Package 'CCI' reference manual

Title:	Computational Test for Conditional Independence
Description:	Tool for performing computational testing for conditional independence between variables in a dataset. 'CCI' implements permutation in combination with Monte Carlo Cross-Validation in generating null distributions and test statistics. For more details see Computational Test for Conditional Independence (2024) <doi:10.3390/a17080323>.
Authors:	Christian Thorjussen [aut, cre] (ORCID: <https://orcid.org/0009-0005-5006-6491>), Kristian Hovde Liland [aut] (ORCID: <https://orcid.org/0000-0001-6468-9423>)
Maintainer:	Christian Thorjussen <[email protected]>
License:	GPL (>= 2)
Version:	0.3.6.1
Built:	2026-05-15 18:17:56 UTC
Source:	https://github.com/khliland/cci

Creates interaction terms for specified variables in a data frame Interaction terms are named as `<var1>_int_<var2>` (e.g., `Z1_int_Z2` for the product of `Z1` and `Z2`).

Description

Creates interaction terms for specified variables in a data frame Interaction terms are named as <var1>_int_<var2> (e.g., Z1_int_Z2 for the product of Z1 and Z2).

Usage

add_interaction_terms(data, Z, mode = c("numeric_only", "mixed"))
add_interaction_terms(data, Z, mode = c("numeric_only", "mixed"))

Arguments

data

Data frame. The data frame containing the variables for which interaction terms are to be created.

Z

Character vector. The names of the variables for which interaction terms are to be created.

mode

Character. Specifies the type of interaction terms to create. Options are: numeric_only (only numeric-numeric interactions as products) or mixed (numeric-numeric as products, factor/character involved as categorical interactions). Default is "numeric_only".

Value

A list with two components:

data: The modified data frame with added interaction terms.
new_terms: A character vector of the names of the added interaction terms (e.g., Z1_int_2).

Examples

data_generator <-  function(N){
Z1 <- rnorm(N,0,1)
Z2 <- rnorm(N,0,1)
X <- rnorm(N, Z1 + Z2, 1)
Y <- rnorm(N, Z1 + Z2, 1)
df <- data.frame(Z1, Z2, X, Y)
return(df)
}
dat <- data_generator(250)
interaction_terms <- add_interaction_terms(data = dat, Z = c("Z1", "Z2"))
head(interaction_terms$data$Z1_int_Z2)

data_generator <-  function(N){
Z1 <- rnorm(N,0,1)
Z2 <- rnorm(N,0,1)
X <- rnorm(N, Z1 + Z2, 1)
Y <- rnorm(N, Z1 + Z2, 1)
df <- data.frame(Z1, Z2, X, Y)
return(df)
}
dat <- data_generator(250)
interaction_terms <- add_interaction_terms(data = dat, Z = c("Z1", "Z2"))
head(interaction_terms$data$Z1_int_Z2)

Creates polynomial terms for specified variables in a data frame Polynomial terms are named as `<variable>_d_<degree>` (e.g., `Z1_d_2` for the square of `Z1`).

Description

Creates polynomial terms for specified variables in a data frame Polynomial terms are named as <variable>_d_<degree> (e.g., Z1_d_2 for the square of Z1).

Usage

add_poly_terms(data, Z, degree = 3, poly = TRUE)
add_poly_terms(data, Z, degree = 3, poly = TRUE)

Arguments

data

Data frame. The data frame containing the variables for which polynomial terms are to be created.

Z

Character vector. The names of the variables for which polynomial terms are to be created.

degree

Integer. The maximum degree of polynomial terms to be created. Default is 3.

poly

Logical. If TRUE, polynomial terms will be created. If FALSE, no polynomial terms will be created. Default is TRUE.

Value

A list with two components:

data: The modified data frame with added polynomial terms.
new_terms: A character vector of the names of the added polynomial terms (e.g., Z1_d_2).

Examples

set.seed(123)
data_generator <-  function(N){
Z1 <- rnorm(N,0,1)
Z2 <- rnorm(N,0,1)
X <- rnorm(N, Z1 + Z2, 1)
Y <- rnorm(N, Z1 + Z2, 1)
df <- data.frame(Z1, Z2, X, Y)
return(df)
}
dat <- data_generator(250)
poly_terms <- add_poly_terms(data = dat, Z = c("Z1", "Z2"), degree = 3, poly = TRUE)
print(poly_terms$new_terms)
set.seed(123)
data_generator <-  function(N){
Z1 <- rnorm(N,0,1)
Z2 <- rnorm(N,0,1)
X <- rnorm(N, Z1 + Z2, 1)
Y <- rnorm(N, Z1 + Z2, 1)
df <- data.frame(Z1, Z2, X, Y)
return(df)
}
dat <- data_generator(250)
poly_terms <- add_poly_terms(data = dat, Z = c("Z1", "Z2"), degree = 3, poly = TRUE)
print(poly_terms$new_terms)

Build an expanded formula with poly and interaction terms

Description

Build an expanded formula with poly and interaction terms

Usage

build_formula(formula, poly_terms = NULL, interaction_terms = NULL)
build_formula(formula, poly_terms = NULL, interaction_terms = NULL)

Arguments

formula

A base formula in the format Y ~ X | Z1 + Z2

poly_terms

Character vector of polynomial term names

interaction_terms

Character vector of interaction term names

Value

A formula object combining all terms

Examples

poly_terms <- c("Z1_d_2", "Z2_d_2")
interaction_terms <- c("Z1_int_Z2")
formula <- Y ~ X | Z1 + Z2
final_formula <- build_formula(formula, poly_terms, interaction_terms)
print(final_formula)
poly_terms <- c("Z1_d_2", "Z2_d_2")
interaction_terms <- c("Z1_int_Z2")
formula <- Y ~ X | Z1 + Z2
final_formula <- build_formula(formula, poly_terms, interaction_terms)
print(final_formula)

Choose Direction for testing for the CCI test

Description

This function selects the best direction for the CCI test based on cross validation. For the condition Y || X | Z, the function return the recommended formula either Y ~ X | Z or X ~ Y | Z .

Usage

CCI.direction(
  formula,
  data,
  method = "rf",
  folds = 4,
  nrounds = 600,
  max_depth = 6,
  eta = 0.3,
  gamma = 0,
  colsample_bytree = 1,
  min_child_weight = 1,
  subsample = 1,
  poly = TRUE,
  degree = 3,
  interaction = TRUE,
  verbose = FALSE,
  ...
)
CCI.direction(
  formula,
  data,
  method = "rf",
  folds = 4,
  nrounds = 600,
  max_depth = 6,
  eta = 0.3,
  gamma = 0,
  colsample_bytree = 1,
  min_child_weight = 1,
  subsample = 1,
  poly = TRUE,
  degree = 3,
  interaction = TRUE,
  verbose = FALSE,
  ...
)

Arguments

formula

A formula object specifying the model to be fitted.

data

A data frame containing the variables specified in the formula.

method

A character string specifying the method to be used for model fitting. Options include "rf" (random forest), "xgboost" (XGBoost), "nnet" (neural network), "gpr" (Gaussian process regression), and "svm" (support vector machine).

folds

An integer specifying the number of folds for cross-validation. Default is 4.

nrounds

Integer. The number of rounds (trees) for methods like xgboost, ranger, and lightgbm. Default is 600.

max_depth

Integer. The maximum depth of the trees for methods like xgboost. Default is 6.

eta

Numeric. The learning rate for methods like xgboost. Default is 0.3.

gamma

Numeric. The minimum loss reduction required to make a further partition on a leaf node of the tree for methods like xgboost. Default is 0.

colsample_bytree

Numeric. The subsample ratio of columns when constructing each tree for methods like xgboost. Default is 1.

min_child_weight

Numeric. The minimum sum of instance weight (hessian) needed in a child for methods like xgboost. Default is 1.

subsample

Numeric. The proportion of the data to be used for subsampling. Default is 1 (no subsampling).

poly

Logical. If TRUE, polynomial terms of the conditioning variables are included in the model. Default is TRUE.

degree

Integer. The degree of polynomial terms to include if poly is TRUE. Default is 3.

interaction

Logical. If TRUE, interaction terms of the conditioning variables are included in the model. Default is TRUE.

verbose

Logical. If TRUE, prints additional information during the execution. Default is FALSE.

...

Additional arguments to be passed to the model fitting function.

Value

A formula object specifying the selected model direction.

CCI tuner function for CCI test

Description

The CCI.tuner function performs a grid search over parameters for a conditional independence test using machine learning model supported by CCI.test. The tuner use the caret package for tuning.

Usage

CCI.pretuner(
  formula,
  data,
  method = "rf",
  metric = "RMSE",
  validation_method = "cv",
  folds = 4,
  training_share = 0.7,
  tune_length = 4,
  random_grid = TRUE,
  samples = 35,
  poly = TRUE,
  degree = 3,
  interaction = TRUE,
  verboseIter = FALSE,
  include_explanatory = FALSE,
  verbose = FALSE,
  parallel = FALSE,
  mtry = 1:10,
  nrounds = c(100, 200, 300, 400, 500, 600, 700, 800, 900, 1000),
  eta = seq(0.01, 0.3, by = 0.05),
  max_depth = 2:6,
  gamma = c(0, 1, 2, 3),
  colsample_bytree = c(0.8, 0.9, 1),
  min_child_weight = c(1, 3),
  subsample = 1,
  sigma = seq(0.1, 2, by = 0.3),
  C = seq(0.1, 2, by = 0.5),
  ...
)
CCI.pretuner(
  formula,
  data,
  method = "rf",
  metric = "RMSE",
  validation_method = "cv",
  folds = 4,
  training_share = 0.7,
  tune_length = 4,
  random_grid = TRUE,
  samples = 35,
  poly = TRUE,
  degree = 3,
  interaction = TRUE,
  verboseIter = FALSE,
  include_explanatory = FALSE,
  verbose = FALSE,
  parallel = FALSE,
  mtry = 1:10,
  nrounds = c(100, 200, 300, 400, 500, 600, 700, 800, 900, 1000),
  eta = seq(0.01, 0.3, by = 0.05),
  max_depth = 2:6,
  gamma = c(0, 1, 2, 3),
  colsample_bytree = c(0.8, 0.9, 1),
  min_child_weight = c(1, 3),
  subsample = 1,
  sigma = seq(0.1, 2, by = 0.3),
  C = seq(0.1, 2, by = 0.5),
  ...
)

Arguments

formula

Model formula specifying the relationship between dependent and independent variables.

data

A data frame containing the variables specified in the formula.

method

Character. Specifies the machine learning method to use. Supported methods are random forest "rf", extreme gradient boosting "xgboost" and Support Vector Machine "svm".

metric

Character. The performance metric to optimize during tuning. Default is "RMSE".

validation_method

Character. Specifies the resampling method. Default is "cv".

folds

Integer. The number of folds for cross-validation during the tuning process. Default is 10.

training_share

Numeric. For leave-group out cross-validation: the training percentage. Default is 0.7.

tune_length

Integer. The number of parameter combinations to try during the tuning process. Default is 10.

random_grid

Logical. If TRUE, a random grid search is performed. If FALSE, a full grid search is performed. Default is TRUE.

samples

Integer. The number of random samples to take from the grid. Default is 30.

poly

Logical. If TRUE, polynomial terms of the conditional variables are included in the model. Default is TRUE.

degree

Integer. The degree of polynomial terms to include if poly is TRUE. Default is 3.

interaction

Logical. If TRUE, interaction terms of the conditional variables are included in the model. Default is TRUE.

verboseIter

Logical. If TRUE, the function will print the tuning process. Default is FALSE.

include_explanatory

Logical. If TRUE, given the condition Y || X | Z, the function will include explanatory variable X in the model for Y. Default is FALSE

verbose

Logical. If TRUE, the function will print the tuning process. Default is FALSE..

parallel

Logical. If TRUE, the function will use parallel processing. Default is TRUE.

mtry

Integer. The number of variables randomly sampled as candidates at each split for random forest. Default is 1:5.

nrounds

Integer. The number of rounds (trees) for methods such as xgboost and random forest. Default is seq(50, 200, by = 25).

eta

Numeric. The learning rate for xgboost. Default is seq(0.01, 0.3, by = 0.05).

max_depth

Integer. The maximum depth of the tree for xgboost. Default is 1:6.

gamma

Numeric. The minimum loss reduction required to make a further partition on a leaf node for xgboost. Default is seq(0, 5, by = 1).

colsample_bytree

Numeric. The subsample ratio of columns when constructing each tree for xgboost. Default is seq(0.5, 1, by = 0.1).

min_child_weight

Integer. The minimum sum of instance weight (hessian) needed in a child for xgboost. Default is 1:5.

subsample

Numeric. The subsample ratio of the training. Default is 1.

sigma

Numeric. The standard deviation of the Gaussian kernel for Gaussian Process Regression. Default is seq(0.1, 2, by = 0.3).

C

Numeric. The regularization parameter for Support Vector Machine. Default is seq(0.1, 2, by = 0.5).

...

Additional arguments to pass to the CCI.tuner function.

Value

A list containing:

best_param: A data frame with the best parameters.
tuning_result: A data frame with all tested parameter combinations and their performance metrics.
warnings: A character vector of warnings issued during tuning.

Examples

set.seed(123)
data <- data.frame(x1 = rnorm(100), x2 = rnorm(100), x3 = rnorm(100), y = rnorm(100))
# Tune random forest parameters
result <- CCI.pretuner(formula = y ~ x1 | x2 + x3,
data = data,
samples = 5,
folds = 3,
method = "rf")
set.seed(123)
data <- data.frame(x1 = rnorm(100), x2 = rnorm(100), x3 = rnorm(100), y = rnorm(100))
# Tune random forest parameters
result <- CCI.pretuner(formula = y ~ x1 | x2 + x3,
data = data,
samples = 5,
folds = 3,
method = "rf")

Computational test for conditional independence based on ML and Monte Carlo Cross Validation

Description

The CCI.test function performs a conditional independence test using a specified machine learning model or a custom model provided by the user. It calculates the test statistic, generates a null distribution via permutations, computes p-values, and optionally generates a plot of the null distribution with the observed test statistic. The 'CCI.test' function serves as a wrapper around the 'perm.test' function

Usage

CCI.test(
  formula = NULL,
  data,
  p = 0.5,
  nperm = 160,
  nrounds = 600,
  mtry = NULL,
  metric = "Auto",
  method = "rf",
  choose_direction = FALSE,
  parametric = FALSE,
  poly = TRUE,
  degree = 3,
  robust = TRUE,
  subsample = "Auto",
  subsample_set,
  min_child_weight = 1,
  colsample_bytree = 1,
  eta = 0.3,
  gamma = 0,
  max_depth = 6,
  interaction = TRUE,
  mode = "numeric_only",
  metricfunc = NULL,
  mlfunc = NULL,
  tail = NA,
  tune = FALSE,
  samples = 35,
  folds = 5,
  tune_length = 10,
  k = 15,
  center = TRUE,
  scale = TRUE,
  eps = 1e-15,
  positive = NULL,
  kernel = "optimal",
  distance = 2,
  seed = NA,
  random_grid = TRUE,
  nthread = 2,
  verbose = FALSE,
  progress = TRUE,
  ...
)
CCI.test(
  formula = NULL,
  data,
  p = 0.5,
  nperm = 160,
  nrounds = 600,
  mtry = NULL,
  metric = "Auto",
  method = "rf",
  choose_direction = FALSE,
  parametric = FALSE,
  poly = TRUE,
  degree = 3,
  robust = TRUE,
  subsample = "Auto",
  subsample_set,
  min_child_weight = 1,
  colsample_bytree = 1,
  eta = 0.3,
  gamma = 0,
  max_depth = 6,
  interaction = TRUE,
  mode = "numeric_only",
  metricfunc = NULL,
  mlfunc = NULL,
  tail = NA,
  tune = FALSE,
  samples = 35,
  folds = 5,
  tune_length = 10,
  k = 15,
  center = TRUE,
  scale = TRUE,
  eps = 1e-15,
  positive = NULL,
  kernel = "optimal",
  distance = 2,
  seed = NA,
  random_grid = TRUE,
  nthread = 2,
  verbose = FALSE,
  progress = TRUE,
  ...
)

Arguments

formula

Model formula specifying the relationship between dependent and independent variables. (Ex: Y ~ X | Z1 + Z2 for Y || X | Z1, Z2)

data

A data frame containing the variables specified in the formula.

p

Numeric. Proportion of data used for training the model. Default is 0.5.

nperm

Integer. The number of permutations to perform. Default is 60.

nrounds

Integer. The number of rounds (trees) for methods 'xgboost' and 'rf' Default is 600.

mtry

Number of variables to possibly split at in each node for method 'rf'. Default is NULL (sqrt of number of variables).

metric

Character. Specifies the type of data: "Auto", "RMSE" or "Kappa". Default is "Auto".

method

Character. Specifies the machine learning method to use. Supported methods are random forest "rf", extreme gradient boosting "xgboost", support vector machine 'svm' and K-nearest neighbour 'KNN'. Default is "rf".

choose_direction

Logical. If TRUE, the function will choose the best direction for testing. Default is FALSE.

parametric

Logical, indicating whether to compute a parametric p-value instead of the empirical p-value. A parametric p-value assumes that the null distribution is gaussian. Default is FALSE.

poly

Logical. If TRUE, polynomial terms of the conditional variables are included in the model. Default is TRUE.

degree

Integer. The degree of polynomial terms to include if poly is TRUE. Default is 3.

robust

Logical. If TRUE, uses a robust method for permutation. Default is TRUE.

subsample

Character. Specifies whether to use automatic subsampling based on sample size ("Auto"), user-defined subsampling ("Yes"), or no subsampling ("No"). Default is "Auto"

subsample_set

Numeric. If subsample is set to "Yes", this parameter defines the proportion of data to use for subsampling. Default is NA.

min_child_weight

Numeric. The minimum sum of instance weight (hessian) needed in a child for methods like xgboost. Default is 1.

colsample_bytree

Numeric. The subsample ratio of columns when constructing each tree for methods like xgboost. Default is 1.

eta

Numeric. The learning rate for methods like xgboost. Default is 0.3.

gamma

Numeric. The minimum loss reduction required to make a further partition on a leaf node of the tree for methods like xgboost. Default is 0.

max_depth

Integer. The maximum depth of the trees for methods like xgboost. Default is 6.

interaction

Logical. If TRUE, interaction terms of the conditional variables are included in the model. Default is TRUE.

mode

Character. Specifies the mode of operation: "numeric_only" or "mixed". Default is "numeric_only".

metricfunc

Optional the user can pass a custom function for calculating a performance metric based on the model's predictions. Default is NULL.

mlfunc

Optional the user can pass a custom machine learning wrapper function to use instead of the predefined methods. Default is NULL.

tail

Character. Specifies whether to calculate left-tailed or right-tailed p-values, depending on the performance metric used. Only applicable if using metricfunc or mlfunc. Default is NA.

tune

Logical. If TRUE, the function will perform hyperparameter tuning for the specified machine learning method. Default is FALSE.

samples

Integer. Number of hyperparameter combinations used in tuning. Default is 35.

folds

Integer. The number of folds for cross-validation during the tuning process. Default is 5.

tune_length

Integer. The number of parameter combinations to try during the tuning process. Default is 10.

k

Integer. The number of nearest neighbors to use for KNN method. Default is 15.

center

Logical. If TRUE, the data will be centered before fitting the model

scale

Logical. If TRUE, the data will be scaled before fitting the model. Default is TRUE.

eps

Numeric. A small value to avoid division by zero in some calculations.

positive

Character. The name of the positive class (KNN) in the data, used for classification tasks. Default is NULL.

kernel

Character. The kernel type to use for KNN method. Default is "optimal".

distance

Numeric. Parameter of Minkowski distance for the "KNN" method. Default is 2.

seed

Integer. Set the seed for reproducing results. Default is NA.

random_grid

Logical. If TRUE, a random grid search is performed. If FALSE, a full grid search is performed. Default is TRUE.

nthread

Integer. The number of threads to use for parallel processing. Default is 1.

verbose

Logical. If TRUE, additional information is printed during the execution of the function. Default is FALSE.

progress

Logical. If TRUE, a progress bar is displayed during the permutation process. Default is TRUE.

...

Additional arguments to pass to the perm.test function.

Value

Invisibly returns the result of perm.test, which is an object of class 'CCI' containing the null distribution, observed test statistic, p-values, the machine learning model used, and the data.

Examples

set.seed(123)
data <- data.frame(x1 = stats::rnorm(100), x2 = stats::rnorm(100), y = stats::rnorm(100))
result <- CCI.test(y ~ x1 | x2, data = data, nperm = 25, interaction = FALSE)
summary(result)
set.seed(123)
data <- data.frame(x1 = stats::rnorm(100), x2 = stats::rnorm(100), y = stats::rnorm(100))
result <- CCI.test(y ~ x1 | x2, data = data, nperm = 25, interaction = FALSE)
summary(result)

Check the formula statement

Description

This function verifies that all variables specified in the formula are present in the provided data frame. If any variables are missing, the function will stop and return an error message listing the missing variables.

Usage

check_formula(formula, data)
check_formula(formula, data)

Arguments

formula

Formula. The model formula that specifies the relationship between the dependent and independent variables.

data

Data frame. The data frame in which to check for the presence of variables specified in the formula.

Value

Invisibly returns NULL if all variables are present. Stops with an error if any variables are missing.

Clean and Reformat Formula String

Description

This function processes and reformats formula string to ensure it is in the correct format for conditional independence testing. The function checks if the formula uses the '+' operator for additive models and transforms it into a format that includes a conditioning variable separated by '|'.

Usage

clean_formula(formula)
clean_formula(formula)

Arguments

formula

Formula. The model formula that specifies the relationship between the dependent and independent variables, and potentially the conditioning variables. The formula is expected to follow the format Y ~ X + Z1 + Z2 or Y ~ X | Z1 + Z2.

Value

A reformatted formula in the correct format for conditional independence testing. The returned formula will either retain the original format or be transformed to include conditioning variables.

Examples

clean_formula(y ~ x | z + v)
clean_formula(y ~ x + z + v)
clean_formula(y ~ x | z + v)
clean_formula(y ~ x + z + v)

Example dataset: ExponentialNoise

Description

A dataset containing simulated conditional independence test results.

Usage

ExponentialNoise
ExponentialNoise

Format

A data frame with 600 rows and 3 variables:

X: Numeric vector
Y: Numeric vector
Z1: Conditioning variable
Z2: Conditioning variable

Source

Simulated data.

P-value Calculation Based on Null Distribution and Test Statistic

Description

This function calculates p-values based on the comparison of a test statistic against a null distribution. It can perform either empirical or parametric p-value calculations and supports both left-tailed and right-tailed tests.

Usage

get_pvalues(
  dist,
  test_statistic,
  parametric = FALSE,
  tail = c("left", "right")
)
get_pvalues(
  dist,
  test_statistic,
  parametric = FALSE,
  tail = c("left", "right")
)

Arguments

dist

Numeric vector. Represents the null distribution of the test statistic.

test_statistic

Numeric. The observed test statistic for which the p-value is to be calculated.

parametric

Logical. If TRUE, calculates parametric p-values assuming the null distribution is normal. If FALSE, calculates empirical p-values. Default is FALSE.

tail

Character. Specifies whether to calculate left-tailed or right-tailed p-values. Must be either "left" or "right". Default is "left".

Value

Numeric. The calculated p-value.

Examples

set.seed(123)
null_dist <- rnorm(1000)
observed_stat <- 1.5
p_value <- get_pvalues(null_dist, observed_stat, parametric = FALSE, tail = "right")
print(p_value)
set.seed(123)
null_dist <- rnorm(1000)
observed_stat <- 1.5
p_value <- get_pvalues(null_dist, observed_stat, parametric = FALSE, tail = "right")
print(p_value)

Get the best parameters after tuning with CCI.tuner

Description

Get the best parameters after tuning with CCI.tuner

Usage

get_tuned_params(tuned_model)
get_tuned_params(tuned_model)

Arguments

tuned_model

A model object returned from the CCI.pretuner function. This object contains the tuned parameters and other relevant information.

Value

A named list of tuned parameters specific to the model method (e.g., mtry for random forest, eta, max_depth for xgboost). Returns NULL for unsupported methods.

Example dataset: HardCase

Description

A dataset containing simulated conditional independence test results.

Usage

HardCase
HardCase

Format

A data frame with 500 rows and 3 variables:

X: Numeric vector
Y: Numeric vector
Z1: Conditioning variable
Z2: Conditioning variable

Source

Simulated data.

Check whether Z contains at least one categorical variable

Description

Categorical is defined as factor (and optionally character and logical). This helper is intentionally strict about inputs and does not modify data.

Usage

is_categorical_Z_any(sub_data, Z, allow_character = TRUE, allow_logical = TRUE)
is_categorical_Z_any(sub_data, Z, allow_character = TRUE, allow_logical = TRUE)

Arguments

sub_data

data.frame containing the Z columns.

Z

character vector of column names defining the conditioning set.

allow_character

logical; treat character as categorical. Default TRUE.

allow_logical

logical; treat logical as categorical. Default TRUE.

Value

logical scalar.

Create strata from the categorical subset of Z

Description

Uses interaction() on the categorical Z columns only. Characters (and logicals, if enabled) are coerced to factor for stable behavior.

Usage

make_strata_from_categorical_Z(
  sub_data,
  Z,
  allow_character = TRUE,
  allow_logical = TRUE
)
make_strata_from_categorical_Z(
  sub_data,
  Z,
  allow_character = TRUE,
  allow_logical = TRUE
)

Arguments

sub_data

data.frame containing Z columns.

Z

character vector of Z column names.

allow_character

logical; treat character as categorical. Default TRUE.

allow_logical

logical; treat logical as categorical. Default TRUE.

Details

Rows with NA in any stratification column will receive NA strata.

Value

A factor defining strata (same length as nrow(sub_data)).

Example dataset: NonLinearCategorization

Description

A dataset containing simulated data from a non-linear transformation followed by categorization.

Usage

NonLinearCategorization
NonLinearCategorization

Format

A data frame with 600 rows and 3 variables:

X: Numeric vector
Y: Numeric vector
Z: Conditioning variable

Source

Simulated data.

Example dataset: NonLinNormal

Description

A dataset containing simulated data from a non-linear transformation of a multivariate normal distribution.

Usage

NonLinNormal
NonLinNormal

Format

A data frame with 500 rows and 4 variables:

X: Numeric vector
Y: Numeric vector
Z1: Conditioning variable
Z2: Conditioning variable

Source

Simulated data.

Example dataset: NonLinNormalZs_d0

Description

A dataset containing simulated data with uniform noise.

Usage

NonLinNormalZs_d0
NonLinNormalZs_d0

Format

A data frame with 1000 rows and 15 variables:

X: Numeric vector
Y: Numeric vector
Z1: Conditioning variable
Z2: Conditioning variable
Z3: Conditioning variable
Z4: Conditioning variable
Z5: Conditioning variable
Z6: Conditioning variable
Z7: Conditioning variable
Z8: Conditioning variable
Z9: Conditioning variable
Z10: Conditioning variable
Z11: Conditioning variable
Z12: Conditioning variable
Z13: Conditioning variable
Z14: Conditioning variable
Z15: Conditioning variable

Source

Simulated data.

Example dataset: NonLinNormalZs_d05

Description

A dataset containing simulated data with uniform noise.

Usage

NonLinNormalZs_d05
NonLinNormalZs_d05

Format

A data frame with 1000 rows and 15 variables:

X: Numeric vector
Y: Numeric vector
Z1: Conditioning variable
Z2: Conditioning variable
Z3: Conditioning variable
Z4: Conditioning variable
Z5: Conditioning variable
Z6: Conditioning variable
Z7: Conditioning variable
Z8: Conditioning variable
Z9: Conditioning variable
Z10: Conditioning variable
Z11: Conditioning variable
Z12: Conditioning variable
Z13: Conditioning variable
Z14: Conditioning variable
Z15: Conditioning variable

Source

Simulated data.

Example dataset: NormalData

Description

A dataset containing simulated data from a multivariate normal distribution.

Usage

NormalData
NormalData

Format

A data frame with 400 rows and 4 variables:

X: Numeric vector
Y: Numeric vector
Z1: Conditioning variable
Z2: Conditioning variable

@source Simulated data.

Permutation Test for Conditional Independence

Description

Permutation Test for Conditional Independence

Usage

perm.test(
  formula,
  data,
  p = 0.5,
  nperm = 160,
  subsample = 1,
  metric = "RMSE",
  method = "rf",
  nrounds = 600,
  mtry = NULL,
  parametric = FALSE,
  tail = NA,
  robust = TRUE,
  metricfunc = NULL,
  mlfunc = NULL,
  nthread = 1,
  progress = TRUE,
  k = 15,
  center = TRUE,
  scale = TRUE,
  eps = 1e-15,
  positive = NULL,
  kernel = "optimal",
  distance = 2,
  ...
)
perm.test(
  formula,
  data,
  p = 0.5,
  nperm = 160,
  subsample = 1,
  metric = "RMSE",
  method = "rf",
  nrounds = 600,
  mtry = NULL,
  parametric = FALSE,
  tail = NA,
  robust = TRUE,
  metricfunc = NULL,
  mlfunc = NULL,
  nthread = 1,
  progress = TRUE,
  k = 15,
  center = TRUE,
  scale = TRUE,
  eps = 1e-15,
  positive = NULL,
  kernel = "optimal",
  distance = 2,
  ...
)

Arguments

formula

Model formula or DAGitty object specifying the relationship between dependent and independent variables.

data

A data frame containing the variables specified in the formula.

p

Proportion of data to use for training the model. Default is 0.5.

nperm

Number of permutations to perform. Default is 160.

subsample

The proportion of the data to be used. Default is 1 (no subsampling).

metric

Type of metric: "RMSE", "Kappa" or "LogLoss". Default is 'RMSE'.

method

The machine learning method to use for the learner. Supported methods include "rf", "xgboost", "KNN" and "svm". Default is "rf".

nrounds

Number of rounds (trees) for methods 'xgboost' and 'rf'. Default is 600.

mtry

Number of variables to possibly split at in each node for method 'rf'. Default is NULL (sqrt of number of variables).

parametric

Logical. If TRUE, a parametric p-value is calculated instead of an empirical p-value. Default is FALSE.

tail

Specifies whether the test is one-tailed ("left" or "right") or two-tailed. Default is NA.

robust

Logical. If TRUE, uses a robust method for permutation. Default is TRUE.

metricfunc

An optional custom function to calculate the performance metric based on the model's predictions. Default is NULL.

mlfunc

An optional custom machine learning function to use instead of the predefined methods. Default is NULL.

nthread

Integer. The number of threads to use for parallel processing for method 'rf' and 'xgboost'. Default is 1.

progress

Logical. If TRUE, a progress bar is displayed during the permutation process. Default is TRUE.

k

Integer. The number of nearest neighbors for the "KNN" method. Default is 15.

center

Logical. If TRUE, the data is centered before model fitting. Default is TRUE.

scale

Logical. If TRUE, the data is scaled before model fitting. Default is TRUE.

eps

Numeric. A small value added to avoid division by zero. Default is 1e-15.

positive

Character vector. Specifies which levels of a factor variable should be treated as positive class in classification tasks. Default is NULL.

kernel

Character string specifying the kernel type for method option "KNN" . Possible choices are "rectangular" (which is standard unweighted knn), "triangular", "epanechnikov" (or beta(2,2)), "biweight" (or beta(3,3)), "triweight" (or beta(4,4)), "cos", "inv", "gaussian" and "optimal". Default is "optimal".

distance

Numeric. Parameter of Minkowski distance for the "KNN" method. Default is 2.

...

Additional arguments to pass to the machine learning model fitting function.

Value

An object of class 'CCI' containing the null distribution, observed test statistic, p-values, the machine learning model used, and the data.

Examples

set.seed(123)
dat <- data.frame(x1 = rnorm(100),
x2 = rnorm(100),
x3 = rnorm(100),
x4 = rnorm(100),
y = rnorm(100))
perm.test(y ~ x1 | x2 + x3 + x4, data = dat, nperm = 25)
set.seed(123)
dat <- data.frame(x1 = rnorm(100),
x2 = rnorm(100),
x3 = rnorm(100),
x4 = rnorm(100),
y = rnorm(100))
perm.test(y ~ x1 | x2 + x3 + x4, data = dat, nperm = 25)

Stratified permutation of x within strata

Description

Permutes values of 'x' within each stratum. By default, rows with NA in 'x' or 'strata' are left unchanged to preserve alignment.

Usage

permute_within_strata(x, strata, seed = NULL, na_action = c("keep", "drop"))
permute_within_strata(x, strata, seed = NULL, na_action = c("keep", "drop"))

Arguments

x

vector to permute.

strata

factor-like vector defining strata.

seed

optional seed for reproducibility.

na_action

how to handle NA rows: "keep" (default) keeps them fixed, "drop" removes them (returns shorter vector).

Value

x_star permuted within strata.

Examples

set.seed(123)
x <- 1:10
strata <- rep(letters[1:2], each = 5)
x_permuted <- permute_within_strata(x, strata, seed = 123)
set.seed(123)
x <- 1:10
strata <- rep(letters[1:2], each = 5)
x_permuted <- permute_within_strata(x, strata, seed = 123)

Plot for CCI testing

Description

Plot for CCI testing

Usage

## S3 method for class 'CCI'
plot(
  x,
  fill_color = "lightblue",
  title.size = 14,
  axis.text.x = 13,
  axis.text.y = 13,
  strip.text.x = 13,
  strip.text.y = 13,
  legend.text = 13,
  legend.title = 13,
  axis.title.x = 13,
  axis.title.y = 13,
  base_size = 13,
  ...
)
## S3 method for class 'CCI'
plot(
  x,
  fill_color = "lightblue",
  title.size = 14,
  axis.text.x = 13,
  axis.text.y = 13,
  strip.text.x = 13,
  strip.text.y = 13,
  legend.text = 13,
  legend.title = 13,
  axis.title.x = 13,
  axis.title.y = 13,
  base_size = 13,
  ...
)

Arguments

x

Object of class 'CCI'

fill_color

Color for the histogram fill

title.size

Size of the plot title

axis.text.x

Size of x-axis text

axis.text.y

Size of y-axis text

strip.text.x

Size of x-axis strip text

strip.text.y

Size of y-axis strip text

legend.text

Size of legend text

legend.title

Size of legend title

axis.title.x

Size of x-axis title

axis.title.y

Size of y-axis title

base_size

Base font size

...

Additional arguments to ggplot2

Value

A plot of the null distribution and the test statistic in ggplot2 format.

Examples

dat <- data.frame(x1 = rnorm(100), x2 = rnorm(100), y = rnorm(100))
cci <- CCI.test(y ~ x1 + x2, data = dat, interaction = FALSE, nperm = 30)
plot(cci)
dat <- data.frame(x1 = rnorm(100), x2 = rnorm(100), y = rnorm(100))
cci <- CCI.test(y ~ x1 + x2, data = dat, interaction = FALSE, nperm = 30)
plot(cci)

Example dataset: PoissonNoise

Description

A dataset containing simulated data from a Poisson distribution.

Usage

PoissonNoise
PoissonNoise

Format

A data frame with 1000 rows and 4 variables:

X: Numeric vector
Y: Numeric vector
Z1: Conditioning variable
Z2: Conditioning variable

Source

Simulated data.

Example dataset: PolyData

Description

A dataset containing simulated data from a polynomial relationship.

Usage

PolyData
PolyData

Format

A data frame with 600 rows and 4 variables:

X: Numeric vector
Y: Numeric vector
Z1: Conditioning variable
Z2: Conditioning variable

Source

Simulated data.

Print and summary methods for the CCI class

Description

Print and summary methods for the CCI class

Usage

## S3 method for class 'summary.CCI'
print(x, ...)
## S3 method for class 'summary.CCI'
print(x, ...)

Arguments

x

Object of class 'CCI'

...

Additional arguments to print/summary

Value

The print methods have no return value, the summary methods return an object of class 'summary.CCI'.

QQ-plot for multiple testing in CCI

Description

QQ-plot for multiple testing in CCI

Usage

QQplot(
  object,
  title.size = 14,
  axis.text.x = 13,
  axis.text.y = 13,
  strip.text.x = 13,
  strip.text.y = 13,
  legend.text = 13,
  legend.title = 13,
  axis.title.x = 13,
  axis.title.y = 13,
  progress = TRUE,
  ...
)
QQplot(
  object,
  title.size = 14,
  axis.text.x = 13,
  axis.text.y = 13,
  strip.text.x = 13,
  strip.text.y = 13,
  legend.text = 13,
  legend.title = 13,
  axis.title.x = 13,
  axis.title.y = 13,
  progress = TRUE,
  ...
)

Arguments

object

Object of class 'CCI'

title.size

Size of the plot title

axis.text.x

Size of x-axis text

axis.text.y

Size of y-axis text

strip.text.x

Size of x-axis strip text

strip.text.y

Size of y-axis strip text

legend.text

Size of legend text

legend.title

Size of legend title

axis.title.x

Size of x-axis title

axis.title.y

Size of y-axis title

progress

Logical indicating whether to show progress during computation

...

Additional arguments to pass to the test.gen function.

Value

A QQ-plot of the p-values in ggplot2 format.

Examples

dat <- data.frame(x1 = rnorm(100), x2 = rnorm(100), y = rnorm(100))
cci <- CCI.test(y ~ x1 | x2,
data = dat,
nperm = 25,
interaction = FALSE)
QQplot(cci)
dat <- data.frame(x1 = rnorm(100), x2 = rnorm(100), y = rnorm(100))
cci <- CCI.test(y ~ x1 | x2,
data = dat,
nperm = 25,
interaction = FALSE)
QQplot(cci)

Generate the Test Statistic or Null Distribution Using Permutation

Description

This function generates the test statistic or a null distribution through permutation for conditional independence testing. It supports various machine learning methods, including random forests, extreme gradient boosting, and allows for custom metric functions and model fitting functions.

Usage

test.gen(
  formula,
  data,
  method = "rf",
  metric = "RMSE",
  nperm = 160,
  subsample = 1,
  p = 0.5,
  nrounds = 600,
  mtry = NULL,
  nthread = 1,
  permutation = FALSE,
  robust = TRUE,
  metricfunc = NULL,
  mlfunc = NULL,
  progress = TRUE,
  center = TRUE,
  scale = TRUE,
  eps = 1e-15,
  k = 15,
  positive = NULL,
  kernel = "optimal",
  distance = 2,
  ...
)
test.gen(
  formula,
  data,
  method = "rf",
  metric = "RMSE",
  nperm = 160,
  subsample = 1,
  p = 0.5,
  nrounds = 600,
  mtry = NULL,
  nthread = 1,
  permutation = FALSE,
  robust = TRUE,
  metricfunc = NULL,
  mlfunc = NULL,
  progress = TRUE,
  center = TRUE,
  scale = TRUE,
  eps = 1e-15,
  k = 15,
  positive = NULL,
  kernel = "optimal",
  distance = 2,
  ...
)

Arguments

formula

Formula specifying the relationship between dependent and independent variables.

data

Data frame. The data containing the variables used.

method

Character. The modeling method to be used. Options include "xgboost" for gradient boosting, or "rf" for random forests or "svm" for Support Vector Machine.

metric

Character. The type of metric: can be "RMSE", "Kappa" or "LogLoss". Default is 'RMSE'

nperm

Integer. The number of generated Monte Carlo samples. Default is 160.

subsample

Numeric. The proportion of the data to be used for subsampling. Default is 1 (no subsampling).

p

Numeric. The proportion of the data to be used for training. The remaining data will be used for testing. Default is 0.5.

nrounds

Integer. The number of rounds (trees) for methods like 'xgboost' and 'rf'. Default is 600.

mtry

Integer. The number of variables to possibly split at in each node for method 'rf'. Default is the rounded down square root of numbers of columns in data.

nthread

Integer. The number of threads to use for parallel processing. Only relevant for methods 'rf' and 'xgboost'. Default is 1.

permutation

Logical. Whether to perform permutation of the 'X' variable. Used to generate a null distribution. Default is FALSE.

robust

Logical. If TRUE, automatically performs stratified permutation if all conditional variables are factor or categorical. Default is TRUE.

metricfunc

Function. A custom metric function provided by the user. It must take arguments: actual, predictions, and optionally ..., and return a single numeric performance value.

mlfunc

Function. A custom machine learning function provided by the user. The function must have the arguments: formula, data, train_indices, test_indices, and ..., and return a single value performance metric. Default is NULL.

progress

Logical. A logical value indicating whether to show a progress bar during when building the null distribution. Default is TRUE.

center

Logical. If TRUE, the data is centered before model fitting. Default is TRUE.

scale

Logical. If TRUE, the data is scaled before model fitting. Default is TRUE.

eps

Numeric. A small value added to avoid division by zero. Only relevant for method 'KNN'. Default is 1e-15.

k

Integer. The number of nearest neighbors for the "KNN" method. Default is 15.

positive

Character vector. Only relevant for method 'KNN'. Specifies which levels of a factor variable should be treated as positive class in classification tasks. Default is NULL.

kernel

Character. Only relevant for method 'KNN'. Specifies the kernel type for method option "KNN" . Possible choices are "rectangular" (which is standard unweighted knn), "triangular", "epanechnikov" (or beta(2,2)), "biweight" (or beta(3,3)), "triweight" (or beta(4,4)), "cos", "inv", "gaussian" and "optimal". Default is "optimal".

distance

Numeric. Parameter of Minkowski distance for the "KNN" method. Default is 2.

...

Additional arguments to pass to the machine learning wrapper functions wrapper_xgboost, wrapper_ranger, wrapper_knn and wrapper_svm, or to a custom-built wrapper function.

Value

A list containing the test distribution.

Examples

set.seed(123)
data <- data.frame(x1 = rnorm(100),
x2 = rnorm(100),
x3 = rnorm(100),
x4 = rnorm(100),
y = rnorm(100))
result <- test.gen(formula = y ~ x1 | x2 + x3 + x4,
                   metric = "RMSE",
                   nperm = 50,
                   data = data)
hist(result$distribution)
set.seed(123)
data <- data.frame(x1 = rnorm(100),
x2 = rnorm(100),
x3 = rnorm(100),
x4 = rnorm(100),
y = rnorm(100))
result <- test.gen(formula = y ~ x1 | x2 + x3 + x4,
                   metric = "RMSE",
                   nperm = 50,
                   data = data)
hist(result$distribution)

Convert CI-style formula Y ~ X | Z into regression-style Y ~ X + Z

Description

Convert CI-style formula Y ~ X | Z into regression-style Y ~ X + Z

Usage

unclean_formula(f)
unclean_formula(f)

Arguments

f

A formula of the form Y ~ X | Z1 + Z2, or already Y ~ X + Z1 + Z2.

Value

A standard formula of the form Y ~ X + Z1 + Z2.

Example dataset: UniformNoise_large

Description

A dataset containing simulated data with uniform noise.

Usage

UniformNoise_large
UniformNoise_large

Format

A data frame with 20000 rows and 4 variables:

X: Numeric vector
Y: Numeric vector
Z1: Conditioning variable
Z2: Conditioning variable

Source

Simulated data.

k-Nearest Neighbors (KNN) wrapper for CCI (kknn-based)

Description

k-Nearest Neighbors (KNN) wrapper for CCI (kknn-based)

Usage

wrapper_knn(
  formula,
  data,
  train_indices,
  test_indices,
  metric,
  metricfunc = NULL,
  k = 15,
  eps = 1e-15,
  positive = NULL,
  kernel = "optimal",
  distance = 2,
  ...
)
wrapper_knn(
  formula,
  data,
  train_indices,
  test_indices,
  metric,
  metricfunc = NULL,
  k = 15,
  eps = 1e-15,
  positive = NULL,
  kernel = "optimal",
  distance = 2,
  ...
)

Arguments

formula

Model formula

data

Data frame

train_indices

Indices for training rows

test_indices

Indices for test rows

metric

Performance metric: "RMSE" (regression), "Kappa" (classification), or "LogLoss" (classification)

metricfunc

Optional custom metric function: function(actual, predictions, ...)

k

Integer, number of neighbors (default 15)

eps

Small value to avoid log(0) in LogLoss calculations. Default is 1e-15.

positive

Character. The positive class label for binary classification (used in LogLoss). Default is NULL.

kernel

Character. Weighting kernel for kknn. Default "optimal".

distance

Numeric. Minkowski distance parameter. 2 = Euclidean. Default 2.

...

Additional arguments passed to kknn::kknn (e.g., ykernel, na.action)

Value

Numeric performance metric

Random Forest wrapper for CCI

Description

Random Forest wrapper for CCI

Usage

wrapper_ranger(
  formula,
  data,
  train_indices,
  test_indices,
  metric,
  metricfunc = NULL,
  nthread = 1,
  mtry = NULL,
  num.trees,
  eps = 1e-15,
  ...
)
wrapper_ranger(
  formula,
  data,
  train_indices,
  test_indices,
  metric,
  metricfunc = NULL,
  nthread = 1,
  mtry = NULL,
  num.trees,
  eps = 1e-15,
  ...
)

Arguments

formula

Model formula specifying the dependent and independent variables.

data

Data frame containing the dataset to be used for training and testing the model.

train_indices

A vector of indices specifying the rows in data to be used as the training set.

test_indices

A vector of indices specifying the rows in data to be used as the test set.

metric

Type of metric ("RMSE", "Kappa" or "Log Loss")

metricfunc

Optional user-defined function to calculate a custom performance metric. This function should take the arguments data, model, and test_indices, and return a numeric value representing the performance metric.

nthread

Integer. The number of threads to use for parallel processing. Default is 1.

mtry

Integer. The number of variables to possibly split at in each node. Default is the square root of the number of columns in data.

num.trees

Integer. The number of trees to grow in the random forest.

eps

Small value to avoid log(0) in LogLoss calculations. Default is 1e-15.

...

Additional arguments passed to the ranger function.

Value

A numeric value representing the performance metric of the model on the test set.

SVM wrapper for CCI

Description

SVM wrapper for CCI

Usage

wrapper_svm(
  formula,
  data,
  train_indices,
  test_indices,
  metric,
  metricfunc = NULL,
  eps = 1e-15,
  ...
)
wrapper_svm(
  formula,
  data,
  train_indices,
  test_indices,
  metric,
  metricfunc = NULL,
  eps = 1e-15,
  ...
)

Arguments

formula

Model formula

data

Data frame

train_indices

Indices for training data

test_indices

Indices for testing data

metric

Type of metric ("RMSE", "Kappa" or "Log Loss")

metricfunc

Optional user-defined function to calculate a custom performance metric.

eps

Small value to avoid log(0) in LogLoss calculations. Default is 1e-15.

...

Additional arguments passed to e1071::svm

Value

Performance metric (RMSE for continuous, Kappa for classification)

Extreme Gradient Boosting wrapper for CCI

Description

Extreme Gradient Boosting wrapper for CCI

Usage

wrapper_xgboost(
  formula,
  data,
  train_indices,
  test_indices,
  metric,
  nrounds = 500,
  metricfunc = NULL,
  nthread = 1,
  eps = 1e-15,
  subsample = 1,
  ...
)
wrapper_xgboost(
  formula,
  data,
  train_indices,
  test_indices,
  metric,
  nrounds = 500,
  metricfunc = NULL,
  nthread = 1,
  eps = 1e-15,
  subsample = 1,
  ...
)

Arguments

formula

Model formula

data

Data frame

train_indices

Indices for training data

test_indices

Indices for training data

metric

Type of metric ("RMSE", "Kappa" or "Log Loss")

nrounds

Number of boosting rounds

metricfunc

A user specific metric function which have the arguments data, model test_indices and test_matrix and returns a numeric value

nthread

Integer. Number of threads to use for parallel computation during model training in XGBoost. Default is 1.

eps

Small value to avoid log(0) in LogLoss calculations. Default is 1e-15.

subsample

Numeric. The proportion of the data to be used for subsampling. Default is 1 (no subsampling).

...

Additional arguments passed to xgb.train

Value

Performance metric

Package 'CCI'

Help Index

Creates interaction terms for specified variables in a data frame Interaction terms are named as <var1>_int_<var2> (e.g., Z1_int_Z2 for the product of Z1 and Z2).

Description

Usage

Arguments

Value

Examples

Creates polynomial terms for specified variables in a data frame Polynomial terms are named as <variable>_d_<degree> (e.g., Z1_d_2 for the square of Z1).

Description

Usage

Arguments

Value

Examples

Build an expanded formula with poly and interaction terms

Description

Usage

Arguments

Value

Examples

Choose Direction for testing for the CCI test

Description

Usage

Arguments

Value

CCI tuner function for CCI test

Description

Usage

Arguments

Value

See Also

Examples

Computational test for conditional independence based on ML and Monte Carlo Cross Validation

Description

Usage

Arguments

Value

See Also

Examples

Check the formula statement

Description

Usage

Arguments

Value

Clean and Reformat Formula String

Description

Usage

Arguments

Value

Examples

Example dataset: ExponentialNoise

Description

Usage

Format

Source

P-value Calculation Based on Null Distribution and Test Statistic

Description

Usage

Arguments

Value

Examples

Get the best parameters after tuning with CCI.tuner

Description

Usage

Arguments

Value

Example dataset: HardCase

Description

Usage

Format

Source

Check whether Z contains at least one categorical variable

Description

Usage

Arguments

Value

Create strata from the categorical subset of Z

Description

Usage

Arguments

Creates interaction terms for specified variables in a data frame Interaction terms are named as `<var1>_int_<var2>` (e.g., `Z1_int_Z2` for the product of `Z1` and `Z2`).

Creates polynomial terms for specified variables in a data frame Polynomial terms are named as `<variable>_d_<degree>` (e.g., `Z1_d_2` for the square of `Z1`).