Package 'HMC'

Anchored test for two-sample mean comparison.

Description

Anchored test for two-sample mean comparison.

Usage

anchored_lasso_testing(
  sample_1,
  sample_2,
  pca_method = "sparse_pca",
  mean_method = "lasso",
  lasso_tuning_method = "min",
  num_latent_factor = 1,
  n_folds = 5,
  verbose = TRUE
)

Arguments

sample_1	Group 1 sample. Each row is a subject and each column corresponds to a feature.
sample_2	Group 2 sample. Each row is a subject and each column corresponds to a feature.
pca_method	Methods used to estimate principle component The default is "sparse_pca", using sparse PCA from package PMA. Other choices are "dense_pca"—the regular PCA; and "hard"— hard-thresholding PCA, which also induces sparsity.
mean_method	Methods used to estimate the discriminant direction. Default is logistic Lasso "lasso". Can also take value "lasso_no_truncation"
lasso_tuning_method	Method for Lasso penalty hyperparameter tuning. Default is "min", the minimizer of cross-validation error; users can also use "1se" for more sparse solutions.
num_latent_factor	The principle component that lasso coefficient anchors at. The default is PC1 = 1.
n_folds	Number of splits when performing cross-fitting. The default is 5, if computational time allows, you can try to set it to 10.
verbose	Print information to the console. Default is TRUE.

Value

A list of test statistics.

test_statistics	Test statistics. Each entry corresponds to the test result of one principle component.
standard_error	Estimated standard error of test_statistics_before_studentization.
test_statistics_before_studentization	Similar to test_statistics but does not have variance = 1.
split_data	Intermediate quantities needed for further assessment and interpretation of the test results.

Examples

sample_size_1 <- sample_size_2 <- 300
true_mean_1 <- matrix(c(rep(1, 10), rep(0, 90)), ncol = 1)
true_mean_2 <- matrix(c(rep(1.5, 10), rep(0, 90)), ncol = 1)

sample_1 <- data.frame(MASS::mvrnorm(sample_size_1,
                               mu = true_mean_1,
                               Sigma = diag(1, 100)))
 sample_2 <- data.frame(MASS::mvrnorm(sample_size_2,
                               mu = true_mean_2,
                               Sigma = diag(1, 100)))
 result <- anchored_lasso_testing(sample_1, sample_2)
 result$test_statistics
 ##the test statistic. It should follow normal(0,1) when there is no difference between the groups.
 summarize_feature_name(result) 
 #summarize which features contribute to discriminant vectors (i.e. logistic lasso)
 extract_pc(result) # extract the estimated discriminant coefficients

---

Debiased one-step test for two-sample mean comparison. A small p-value tells us not only there is difference in the mean vectors, but can also indicates which principle component the difference aligns with.

Description

Debiased one-step test for two-sample mean comparison. A small p-value tells us not only there is difference in the mean vectors, but can also indicates which principle component the difference aligns with.

Usage

debiased_pc_testing(
  sample_1,
  sample_2 = NULL,
  pca_method = "sparse_pca",
  mean_method = "naive",
  num_latent_factor = 1,
  n_folds = 5,
  verbose = TRUE
)

Arguments

sample_1	Group 1 sample. Each row is a subject and each column corresponds to a feature.
sample_2	Group 2 sample. Each row is a subject and each column corresponds to a feature.
pca_method	Methods used to estimate principle component The default is "sparse_pca", using sparse PCA from package PMA. Other choices are "dense_pca"—the regular PCA; and "hard"— hard-thresholding PCA, which also induces sparsity.
mean_method	Methods used to estimate the mean vector. Default is sample mean "naive". There is also a hard-thresholding sparse estiamtor "hard".
num_latent_factor	Number of principle to be estimated/tested. Default is 1.
n_folds	Number of splits when performing cross-fitting. The default is 5, if computational time allows, you can try to set it to 10.
verbose	Print information to the console. Default is TRUE.

Value

A list of test statistics.

test_statistics	Test statistics. Each entry corresponds to the test result of one principle component.
standard_error	Estimated standard error of test_statistics_before_studentization.
test_statistics_before_studentization	Similar to test_statistics but does not have variance = 1.
split_data	Intermediate quantities needed for further assessment and interpretation of the test results.

Examples

sample_size_1 <- sample_size_2 <- 300

true_mean_1 <- matrix(c(rep(1, 10), rep(0, 90)), ncol = 1)
true_mean_2 <- matrix(c(rep(1.5, 10), rep(0, 90)), ncol = 1)
pc1 <- c(rep(1, 10), rep(0, 90))
pc1 <- pc1/norm(pc1, type = '2')

simulation_covariance <- 10 * pc1 %*% t(pc1)
simulation_covariance <- simulation_covariance + diag(1, 100)

sample_1 <- data.frame(MASS::mvrnorm(sample_size_1,
                               mu = true_mean_1,
                               Sigma = simulation_covariance))
 sample_2 <- data.frame(MASS::mvrnorm(sample_size_2,
                               mu = true_mean_2,
                               Sigma = simulation_covariance))
 result <- debiased_pc_testing(sample_1, sample_2)
 result$test_statistics
 ##these are test statistics. Each one of them corresponds to one PC.
 summarize_pc_name(result, latent_fator_index = 1) #shows which features contribute to PC1
 extract_pc(result) # extract the estimated leading PCs.

---

The function for nuisance parameter estimation in anchored_lasso_testing().

Description

The function for nuisance parameter estimation in anchored_lasso_testing().

Usage

estimate_nuisance_parameter_lasso(
  nuisance_sample_1,
  nuisance_sample_2,
  pca_method = "sparse_pca",
  mean_method = "lasso",
  lasso_tuning_method = "min",
  num_latent_factor = 1,
  local_environment = local_environment,
  verbose = TRUE
)

Arguments

nuisance_sample_1	Group 1 sample. Each row is a subject and each column corresponds to a feature.
nuisance_sample_2	Group 2 sample. Each row is a subject and each column corresponds to a feature.
pca_method	Methods used to estimate principle component The default is "sparse_pca", using sparse PCA from package PMA. Other choices are "dense_pca"—the regular PCA; and "hard"— hard-thresholding PCA, which also induces sparsity.
mean_method	Methods used to estimate the discriminant direction. Default is logistic Lasso "lasso". Can also take value "lasso_no_truncation"
lasso_tuning_method	Method for Lasso penalty hyperparameter tuning. Default is "min", the minimizer of cross-validation error; users can also use "1se" for more sparse solutions.
num_latent_factor	The principle component that lasso coefficient anchors at. The default is PC1 = 1.
local_environment	An environment for hyperparameters shared between folds.
verbose	Print information to the console. Default is TRUE.

Value

A list of estimated nuisance quantities.

estimate_leading_pc	Leading principle components
estimate_mean_1	Sample mean for group 1
estimate_mean_2	Sample mean for group 1
estimate_lasso_beta	Logistic Lasso regression coefficients.
estimate_projection_direction	Anchored projection direction. It is similar to PC1 when signal is weak but similar to estimate_optimal_direction when the signal is moderately large.
estimate_optimal_direction	Discriminant direction.

---

The function for nuisance parameter estimation in simple_pc_testing() and debiased_pc_testing().

Description

The function for nuisance parameter estimation in simple_pc_testing() and debiased_pc_testing().

Usage

estimate_nuisance_pc(
  nuisance_sample_1,
  nuisance_sample_2 = NULL,
  pca_method = "sparse_pca",
  mean_method = "naive",
  num_latent_factor = 1,
  local_environment = NA
)

Arguments

nuisance_sample_1	Group 1 sample. Each row is a subject and each column corresponds to a feature.
nuisance_sample_2	Group 2 sample. Each row is a subject and each column corresponds to a feature.
pca_method	Methods used to estimate principle component The default is "sparse_pca", using sparse PCA from package PMA. Other choices are "dense_pca"—the regular PCA; and "hard"— hard-thresholding PCA, which also induces sparsity.
mean_method	Methods used to estimate the mean vector. Default is sample mean "naive". There is also a hard-thresholding sparse estiamtor "hard".
num_latent_factor	Number of principle to be estimated/tested. Default is 1.
local_environment	A environment for hyperparameters shared between folds.

Value

A list of estimated nuisance quantities.

estimate_leading_pc	Leading principle components
estimate_mean_1	Sample mean for group 1
estimate_mean_2	Sample mean for group 1
estimate_eigenvalue	Eigenvalue for each principle compoenent.
estimate_noise_variance	Noise variance, I need this to construct block-diagonal estimates of the covariance matrix.

---

Calculate the test statistics on the left-out samples. Called in debiased_pc_testing().

Description

Calculate the test statistics on the left-out samples. Called in debiased_pc_testing().

Usage

evaluate_influence_function_multi_factor(
  cross_fitting_sample_1,
  cross_fitting_sample_2 = NULL,
  nuisance_collection,
  num_latent_factor = 1
)

Arguments

cross_fitting_sample_1	Group 1 sample. Each row is a subject and each column corresponds to a feature.
cross_fitting_sample_2	Group 2 sample. Each row is a subject and each column corresponds to a feature.
nuisance_collection	A collection of nuisance quantities estimated using "nuisance" samples. It is the output of estimate_nuisance_pc().
num_latent_factor	Number of principle components to be considered.

Value

A list of test statistics.

inner_product_1	Simple inner products for sample 1.
inner_product_2	Simple inner products for sample 2.
influence_eigenvector_each_subject_1	Debiased test statistics, sample 1.
influence_eigenvector_each_subject_2	Debiased test statistics, sample 1.
for_variance_subject_1	Statistics for variance calculation, sample 1.
for_variance_subject_2	Statistics for variance calculation, sample 2.

---

Calculate the test statistics on the left-out samples. Called in anchored_lasso_testing().

Description

Calculate the test statistics on the left-out samples. Called in anchored_lasso_testing().

Usage

evaluate_pca_lasso_plug_in(
  cross_fitting_sample_1,
  cross_fitting_sample_2,
  nuisance_collection,
  mean_method = "lasso"
)

Arguments

cross_fitting_sample_1	Group 1 sample. Each row is a subject and each column corresponds to a feature.
cross_fitting_sample_2	Group 2 sample. Each row is a subject and each column corresponds to a feature.
nuisance_collection	A collection of nuisance quantities estimated using "nuisance" samples. It is the output of estimate_nuisance_pc().
mean_method	Methods used to estimate the discriminant direction. Default is logistic Lasso "lasso". Can also take value "lasso_no_truncation"

Value

A list of test statistics.

influence_each_subject_1	Test statistics for sample 1.
influence_each_subject_1	Test statistics for sample 2.
for_variance_each_subject_1	Statistics for variance calculation, sample 1.
for_variance_each_subject_2	Statistics for variance calculation, sample 2.

---

Calculate the test statistics on the left-out samples. Called in simple_pc_testing().

Description

Calculate the test statistics on the left-out samples. Called in simple_pc_testing().

Usage

evaluate_pca_plug_in(
  cross_fitting_sample_1,
  cross_fitting_sample_2 = NULL,
  nuisance_collection
)

Arguments

cross_fitting_sample_1	Group 1 sample. Each row is a subject and each column corresponds to a feature.
cross_fitting_sample_2	Group 2 sample. Each row is a subject and each column corresponds to a feature.
nuisance_collection	A collection of nuisance quantities estimated using "nuisance" samples. It is the output of estimate_nuisance_pc().

Value

A list of test statistics.

influence_each_subject_1	Statistics for sample 1.
influence_each_subject_2	Statistics for sample 2.

---

Extract the lasso estimate from the output of anchored_lasso_testing().

Description

Extract the lasso estimate from the output of anchored_lasso_testing().

Usage

extract_lasso_coef(testing_result)

Arguments

testing_result

The output/test result list from anchored_lasso_testing().

Value

A list, whose elements are the estimated discriminant directions for each split—the length of the output list is the same as n_folds.

The discriminant vectors for each split.

---

Extract the principle components from the output of simple_pc_testing() and debiased_pc_testing().

Description

Extract the principle components from the output of simple_pc_testing() and debiased_pc_testing().

Usage

extract_pc(testing_result)

Arguments

testing_result

The output/test result list from simple_pc_testing() or debiased_pc_testing().

Value

A list, whose elements are the estimated PC for each split—the length of the output list is the same as n_folds.

The PC vectors for each split.

---

Split the sample index into n_folds many groups so that we can perform cross-fitting

Description

Split the sample index into n_folds many groups so that we can perform cross-fitting

Usage

index_spliter(array, n_folds = 5)

Arguments

array	Sample index. Usually just an array from 1 to the number of samples in one group.
n_folds	Number of splits

Value

A list indicates the sample indices in each split.

---

Simple plug-in test for two-sample mean comparison.

Description

Simple plug-in test for two-sample mean comparison.

Usage

simple_pc_testing(
  sample_1,
  sample_2 = NULL,
  pca_method = "sparse_pca",
  mean_method = "naive",
  num_latent_factor = 1,
  n_folds = 5,
  verbose = TRUE
)

Arguments

sample_1	Group 1 sample. Each row is a subject and each column corresponds to a feature.
sample_2	Group 2 sample. Each row is a subject and each column corresponds to a feature.
pca_method	Methods used to estimate principle component The default is "sparse_pca", using sparse PCA from package PMA. Other choices are "dense_pca"—the regular PCA; and "hard"— hard-thresholding PCA, which also induces sparsity.
mean_method	Methods used to estimate the mean vector. Default is sample mean "naive". There is also a hard-thresholding sparse estiamtor "hard".
num_latent_factor	Number of principle to be estimated/tested. Default is 1.
n_folds	Number of splits when performing cross-fitting. The default is 5, if computational time allows, you can try to set it to 10.
verbose	Print information to the console. Default is TRUE.

Value

A list of test statistics.

test_statistics	Test statistics. Each entry corresponds to the test result of one principle component.
standard_error	Estimated standard error of test_statistics_before_studentization.
test_statistics_before_studentization	Similar to test_statistics but does not have variance = 1.
split_data	Intermediate quantities needed for further assessment and interpretation of the test results.

Examples

sample_size_1 <- sample_size_2 <- 300
true_mean_1 <- matrix(c(rep(1, 10), rep(0, 90)), ncol = 1)
true_mean_2 <- matrix(c(rep(1.5, 10), rep(0, 90)), ncol = 1)
pc1 <- c(rep(1, 10), rep(0, 90))
pc1 <- pc1/norm(pc1, type = '2')

simulation_covariance <- 10 * pc1 %*% t(pc1)
simulation_covariance <- simulation_covariance + diag(1, 100)

sample_1 <- data.frame(MASS::mvrnorm(sample_size_1,
                                     mu = true_mean_1,
                                     Sigma = simulation_covariance))
sample_2 <- data.frame(MASS::mvrnorm(sample_size_2,
                                     mu = true_mean_2,
                                     Sigma = simulation_covariance))
result <- simple_pc_testing(sample_1, sample_2)
result$test_statistics
##these are test statistics. Each one of them corresponds to one PC.
summarize_pc_name(result, latent_fator_index = 1) #shows which features contribute to PC1
extract_pc(result) # extract the estimated leading PCs.

---

Summarize the features (e.g. genes) that contribute to the test result, i.e. those features consistently show up in Lasso vectors.

Description

Summarize the features (e.g. genes) that contribute to the test result, i.e. those features consistently show up in Lasso vectors.

Usage

summarize_feature_name(testing_result, method = "majority voting")

Arguments

testing_result	The output/test result list from anchored_lasso_testing().
method	How to combine the feature list across different splits. Default is 'majority voting'—features that show up more than 50% of the splits are considered active/useful. It can be 'union'—all the features pooled together; or 'intersection'—only include features showing up in all splits.

Value

A list of names of features (your very original input data need to have column names!) that contribute to the test result. An empty list means there is barely any difference between the two groups.

Feature names that consistently showing up in the discriminant vectors.

---

Summarize the features (e.g. genes) that contribute to the test result, i.e. those features consistently show up in the sparse principle components.

Description

Summarize the features (e.g. genes) that contribute to the test result, i.e. those features consistently show up in the sparse principle components.

Usage

summarize_pc_name(
  testing_result,
  latent_fator_index = 1,
  method = "majority voting"
)

Arguments

testing_result	The output/test result list from simple_pc_testing() or debiased_pc_testing().
latent_fator_index	Which principle component should the algorithm summarize? Default is PC1.
method	How to combine the feature list across different splits. Default is 'majority voting'—features that show up more than 50% of the splits are considered active/useful. It can be 'union'—all the features pooled together; or 'intersection'—only include features showing up in all splits.

Value

A list of names of features (your very original input data need to have column names!) that contribute to the test result.

Feature names that consistently showing up in the estimated PC vectors.

---