np.loc.test: Nonparametric Tests of Location Parameters

Description

Performs one and two sample nonparametric (randomization) tests of location parameters, i.e., means and medians. Implements univariate and multivariate tests using eight different test statistics: Student's one-sample t-test, Johnson's modified t-test, Wilcoxon's Signed Rank test, Fisher's Sign test, Student's two-sample t-test, Welch's t-test, Wilcoxon's Rank Sum test (i.e., Mann-Whitney's \(U\) test), and a studentized Wilcoxon test for unequal variances.

Usage

np.loc.test(x, y = NULL,
            alternative = c("two.sided", "less", "greater"),
            mu = 0, paired = FALSE, var.equal = FALSE, 
            median.test = FALSE, symmetric = TRUE,
            R = 9999, parallel = FALSE, cl = NULL,
            perm.dist = TRUE, na.rm = TRUE)

Value

statistic: Test statistic value.
p.value: p-value for testing \(H_0: \mu = \mu_0\).
perm.dist: Permutation distribution of statistic.
alternative: Alternative hypothesis.
null.value: Null hypothesis value for \(\mu\).
var.equal: Assuming equal variances? Only for two sample tests.
median.test: Testing the median?
symmetric: Assuming symmetry? Only for one sample and paired tests.
R: Number of resamples.
exact: Exact permutation test? See Note.
estimate: Estimate of parameter \(\mu\).
univariate: Univariate test statistic value for \(j\)-th variable (for multivariate input).
adj.p.value: Adjusted p-value for testing significance of \(j\)-th variable (for multivariate input).
method: Method used for permutation test. See Details.

Arguments

x: Numeric vector (or matrix) of data values.
y: Optional numeric vector (or matrix) of data values.
alternative: Alternative hypothesis. Must be either "two.sided" (\(H_1: \mu \neq \mu_0\)), "less" (\(H_1: \mu < \mu_0\)), or "greater" (\(H_1: \mu > \mu_0\)).
mu: Null hypothesis value \(\mu_0\). Defaults to zero.
paired: Logical indicating whether you want a paired location test.
var.equal: Logical indicating whether to treat the two 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.
symmetric: Logical indicating if the distribution of x should be assumed to be symmetric around \(\mu\). Only used for one (or paired) sample tests.
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 y if provided) are passed to the na.omit function to remove cases with missing data.

Author

Nathaniel E. Helwig <helwig@umn.edu>

Multivariate Tests

If the input x (and possibly y) is a matrix with \(m > 1\) columns, the multivariate test statistic is defined as

`alternative`	`statistic`
`two.sided`	`max(abs(univariate))`
`less`	`min(univariate)`
`greater`	`max(univariate)`

The global null hypothesis (across all \(m\) variables) is tested by comparing the observed statistic to the permutation distribution perm.dist. This produces the p.value for testing the global null hypothesis.

The local null hypothesis (separately for each variable) is tested by comparing the univariate test statistic to perm.dist. This produces the adjusted p-values (adj.p.values), which control the familywise Type I error rate across the \(m\) tests.

Details

One sample	\(\mu\) is the mean (or median) of \(X\)
Paired	\(\mu\) is the mean (or median) of \(X - Y\)
Two sample	\(\mu\) is the mean difference \(E(X) - E(Y)\)
	or the median of the differences \(X - Y\)

For one (or paired) sample tests, the different test statistics can be obtained using

	`median.test = F`	`median.test = T`
`symmetric = F`	Johnson t test	Fisher sign test
`symmetric = T`	Student t test	Wilcoxon signed rank test

For two sample tests, the different test statistics can be obtained using

	`median.test = F`	`median.test = T`
`var.equal = F`	Welch t test	Studentized Wilcoxon test
`var.equal = T`	Student t test	Wilcoxon rank sum test

References

Blair, R. C., Higgins, J. J., Karniski, W., & Kromrey, J. D. (1994). A study of multivariate permutation tests which may replace Hotelling's T2 test in prescribed circumstances. Multivariate Behavioral Research, 29(2), 141-163. doi: 10.1207/s15327906mbr2902_2

Chung, E., & Romano, J. P. (2016). Asymptotically valid and exact permutation tests based on two-sample U-statistics. Journal of Statistical Planning and Inference, 168, 97-105. doi: 10.1016/j.jspi.2015.07.004

Fisher, R. A. (1925). Statistical methods for research workers. Edinburgh: Oliver and Boyd.

Helwig, N. E. (2019). 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

Janssen, A. (1997). Studentized permutation tests for non-i.i.d. hypotheses and the generalized Behrens-Fisher problem. Statistics & Probability Letters , 36 (1), 9-21. doi: 10.1016/S0167-7152(97)00043-6

Johnson, N. J. (1978). Modified t tests and confidence intervals for asymmetrical populations. Journal of the American Statistical Association, 73 (363), 536-544. doi: 10.2307/2286597

Mann, H. B., & Whitney, D. R. (1947). On a test of whether one of two random variables is stochastically larger than the other. Annals Of Mathematical Statistics, 18(1), 50-60. doi: 10.1214/aoms/1177730491

Pitman, E. J. G. (1937a). Significance tests which may be applied to samples from any populations. Supplement to the Journal of the Royal Statistical Society, 4(1), 119-130. doi: 10.2307/2984124

Romano, J. P. (1990). On the behavior of randomization tests without a group invariance assumption. Journal of the American Statistical Association, 85(411), 686-692. doi: 10.1080/01621459.1990.10474928

Student. (1908). The probable error of a mean. Biometrika, 6(1), 1-25. doi: 10.2307/2331554

Welch, B. L. (1938). The significance of the difference between two means when the population variances are unequal. Biometrika, 39(3/4), 350-362. doi: 10.2307/2332010

Wilcoxon, F. (1945). Individual comparisons by ranking methods. Biometrics Bulletin, 1(6), 80-83. doi: 10.2307/3001968

Examples

Run this code


######******######   UNIVARIATE   ######******######

###***###   ONE SAMPLE   ###***###

# generate data
set.seed(1)
n <- 10
x <- rnorm(n, mean = 0.5)

# one sample t-test
set.seed(0)
np.loc.test(x)

# Johnson t-test
set.seed(0)
np.loc.test(x, symmetric = FALSE)

# Wilcoxon signed rank test
set.seed(0)
np.loc.test(x, median.test = TRUE)

# Fisher sign test
set.seed(0)
np.loc.test(x, median.test = TRUE, symmetric = FALSE)


###***###   PAIRED SAMPLE   ###***###

# generate data
set.seed(1)
n <- 10
x <- rnorm(n, mean = 0.5)
y <- rnorm(n)

# paired t-test
set.seed(0)
np.loc.test(x, y, paired = TRUE)

# paired Johnson t-test
set.seed(0)
np.loc.test(x, y, paired = TRUE, symmetric = FALSE)

# paired Wilcoxon signed rank test
set.seed(0)
np.loc.test(x, y, paired = TRUE, median.test = TRUE)

# paired Fisher sign test
set.seed(0)
np.loc.test(x, y, paired = TRUE, median.test = TRUE, symmetric = FALSE)


###***###   TWO SAMPLE   ###***###

# generate data
set.seed(1)
m <- 7
n <- 8
x <- rnorm(m, mean = 0.5)
y <- rnorm(n)

# Welch t-test
set.seed(0)
np.loc.test(x, y)

# Student t-test
set.seed(0)
np.loc.test(x, y, var.equal = TRUE)

# Studentized Wilcoxon test
set.seed(0)
np.loc.test(x, y, median.test = TRUE)

# Wilcoxon rank sum test
set.seed(0)
np.loc.test(x, y, var.equal = TRUE, median.test = TRUE)



if (FALSE) {

######******######   MULTIVARIATE   ######******######

###***###   ONE SAMPLE   ###***###

# generate data
set.seed(1)
n <- 10
x <- cbind(rnorm(n, mean = 0.5), 
           rnorm(n, mean = 1), 
           rnorm(n, mean = 1.5))

# multivariate one sample t-test
set.seed(0)
ptest <- np.loc.test(x)
ptest
ptest$univariate
ptest$adj.p.values


###***###   PAIRED SAMPLE   ###***###

# generate data
set.seed(1)
n <- 10
x <- cbind(rnorm(n, mean = 0.5), 
           rnorm(n, mean = 1), 
           rnorm(n, mean = 1.5))
y <- matrix(rnorm(n * 3), nrow = n, ncol = 3)

# multivariate paired t-test
set.seed(0)
ptest <- np.loc.test(x, y, paired = TRUE)
ptest
ptest$univariate
ptest$adj.p.values


###***###   TWO SAMPLE   ###***###

# generate data
set.seed(1)
m <- 7
n <- 8
x <- cbind(rnorm(m, mean = 0.5), 
           rnorm(m, mean = 1), 
           rnorm(m, mean = 1.5))
y <- matrix(rnorm(n * 3), nrow = n, ncol = 3)

# multivariate Welch t-test
set.seed(0)
ptest <- np.loc.test(x, y)
ptest
ptest$univariate
ptest$adj.p.values
}

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