# Nonparametric Tests of Location Parameters 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. ```r 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 - `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. ## 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| ## Returns - **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. ## 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 ## Author(s) Nathaniel E. Helwig ## 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. ## 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. ## See Also `plot.np.loc.test` S3 plotting method for visualizing the results ## Examples ```r ######******###### 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) ```

np.loc.test function