MaxIntTools: Tools for Detecting and Understanding Maximal Interaction Two-Mode Clustering

Description

MaxIntTools provides functions for performing maximal interaction two-mode clustering and related visualization techniques. It facilitates detection and understanding of latent interaction in complex two-mode data sets with applications in psychology, marketing, neuroscience, and other fields.

Details

The package includes the following key functions:

• MaxInt_data_analysis: Analysis two-mode data using this function.

• MaxInt.Screeplot: Plot all (P, Q) clustering combinations so that we can decide the optimal ones.

• MaxInteraction.plot: Plot the resulting of maximal interaction two-mode clustering for optimal choice.

• MaxInt.heatmap.plot: Generate heatmaps annotated with optimal clustering results.

These tools support exploratory data analysis and interpretation of complex interaction structures in two-mode data.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de) Maintainer: Zaheer Ahmed (ahmed@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

MaxInt_data_analysis, screeplot.MaxInt_data_analysis, MaxInteraction.plot, MaxInt.heatmap.plot

Two-Mode Clustering via the E_ReMI Method

Description

Performs row-by-column clustering using the E-ReMI (Extended REMAXINT) algorithm. It estimates latent cluster structures in rows and columns of a data matrix with unequal row cluster proportions using a REMAXINT-like procedure.

Usage

E_ReMI(DC, P, Q, Nruns, initR, initC, initG)

Arguments

Details

The E_ReMI function implements an EM-style algorithm for biclustering, based on maximizing the likelihood of a Gaussian interaction model with unequal row cluster sizes. Starting from user-provided initial values for cluster assignments and the interaction matrix, the algorithm iteratively updates row clusters, column clusters, the bi-cluster interaction matrix, and estimates the proportions of row clusters (Omega) and residual variance. Unlike REMAXINT clustering models, E_ReMI accounts for unequal cluster sizes in expectation, allowing for more flexible modeling of heterogeneous block-structured data.

Value

A list containing:

Note

This function is useful for clustering data with unequal row cluster sizes and estimating interaction structures between biclusters.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation-based alternative. Available on OSF (submitted to ADAC). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331. Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013.

See Also

Permutation_Function, Randompartition_function, Unequal_Randompartition_Function, Update_row_clusters_E_ReMI, Update_column_clusters, Update_G_Omega, Log_Likelihood_function_E_ReMI, pinv

Examples

I <- 10
P <- 4
J <- 5 
Q <- 2
Nruns <- 5
DC <- matrix(rnorm(I * J), I, J)
initR <- diag(1, nrow = I, ncol = P)
initC <- diag(1, nrow = J, ncol = Q)
R_inv <- pracma::pinv(t(initR) %*% initR)
C_inv <- pracma::pinv(t(initC) %*% initC)
initG <- R_inv %*% t(initR) %*% DC %*% initC %*% C_inv
initG <- as.matrix(initG)
result <- E_ReMI(DC, P, Q, Nruns, initR, initC, initG)
result

Compute Log-Likelihood Ratio test statistics for permutation based Extended Random Effect Maximal Interaction (E-ReMI) Two-Mode Clustering Model

Description

Computes the log-likelihood ratio statistic used for hypothesis testing in the E-ReMI for Two-Mode Clustering model. The statistic compares the fitted model (some intercation) against a null model (no intercation).

Usage

Log_LR_Test_Statistic(DC, I, J, initM, initOmega_hat)

Arguments

Details

This function evaluates the log-likelihood ratio (LLR) test statistic by comparing the log-likelihood under the fitted model (presence of row by colum cluster intercation) to that of a baseline model with no intercation. The test statistic can be used in permutation-based inference procedures to assess significance of row by colum cluster intercation in two-mode data.

Value

Returns a single numeric value:

Note

Used internally in permutation-based hypothesis testing procedures for the E-ReMI Two-Mode Clustering framework.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

Permutation_Function, diag, pinv

Examples

I <- 10; P <- 4; J <- 5; Q <- 2
DC <- matrix(rnorm(I*J), I, J)
R  <- diag(1, I, P)
C  <- diag(1, J, Q)
R_inv <- pracma::pinv(t(R) %*% R)
C_inv <- pracma::pinv(t(C) %*% C)
G <- R_inv %*% t(R) %*% DC %*% C %*% C_inv
initM <- R %*% G %*% t(C)
initOmega_hat <- c(0.5, rep((1-0.5)/P, P))
result <- Log_LR_Test_Statistic(DC, I, J, initM, initOmega_hat)
result

Log-Likelihood Calculation for E-ReMI Clustering

Description

Computes the log-likelihood of the observed data in the E-ReMI clustering framework.

Usage

Log_Likelihood_function_E_ReMI(DC, I, J, initM, initOmega_hat)

Arguments

Details

This function is used internally in the E-ReMI algorithm to monitor model fit by calculating the log-likelihood of the observed data under the current clustering configuration and model estimates. The error matrix is computed by subtracting the reconstructed matrix initM from the observed data DC, and the log-likelihood is derived from the Gaussian likelihood function with known variance.

Value

Returns a numeric scalar:

Note

This function is intended for internal use in model monitoring and convergence checking during E-ReMI algorithm execution.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

E_ReMI, Update_row_clusters_E_ReMI, pinv

Examples

I <- 10; P <- 4; J <- 5; Q <- 2
DC <- matrix(rnorm(I*J), I, J)
R  <- diag(1, I, P)
C  <- diag(1, J, Q)
R_inv <- pracma::pinv(t(R) %*% R)
C_inv <- pracma::pinv(t(C) %*% C)
G <- R_inv %*% t(R) %*% DC %*% C %*% C_inv
initM <- R %*% G %*% t(C)
initOmega_hat <- c(0.5, rep((1-0.5)/P, P))
result <- Log_Likelihood_function_E_ReMI(DC, I, J, initM, initOmega_hat)
result

Compute Log-Likelihood for REMAXINT method

Description

Computes the squared error log-likelihood for a given data matrix under the REMAXINT clustering framework.. The function evaluates the fit of the model by comparing the observed data to the reconstructed (fitted) matrix.

Usage

Log_Likelihood_function_REMAXINT(DC, I, J, initM)

Arguments

Details

This function is used within the REMAXINT framework to compute the log-likelihood of a model by comparing the original data matrix (DC) to the estimated matrix (initM). The error matrix is calculated as the element-wise difference between these two matrices, and the log-likelihood is computed as the sum of squared errors.

Value

A single numeric value representing the log-likelihood (sum of squared residuals).

Note

This function assumes that both input matrices DC and initM are of the same dimensions and numeric type.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

REMAXINT, Update_row_clusters_REMAXINT, Update_column_clusters, matrix, sum, pinv

Examples

I <- 10; P <- 4; J <- 5; Q <- 2
DC <- matrix(rnorm(I*J), I, J)
R  <- diag(1, I, P)
C  <- diag(1, J, Q)
R_inv <- pracma::pinv(t(R) %*% R)
C_inv <- pracma::pinv(t(C) %*% C)
G <- R_inv %*% t(R) %*% DC %*% C %*% C_inv
initM <- R %*% G %*% t(C)
result <- Log_Likelihood_function_REMAXINT(DC, I, J, initM)
result

Heatmap for Visualizing Row by Column (Participant-by-Task) Clusters Interaction

Description

This function generates a heatmap to visualize the estimated interaction values (\Gamma) between row clusters (e.g., participant groups) and column clusters (e.g., task groups) obtained from the maximal interaction two-mode clustering approach (REMAXINT and/or E-ReMI).

Usage

MaxInt.heatmap.plot(Gamma_matrix, row_counts = NULL)

Arguments

Details

The function converts the Gamma_matrix into a long format and visualizes it as a heatmap using ggplot2. The heatmap tiles are colored based on the magnitude of the estimated \Gamma values, and the corresponding values are displayed within the tiles.

If row_counts is provided, the counts for each row cluster are displayed in the plot caption to provide additional context on cluster sizes.

Value

Returns a ggplot object containing the heatmap visualization. The plot includes:

• Heatmap tiles filled based on the \Gamma values.

• Numeric labels inside each tile showing the rounded \Gamma value.

• Optional caption with participant counts for each row cluster.

Note

This function is intended to be used after obtaining \Gamma matrices from the MaxInt_data_analysis function.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

MaxInt_data_analysis, MaxInteraction.plot, screeplot.MaxInt_data_analysis

Examples

Gamma_matrix <- matrix(c(0.3, -0.2, 0.5, 0.1, 0.4, -0.1), nrow = 2, byrow = TRUE)
rownames(Gamma_matrix) <- c("Cluster 1", "Cluster 2")
colnames(Gamma_matrix) <- c("Task A", "Task B", "Task C")
MaxInt.heatmap.plot(Gamma_matrix, row_counts = c(50, 40))

Permutation-Based Inference and Interaction Estimation for Maximal Interaction Two-Mode Clustering

Description

Performs permutation-based statistical inference for the REMAXINT and E-ReMI biclustering models across multiple combinations of row and column clusters. It computes test statistics, p-values, critical values, and estimates interaction parameters for each specified combination of clusters. Furthermore, it provides us optimal row and column partitioning structure for REMAXINT and E-ReMI biclustering methods.

Usage

MaxInt_data_analysis(Data, Clus, Nruns, permutations, alpha_level, verbose = TRUE)

Arguments

Details

The function iterates over all specified (P,Q) cluster combinations, applies the Permutation_Function for each combination, and stores the resulting test statistics, p-values, critical values, and estimated interaction matrices. It further computes a Gamma value for each (P,Q) pair, which quantifies the overall interaction strength using row and column cluster cardinalities and the squared interaction parameters within each bicluster.

Value

A list of class "MaxInt_data_analysis" with the following components:

Note

This function serves as a wrapper for repeated calls to Permutation_Function to facilitate data analysis for various (P,Q) combinations. This function is computationally intensive and may take time depending on the number of cluster configurations, permutations, and runs specified. Use parallel computing where possible to speed up execution.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

Permutation_Function, screeplot.MaxInt_data_analysis, MaxInteraction.plot

Examples

set.seed(123)
Data <- matrix(rnorm(60), nrow = 10, ncol = 6)
Clus <- matrix(c(2,2, 2,3, 3,2), ncol = 2, byrow = TRUE)

# Run with default verbose output
result <- MaxInt_data_analysis(Data, Clus, Nruns = 5, permutations = 10, 
					alpha_level = 0.05, verbose = FALSE)
result

Interaction Plot to understand the nature of row-by-column clusters interaction against the optimal choice of (P) and (Q).

Description

Generates an interaction plot based on a given interaction matrix (Gamma_matrix) resulting from a two-mode clustering analysis. The plot displays estimated interaction effects for the optimal combination of row and column clusters.

Usage

MaxInteraction.plot(Gamma_matrix, row_counts = NULL)

Arguments

Details

The function creates a line plot for each row cluster, where the x-axis corresponds to column clusters and the y-axis to estimated interaction values (\Gamma). The function automatically assigns generic names (Row Cluster 1, Row Cluster 2, ...) and (Column Cluster 1, Column Cluster 2, ...) if row or column names are not provided.

If row_counts is supplied, a caption is added to indicate the number of observations per row cluster.

Value

A ggplot object displaying the interaction plot. The function does not return any additional values.

Note

This function is intended for visualizing the interaction matrix obtained after selecting the optimal number of row and column clusters (P, Q) using MaxInt_data_analysis and MaxInt.Screeplot.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

MaxInt_data_analysis, screeplot.MaxInt_data_analysis

Examples

Gamma_matrix <- matrix(c(0.5, 0.8, -0.2,
                         0.1, 0.3, 0.6,
                         -0.4, 0.2, 0.7),
                       nrow = 3, byrow = TRUE)
rownames(Gamma_matrix) <- c("High", "Low", "Intermediate")
colnames(Gamma_matrix) <- c("Task 1", "Task 2", "Task 3")
MaxInteraction.plot(Gamma_matrix, row_counts = c(50, 40, 60))

Permutation-based hypothesis testing procedure for REMAXINT and E-REMI based test statistic(s).

Description

Conducts a permutation test to assess the significance of a latent interaction structure in a data matrix using two model-based clustering approaches: REMAXINT and E-ReMI. It compares the observed likelihood ratio statistics to the distribution under the null hypothesis derived from permutations.

Usage

Permutation_Function(D, P, Q, Nruns, permutations, alpha_level)

Arguments

Details

This function applies a model-based biclustering algorithm (REMAXINT and E-ReMI) to the original and permuted data matrices to test the presence of an underlying block mean (interaction) structure. For each method, it computes the observed log-likelihood ratio (LLR) statistic and compares it to a distribution of LLRs under the null hypothesis generated through permutation. The procedure yields empirical \( p \)-values and critical values, allowing for hypothesis testing on whether the data contains a meaningful block structure or not.

Value

A list containing the following components:

Note

Permutation function ensures input matrix D is preprocessed appropriately (e.g., doubly centered) within this function.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

REMAXINT, E_ReMI, Log_LR_Test_Statistic

Examples

I <- 10; P <- 4; J <- 5; Q <- 2; Nruns <- 5; permutations <- 10; alpha_level <- 0.05
D <- matrix(rnorm(I*J), I, J)
result <- Permutation_Function(D, P, Q, Nruns, permutations, alpha_level)
result

Two-Mode Clustering via the REMAXINT Method

Description

Performs two-mode clustering (row and column) of a data matrix using the REMAXINT (Random Effect Maximal Interaction Two-mode Clustering) algorithm. It iteratively updates row and column cluster memberships to maximize a Gaussian log-likelihood criterion and ultimately provides partitions of the data that maximize interaction between row and column clusters. Furthermore, REMAXINT method assumes row and column clusters sizes are equal in expectation.

Usage

REMAXINT(DC, P, Q, Nruns)

Arguments

Details

The REMAXINT function implements a likelihood-based biclustering approach. For each of the specified Nruns, it initializes row and column cluster assignments randomly and iteratively updates them to maximize the log-likelihood of the data under a block interaction model. The algorithm uses alternating updates of the row and column cluster assignments, along with an estimation of the latent cluster interaction matrix G and noise variance. To avoid poor local maxima, multiple random starts are performed, and the result with the highest final log-likelihood is selected.

Value

A list with the following components:

Note

This method is particularly useful when the data shows block-wise latent interaction structures between row and column clusters.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

Permutation_Function, Randompartition_function, Update_row_clusters_REMAXINT, Update_column_clusters, Update_G_Omega, Log_Likelihood_function_REMAXINT,

Examples

I <- 10; P <- 4; J <- 5; Q <- 2; Nruns <- 5
DC <- matrix(rnorm(I*J), I, J)
result <- REMAXINT(DC, P, Q, Nruns)
result

Generate a Random Partition Matrix of Equally sized row and column clusters. These clusters are mutually exclusive and there is no empty row and column cluster.

Description

Generates a binary matrix representing a random partition of i items into p latent groups. Each row indicates group membership in a one-hot encoded format.

Usage

Randompartition_function(i, p)

Arguments

Details

This function creates a binary matrix of size i x p, where each row corresponds to a row cluster. The function first ensures that each cluster has at least one initial member (by using an identity matrix), and then assigns the remaining items randomly to clusters. Finally, the matrix rows are shuffled to ensure randomization of the assignment order.

Value

A binary matrix of dimensions i x p:

Note

This function is primarily used for generating initial row or column cluster membership matrices in the REMAXINT and E-ReMI methods.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

REMAXINT, E_ReMI, diag

Examples

result <- Randompartition_function(i = 10, p = 3)
result

Generate a Random Partition Matrix of Unequally sized row clusters. Therefore, it assign higher weight or probability to first the row cluster and the remaining row clusters are evenly distributed.

Description

Generates a binary matrix representing a random partition of i rows into p row groups, ensuring each group is represented at least once, and allowing the first group to have a higher probability.

Usage

Unequal_Randompartition_Function(i, p, firstp)

Arguments

Details

The function first initializes a p x p identity matrix to ensure each group is selected at least once. The rest of the rows (i - p) are assigned to groups based on a multinomial distribution with a specified probability vector, where the first group's probability is firstp and the rest share the remaining 1 - firstp equally.

Value

A binary matrix of dimension i x p, where each row contains a single 1 indicating the assigned group, and 0s elsewhere.

Note

This function is primarily used for generating initial row cluster membership matrix in the E-ReMI methods.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

E_ReMI, rmultinom, diag, sample

Examples

result <- Unequal_Randompartition_Function(i = 20, p = 4, firstp = 0.5)
result

Updates Row Cluster Membership Probabilities

Description

Updates the Row Cluster Membership Probabilities based on the current assignments of rows to row clusters in E_ReMI clustering algorithm.

Usage

Update_G_Omega(DC, I, J, Updated_C, Updated_R)

Arguments

Details

This function is typically called within each iteration of a E-ReMI clustering algorithm, where the goal is to maximize the log-likelihood or minimize least squared of DC via row and column cluster memberships. Thus it updates model parameters (row cluster membership probabilities) under E-ReMI clustering framework.

Value

A list with the following components:

Note

This function assumes binary (one-hot encoded) cluster membership matrices Updated_C and Updated_R.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

E_ReMI, REMAXINT, pinv

Examples

I <- 10; P <- 4; J <- 5; Q <- 2
DC <- matrix(rnorm(I*J), I, J)
Updated_R <- diag(1, I, P)
Updated_C <- diag(1, J, Q)
result <- Update_G_Omega(DC, I, J, Updated_C, Updated_R)
result

Update Column Cluster Assignments in REMAXINT and E-ReMI clustering framework.

Description

Updates column cluster assignments (C matrix) and corresponding G matrix in a two-mode clustering model based on a likelihood maximization procedure.

Usage

Update_column_clusters(DC, I, J, initR, initC, initG, Q)

Arguments

Details

This function updates the column cluster assignments by iteratively testing all possible cluster configurations for each column and selecting the one that maximizes the log-likelihood. After reassigning the clusters, it checks for empty clusters and reassigns the least likely columns to ensure all clusters are non-empty. The function then updates the G matrix accordingly.

Value

A list with the following components:

Note

This function assumes that the input cluster assignment matrices are binary and mutually exclusive for each item (i.e., hard clustering).

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

REMAXINT, E_ReMI, pinv

Examples

I <- 10
P <- 4
J <- 5 
Q <- 2
Nruns <- 5
DC <- matrix(rnorm(I * J), I, J)
initR <- diag(1, nrow = I, ncol = P)
initC <- diag(1, nrow = J, ncol = Q)
R_inv <- pracma::pinv(t(initR) %*% initR)
C_inv <- pracma::pinv(t(initC) %*% initC)
initG <- R_inv %*% t(initR) %*% DC %*% initC %*% C_inv
initG <- as.matrix(initG)
result <- Update_column_clusters(DC, I, J, initR, initC, initG, Q)
result

Update Row Cluster Assignments in E-ReMI clustering framework.

Description

Updates the row cluster membership matrix within the E-ReMI algorithm by maximizing the log-likelihood for each row based on the current estimates of model parameters.

Usage

Update_row_clusters_E_ReMI(DC, I, J, P, 
Updated_R, Updated_C, Updated_omegahat, 
Updated_G, 
Updated_sigma)

Arguments

Details

This function iteratively updates the row cluster assignments for each row in the data matrix DC. For each row, it evaluates the log-likelihood of assignment to all possible clusters and updates the cluster membership to the one that maximizes the likelihood. Singleton constraints are respected to avoid assigning rows to clusters that would result in empty or singleton groups.

Value

A binary matrix of the same dimensions as Updated_R, representing the updated row cluster membership after one iteration of the E-ReMI update step.

Note

This function is a part of the E-ReMI clustering framework.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

E_ReMI, pinv

Examples

I <- 10; P <- 4; J <- 5; Q <- 2
DC <- matrix(rnorm(I*J), I, J)
Updated_R <- diag(1, I, P)
Updated_C <- diag(1, J, Q)
Updated_omegahat <- c(0.5, rep((1-0.5)/P, P))
R_pinv <- pracma::pinv(t(Updated_R) %*% Updated_R)
C_pinv <- pracma::pinv(t(Updated_C) %*% Updated_C)
Updated_G <- R_pinv %*% t(Updated_R) %*% DC %*% Updated_C %*% C_pinv
Updated_G <- as.matrix(Updated_G)
Updated_M <- Updated_R %*% Updated_G %*% t(Updated_C)
Updated_sigma <- sum((DC - Updated_M)^2)/(I*J)
result <- Update_row_clusters_E_ReMI(
  DC, I, J, P,
  Updated_R, Updated_C,
  Updated_omegahat, Updated_G,
  Updated_sigma
)
result

Update Row Cluster Assignments in REMAXINT clustering method

Description

Updates the row cluster assignments for a fixed column clustering and G matrix in the REMAXINT clustering framework, using a maximum likelihood strategy.

Usage

Update_row_clusters_REMAXINT(DC, I, J, P, Updated_R, Updated_C, Updated_G)

Arguments

Details

The function iteratively evaluates each row in the data matrix and reassigns it to the cluster that maximizes the log-likelihood under the REMAXINT model. Rows assigned to singleton clusters are not re-evaluated. The function avoids assigning rows to clusters with only one member (singleton clusters) to prevent degenerate solutions.

Value

Returns an updated row cluster membership matrix of the same dimension as Updated_R, where each row is reassigned based on the maximum likelihood estimate given the current column clustering and latent interactions between row and column clusters of size size P × Q. This matrix has:

Note

Used internally by the REMAXINT model fitting procedure. Assumes fixed column cluster membership and updates row assignments to maximize model fit.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

REMAXINT, pinv

Examples

I <- 10; P <- 4; J <- 5; Q <- 2
DC <- matrix(rnorm(I*J), I, J)
Updated_R <- diag(1, I, P)
Updated_C <- diag(1, J, Q)
R_pinv <- pracma::pinv(t(Updated_R) %*% Updated_R)
C_pinv <- pracma::pinv(t(Updated_C) %*% Updated_C)
Updated_G <- R_pinv %*% t(Updated_R) %*% DC %*% Updated_C %*% C_pinv
Updated_G <- as.matrix(Updated_G)
result <- Update_row_clusters_REMAXINT(DC, I, J, P, Updated_R, Updated_C, Updated_G)
result

Extract Estimated Gamma Matrix for a Specific Cluster Combination

Description

Extracts the estimated Gamma matrix corresponding to a specific (P, Q) cluster solution from the output of the E-ReMI algorithm.

Usage

extract_Gamma_EReMI(x, selection)

Arguments

Details

This function simplifies access to the estimated Gamma matrix for a user-specified combination of row (P) and column (Q) clusters, as identified in the output of the E-ReMI algorithm. It automatically reformats the input if the selection is provided in character form (e.g., "(3,4)").

Value

A matrix of estimated Gamma values corresponding to the specified cluster combination.

If the cluster combination is invalid or not found, NULL is returned.

Note

Used internally by visualization and post-processing functions for E-ReMI output.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation-based alternative. Available on OSF (submitted to ADAC). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331. Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013.

See Also

E_ReMI, Permutation_Function, MaxInt_data_analysis, screeplot.MaxInt_data_analysis, MaxInteraction.plot

Print Summary for MaxInt Data Analysis Results

Description

Displays a concise summary of the results from a MaxInt_data_analysis object, including inference statistics and interaction Explained Sum of Squares (iESS).

Usage

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

Arguments

Details

This method provides a simple print interface for viewing the key components of a MaxInt_data_analysis object. It prints the results of the permutation-based inference tests and the computed Gamma_results, which contain the interaction Explained Sum of Squares (iESS) for each cluster combinations.

Value

Returns invisible(x). Called primarily for its side effects (printing to the console).

Note

This is the default print method for objects of class "MaxInt_data_analysis". Use it to quickly inspect inference and iESS results.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation-based alternative. Available on OSF (submitted to ADAC). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331. Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013.

See Also

MaxInt_data_analysis

Examples

set.seed(123)
Data <- matrix(rnorm(60), nrow = 10, ncol = 6)
Clus <- matrix(c(2,2, 2,3, 3,2), ncol = 2, byrow = TRUE)
result <- MaxInt_data_analysis(Data, Clus, Nruns = 5, permutations = 10, alpha_level = 0.05)
result

Scree Plot for Maximal Interaction Two-Mode Clustering Results

Description

Generates a scree plot showing the interaction Explained Sum of Squares (iESS) for different combinations of latent row and column clusters in the Maximal Interaction two-mode clustering analysis.

Usage

## S3 method for class 'MaxInt_data_analysis'
screeplot(x, ...)

Arguments

Details

This function helps visualize how the interaction Explained Sum of Squares (iESS) varies across different combinations of latent row and column cluster numbers, P and Q, by plotting their product (P - 1)(Q - 1) on the x-axis and the corresponding iESS on the y-axis. This plot aids in selecting the optimal number of latent row by column clusters.

Value

Returns a base R plot object for visual inspection. No return value is assigned invisibly.

If needed, additional customization can be applied using graphical parameters through the ... argument.

Note

Ensure that the input object contains a properly structured Gamma_results data frame. Use this plot to inform your choice of latent clusters.

Author(s)

Zaheer Ahmed (ahmed@ifado.de), Jan Schepers (jan.schepers@maastrichtuniversity.nl), Uwe Ligges (ligges@statistik.tu-dortmund.de), Alberto Cassese (alberto.cassese@unifi.it), Gerard van Breukelen (gerard.vbreukelen@maastrichtuniversity.nl), Katja Ickstadt (ickstadt@statistik.tu-dortmund.de), Edmund Wascher (wascher@ifado.de)

References

Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2025). Robustness study of normality-based likelihood ratio tests for testing maximal interaction two-mode clustering and a permutation based alternative. Available on OSF (submitted to Advances in Data Analysis and Classification). Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2023). E-ReMI: extended maximal interaction two-mode clustering. Journal of Classification, 40, 298-331 Ahmed, Z., van Breukelen, G. J. P., Schepers, J., & Cassese, A. (2021). REMAXINT: a two-mode clustering-based method for statistical inference on two-way interaction. Advances in Data Analysis and Classification, 15(4), 987-1013

See Also

MaxInt_data_analysis, MaxInteraction.plot

Examples

set.seed(123)
set.seed(123)
Data <- matrix(rnorm(60), nrow = 10, ncol = 6)
Clus <- matrix(c(2,2, 2,3, 3,2), ncol = 2, byrow = TRUE)
result <- MaxInt_data_analysis(Data, Clus, Nruns = 5, permutations = 10, alpha_level = 0.05)
screeplot(result)