APCinteraction: Nonparametric Interaction Tests in Balanced Two-Way ANOVA Models

Description

Provides nonparametric tests for interaction in two-way ANOVA designs with balanced replications using All Possible Comparisons (APC). The APCSSA and APCSSM statistics extend previous methods, allow greater flexibility, and demonstrate higher power in detecting interactions for non-normal data. The package includes optimized functions for computing these test statistics, generating interaction plots, and simulating their null distributions.

Main functions

APCSSA

Compute the APCSSA interaction test statistic.

sim_nullAPCSSA

Simulate the null distribution for the APCSSA statistic.

APCSSM

Compute the APCSSM interaction test statistic.

sim_nullAPCSSM

Simulate the null distribution for the APCSSM statistic.

save_null

Save the simulated null distributions locally for future usage.

Author(s)

Bao Khue Tran (maintainer) baokhuetran@outlook.com, Andrew Nguyen, Amy Wagaman, and Bradley Hartlaub

See Also

Useful links:

• https://github.com/tranbaokhue/APCinteraction

• Report bugs at https://github.com/tranbaokhue/APCinteraction/issues

APCSSA

Description

This function tests for interaction in a two-way layout using all possible crossed comparisons (APC) based on aligned ranks, with alignment performed using the mean.

Usage

APCSSA(formula, data, numSim = 1e+05)

Arguments

Details

APCSSA performs a nonparametric test for interaction in a two-way layout with balanced replication. Two statistics, APCCRA and APCRCA, are computed by aligning the data with the averages and ranking within the rows or columns.

These statistics use all possible crossed comparisons 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.

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

If the design is not covered by the pre-computed distributions or if a higher precision is desired (numSim > 100,000), users must first generate the null distribution using sim_nullAPCSSA before using this function to evaluate the interaction effect.

For more information, see the referenced article.

Value

An object of class "APC", which is a data frame with columns:

• Interaction: The interaction term.

• Statistic: The APCSSA test statistic (maximum of APCCRA and APCRCA).

• p.value: The estimated p-value.

Use summary.APC to print a formatted summary table and generate an interaction plot.

References

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. https://arxiv.org/abs/2410.04700

See Also

summary.APC, sim_nullAPCSSA, APCSSM

Examples

# 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(result)

APCSSM

Description

This function tests for interaction in a two-way layout using all possible crossed comparisons (APC) based on aligned ranks, with alignment performed using the median.

Usage

APCSSM(formula, data, numSim = 1e+05)

Arguments

Details

APCSSM performs a nonparametric test for interaction in a two-way layout with balanced replication. Two statistics, APCCRM and APCRCM, are computed by aligning the data with the medians and ranking within the rows or columns.

These statistics use all possible crossed comparisons 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.

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

If the design is not covered by the pre-computed distributions or if a higher precision is desired (numSim > 100,000), users must first generate the null distribution using sim_nullAPCSSM before using this function to evaluate the interaction effect.

For more information, see the referenced article.

Value

An object of class "APC", which is a data frame with columns:

• Interaction: The interaction term.

• Statistic: The APCSSM test statistic (maximum of APCCRM and APCRCM).

• p.value: The estimated p-value.

Use summary.APC to print a formatted summary table and generate an interaction plot.

References

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. https://arxiv.org/abs/2410.04700

See Also

summary.APC, sim_nullAPCSSM, APCSSA

Examples

# 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(result)

Save null distribution to a directory

Description

Saves a null distribution from the package's cache to RData files in the specified directory.

Usage

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

Arguments

Value

Invisibly returns the file paths to where the data were saved.

See Also

sim_nullAPCSSA, sim_nullAPCSSM

Examples

## Not run: 
# First generate the desired null distribution
sim_nullAPCSSA(2, 2, 2, 50000)

# Save to a temporary directory
save_null("APCSSA", 2, 2, 2, path = tempdir())

# Or save to a custom location
save_null("APCSSA", 2, 2, 2, path = "path/to/directory")

## End(Not run)

Simulate the null distribution for APCSSA

Description

A function for simulating the null distribution of APCSSA.

Usage

sim_nullAPCSSA(i, j, k, numSim = 1e+05, parallel = TRUE, verbose = TRUE)

Arguments

Details

While we have provided null distributions for various settings in this package, there are still two-way settings that we don't readily have the null distribution to evaluate the significance of the test statistics (see References for the complete list of settings available). For example, one might wish to estimate the p-value with a higher numSim, e.g., 250,000.

Accompanying this function is our customized save_null to help researchers save the null distribution generated for future reference and calculations of the statistics in similar experiment design.

Value

A data frame with the null mean and standard deviation for the two test statistics (APCCRA and APCRCA) that will get standardized into APCSSA and a numeric vector with length equal to the numSim of all APCSSA statistics on the null data sets.

Note

Benchmarks for 100,000 simulations (no bias toward any chip type):

• 3×3×3 (243 summands): ~ 5 hours on an Intel i5

• 2×5×5 (625 summands): < 5 hours on Apple M1 (7 cores, parallel)

• 6×6×5 (9375 summands): ~ 108 hours on Apple M1 (7 cores, parallel) Factor levels (i,j) have some impact, but total summands scale as k^4 (see References for more information), so replication depth (k) dominates run time.

Tip: leave parallel = TRUE to shorten run times dramatically.

Results are stored in the package's cache. The customized function save_null is highly recommended for saving the null distribution to your local storage for future reference and testing in the same design.

References

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. https://arxiv.org/abs/2410.04700

See Also

save_null, APCSSA, sim_nullAPCSSM

Examples

## Not run: 
sim_nullAPCSSA(2, 2, 2, 50000)

## End(Not run)

Simulate the null distribution for APCSSM

Description

A function for simulating the null distribution of APCSSM

Usage

sim_nullAPCSSM(i, j, k, numSim = 1e+05, parallel = TRUE, verbose = TRUE)

Arguments

Details

While we have provided null distributions for various settings in this package, there are still two-way settings that we don't readily have the null distribution to evaluate the significance of the test statistics (see References for the complete list of settings available). For example, one might wish to estimate the p-value with a higher numSim, e.g., 250,000.

#' Accompanying this function is our customized save_null to help researchers save the null distribution generated for future reference and calculations of the statistics in similar experiment design.

Value

A data frame with the null mean and standard deviation for the two test statistics (APCCRM and APCRCM) that will get standardized into APCSSM and a numeric vector with length equal to the numSim of all APCSSM statistics on the null data sets.

Note

Benchmarks for 100,000 simulations (no bias toward any chip type):

• 3×3×3 (243 summands): ~5 hours on an Intel i5

• 2×5×5 (625 summands): < 5 hours on Apple M1 (7 cores, parallel)

• 6×6×5 (9375 summands): ~108 hours on Apple M1 (7 cores, parallel) Factor levels (i,j) have some impact, but total summands scale as k^4 (see References for more information), so replication depth (k) dominates run time.

Tip: leave parallel = TRUE to shorten run times dramatically.

Results are stored in the package's cache. The customized function save_null is highly recommended for saving the null distribution to your local storage for future reference and testing in the same design.

References

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. https://arxiv.org/abs/2410.04700

See Also

save_null, APCSSM, sim_nullAPCSSA

Examples

## Not run: 
sim_nullAPCSSM(2, 2, 2, 50000)

## End(Not run)

Summary method for APC objects

Description

Prints a formatted summary table and generates an interaction plot for objects returned by APCSSA or APCSSM.

Usage

## S3 method for class 'APC'
summary(object, ...)

Arguments

Value

Invisibly returns the input object.

See Also

APCSSA, APCSSM