Startup Message for nptest

Description

Prints the startup message when nptest is loaded. Not intended to be called by the user.

Details

The ‘nptest’ ascii start-up message was created using the taag software.

References

https://patorjk.com/software/taag/

Generate All Sign-Flips of n Elements

Description

Generates all 2^n vectors of length n consisting of the elements -1 and 1.

Usage

flipn(n)

Arguments

Details

Adapted from the "bincombinations" function in the e1071 R package.

Value

Matrix of dimension n by 2^n where each column contains a unique sign-flip vector.

Warning

For large n this function will consume a lot of memory and may even crash R.

Note

Used for exact tests in np.loc.test and np.reg.test.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

Meyer, D., Dimitriadou, E., Hornik, K., Weingessel, A., & Leisch, F. (2018). e1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien. R package version 1.7-0. https://CRAN.R-project.org/package=e1071

Examples

flipn(2)
flipn(3)

Monte Carlo Standard Errors for Tests

Description

This function calculates Monte Carlo standard errors for (non-exact) nonparametric tests. The MCSEs can be used to determine (i) the accuracy of a test for a given number of resamples, or (ii) the number of resamples needed to achieve a test with a given accuracy.

Usage

mcse(R, delta, conf.level = 0.95, sig.level = 0.05,
     alternative = c("two.sided", "one.sided"))

Arguments

Details

Note: either R or delta must be provided.

Let F(x) denote the distribution function for the full permutation distribution, and let G(x) denote the approximation obtained from R resamples. The Monte Carlo standard error is given by

\sigma(x) = \sqrt{ F(x) [1 - F(x)] / R }

which is the standard deviation of G(x).

A symmetric confidence interval for F(x) can be approximated as

G(x) +/- C \sigma(x)

where C is some quantile of the standard normal distribution. Note that the critical value C corresponds to the confidence level (conf.level) of the approximation.

Let \alpha denote the significance level (sig.level) for a one-sided test (\alpha is one-half the significance level for two-sided tests). Define a to be the value of the test statistic such that F(a) = \alpha.

The parameter \delta (delta) quantifies the accuracy of the approximation, such that

|G(a) - \alpha| < \alpha \delta

with a given confidence, which is controlled by the conf.level argument.

Value

Note

This function is only relevant for non-exact tests. For exact tests, F(x) = G(x) so the Monte Carlo standard error is zero.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

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

See Also

np.cor.test, np.loc.test, np.reg.test

Examples

###***###   EXAMPLE 1   ###***###

# get the Monte Carlo standard error and the 
# accuracy (i.e., delta) for given R = 10000
# using the default two-sided alternative hypothesis,
# the default confidence level (conf.level = 0.95),
# and the default significance level (sig.level = 0.05)

mcse(R = 10000)

# se = 0.0016
# delta = 0.1224


###***###   EXAMPLE 2   ###***###

# get the Monte Carlo standard error and the 
# number of resamples (i.e., R) for given delta = 0.01
# using a one-sided alternative hypothesis,
# the default confidence level (conf.level = 0.95),
# and the default significance level (sig.level = 0.05)

mcse(delta = 0.1, alternative = "one.sided")

# se = 0.0026
# R = 7299

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)

Arguments

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.

Value

Note

For the one-way ANOVA, the number of elements of the exact (i.e., fully enumerated) permutation distribuion is given by the multinomial coefficient:

\frac{N!}{n_1! n_2! \cdots n_k!}

which will be quite large (much larger than typcailly choices for R) for any non-trivial sample sizes. Consequently, exact tests are **not** implemented by this function.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

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

See Also

plot.np.aov.test S3 plotting method for visualizing the results

Examples

###***###   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)


## Not run: 

###***###   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)

## End(Not run)

Nonparametric Bootstrap Resampling

Description

Nonparametric bootstrap resampling for univariate and multivariate statistics. Computes bootstrap estimates of the standard error, bias, and covariance. Also computes five different types of bootstrap confidence intervals: normal approximation interval, basic (reverse percentile) interval, percentile interval, studentized (bootstrap-t) interval, and bias-corrected and accelerated (BCa) interval.

Usage

np.boot(x, statistic, ..., R = 9999, level = c(0.9, 0.95, 0.99),
        method = c("norm", "basic", "perc", "stud", "bca")[-4], 
        sdfun = NULL, sdrep = 99, jackknife = NULL, 
        parallel = FALSE, cl = NULL, boot.dist = TRUE)

Arguments

Details

The first three intervals (normal, basic, and percentile) are only first-order accurate, whereas the last two intervals (studentized and BCa) are both second-order accurate. Thus, the results from the studentized and BCa intervals tend to provide more accurate coverage rates.

Unless the standard deviation function for the studentized interval is input via the sdfun argument, the studentized interval can be quite computationally costly. This is because an inner bootstrap is needed to estimate the standard deviation of the statistic for each (outer) bootstrap replicate—and you may want to increase the default number of inner bootstrap replicates (see Note).

The efficiency of the BCa interval will depend on the sample size n and the computational complexity of the (jackknife) statistic estimate. Assuming that n is not too large and the jackknife statistic is not too difficult to compute, the BCa interval can be computed reasonably quickly—especially in comparison the studentized interval with an inner bootstrap.

Computational details of the various confidence intervals are described in Efron and Tibshirani (1994) and in Davison and Hinkley (1997). For a useful and concise discussion of the various intervals, see Carpenter and Bithell (2000).

Value

Note

If boot.dist = TRUE, the output boot.dist will be a matrix of dimension R by length(statistic(x, ...)) if the statistic is multivariate. Otherwise the bootstrap distribution will be a vector of length R.

For the "stud" method, the default of sdrep = 99 may produce a crude estimate of the standard deviation of the statistic(s). For more accurate estimates, the value of sdrep may need to be set substantially larger, e.g., sdrep = 999.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

Carpenter, J., & Bithell, J. (2000). Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians. Statistics in Medicine, 19(9), 1141-1164. doi: 10.1002/(SICI)1097-0258(20000515)19:9%3C1141::AID-SIM479%3E3.0.CO;2-F

Davison, A. C., & Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge University Press. doi: 10.1017/CBO9780511802843

Efron, B., & Tibshirani, R. J. (1994). An Introduction to the Boostrap. Chapman & Hall/CRC. doi: 10.1201/9780429246593

Examples

######***######   UNIVARIATE DATA   ######***######

### Example 1: univariate statistic (median)

# generate 100 standard normal observations
set.seed(1)
n <- 100
x <- rnorm(n)

# nonparametric bootstrap
npbs <- np.boot(x = x, statistic = median)
npbs


### Example 2: multivariate statistic (quartiles)

# generate 100 standard normal observations
set.seed(1)
n <- 100
x <- rnorm(n)

# nonparametric bootstrap
npbs <- np.boot(x = x, statistic = quantile, 
                probs = c(0.25, 0.5, 0.75))
npbs



## Not run: 

######***######   MULTIVARIATE DATA   ######***######

### Example 1: univariate statistic (correlation)

## Generate bivariate data with population var = 1 and cor = 0.5

# correlation matrix square root (with rho = 0.5)
rho <- 0.5
val <- c(sqrt(1 + rho), sqrt(1 - rho))
corsqrt <- matrix(c(val[1], -val[2], val), 2, 2) / sqrt(2)

# generate 100 bivariate observations (with rho = 0.5)
n <- 100
set.seed(1)
data <- cbind(rnorm(n), rnorm(n)) %*% corsqrt

## Method A: x = data frame;  statistic of x

# define statistic function
statfun <- function(data) cor(data[,1], data[,2])

# nonparametric bootstrap
set.seed(2)
npbs <- np.boot(x = data, statistic = statfun)
npbs

## Method B: x = 1:n;  statistic of data[ix,] with data passed via ...

# define statistic function
statfun <- function(ix, data) cor(data[ix,1], data[ix,2])

# nonparametric bootstrap
set.seed(2)
npbs <- np.boot(x = 1:n, statistic = statfun, data = data)
npbs


### Example 2: multivariate statistic (variances and covariance)

## Generate bivariate data with population var = 1 and cor = 0.5

# correlation matrix square root (with rho = 0.5)
rho <- 0.5
val <- c(sqrt(1 + rho), sqrt(1 - rho))
corsqrt <- matrix(c(val[1], -val[2], val), 2, 2) / sqrt(2)

# generate 100 bivariate observations (with rho = 0.5)
n <- 100
set.seed(1)
data <- cbind(rnorm(n), rnorm(n)) %*% corsqrt

## Method A: x = data frame;  statistic of x

# define statistic function
statfun <- function(data) {
  cmat <- cov(data)
  ltri <- lower.tri(cmat, diag = TRUE)
  cvec <- cmat[ltri]
  names(cvec) <- c("var(x1)", "cov(x1,x2)", "var(x2)")
  cvec
}

# nonparametric bootstrap
set.seed(2)
npbs <- np.boot(x = data, statistic = statfun)
npbs

## Method B: x = 1:n;  statistic of data[ix,] with data passed via ...

# define statistic function
statfun <- function(ix, data) {
  cmat <- cov(data[ix,])
  ltri <- lower.tri(cmat, diag = TRUE)
  cvec <- cmat[ltri]
  names(cvec) <- c("var(x1)", "cov(x1,x2)", "var(x2)")
  cvec
}

# nonparametric bootstrap
set.seed(2)
npbs <- np.boot(x = 1:n, statistic = statfun, data = data)
npbs



######***######   REGRESSION   ######***######

### Example 1: bootstrap cases

## Generate bivariate data with E(y|x) = 1 + 2 * x

# generate 100 observations
n <- 100
set.seed(1)
x <- seq(0, 1, length.out = n)
y <- 1 + 2 * x + rnorm(n)
data <- data.frame(x = x, y = y)

## Method A: x = data frame;  statistic of x

# define statistic function
statfun <- function(data) {
  lm(y ~ x, data = data)$coefficients
}

# nonparametric bootstrap
set.seed(2)
npbs <- np.boot(x = data, statistic = statfun)
npbs

## Method B: x = 1:n;  statistic of data[ix,] with data passed via ...

# define statistic function
statfun <- function(ix, data) {
  lm(y ~ x, data = data[ix,])$coefficients
}

# nonparametric bootstrap
set.seed(2)
npbs <- np.boot(x = 1:n, statistic = statfun, data = data)
npbs


### Example 2: bootstrap residuals

# generate 100 observations
n <- 100
set.seed(1)
x <- seq(0, 1, length.out = n)
y <- 1 + 2 * x + rnorm(n)
data <- data.frame(x = x, y = y)

# prepare data
mod0 <- lm(y ~ x, data = data)
df <- data.frame(x = x, y = y, fit = mod0$fitted.values, resid = mod0$residuals)

# define statistic function
statfun <- function(ix, data) {
  data$y <- data$fit + data$resid[ix]
  lm(y ~ x, data = data)$coefficients
}

# define jackknife function
jackfun <- function(ix, data){
  lm(y ~ x, data = data[ix,])$coefficients
}

# nonparametric bootstrap
npbs <- np.boot(x = 1:n, statistic = statfun, data = df, jackknife = jackfun)
npbs

## End(Not run)

Nonparametric Distribution Tests

Description

Peforms one- or two-sample nonparametric (randomization) tests of cumulative distribution functions. Implements Anderson-Darling, Cramer-von Mises, and Kolmogorov-Smirnov test statistics.

Usage

np.cdf.test(x, y = NULL,
            method = c("AD", "CVM", "KS"),
            R = 9999, parallel = FALSE, cl = NULL,
            perm.dist = TRUE, na.rm = TRUE)

Arguments

Details

One-sample statistics:

where F_n(x) is the empirical cumulative distribution function (estimated by ecdf) and F_0 is the null hypothesized distribution (specified by the y argument).

Two-sample statistics:

where F_x and F_y are the groupwise ECDF functions (estimated by applying ecdf separately to x and y) and F_0 is the joint ECDF (estimated by applying ecdf to z = c(x,y) ).

Value

Note

For one-sample tests, the y argument should satisfy:
paste("p", y) gives the name of a function specifying the CDF
paste("r", y) gives the name of a function sampling from the distribution

If y = NULL, the default sets y = "norm", which tests the null hypothesis that x follows a standard normal distribution. See the examples for how to test a user-specified distribution.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

Anderson, T. W. and Darling., D. A. (1952). Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes. Annals of Mathematical Statistics, 23(2), 193-212. doi:10.1214/aoms/1177729437

Anderson, T. W., and Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association, 49(268), 765-769. doi:10.1080/01621459.1954.10501232

Anderson, T. W. (1962). On the distribution of the two-sample Cramer-von Mises criterion. Annals of Mathematical Statistics, 33(3) 1148-1159. doi:10.1214/aoms/1177704477

Cramer, H. (1928). On the composition of elementary errors: First paper: Mathematical deductions. Scandinavian Actuarial Journal, 1928(1), 13-74. doi:10.1080/03461238.1928.10416862

Kolmogorov, A. N. (1933). Sulla determinazione empirica di una legge di distribuzione. Giornale dell'Istituto Italiano degli Attuari 4, 83-91.

Kolmogorov, A. N. (1941). Confidence limits for an unknown distribution function. Annals of Mathematical Statistics 12(4), 461-483. doi:10.1214/aoms/1177731684

Smirnov, N. (1948). Table for estimating the goodness of fit of empirical distributions. Annals of Mathematical Statistics 19(2) 279-281. doi:10.1214/aoms/1177730256

von Mises, R. (1928). Wahrscheinlichkeit, Statistik und Wahrheit. Julius Springer.

See Also

plot.np.cdf.test S3 plotting method for visualizing the results

Examples

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

## generate standard normal data
n <- 100
set.seed(0)
x <- rnorm(n)


## Example 1: Fn = norm,  F0 = norm

# Anderson-Darling test of H0: Fx = pnorm
set.seed(1)
np.cdf.test(x, y = "norm")

## Not run: 

# Cramer-von Mises test of H0: Fx = pnorm
set.seed(1)
np.cdf.test(x, y = "norm", method = "CVM")

# Kolmogorov-Smirnov test of H0: Fx = pnorm
set.seed(1)
np.cdf.test(x, y = "norm", method = "KS")


## Example 2: Fn = norm,  F0 = t3

# user-defined distribution (Student's t with df = 3)
pt3 <- function(q) pt(q, df = 3)      # cdf = paste("p", y)
rt3 <- function(n) rt(n, df = 3)      # sim = paste("r", y)

# Anderson-Darling test of H0: Fx = t3
set.seed(1)
np.cdf.test(x, y = "t3")

# Cramer-von Mises test of H0: Fx = t3
set.seed(1)
np.cdf.test(x, y = "t3", method = "CVM")

# Kolmogorov-Smirnov test of H0: Fx = t3
set.seed(1)
np.cdf.test(x, y = "t3", method = "KS")



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

# generate N(0, 1) and N(2/3, 1) data
m <- 25
n <- 25
set.seed(0)
x <- rnorm(m)
y <- rnorm(n, mean = 2/3)

# Anderson-Darling test of H0: Fx = Fy
set.seed(1)
np.cdf.test(x, y)

# Cramer-von Mises test of H0: Fx = Fy
set.seed(1)
np.cdf.test(x, y, method = "CVM")

# Kolmogorov-Smirnov test of H0: Fx = Fy
set.seed(1)
np.cdf.test(x, y, method = "KS")

## End(Not run)

Nonparametric Tests of Correlation Coefficients

Description

Denoting the Pearson product-moment correlation coefficient as

\rho = Cov(X, Y) / \sqrt{Var(X) Var(Y)}

this function implements permutation tests of H_0: \rho = \rho_0 where \rho_0 is the user-specified null value. Can also implement tests of partial correlations, semi-partial (or part) correlations, and independence.

Usage

np.cor.test(x, y, z = NULL,
            alternative = c("two.sided", "less", "greater"),
            rho = 0, independent = FALSE, partial = TRUE,
            R = 9999, parallel = FALSE, cl = NULL,
            perm.dist = TRUE, na.rm = TRUE)

Arguments

Details

Default use of this function tests the Pearson correlation between X and Y using the studentized test statistic proposed by DiCiccio and Romano (2017). If independent = TRUE, the classic (unstudentized) test statistic is used to test the null hypothesis of independence.

If Z is provided, the partial or semi-partial correlation between X and Y controlling for Z is tested. For the semi-partial correlation, the effect of Z is partialled out of X.

Value

Note

The permutation test will be exact when the requested number of resamples R is greater than factorial(n) minus one. In this case, the permutation distribution perm.dist contains all factorial(n) possible values of the test statistic.

If z = NULL, the result will be the same as using np.reg.test with method = "perm".

If z is supplied and partial = TRUE, the result will be the same as using np.reg.test with method = "KC" and homosced = FALSE.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

DiCiccio, C. J., & Romano, J. P. (2017). Robust permutation tests for correlation and regression coefficients. Journal of the American Statistical Association, 112(519), 1211-1220. doi: 10.1080/01621459.2016.1202117

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

Pitman, E. J. G. (1937b). Significance tests which may be applied to samples from any populations. ii. the correlation coefficient test. Supplement to the Journal of the Royal Statistical Society, 4(2), 225-232. doi: 10.2307/2983647

See Also

plot.np.cor.test S3 plotting method for visualizing the results

Examples

# generate data
rho <- 0.5
val <- c(sqrt(1 + rho), sqrt(1 - rho))
corsqrt <- matrix(c(val[1], -val[2], val), 2, 2) / sqrt(2)
set.seed(1)
n <- 10
z <- cbind(rnorm(n), rnorm(n)) %*% corsqrt
x <- z[,1]
y <- z[,2]

# test H0: rho = 0
set.seed(0)
np.cor.test(x, y)

# test H0: X and Y are independent
set.seed(0)
np.cor.test(x, y, independent = TRUE)

Nonparametric Tests of Linear Model Terms

Description

Performs type III sums-of-squares tests of linear model terms or coefficients.

Usage

np.lm.test(formula, data, ..., anova.test = TRUE,
           method = "perm", homosced = FALSE, lambda = 0, 
           R = 9999, parallel = FALSE, cl = NULL,
           perm.dist = TRUE, na.rm = TRUE)

Arguments

Details

The recommended default of method = "perm" is equivalent to using Manly's (1986) permutation method separately for each of the model terms. Assuming that the random seed is set the same for each variable's test, equivalent results could be obtained from repeated calls to np.reg.test where a different term/coefficient is tested each time (see Example 2). This implementation is more efficient than repeated calls to np.reg.test because this function computes all of the type III SS tests simultaneously for each permutation.

Value

Note

The "perm" method should be sufficient for most applications. Note that the "flip" and "both" methods require adding an additional (symmetry) assumption, which should be avoided unless one is reasonably certain the error distribution is symmetric. See np.reg.test or the below references (Helwig, 2019a,b) for details.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

DiCiccio, C. J., & Romano, J. P. (2017). Robust permutation tests for correlation and regression coefficients. Journal of the American Statistical Association, 112(519), 1211-1220. doi: 10.1080/01621459.2016.1202117

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

Manly, B. (1986). Randomization and regression methods for testing for associations with geographical, environmental and biological distances between populations. Researches on Population Ecology, 28(2), 201-218. doi:10.1007/BF02515450

See Also

plot.np.lm.test S3 plotting method for visualizing the results

Examples

### Example 1:  anova.test and homosced options

# data generation design
n <- 90
z <- factor(rep(LETTERS[1:3], times = 30))
x <- seq(-1, 1, length.out = n)
tau <- c(-1, 0, 1)

# generate data
set.seed(0)
y <- tau[z] + 2 * x + rnorm(n)
data <- data.frame(x = x, y = y, z = z)

# test of model terms (heteroscedastic)
set.seed(1)
np.lm.test(y ~ x + z, data = data)

# test of coefficients (heteroscedastic)
set.seed(1)
np.lm.test(y ~ x + z, data = data, anova.test = FALSE)

# test of model terms (homoscedastic)
set.seed(1)
np.lm.test(y ~ x + z, data = data, homosced = TRUE)

# test of coefficients (homoscedastic)
set.seed(1)
np.lm.test(y ~ x + z, data = data, homosced = TRUE, anova.test = FALSE)


### Example 2:  equivalence with np.reg.test()

# type III tests of all coefficients
set.seed(1)
mod.lm <- np.lm.test(y ~ x + z, data = data, anova.test = FALSE)

# make design matrix
xmat <- model.matrix(y ~ x + z, data = data)[,-1]

# test effect of x given zB and zC
set.seed(1)
mod.x <- np.reg.test(x = xmat[,1], y = y, z = xmat[,2:3], method = "MA")

# test effect of zB given x and zC
set.seed(1)
mod.zB <- np.reg.test(x = xmat[,2], y = y, z = xmat[,c(1,3)], method = "MA")

# test effect of zC given x and zB
set.seed(1)
mod.zC <- np.reg.test(x = xmat[,3], y = y, z = xmat[,1:2], method = "MA")

# compare np.lm.test() and np.reg.test() results --- identical!
mod.reg <- data.frame(terms = colnames(xmat), df = rep(1, 3), 
                      statistic = c(mod.x$stat, mod.zB$stat, mod.zC$stat),
                      p.value = c(mod.x$p.valu, mod.zB$p.valu, mod.zC$p.valu))
mod.lm$signif.table
mod.reg

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)

Arguments

Details

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

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

Value

Multivariate Tests

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

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.

Note

For one sample (or paired) tests, the permutation test will be exact when the requested number of resamples R is greater than 2^n minus one. In this case, the permutation distribution perm.dist contains all 2^n possible values of the test statistic.

For two sample tests, the permutation test will be exact when the requested number of resamples R is greater than choose(N, n) minus one, where m = length(x), n = length(y), and N = m + n. In this case, the permutation distribution perm.dist contains all choose(N, n) possible values of the test statistic.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

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

See Also

plot.np.loc.test S3 plotting method for visualizing the results

Examples

######******######   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)



## Not run: 

######******######   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

## End(Not run)

Nonparametric Tests of Regression Coefficients

Description

Assuming a linear model of the form

Y = \alpha + X \beta + \epsilon

or

Y = \alpha + X \beta + Z \gamma + \epsilon

this function implements permutation tests of H_0: \beta = \beta_0 where \beta_0 is the user-specified null vector.

Usage

np.reg.test(x, y, z = NULL, method = NULL,
            beta = NULL, homosced = FALSE, lambda = 0, 
            R = 9999, parallel = FALSE, cl = NULL,
            perm.dist = TRUE, na.rm = TRUE)

Arguments

Details

With no nuisance variables in the model (i.e., z = NULL), there are three possible options for the method argument:

where P is a permutation matrix and S is a sign-flipping matrix.

With nuisance variables in the model, there are eight possible options for the method argument:

where P is permutation matrix and Q is defined as R_z = Q Q' with Q'Q = I.

Note that H_z is the hat matrix for the nuisance variable design matrix, and R_z = I - H_z is the corresponding residual forming matrix. Similarly, H_m and R_m are the hat and residual forming matrices for the full model including the predictor and nuisance variables.

Value

Multivariate Tests

If the input y is a matrix with m > 1 columns, the multivariate test statistic is defined as statistic = max(univariate) given that the univariate test statistics are non-negative.

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.

Note

If method = "flip", the permutation test will be exact when the requested number of resamples R is greater than 2^n minus one. In this case, the permutation distribution perm.dist contains all 2^n possible values of the test statistic.

If method = "both", the permutation test will be exact when the requested number of resamples R is greater than factorial(n) * (2^n) minus one. In this case, the permutation distribution perm.dist contains all factorial(n) * (2^n) possible values of the test statistic.

If method = "HJ", the permutation test will be exact when the requested number of resamples R is greater than factorial(n-q-1) minus one. In this case, the permutation distribution perm.dist contains all factorial(n-q-1) possible values of the test statistic.

Otherwise the permutation test will be exact when the requested number of resamples R is greater than factorial(n) minus one. In this case, the permutation distribution perm.dist contains all factorial(n) possible values of the test statistic.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

DiCiccio, C. J., & Romano, J. P. (2017). Robust permutation tests for correlation and regression coefficients. Journal of the American Statistical Association, 112(519), 1211-1220. doi: 10.1080/01621459.2016.1202117

Draper, N. R., & Stoneman, D. M. (1966). Testing for the inclusion of variables in linear regression by a randomisation technique. Technometrics, 8(4), 695-699. doi: 10.2307/1266641

Freedman, D., & Lane, D. (1983). A nonstochastic interpretation of reported significance levels. Journal of Business and Economic Statistics, 1(4), 292-298. doi: 10.2307/1391660

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

Huh, M.-H., & Jhun, M. (2001). Random permutation testing in multiple linear regression. Communications in Statistics - Theory and Methods, 30(10), 2023-2032. doi: 10.1081/STA-100106060

Kennedy, P. E., & Cade, B. S. (1996). Randomization tests for multiple regression. Communications in Statistics - Simulation and Computation, 25(4), 923-936. doi: 10.1080/03610919608813350

Manly, B. (1986). Randomization and regression methods for testing for associations with geographical, environmental and biological distances between populations. Researches on Population Ecology, 28(2), 201-218. doi: 10.1007/BF02515450

Nichols, T. E., Ridgway, G. R., Webster, M. G., & Smith, S. M. (2008). GLM permutation: nonparametric inference for arbitrary general linear models. NeuroImage, 41(S1), S72.

O'Gorman, T. W. (2005). The performance of randomization tests that use permutations of independent variables. Communications in Statistics - Simulation and Computation, 34(4), 895-908. doi: 10.1080/03610910500308230

Still, A. W., & White, A. P. (1981). The approximate randomization test as an alternative to the F test in analysis of variance. British Journal of Mathematical and Statistical Psychology, 34(2), 243-252. doi: 10.1111/j.2044-8317.1981.tb00634.x

ter Braak, C. J. F. (1992). Permutation versus bootstrap significance tests in multiple regression and ANOVA. In K. H. Jockel, G. Rothe, & W. Sendler (Eds.), Bootstrapping and related techniques. Lecture notes in economics and mathematical systems, vol 376 (pp. 79-86). Springer.

White, H. (1980). A heteroscedasticity-consistent covariance matrix and a direct test for heteroscedasticity. Econometrica, 48(4), 817-838. doi: 10.2307/1912934

Winkler, A. M., Ridgway, G. R., Webster, M. A., Smith, S. M., & Nichols, T. E. (2014). Permutation inference for the general linear model. NeuroImage, 92, 381-397. doi: 10.1016/j.neuroimage.2014.01.060

See Also

plot.np.reg.test S3 plotting method for visualizing the results

Examples

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

###***###   TEST ALL COEFFICIENTS   ###***###

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

# Wald test (method = "perm")
set.seed(0)
np.reg.test(x, y)

# F test (method = "perm")
set.seed(0)
np.reg.test(x, y, homosced = TRUE)


###***###   TEST SUBSET OF COEFFICIENTS   ###***###

# generate data
set.seed(1)
n <- 10
x <- rnorm(n)
z <- rnorm(n)
y <- 3 + 2 * z + rnorm(n)

# Wald test (method = "HJ")
set.seed(0)
np.reg.test(x, y, z)

# F test (method = "HJ")
set.seed(0)
np.reg.test(x, y, z, homosced = TRUE)


## Not run: 

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

###***###   TEST ALL COEFFICIENTS   ###***###

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

# multivariate Wald test (method = "perm")
set.seed(0)
np.reg.test(x, y)

# multivariate F test (method = "perm")
set.seed(0)
np.reg.test(x, y, homosced = TRUE)


###***###   TEST SUBSET OF COEFFICIENTS   ###***###

# generate data
set.seed(1)
n <- 10
x <- rnorm(n)
z <- rnorm(n)
y <- cbind(1 + 3 * z + rnorm(n),
           2 + 2 * z + rnorm(n),
           3 + 1 * z + rnorm(n))
           
# multivariate Wald test (method = "HJ")
set.seed(0)
np.reg.test(x, y, z)

# multivariate F test (method = "HJ")
set.seed(0)
np.reg.test(x, y, z, homosced = TRUE)


## End(Not run)

Internal 'nptest' Functions

Description

Internal functions for 'nptest' package.

Details

These functions are not to be called by the user.

Generate All Permutations of n Elements

Description

Generates all n! vectors of length n consisting of permutations of the integers 1 to n.

Usage

permn(n)

Arguments

Details

Adapted from the "permutations" function in the e1071 R package.

Value

Matrix of dimension n by n! where each column contains a unique permutation vector.

Warning

For large n this function will consume a lot of memory and may even crash R.

Note

Used for exact tests in np.cor.test and np.reg.test.

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

Meyer, D., Dimitriadou, E., Hornik, K., Weingessel, A., & Leisch, F. (2018). e1071: Misc Functions of the Department of Statistics, Probability Theory Group (Formerly: E1071), TU Wien. R package version 1.7-0. https://CRAN.R-project.org/package=e1071

Examples

permn(2)
permn(3)

Plots Permutation Distribution for Nonparametric Tests

Description

plot methods for object classes "np.cor.test", "np.loc.test", and "np.reg.test"

Usage

## S3 method for class 'np.aov.test'
plot(x, alpha = 0.05, col = "grey", col.rr = "red",
     col.stat = "black", lty.stat = 2, lwd.stat = 2, 
     xlab = "Test Statistic", main = "Permutation Distribution", 
     breaks = "scott", border = NA, box = TRUE, ...)

## S3 method for class 'np.cdf.test'
plot(x, alpha = 0.05, col = "grey", col.rr = "red",
     col.stat = "black", lty.stat = 2, lwd.stat = 2, 
     xlab = "Test Statistic", main = "Permutation Distribution", 
     breaks = "scott", border = NA, box = TRUE, ...)     

## S3 method for class 'np.cor.test'
plot(x, alpha = 0.05, col = "grey", col.rr = "red",
     col.stat = "black", lty.stat = 2, lwd.stat = 2, 
     xlab = "Test Statistic", main = "Permutation Distribution", 
     breaks = "scott", border = NA, box = TRUE, ...)
           
## S3 method for class 'np.loc.test'
plot(x, alpha = 0.05, col = "grey", col.rr = "red",
     col.stat = "black", lty.stat = 2, lwd.stat = 2, 
     xlab = "Test Statistic", main = "Permutation Distribution", 
     breaks = "scott", border = NA, box = TRUE, ...)
           
## S3 method for class 'np.lm.test'
plot(x, which = 1, alpha = 0.05, col = "grey", col.rr = "red",
     col.stat = "black", lty.stat = 2, lwd.stat = 2, 
     xlab = "Test Statistic", main = "Permutation Distribution", 
     breaks = "scott", border = NA, box = TRUE, SQRT = TRUE, ...)

## S3 method for class 'np.reg.test'
plot(x, alpha = 0.05, col = "grey", col.rr = "red",
     col.stat = "black", lty.stat = 2, lwd.stat = 2, 
     xlab = "Test Statistic", main = "Permutation Distribution", 
     breaks = "scott", border = NA, box = TRUE, SQRT = TRUE, ...)

Arguments

Details

Plots a histogram of the permutation distribution and the observed test statistic. The argument 'alpha' controls the rejection region of the nonparametric test, which is plotted using a separate color (default is red).

Author(s)

Nathaniel E. Helwig <helwig@umn.edu>

References

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

See Also

np.aov.test for information on nonparametric ANOVA tests

np.aov.test for information on nonparametric distribution tests

np.cor.test for information on nonparametric correlation tests

np.loc.test for information on nonparametric location tests

np.lm.test for information on nonparametric linear model tests

np.reg.test for information on nonparametric regression tests

Examples

######******######   np.cor.test   ######******######

# generate data
rho <- 0.5
val <- c(sqrt(1 + rho), sqrt(1 - rho))
corsqrt <- matrix(c(val[1], -val[2], val), 2, 2) / sqrt(2)
set.seed(1)
n <- 50
z <- cbind(rnorm(n), rnorm(n)) %*% corsqrt
x <- z[,1]
y <- z[,2]

# test H0: rho = 0
set.seed(0)
test <- np.cor.test(x, y)

# plot results
plot(test)


######******######   np.loc.test   ######******######

# generate data
set.seed(1)
n <- 50
x <- rnorm(n, mean = 0.5)

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

# plot results
plot(test)


######******######   np.reg.test   ######******######

# generate data
set.seed(1)
n <- 50
x <- cbind(rnorm(n), rnorm(n))
beta <- c(0.25, 0.5)
y <- x %*% beta + rnorm(n)

# Wald test (method = "perm")
set.seed(0)
test <- np.reg.test(x, y)

# plot results
plot(test)