A nonparametric R package for detecting interaction in two-way ANOVA designs with balanced replication using all possible crossed comparisons (APC). The test statistics in this package extend the methods of Hartlaub, Dean, and Wolfe (1999) to cases with multiple replications, using rank-based alignment techniques.

You can install the development version of APCinteraction from GitHub with:

# Install devtools if you don't have it
install.packages("devtools")

# Install APCinteraction from GitHub
devtools::install_github("tranbaokhue/APCinteraction")

# Load the package
library(APCinteraction)

The companion package APCinteractionData provides 90 pre-computed null distributions for each test statistic (100,000 simulations each), covering two-way designs with 2 to 6 factor levels and 1 to 10 replications. Install it for fast p-value estimation:

devtools::install_github("tranbaokhue/APCinteractionData")

This companion package is optional. Two null distribution files are bundled in APCinteraction for the examples, and users can generate their own with sim_nullAPCSSA() / sim_nullAPCSSM().

The APCinteraction R package provides nonparametric tests for detecting interaction in two-way ANOVA designs with balanced replications. The core test statistics — APCSSA and APCSSM — are based on All Possible Comparisons (APC) and extend the methods of Hartlaub, Dean, and Wolfe (1999) to settings with replication, as recommended by Salazar-Alvarez et al. (2014).

These tests are designed to be robust and flexible, avoiding reliance on restrictive parametric assumptions. The methods utilize rank-based procedures with alignment to eliminate nuisance effects, offering enhanced power in detecting interactions.

Key functions the package provide are:

These statistics use All Possible Comparisons (APC) to detect interaction effects. The final test statistic is the maximum of the two standardized statistics. A p-value is estimated by comparing this statistic to a pre-simulated null distribution specific to the design dimensions. In addition to providing the statistics and estimated p-value in the summary table, the function will also output an interaction plot to help visualize potential interaction.

# Test using mean alignment
APCSSA(formula, data, numSim = 1e+05)

# Test using median alignment
APCSSM(formula, data, numSim = 1e+05)

If the design setting you are hoping to check for interaction is not among the pre-simulated ones in APCinteractionData, we also provide functions to simulate the null distribution on your own device.

While the companion package APCinteractionData provides an extensive collection of 90 pre-simulated null distributions (with 100,000 replications), there are definitely still scenarios where that is not sufficient. For example, a researcher might have an experiment with more factor levels or want to guarantee the accuracy of the estimated p-value by comparing it with null distributions generated from 150,000 and 250,000 replications.

Thus, we also provide functions for generating your own null distributions given a design and replication number. These functions are designed with the same structure as the function to calculate the test statistics for ease of use.

# Simulation for APCSSA null distributions
sim_nullAPCSSA(formula, data, numSim = 1e+05)

# Simulation for APCSSM null distributions
sim_nullAPCSSM(formula, data, numSim = 1e+05)

Accompanying the two simulation functions above, we provide users a function to quickly save the simulated values to a specified directory so that they are accessible by APCSSA/APCSSM even in other R sessions. This is optional, but is highly recommended!

save_null(type, i, j, k, path)

The following are basic examples which show you how to solve a common problem of determining whether there is interaction within a dataset.

# Set the seed for reproducibility
set.seed(206)

# Parameters - number of levels for factors A and B, and replications
nA <- 3; nB <- 3; nrep <- 3

# Generate levels and create the full design
A_vals <- 1:nA
B_vals <- 1:nB
design <- expand.grid(A = A_vals, B = B_vals)
design <- design[rep(seq_len(nrow(design)), each = nrep), ]

# Compute response as interaction (product of numeric factor levels)
mu <- A_vals[design$A] * B_vals[design$B]
value <- mu + rnorm(length(mu))  # Add normal noise

# Assemble the data
data <- data.frame(
  value = value,
  A = factor(design$A),
  B = factor(design$B)
)

# Run the APCSSA test and view summary
result <- APCSSA(value ~ A + B, data = data)
summary.APC(result)

# Set seed for reproducibility
set.seed(36)

# Parameters - number of levels for factos A and B, replications, and interaction effect
nA <- 3; nB <- 4; nrep <- 2; c <- 1.25

# Generate levels and create the full design
A_vals <- seq(-2, 2, length.out = nA)
B_vals <- seq(-1.5, 1.5, length.out = nB)
design <- expand.grid(A = 1:nA, B = 1:nB)
design <- design[rep(seq_len(nrow(design)), each = nrep), ]

# Create the specific interaction matrix: alternate ±c in top rows
specInt <- matrix(0, nA, nB)
specInt[1:2, 1:nB] <- matrix(c(c, -c, -c, c), nrow = 2, byrow = TRUE)

# Compute response
mu <- A_vals[design$A] + B_vals[design$B] + specInt[cbind(design$A, design$B)]
value <- mu + rt(length(mu), df = 1)

# Assemble the data
data <- data.frame(
  value = value,
  A = factor(design$A, labels = round(A_vals, 2)),
  B = factor(design$B, labels = round(B_vals, 2))
)

# With Cauchy errors, we opt for APCSSM to check for interaction
result <- APCSSM(value ~ A + B, data = data)
summary.APC(result)

Tran, B. K., Wagaman, A. S., Nguyen, A., Jacobson, D., & Hartlaub, B. (2024). Nonparametric tests for interaction in two-way ANOVA with balanced replications. arXiv preprint arXiv:2410.04700.