‘WAnova’ is an R package that provides functions for conducting Welch’s ANOVA and Games-Howell post hoc tests based on summary statistics. These tests are particularly useful when working with unequal variances and sample sizes. Additionally, the package includes a Monte Carlo simulation to assess residual normality and homoscedasticity, with the option to apply a continuity correction for the resulting proportions.
You can install the latest version of ‘WAnova’ from GitHub with:
# If you don't have devtools installed, first install it:
install.packages("devtools")
# Then, install the WAnova package from GitHub:
devtools::install_github("niklasburgard/WAnova")
# Import packages if not already installed:
packages <- c("car", "utils", "stats", "SuppDists")
install.packages(setdiff(packages, rownames(installed.packages())))fmax.test(levels,n,sd)
welch_anova.test(levels, n, means, sd, effsize =
c(“AnL”,“Kirk”,“CaN”)
games_howell.test(levels, n, means, sd, conf.level = 0.95)
wanova_pwr.test(n, means, sd, power = 0.90, alpha = 0.05)
welch_anova.mc(n,means, sd, n_sim = 1000, alpha = 0.05, adj = TRUE)
levels Vector with level names of the
independent variable
n Vector with sample size for each level
means Vector with sample mean for each
level
sd Vector with sample standard deviation for
each level
effsize Options “AnL”, “Kirk”, “CaN”
conf.level Confidence level used in the
computation power Desired power of the test
alpha Significance level for the test
adj Logical, applies to continuity correction
if TRUE n_sim Number of Monte Carlo
Simulations
library(WAnova)
library(SuppDists)
# Example data
probe_data <- data.frame(
group = c("probe_a", "probe_b", "probe_c"),
size = c(10, 10, 10), # Equal sample sizes
mean = c(43.00000, 33.44444, 35.75000),
sd = c(4.027682, 9.302031, 16.298554)
)
# Perform Hartley's Fmax
result <- fmax_test(
levels = probe_data$group,
n = probe_data$size,
sd = probe_data$sd
)
#Print results
print(result)Note: Applicable results assume normally distributed data with equal sample sizes.
Null Hypothesis: Assumes homogeneity of variances, which means all
groups have the same variance.
Alternative Hypothesis: Assumes that not all group variances are equal.
This hypothesis is supported if the p-value is below the significance
level.
library(WAnova)
# Example data
probe_data <- data.frame(
group = c("probe_a", "probe_b", "probe_c"),
size = c(10, 9, 8),
mean = c(43.00000, 33.44444, 35.75000),
sd = c(4.027682, 9.302031, 16.298554)
)
# Perform Welch's ANOVA
result <- welch_anova.test(
levels = probe_data$group,
n = probe_data$size,
means = probe_data$mean,
sd = probe_data$sd,
effsize = "Kirk"
)
# Print summary
summary(result)Note: Omega squared can range from -1 to 1, with zero indicating no effect. When the observed F is less than one, omega squared will be negative. It has been suggested that values of .01, .06 and .14 represent small, medium and large effects, respectively (Kirk 1996).
Note: Traditional omega squared assumes
homogeneity of variance, using parameters calculated in a traditional
ANOVA with unweighted means. There are three methods—Kirk
(“Kirk”), Carroll and Nordholm
(“CaN”), and Albers and Lakens
(“AnL”)—to estimate omega squared using
summary statistics, all of which yield the same result when based on
unweighted means.
When applying parameters derived from Welch’s ANOVA, the
Kirk and CaN methods
produce an adjusted omega squared that reflects the weighted means from
the F-statistic but do not account for the corrected within-group
degrees of freedom associated with those weighted means.
The AnL method further adjusts omega squared
to incorporate these corrected degrees of freedom, aligning with the
design of Welch’s ANOVA and providing a more accurate measure, making it
the preferred approach.
References:
Welch, B. L. (1951). On the comparison of
several mean values: an alternative approach. Biometrika 38.3/4,
330-336.
Hays, W. L. (1973). Statistics for the social sciences (2nd ed.). Holt,
Rinehart and Winston, 486.
Kirk, R. E. (1996). Practical significance: A
concept whose time has come. Educational and Psychological Measurement,
56(5), 746-759.
Carroll, R. M., & Nordholm, L. A. (1975).
Sampling characteristics of Kelley’s epsilon and Hays’ omega Educational
and Psychological Measurement, 35(3), 541-554.
Albers, C., & Lakens, D. (2018). When
power analyses based on pilot data are biased: Inaccurate effect size
estimators and follow-up bias. Journal of Experimental Social
Psychology, 74, 187–195.
library(WAnova)
# Conduct Games-Howell post hoc test
posthoc_result <- games_howell.test(
levels = probe_data$group,
n = probe_data$size,
means = probe_data$mean,
sd = probe_data$sd
)
# Print results
print(posthoc_result)References:
Games, P. A., & Howell, J. F. (1976).
Pairwise Multiple Comparison Procedures with Unequal N’s and/or
Variances: A Monte Carlo Study. Journal of Educational and Behavioural
Statistics, 1, 113-125.
library(WAnova)
n <- c(10, 10, 10, 10)
means <- c(1, 0, 0, -1)
sd <- c(1, 1, 1, 1)
result <- wanova_pwr.test(n, means, sd, power = 0.90, alpha = 0.05)
print(result)References:
Levy, K. J. (1978a). Some empirical power
results associated with Welch’s robust analysis of variance technique.
Journal of Statistical Computation and Simulation, 8, 43–48.
Show-Li, J., & Gwowen, S. (2014). Sample
size determinations for Welch’s test in one-way heteroscedastic ANOVA .
British Psychological Society, 67(1), 72-93.
library(WAnova)
means <- c(50, 55, 60)
sd <- c(10, 12, 15)
n <- c(30, 35, 40)
# Perform Monte Carlo simulation
result <- welch_anova.mc(means = means, sd = sd, n = n, n_sim = 1000, alpha = 0.05)
# Print results
print(result)Note: You can apply a continuity correction (r+1)/(N+1) to the resulting proportions by setting adj = TRUE (default). This is useful for improving the accuracy of estimates when proportions are small or close to 1. Without the correction, the original proportion estimate p is calculated as the ratio of simulations where the residuals meet the assumption of normality or homoscedasticity (r) to the total number of simulations (N), with no adjustment applied.
References: Davison AC, Hinkley DV (1997). Bootstrap methods and their application. Cambridge University Press, Cambridge, United Kingdom
To cite the WAnova package in publications, please use:
Niklas Burgard (2023). WAnova: Welch's Anova from Summary Statistics. R package version 0.4.0. https://github.com/niklasburgard/WAnovaYou can also find a BibTeX entry for LaTeX users:
@Manual{,
title = {WAnova: Welch's Anova from Summary Statistics},
author = {Niklas Burgard},
year = {2023},
note = {R package version 0.4.0},
url = {https://github.com/niklasburgard/WAnova},
}