 1000
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * (X)^2 + rnorm(n)
InvariantEnvironmentPrediction(Y, as.factor(E), X)

# Example 2
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * E + rnorm(n)
InvariantEnvironmentPrediction(Y, as.factor(E), X)

# Example 3
E <- rnorm(n)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * (X)^2 + rnorm(n)
InvariantEnvironmentPrediction(Y, E, X, test = wilcoxTestTargetY)
InvariantEnvironmentPrediction(Y, X, E, test = wilcoxTestTargetY)

Invariant residual distribution test.

Description

Tests the null hypothesis that Y and E are independent given X.

Usage

InvariantResidualDistributionTest(Y, E, X, alpha = 0.05,
  verbose = FALSE, fitWithGam = TRUE,
  test = leveneAndWilcoxResidualDistributions, colNameNoSmooth = NULL,
  mtry = sqrt(NCOL(X)), ntree = 100, nodesize = 5, maxnodes = NULL,
  returnModel = FALSE)

Arguments

Y

An n-dimensional vector.

E

An n-dimensional vector. E needs to be a factor.

X

A matrix or dataframe with n rows and p columns.

alpha

Significance level. Defaults to 0.05.

verbose

If TRUE, intermediate output is provided. Defaults to FALSE.

fitWithGam

If TRUE, a GAM is used for the nonlinear regression, else a random forest is used. Defaults to TRUE.

test

Unconditional independence test that tests whether residual distribution is invariant across different levels of E. Defaults to leveneAndWilcoxResidDistributions.

colNameNoSmooth

Gam parameter: Name of variables that should enter linearly into the model. Defaults to NULL.

mtry

Random forest parameter: Number of variables randomly sampled as candidates at each split. Defaults to sqrt(NCOL(X)).

ntree

Random forest parameter: Number of trees to grow. Defaults to 100.

nodesize

Random forest parameter: Minimum size of terminal nodes. Defaults to 5.

maxnodes

Random forest parameter: Maximum number of terminal nodes trees in the forest can have. Defaults to NULL.

returnModel

If TRUE, the fitted quantile regression forest model will be returned. Defaults to FALSE.

Value

A list with the following entries:

Examples


# Example 1
n <- 1000
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * (X)^2 + rnorm(n)
InvariantResidualDistributionTest(Y, as.factor(E), X)
InvariantResidualDistributionTest(Y, as.factor(E), X, test = ksResidualDistributions)

# Example 2
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * E + rnorm(n)
InvariantResidualDistributionTest(Y, as.factor(E), X)
InvariantResidualDistributionTest(Y, as.factor(E), X, test = ksResidualDistributions)

Invariant target prediction.

Description

Tests the null hypothesis that Y and E are independent given X.

Usage

InvariantTargetPrediction(Y, E, X, alpha = 0.05, verbose = FALSE,
  fitWithGam = TRUE, trainTestSplitFunc = caTools::sample.split,
  argsTrainTestSplitFunc = NULL, test = fTestTargetY,
  colNameNoSmooth = NULL, mtry = sqrt(NCOL(X)), ntree = 100,
  nodesize = 5, maxnodes = NULL, permute = TRUE,
  returnModel = FALSE)

Arguments

Y

An n-dimensional vector.

E

An n-dimensional vector or an nxq dimensional matrix or dataframe.

X

A matrix or dataframe with n rows and p columns.

alpha

Significance level. Defaults to 0.05.

verbose

If TRUE, intermediate output is provided. Defaults to FALSE.

fitWithGam

If TRUE, a GAM is used for the nonlinear regression, else a random forest is used. Defaults to TRUE.

trainTestSplitFunc

Function to split sample. Defaults to stratified sampling using caTools::sample.split, assuming E is a factor.

argsTrainTestSplitFunc

Arguments for sampling splitting function.

test

Unconditional independence test that tests whether the out-of-sample prediction accuracy is the same when using X only vs. X and E as predictors for Y. Defaults to fTestTargetY.

colNameNoSmooth

Gam parameter: Name of variables that should enter linearly into the model. Defaults to NULL.

mtry

Random forest parameter: Number of variables randomly sampled as candidates at each split. Defaults to sqrt(NCOL(X)).

ntree

Random forest parameter: Number of trees to grow. Defaults to 100.

nodesize

Random forest parameter: Minimum size of terminal nodes. Defaults to 5.

maxnodes

Random forest parameter: Maximum number of terminal nodes trees in the forest can have. Defaults to NULL.

permute

Random forest parameter: If TRUE, model that would use X only for predicting Y also includes a random permutation of E. Defaults to TRUE.

returnModel

If TRUE, the fitted quantile regression forest model will be returned. Defaults to FALSE.

Value

A list with the following entries:

Examples

# Example 1
n <- 1000
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * (X)^2 + rnorm(n)
InvariantTargetPrediction(Y, as.factor(E), X)
InvariantTargetPrediction(Y, as.factor(E), X, test = wilcoxTestTargetY)

# Example 2
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * E + rnorm(n)
InvariantTargetPrediction(Y, as.factor(E), X)
InvariantTargetPrediction(Y, as.factor(E), X, test = wilcoxTestTargetY)

# Example 3
E <- rnorm(n)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * (X)^2 + rnorm(n)
InvariantTargetPrediction(Y, E, X)
InvariantTargetPrediction(Y, X, E)
InvariantTargetPrediction(Y, E, X, test = wilcoxTestTargetY)
InvariantTargetPrediction(Y, X, E, test = wilcoxTestTargetY)

Kernel conditional independence test.

Description

Tests the null hypothesis that Y and E are independent given X. The distribution of the test statistic under the null hypothesis equals an infinite weighted sum of chi squared variables. This distribution can either be approximated by a gamma distribution or by a Monte Carlo approach. This version includes an implementation of choosing the hyperparameters by Gaussian Process regression.

Usage

KCI(Y, E, X, width = 0, alpha = 0.05, unbiased = FALSE,
  gammaApprox = TRUE, GP = TRUE, nRepBs = 5000, lambda = 0.001,
  thresh = 1e-05, numEig = NROW(Y), verbose = FALSE)

Arguments

Y

A vector of length n or a matrix or dataframe with n rows and p columns.

E

A vector of length n or a matrix or dataframe with n rows and p columns.

X

A matrix or dataframe with n rows and p columns.

width

Kernel width; if it is set to zero, the width is chosen automatically (default: 0).

alpha

Significance level (default: 0.05).

unbiased

A boolean variable that indicates whether a bias correction should be applied (default: FALSE).

gammaApprox

A boolean variable that indicates whether the null distribution is approximated by a Gamma distribution. If it is FALSE, a Monte Carlo approach is used (default: TRUE).

GP

Flag whether to use Gaussian Process regression to choose the hyperparameters

nRepBs

Number of draws for the Monte Carlo approach (default: 500).

lambda

Regularization parameter (default: 1e-03).

thresh

Threshold for eigenvalues. Whenever eigenvalues are computed, they are set to zero if they are smaller than thresh times the maximum eigenvalue (default: 1e-05).

numEig

Number of eigenvalues computed (only relevant for computing the distribution under the hypothesis of conditional independence) (default: length(Y)).

verbose

If TRUE, intermediate output is provided. (default: FALSE).

Value

A list with the following entries:

Examples

# Example 1
n <- 100
E <- rnorm(n)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * (X)^2 + rnorm(n)
KCI(Y, E, X)
KCI(Y, X, E)

Residual prediction test.

Description

Tests the null hypothesis that Y and E are independent given X.

Usage

ResidualPredictionTest(Y, E, X, alpha = 0.05, verbose = FALSE,
  degree = 4, basis = c("nystrom", "nystrom_poly", "fourier",
  "polynomial", "provided")[1], resid_type = "OLS", XBasis = NULL,
  noiseMat = NULL, getnoiseFct = function(n, ...) {     rnorm(n) },
  argsGetNoiseFct = NULL, nSim = 100, funcOfRes = function(x) {    
  abs(x) }, useX = TRUE, returnXBasis = FALSE,
  nSub = ceiling(NROW(X)/4), ntree = 100, nodesize = 5,
  maxnodes = NULL)

Arguments

Y

An n-dimensional vector.

E

An n-dimensional vector or an nxq dimensional matrix or dataframe.

X

A matrix or dataframe with n rows and p columns.

alpha

Significance level. Defaults to 0.05.

verbose

If TRUE, intermediate output is provided. Defaults to FALSE.

degree

Degree of polynomial to use if basis="polynomial" or basis="nystrom_poly". Defaults to 4.

basis

Can be one of "nystrom","nystrom_poly","fourier","polynomial","provided". Defaults to "nystrom".

resid_type

Can be "Lasso" or "OLS". Defaults to "OLS".

XBasis

Basis if basis="provided". Defaults to NULL.

noiseMat

Matrix with simulated noise. Defaults to NULL in which case the simulation is performed inside the function.

getnoiseFct

Function to use to generate the noise matrix. Defaults to function(n, ...){rnorm(n)}.

argsGetNoiseFct

Arguments for getnoiseFct. Defaults to NULL.

nSim

Number of simulations to use. Defaults to 100.

funcOfRes

Function of residuals to use in addition to predicting the conditional mean. Defaults to function(x){abs(x)}.

useX

Set to TRUE if the predictors in X should also be used when predicting the scaled residuals with E. Defaults to TRUE.

returnXBasis

Set to TRUE if basis expansion should be returned. Defaults to FALSE.

nSub

Number of random features to use if basis is one of "nystrom","nystrom_poly" or "fourier". Defaults to ceiling(NROW(X)/4).

ntree

Random forest parameter: Number of trees to grow. Defaults to 500.

nodesize

Random forest parameter: Minimum size of terminal nodes. Defaults to 5.

maxnodes

Random forest parameter: Maximum number of terminal nodes trees in the forest can have. Defaults to NULL.

Value

A list with the following entries:

Examples

# Example 1
n <- 100
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * (X)^2 + rnorm(n)
ResidualPredictionTest(Y, as.factor(E), X)

# Example 2
E <- rbinom(n, size = 1, prob = 0.2)
X <- 4 + 2 * E + rnorm(n)
Y <- 3 * E + rnorm(n)
ResidualPredictionTest(Y, as.factor(E), X)

# not run:
# # Example 3
# E <- rnorm(n)
# X <- 4 + 2 * E + rnorm(n)
# Y <- 3 * (X)^2 + rnorm(n)
# ResidualPredictionTest(Y, E, X)
# ResidualPredictionTest(Y, X, E)

F-test for a nested model comparison.

Description

Used as a subroutine in InvariantTargetPrediction to test whether out-of-sample prediction performance is better when using X and E as predictors for Y, compared to using X only.

Usage

fTestTargetY(Y, predictedOnlyX, predictedXE, verbose, ...)

Arguments

Y

An n-dimensional vector.

predictedOnlyX

Predictions for Y based on predictors in X only.

predictedXE

Predictions for Y based on predictors in X and E.

verbose

Set to TRUE if output should be printed.

...

The dimensions of X (df) and E (dimE) need to be passed via the ... argument to allow for coherent interface of fTestTargetY and wilcoxTestTargetY.

Value

A list with the p-value for the test.

Fishers test to test whether the exceedance of the conditional quantiles is independent of the categorical variable E.

Description

Used as a subroutine in InvariantConditionalQuantilePrediction to test whether the exceedance of the conditional quantiles is independent of the categorical variable E.

Usage

fishersTestExceedance(Y, predicted, E, verbose)

Arguments

Y

An n-dimensional vector.

predicted

A matrix with n rows. The columns contain predictions for different conditional quantiles of Y|X.

E

An n-dimensional vector. E needs to be a factor.

verbose

Set to TRUE if output should be printed.

Value

A list with the p-value for the test.

Kolmogorov-Smirnov test to compare residual distributions

Description

Used as a subroutine in InvariantResidualDistributionTest to test whether residual distribution remains invariant across different levels of E.

Usage

ksResidualDistributions(Y, predicted, E, verbose)

Arguments

Y

An n-dimensional vector.

predicted

An n-dimensional vector of predictions for Y.

E

An n-dimensional vector. E needs to be a factor.

verbose

Set to TRUE if output should be printed.

Value

A list with the p-value for the test.

Levene and wilcoxon test to compare first and second moments of residual distributions

Description

Used as a subroutine in InvariantResidualDistributionTest to test whether residual distribution remains invariant across different levels of E.

Usage

leveneAndWilcoxResidualDistributions(Y, predicted, E, verbose)

Arguments

Y

An n-dimensional vector.

predicted

An n-dimensional vector of predictions for Y.

E

An n-dimensional vector. E needs to be a factor.

verbose

Set to TRUE if output should be printed.

Value

A list with the p-value for the test.

Proportion test to compare two misclassification rates.

Description

Used as a subroutine in InvariantEnvironmentPrediction to test whether out-of-sample performance is better when using X and Y as predictors for E, compared to using X only.

Usage

propTestTargetE(E, predictedOnlyX, predictedXY, verbose)

Arguments

E

An n-dimensional vector.

predictedOnlyX

Predictions for E based on predictors in X only.

predictedXY

Predictions for E based on predictors in X and Y.

verbose

Set to TRUE if output should be printed.

Value

A list with the p-value for the test.

Wilcoxon test to compare two mean squared error rates.

Description

Used as a subroutine in InvariantTargetPrediction to test whether out-of-sample performance is better when using X and E as predictors for Y, compared to using X only.

Usage

wilcoxTestTargetY(Y, predictedOnlyX, predictedXE, verbose, ...)

Arguments

Y

An n-dimensional vector.

predictedOnlyX

Predictions for Y based on predictors in X only.

predictedXE

Predictions for Y based on predictors in X and E.

verbose

Set to TRUE if output should be printed.

...

Argument to allow for coherent interface of fTestTargetY and wilcoxTestTargetY.

Value

A list with the p-value for the test.

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