np.aov.test: Nonparametric One-Way and RM ANOVA Tests

Description

Assuming a one-way (fixed effects) ANOVA model of the form $$Y_{ij} = \mu + \tau_j + \epsilon_{ij}$$ or a one-way repeated measures ANOVA model of the form $$Y_{ij} = \mu + \beta_i + \tau_j + \epsilon_{ij}$$ this function implements permutation tests of $H_0: (\forall j) \tau_j = \tau$ versus $H_1: (\exists j) \tau_j \neq \tau$. Note that $\mu$ is the overall mean/median ignoring block and group, $\beta_i$ is the $i$-th subject's block effect, $\tau_j$ is the $j$-th group's treatment effect, and $\epsilon_{ij}$ is an error term with mean/median zero.

Usage

np.aov.test(x, groups, blocks = NULL, 
            var.equal = FALSE, median.test = FALSE, 
            R = 9999, parallel = FALSE, cl = NULL,
            perm.dist = TRUE, na.rm = TRUE)

Value

statistic: Test statistic value.
p.value: p-value for testing $H_0: (\forall j) \tau_j = \tau$.
perm.dist: Permutation distribution of statistic.
repeated: Repeated-measures ANOVA?
var.equal: Assuming equal variances?
median.test: Testing the median?
R: Number of resamples.
method: Method used for permutation test. See Examples.
ngroups: Number of groups = nlevels(group)
nblocks: Number of blocks = nlevels(blocks) (if applicable)

Arguments

x: Numeric vector (or matrix) of data values (see Details).
groups: Factor vector giving the treatment group for each element/row of x.
blocks: Factor vector giving the block identification for each element/row of x.
var.equal: Logical indicating whether to treat the $k$ group's variances as being equal.
median.test: Logical indicating whether the location test is for the median. Default is FALSE, i.e., $\mu$ is the mean.
R: Number of resamples for the permutation test (positive integer).
parallel: Logical indicating if the parallel package should be used for parallel computing (of the permutation distribution). Defaults to FALSE, which implements sequential computing.
cl: Cluster for parallel computing, which is used when parallel = TRUE. Note that if parallel = TRUE and cl = NULL, then the cluster is defined as makeCluster(2L) to use two cores. To make use of all available cores, use the code cl = makeCluster(detectCores()).
perm.dist: Logical indicating if the permutation distribution should be returned.
na.rm: If TRUE (default), the arguments x and groups (and blocks if provided) are passed to the na.omit function to remove cases with missing data.

Author

Nathaniel E. Helwig <helwig@umn.edu>

Details

One-way ANOVA: the input x should be of length $N = \sum_{j=1}^k n_j$ where $n_j$ is size of $j$-th group.

RM ANOVA: the input x should be of length $N = n k$ where $n$ is number of blocks and $k$ is number of groups.

For multivariate models, the input x should be a matrix with $N$ rows and $m$ columns, where each column has $N = \sum_{j=1}^k n_j$ or $N = n k$ observations.

References

Helwig, N. E. (2019a). Statistical nonparametric mapping: Multivariate permutation tests for location, correlation, and regression problems in neuroimaging. WIREs Computational Statistics, 11(2), e1457. doi: 10.1002/wics.1457

Helwig, N. E. (2019b). Robust nonparametric tests of general linear model coefficients: A comparison of permutation methods and test statistics. NeuroImage, 201, 116030. doi: 10.1016/j.neuroimage.2019.116030

Examples

Run this code


###***###   ONE-WAY ANOVA   ###***###

# data generation design
N <- 90
k <- 3
g <- factor(rep(LETTERS[1:k], each = N/k))
tau <- c(-1/2, 0, 1/2)
sd <- c(1/2, 1, 2)

# generate data
set.seed(0)
x <- rnorm(N, mean = tau[g], sd = sd[g]) 

# mean test with unequal variances (robust W statistic)
set.seed(1)
np.aov.test(x, g)

# mean test with equal variances (classic F statistic)
set.seed(1)
np.aov.test(x, g, var.equal = TRUE)

# median test with unequal variances (robust Kruskal-Wallis statistic)
set.seed(1)
np.aov.test(x, g, median.test = TRUE)

# median test with equal variances (classic Kruskal-Wallis test)
set.seed(1)
np.aov.test(x, g, var.equal = TRUE, median.test = TRUE)

# Kruskal-Wallis test (asymptotic p-value)
kruskal.test(x, g)


if (FALSE) {

###***###   REPEATED MEASURES ANOVA   ###***###

# data generation design
N <- 90
k <- 3
n <- 30
g <- factor(rep(LETTERS[1:k], each = N/k))
b <- factor(rep(paste0("sub", 1:n), times = k),
            levels = paste0("sub", 1:n))
tau <- c(-1/2, 0, 1/2)
sd <- c(1/2, 1, 2)

# generate random block effects
set.seed(773)
beta <- runif(30, -1, 1)

# generate data
set.seed(0)
x <- rnorm(N, mean = tau[g] + beta[b], sd = sd[g]) 

# mean test with unequal variances (robust W statistic)
set.seed(1)
np.aov.test(x, g, b)

# mean test with equal variances (classic F statistic)
set.seed(1)
np.aov.test(x, g, b, var.equal = TRUE)

# median test with unequal variances (robust Friedman statistic)
set.seed(1)
np.aov.test(x, g, b, median.test = TRUE)

# median test with equal variances (classic Friedman test)
set.seed(1)
np.aov.test(x, g, b, var.equal = TRUE, median.test = TRUE)

# Friedman test (asymptotic p-value)
friedman.test(x, g, b)
}

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