Type:	Package
Version:	1.0.0
Title:	Distributed Linear Regression Models with Response Missing Variables
Depends:	R (≥ 4.1.0)
Description:	As a distributed imputation strategy, the Distributed full information Multiple Imputation method is developed to impute missing response variables in distributed linear regression. The philosophy of the package is described in 'Guo' (2025) <doi:10.1038/s41598-025-93333-6>.
License:	Apache License (== 2.0)
RoxygenNote:	7.3.2
Encoding:	UTF-8
Imports:	stats,MASS,glmnet
Packaged:	2025-07-22 15:24:38 UTC; lenovo
Date/Publication:	2025-07-24 04:30:16 UTC
Config/testthat/edition:	3
NeedsCompilation:	no
Author:	Guangbao Guo [aut, cre], Limin Song [aut]
Maintainer:	Guangbao Guo <ggb11111111@163.com>
Repository:	CRAN

Averaged Generalized Method of Moments Imputation (AVGM)

Description

This function performs multiple imputations on missing values in the response variable Y, using AVGMMI logic with support for grouped data. It is fully self-contained.

Usage

AVGM(data, M, midx = 1)

Arguments

Value

A list containing:

Examples

set.seed(123)
data <- data.frame(
  y = c(rnorm(50), rep(NA, 10)),
  x1 = rnorm(60),
  x2 = rnorm(60)
)
result <- AVGM(data, M = 10)
head(result$Yhat)

CSLMI: Consensus-based Stochastic Linear Multiple Imputation (Simplified Version)

Description

Performs multiple imputation and parameter estimation using a consensus-based approach. The response variable is in the first column, all other columns are predictors, missing values are automatically detected, the whole dataset is treated as one block.

Usage

CSLMI(data, M)

Arguments

Value

A list containing:

A list containing the following components:

Examples

set.seed(123)
data <- data.frame(
  y = c(rnorm(50), rep(NA, 10)),
  x1 = rnorm(60),
  x2 = rnorm(60)
)
result <- CSLMI(data = data, M = 10)
head(result$Yhat)
print(result$betahat)
print(result$comm)

Impute Missing Values in Response Variable Y Using Distributed AVGMMI Method (With Grouping)

Description

This function implements the Distributed Averaged Generalized Method of Moments Imputation (DAVGMMI) to fill in missing values in the response variable Y based on observed covariates X. Assumes a single group structure and does not require group size input ('n').

Usage

DAVGMMI(data, R, M)

Arguments

Value

A list containing:

Examples

set.seed(123)
data <- data.frame(
  y = c(rnorm(50), rep(NA, 10)),
  x1 = rnorm(60),
  x2 = rnorm(60)
)
result <- DAVGMMI(data, R = 50, M = 10)
head(result$Yhat)

Distributed and Consensus-Based Stochastic Linear Multiple Imputation (DCSLMI)

Description

Performs multiple imputation for missing response variables in linear regression models. This method iteratively updates parameter estimates using ordinary least squares (OLS) and generates M complete datasets by imputing missing values with different parameter draws.

Usage

DCSLMI(data, R = 1000, M = 20)

Arguments

Value

A list containing:

Yhat

A matrix of size n x M, where each column is a completed response vector.

betahat

A matrix of size (p+1) x M, where each column contains the estimated regression coefficients.

missing_count

The number of missing values in the original response variable.

Examples

# Simulate data with missing responses
set.seed(123)
data <- data.frame(
  y = c(rnorm(50), rep(NA, 10)),
  x1 = rnorm(60),
  x2 = rnorm(60)
)

# Perform multiple imputation
result <- DCSLMI(data, R = 500, M = 10)

# View imputed response values
head(result$Yhat)

# View coefficient estimates
apply(result$betahat, 1, mean)  # average estimates
apply(result$betahat, 1, sd)    # uncertainty across imputations

Distributed Exponentially Weighted Recursive Least Squares (DERLS)

Description

Impute missing values in the response variable Y using distributed ERLS method. Multiple independent runs are performed to stabilize coefficient estimates. Missing values are imputed recursively and refined over multiple iterations.

Usage

DERLS(data, rho, lambda, R, nb)

Arguments

Details

This function implements the Distributed Exponentially Weighted Recursive Least Squares (DERLS) method for imputing missing values in the response variable Y. The key steps include:

1. Initial imputation of missing values.

2. Recursive updates of the regression coefficients using the ERLS algorithm.

3. Multiple independent runs to stabilize the coefficient estimates.

4. Final prediction of missing values using the averaged coefficients.

The ERLS algorithm is particularly useful for online learning and adaptive filtering.

Value

A list containing:

Yhat

Imputed response vector.

betahat

Estimated coefficient vector.

Examples

set.seed(123)
n <- 60
data <- data.frame(
  Y = c(rnorm(n - 10), rep(NA, 10)),  # 50 observed + 10 missing
  X1 = rnorm(n),
  X2 = rnorm(n)
)
result <- DERLS(data, rho = 0.01, lambda = 0.95, R = 3, nb = 50)
head(result$Yhat)  # inspect imputed Y
result$betahat      # inspect estimated coefficients

Distributed Exponentially Weighted Recursive Least Squares (DERLS) using Information Filter

Description

Impute missing values in the response variable Y using a distributed Exponentially Weighted Recursive Least Squares (DERLS) method that employs an Information Filter. Multiple independent runs are performed to stabilize coefficient estimates, and missing values are imputed recursively and refined over multiple iterations.

Usage

DERLS_InfoFilter(data, rho, lambda, R, nb)

Arguments

Value

A list with two components:

Examples

set.seed(123)
n <- 60
data <- data.frame(
  Y = c(rnorm(n - 10), rep(NA, 10)),
  X1 = rnorm(n),
  X2 = rnorm(n)
)
result <- DERLS_InfoFilter(data, rho = 0.01, lambda = 0.95, R = 3, nb = 50)
head(result$Yhat)  # inspect imputed Y
result$betahat     # inspect estimated coefficients

Distributed Exponentially Weighted Recursive Least Squares (DERLS) using Woodbury Identity

Description

Impute missing values in the response variable Y using the distributed ERLS method with the Woodbury Identity. Multiple independent runs are performed to stabilize coefficient estimates. Missing values are imputed recursively and refined over multiple iterations.

Usage

DERLS_Woodbury(data, rho, lambda, R, nb)

Arguments

Details

This function implements the Distributed Exponentially Weighted Recursive Least Squares (DERLS) method using the Woodbury Identity for efficient updates of the covariance matrix. The key steps include:

1. Initial imputation of missing values.

2. Recursive updates of the regression coefficients using the ERLS algorithm with the Woodbury Identity.

3. Multiple independent runs to stabilize the coefficient estimates.

4. Final prediction of missing values using the averaged coefficients.

The Woodbury Identity is used to efficiently update the covariance matrix Pstar during each iteration, making the algorithm computationally efficient and suitable for large datasets.

Value

A list containing:

Yhat

Imputed response vector.

betahat

Estimated coefficient vector.

Examples

set.seed(123)
n <- 60
data <- data.frame(
  Y = c(rnorm(n - 10), rep(NA, 10)),  # 50 observed+10 missing
  X1 = rnorm(n),
  X2 = rnorm(n)
)
result <- DERLS_Woodbury(data, rho = 0.01, lambda = 0.95, R = 3, nb = 50)
head(result$Yhat)  # inspect imputed Y
result$betahat      # inspect estimated coefficients

Distributed Monte Carlo Expectation-Maximization (DMCEM) Algorithm

Description

Implements a distributed version of the Monte Carlo EM algorithm for handling missing response variables in linear regression models. By running multiple simulations and averaging the results, it provides more stable parameter estimates compared to standard EM.

Usage

DMCEM(data, R = 50, tol = 0.01, nb = 50)

Arguments

Details

The DMCEM algorithm works by:

1. Splitting data into observed and missing response subsets.

2. Running multiple MCEM simulations with random imputations.

3. Averaging results across simulations to reduce variance.

4. Using robust matrix inversion to handle near-singular designs.

This approach is particularly useful for datasets with a large proportion of missing responses or high variability in the data.

Value

A list containing:

Yhat

A vector of imputed response values with missing data filled in.

betahat

A vector of final regression coefficients, averaged across simulations.

Examples

# Generate data with 20% missing responses
set.seed(123)
data <- data.frame(
  Y = c(rnorm(80), rep(NA, 20)),
  X1 = rnorm(100),
  X2 = runif(100)
)

# Run DMCEM with 50 simulations
result <- DMCEM(data, R = 50, tol = 0.001, nb = 100)

# View imputed values and coefficients
head(result$Yhat)
result$betahat

# Check convergence and variance
result$converged_ratio
result$sigma2

Distributed Full-information Multiple Imputation (DfiMI)

Description

Perform multiple imputation of the response variable Y via R independent runs and M stochastic imputations per run. Missing values in Y are imputed by means of (intercept-adjusted) OLS regression on the complete predictors.

Usage

DfiMI(data, R, M)

Arguments

Details

This function implements a distributed full-information multiple imputation (DfiMI) approach. It iteratively imputes missing values in the response variable Y using OLS regression on the complete predictors. The process is repeated R times to stabilise the coefficient estimates, and within each run, M imputations are performed to account for the uncertainty in the imputation process.

Value

A named list with components:

Yhat

Numeric vector – the original Y with missing values replaced by their imputed counterparts.

betahat

Numeric vector – final regression coefficients (including intercept).

Examples

set.seed(123)
n  <- 60
data <- data.frame(
  Y  = c(rnorm(n - 10), rep(NA, 10)),  # 50 observed + 10 missing
  X1 = rnorm(n),
  X2 = rnorm(n)
)

res <- DfiMI(data, R = 3, M = 5)
head(res$Yhat)   # inspect imputed Y
res$betahat      # inspect coefficients

Distributed Full-information Multiple Imputation (DfiMI) using LASSO

Description

Performs multiple imputation of the response variable Y via R independent runs and M stochastic imputations per run. Missing Y values are imputed using LASSO regression on predictors.

Usage

DfiMI_lasso(data, R, M)

Arguments

Details

This function extends the Distributed Full-information Multiple Imputation (DfiMI) approach by using LASSO regression for imputing missing values in the response variable Y. LASSO regression is particularly useful for high-dimensional predictor spaces and can handle multicollinearity among predictors. The function performs the following steps:

1. Initialize missing values in Y.

2. Fit LASSO regression models on complete cases.

3. Average coefficients across multiple imputations and runs.

4. Predict missing values using the final averaged coefficients.

The function requires the glmnet package for LASSO regression.

Value

A named list containing:

Yhat

Numeric vector – original Y values with missing values replaced by imputations.

betahat

Numeric vector – final regression coefficients.

Examples

set.seed(123)
data <- data.frame(
  Y = c(rnorm(50), rep(NA, 10)),  # 50 observed + 10 missing
  X1 = rnorm(60),
  X2 = rnorm(60)
)
res <- DfiMI_lasso(data, R = 3, M = 5)
head(res$Yhat)

EM Algorithm for Linear Regression with Missing Data

Description

EM Algorithm for Linear Regression with Missing Data

Usage

EMRE(data, d = 1, tol = 1e-06, nb = 100, niter = 1)

Arguments

Value

List containing:

Examples

# Generate data with 20% missing Y values
set.seed(123)
data <- data.frame(Y=c(rnorm(80),rep(NA,20)), X1=rnorm(100), X2=rnorm(100))

# Run EM algorithm
result <- EMRE(data, d=1, tol=1e-5, nb=50)
print(result$betahat) # View coefficients

Exponentially Weighted Recursive Least Squares with Missing Value Imputation

Description

Exponentially Weighted Recursive Least Squares with Missing Value Imputation

Usage

ERLS(data, rho = 0.01, lambda = 0.95, nb = 100, niter = 1)

Arguments

Value

List containing:

Examples

set.seed(123)
data <- data.frame(
  y = c(rnorm(50), rep(NA, 10)),
  x1 = rnorm(60),
  x2 = rnorm(60)
)
result <- ERLS(data, rho = 0.01, lambda = 0.95, nb = 100, niter = 1)
head(result$Yhat)

FimIMI: Multiple Runs of Improved Multiple Imputation (IMI)

Description

This function performs multiple runs of the Improved Multiple Imputation (IMI) estimation and collects the results. It is designed to facilitate batch processing and repeated runs of IMI.

Usage

FimIMI(d, R, n, M, batch = 0)

Arguments

Details

This function assumes that the data structure d is properly defined and contains the necessary information. The function repeatedly calls the IMI function and collects the regression coefficients and indicator variables.

Value

A list containing:

Examples

# Example data
set.seed(123)
n <- c(300, 300, 400)  # Sample sizes for each group
p <- 5  # Number of independent variables
d <- list(p = p, Y = rnorm(sum(n)), X0 = matrix(rnorm(sum(n) * p), ncol = p))

# Call FimIMI function
result <- FimIMI(d = d, R = 10, n = n, M = 20, batch = 1)

# View results
print(result$Beta)  # Regression coefficients for each run

Generate Missing Data function

Description

This function generates missing data in a specified column of a data frame according to a given missing ratio.

Usage

GMD(data, ratio)

Arguments

Value

Examples

set.seed(123) # for reproducibility
data <- data.frame(x = 1:10, y = rnorm(10))
modified_data <- GMD(data, ratio = 0.5)
summary(modified_data)

Improved Multiple Imputation (IMI) Estimation

Description

This function performs Improved Multiple Imputation (IMI) estimation for grouped data with missing values. It iteratively imputes missing values using the LS function and estimates regression coefficients using the PPLS function. The final regression coefficients are averaged across multiple imputations.

Usage

IMI(d, M, midx, n)

Arguments

Details

The function assumes the data is grouped and contains missing values in specified columns (midx). It uses the LS function to impute missing values and the PPLS function to estimate regression coefficients. The process is repeated M times, and the final regression coefficients are averaged.

Value

A list containing the following elements:

Examples

# Example data

set.seed(123)
n <- c(300, 300, 400)  # Sample sizes for each group
p <- 5  # Number of independent variables
Y <- rnorm(sum(n))  # Dependent variable
X0 <- matrix(rnorm(sum(n) * p), ncol = p)  # Independent variables matrix
d <- list(p = p, Y = Y, X0 = X0)  # Data list
d$all <- cbind(Y, X0)
# Indices of missing variables (assuming some variables are missing)
midx <- c(2, 3)  # For example, the second and third variables are missing
# Call IMI function
result <- IMI(d, M = 5, midx = midx, n = n)
# View results
print(result$betahat)  # Average regression coefficients

Least Squares Estimation for Grouped Data with Ridge Regularization

Description

This function implements the least squares estimation for grouped data, supporting ridge regression regularization. It can handle missing data and returns regression coefficients and the sum of squared residuals for each group.

Usage

LS(d, yidx, Xidx, n, lam = 0.005)

Arguments

Value

A list containing the following elements:

Examples

# Example data
set.seed(123)
n <- 1000
p <- 5
d <- list(all = cbind(rnorm(n), matrix(rnorm(n*p), ncol=p)))

# Call the LS function
result <- LS(d, yidx = 1, Xidx = 2:(p + 1), n = c(1, 300, 600, 1000))

# View the results
print(result$beta)  # Regression coefficients
print(result$SSE)   # Sum of squared residuals

MCEM Algorithm for Missing Response Variables

Description

Implements the Monte Carlo EM algorithm for handling missing response data in linear regression models.

Usage

MCEM(data, d = 5, tol = 0.01, nb = 50)

Arguments

Details

This function implements the Monte Carlo Expectation-Maximization (MCEM) algorithm to handle missing response variables in linear regression models. The algorithm iteratively imputes missing responses and updates regression coefficients until convergence.

Value

A list containing the following components:

Examples

  # Create dataset with 20% missing responses
  set.seed(123)
  data <- data.frame(
    Y = c(rnorm(80), rep(NA, 20)),
    X1 = rnorm(100),
    X2 = runif(100)
  )
  result <- MCEM(data, d = 5, tol = 0.001, nb = 100)
  print(result$Yhat)  # Imputed response vector
  print(result$betahat)  # Final regression coefficients
  print(result$iterations)  # Number of iterations performed

Predictive Mean Matching with Multiple Imputation

Description

Implements PMM algorithm for handling missing data in linear regression models. Uses chained equations approach to generate multiple imputed datasets and pools results using Rubin's rules.

Usage

PMMI(data, k = 5, m = 5)

Arguments

Value

List containing:

Examples

# Create dataset with 30% missing values
data <- data.frame(Y=c(rnorm(70),rep(NA,30)), X1=rnorm(100))
results <- PMMI(data, k=5, m=5)

Penalized Partial Least Squares (PPLS) Estimation

Description

This function performs Penalized Partial Least Squares (PPLS) estimation for grouped data. It supports ridge regression regularization and handles missing data by excluding incomplete cases. The function returns regression coefficients, residual sum of squares, and other diagnostic information.

Usage

PPLS(d, yidx, Xidx, n, lam = 0.005)

Arguments

Details

This function assumes that the data is grouped and that the sample sizes for each group are provided. It excludes cases with missing values in the dependent or independent variables. The function uses Cholesky decomposition to solve the regularized least squares problem.

Value

A list containing the following elements:

Examples

# Example data
set.seed(123)
n_total <- 1000
p <- 5
n_groups <- c(300, 300, 400)
d <- list(all = cbind(rnorm(n_total), matrix(rnorm(n_total*p), ncol=p)),p = p)

# Call PPLS function
result <- PPLS(d, yidx=1, Xidx=2:(p+1), n=n_groups)

# View results
print(result$beta)  # Regression coefficients
print(result$SSE)   # Residual sum of squares

fiMI: Predict Missing Response Variables using Multiple Imputation

Description

This function predicts missing response variables in a linear regression dataset using multiple imputation. It leverages the FimIMI function to perform multiple runs of improved multiple imputation and averages the regression coefficients to predict the missing response values.

Usage

fiMI(data, R, n, M)

Arguments

Details

This function assumes that the first column of data is the response variable and the remaining columns are the independent variables. The function uses the FimIMI function to perform multiple runs of improved multiple imputation and averages the regression coefficients to predict the missing response values.

Value

A list containing:

Examples

# Example data
set.seed(123)
n <- 1000  # Number of rows
p <- 5  # Number of independent variables
data <- data.frame(Y = rnorm(n), X1 = rnorm(n), X2 = rnorm(n))
data[sample(n, 100), 1] <- NA  # Introduce missing response values

# Call fiMI function
result <- fiMI(data, R = 10, n = n, M = 20)

# View results
print(result$Yhat)  # Predicted response values