| Title: | Rank-Based Tests for General Factorial Designs |
|---|---|
| Description: | The rankFD() function calculates the Wald-type statistic (WTS) and the ANOVA-type statistic (ATS) for nonparametric factorial designs, e.g., for count, ordinal or score data in a crossed design with an arbitrary number of factors. Brunner, E., Bathke, A. and Konietschke, F. (2018) <doi:10.1007/978-3-030-02914-2>. |
| Authors: | Frank Konietschke, Sarah Friedrich, Edgar Brunner, Markus Pauly |
| Maintainer: | Frank Konietschke <[email protected]> |
| License: | GPL-2 | GPL-3 |
| Version: | 0.1.1 |
| Built: | 2024-12-08 06:55:10 UTC |
| Source: | CRAN |
Coal acidity values determined under each of three NaOH concentration levels for two different samples from each type of coal
data(Coal)data(Coal)
A data frame with 18 rows and 3 variables:
resulting acidity values
the NaOH concentration
three different types of coal: "Morwell", "Yallourn" and "Maddingley"
Hollander, M., Wolfe, D. A., Chicken, E. (2014) Nonparametric Statistical Methods. Wiley Series in Probability and Statistics.
Sternhell, S. (1958) Chemistry of brown coals VI: Further aspects of the chemistry of hydroxyl groups in Victorian brown coals. Australian Journal of Applied Science 9, 375–379.
Mucociliary efficiency was assessed from the rate of removal of dust in three different groups of subjects
data(Muco)data(Muco)
A data frame with 14 rows and 2 variables:
Half-Time of Mucociliary clearance, assessed from the rate of removal of dust
normal subjects, subjects with obstructive airways disease (OAD) and subjects with asbestosis
Hollander, M., Wolfe, D. A., Chicken, E. (2014) Nonparametric Statistical Methods. Wiley Series in Probability and Statistics.
Thomson, M. L. and Short, M. D.(1969) Mucociliary function in health, chronic obstructive airway disease, and asbestosis. Journal of Applied Physiology 26, 535–539.
Damage of two gaseous substances on nasal mucosa membrane of mice
data(nms)data(nms)
A data frame with 150 rows and 3 variables:
The substance given, either 1 or 2
the concentration in which the substance was given, 1, 2 or 5 ppm
degree of irritation assessed using an ordinal score ranging from 0 to 4 with 0 = “no irritation”, 1 = “mild irritation”, 2 = “strong irritation”, 3 = “severe irritation” and 4 = “irreversible damage”
Brunner, E., Bathke, A.C., Konietschke, F. Rank and Pseudo-Rank Procedures for Independent Observations in Factorial Designs. Springer International Publishing, 2018.
Sample size calculation for the Wilcoxon-Mann-Whitney test using the Noether formula. The function estimates the sample size needed to detect the effect with pre-defined power at significance level alpha using Noether's formula'.
noether(alpha, power, t, p, x1 = c(0), ties = FALSE)noether(alpha, power, t, p, x1 = c(0), ties = FALSE)
alpha |
two sided type I error rate |
power |
power: detect a relative effect p at least with probability power |
t |
proportion of subjects in the first group (between 0 and 1) |
p |
relative effect |
x1 |
advance information is only needed in case of ties |
ties |
TRUE if ties are possible (non continuous distribution), otherwise FALSE |
Returns a data frame with the sample sizes for each group
Noether, G. E. (1987). Sample Size Determination for Some Common Nonparametric Tests. Journal of the American Statistical Association 85, 645.647.
noether(0.05,0.8,1/2, 0.75)noether(0.05,0.8,1/2, 0.75)
The psr() function calculates pseuo-ranks of data in general factorial designs. It returns the input data set complemented by an additional variable containing the pseudo-ranks. We note that more efficient algorithms for the computation of pseudo-ranks are implemented within the package pseudorank.
psr(formula, data, psranks = "pseudorank")psr(formula, data, psranks = "pseudorank")
formula |
A model |
data |
A data.frame, list or environment containing the variables in
|
psranks |
A header specifying the name of the pseudo ranks in the output data set. |
The pseudo-ranks are exported within a new column attached to the given data set.
Konietschke, F., Hothorn, L. A., & Brunner, E. (2012). Rank-based multiple test procedures and simultaneous confidence intervals. Electronic Journal of Statistics, 6, 738-759.
Brunner, E., Bathke, A. C., Konietschke, F. (2018). Rank and pseudo-rank procedures for independent observations in factorial designs. Springer International Publishing.
Happ, M., Zimmermann, G., Brunner, E., Bathke, A. C. (2020). Pseudo-ranks: How to calculate them efficiently in R. Journal of Statistical Software, 95(1), 1-22.
data(Muco) Muco2 <- psr(HalfTime~Disease,data=Muco, psranks="Mypseudos")data(Muco) Muco2 <- psr(HalfTime~Disease,data=Muco, psranks="Mypseudos")
The rank.two.samples() function calculates purely nonparametric rank-based
methods for the analysis of two independent samples. Specifically, it
implements the Brunner-Munzel test and its generalizations for the Nonparametric Behrens-Fisher Problem,
that is, testing whether the relative effect p=P(X<Y)+1/2*P(X=Y) of the two independent samples X and Y
is equal to 1/2. Range preserving confidence intervals (and corresponding test statistics)
are available using Logit or Probit transformations. The function also implements studentized permutation
tests and permutation based confidence intervals for p using any of the method above (see the details below).
Furthermore, the Wilcoxon-Mann-Whitney test (exact and asymptotic) can be used to test the equality of
the two distribution functions of the two samples. The user can specify whether confidence intervals for shift
effects shall be computed. The rank.two.samples() function implements one-sided and two-sided tests
and confidence intervals. You can plot the confidence intervals (for the relative
effects) with the plot() function.
rank.two.samples( formula, data, conf.level = 0.95, alternative = c("two.sided", "less", "greater"), rounds = 4, method = c("t.app", "logit", "probit", "normal"), permu = TRUE, info = TRUE, wilcoxon = c("asymptotic", "exact"), shift.int = TRUE, nperm = 10000 )rank.two.samples( formula, data, conf.level = 0.95, alternative = c("two.sided", "less", "greater"), rounds = 4, method = c("t.app", "logit", "probit", "normal"), permu = TRUE, info = TRUE, wilcoxon = c("asymptotic", "exact"), shift.int = TRUE, nperm = 10000 )
formula |
A model |
data |
A data.frame, list or environment containing the variables in
|
conf.level |
A number specifying the confidence level; the default is 0.95. |
alternative |
A character string specifying the alternative hypothesis. One of "two.sided", "less", "greater". You can specify just the initial letter. |
rounds |
Value specifying the number of digits the results are rounded to. Default is 4 decimals. |
method |
specifying the method used for calculation of the confidence intervals. One of "t.app", "logit", "probit" or "normal". |
permu |
A logical variable indicating whether you want to compute a studentized permutation test. |
info |
A logical variable. Here, info = FALSE suppresses the output of additional information concerning e.g. the interpretation of the test results. |
wilcoxon |
asymptotic or exact calculation of Wilcoxon test. |
shift.int |
Logical, indicating whether or not shift effects should be considered. |
nperm |
Number of permutations used, default is 10000. |
The rank.two.samples() function calculates both transformed (logit or probit) and untransformed statistics
(normal or t.app) for testing the null hypothesis p=1/2. If a studentized permutation test is performed, then the
permutation distribution of the respective statistics are computed, see Pauly et
al.(2016) for details. In any case, the function reports the point estimator and its estimated standard error,
value of the test statistic, confidence interval and p-value. In case of separated samples, point estimator and standard error
would be 0 and thus, test statistics would not be defined. In such a case, point
estimator and its standard error are replaced by the numbers one would obtain if samples overlapped in a single point.
A plot of the confidence interval can be obtained with the plot function.
Frank Konietschke Brunner, E., Bathke, A. C., Konietschke, F. (2018). Rank and Pseudo-Rank Procedures for Independent Observations in Factorial Designs. Springer International Publishing.
Brunner, E. and Munzel, U. (2000). The nonparametric Behrens-Fisher problem: Asymptotic theory and a small-sample approximation. Biometrical Journal 1, 17 - 21.
Kaufmann, J., Werner, C., and Brunner, E. (2005). Nonparametric methods for analysing the accuracy of diagnostic tests with multiple readers. Statistical Methods in Medical Research 14, 129 - 146
Pauly, M., Asendorf, T., Konietschke, F. (2016). Permutation-based inference for the AUC: a unified approach for continuous and discontinuous data.##' Biometrical Journal, 58(6), 1319 – 1337.
data(Muco) Muco2 <- subset(Muco, Disease != "OAD") Muco2$Disease <- droplevels(Muco2$Disease) twosample <- rank.two.samples(HalfTime ~ Disease, data = Muco2, wilcoxon = "exact", permu = TRUE, shift.int = TRUE, nperm = 1000) twosample <- rank.two.samples(HalfTime ~ Disease, data = Muco2, alternative = "greater", method = "probit", wilcoxon = "exact", permu = TRUE, shift.int = FALSE, nperm = 1000) plot(twosample)data(Muco) Muco2 <- subset(Muco, Disease != "OAD") Muco2$Disease <- droplevels(Muco2$Disease) twosample <- rank.two.samples(HalfTime ~ Disease, data = Muco2, wilcoxon = "exact", permu = TRUE, shift.int = TRUE, nperm = 1000) twosample <- rank.two.samples(HalfTime ~ Disease, data = Muco2, alternative = "greater", method = "probit", wilcoxon = "exact", permu = TRUE, shift.int = FALSE, nperm = 1000) plot(twosample)
The function implements purely nonparametric rank-based methods for the analysis
of general factorial designs. You can chose to use either classical ranks (mid-ranks)
(effect="weighted") or pseudo-ranks (effect="unweighted")
for making inference. Pseudo-ranks are used by default.
The package implements point estimators of relative effects (weighted and unweighted) as
well as test procedures (Wald-Type and ANOVA-Type statistics) for testing global null hypotheses
formulated in either (i) distribution functions hypothesis="H0F"
or (ii) relative effects hypothesis="H0p". In case of one-way factorial
designs, the function additionally computes
the Kruskal-Wallis test either with ranks or pseudo-ranks. In addition, multiple
contrast tests (and simultaneous confidence intervals)
for the main or interaction effects are implemented within the contrast
statement. You can either choose from pre-defined contrasts (options see below) or
you can provide your own user-defined contrast matrix. Both the
Fisher-transformation (sci.method="fisher") as well as a multivariate
t-approximation (sci.method="multi.t") are implemented.
The Fisher approximation is used by default. To visualize the results, you can plot
the simultaneous confidence intervals using the plot.sci function.
Furthermore, confidence interval plots for the main or interaction relative effects
(not simultaneous) are available within the plot function.
rankFD( formula, data, alpha = 0.05, CI.method = c("logit", "normal"), effect = c("unweighted", "weighted"), hypothesis = c("H0F", "H0p"), Factor.Information = FALSE, contrast = NULL, sci.method = c("fisher", "multi.t"), info = TRUE, covariance = FALSE, rounds = 4 )rankFD( formula, data, alpha = 0.05, CI.method = c("logit", "normal"), effect = c("unweighted", "weighted"), hypothesis = c("H0F", "H0p"), Factor.Information = FALSE, contrast = NULL, sci.method = c("fisher", "multi.t"), info = TRUE, covariance = FALSE, rounds = 4 )
formula |
A model |
data |
A data.frame, list or environment containing the variables in
|
alpha |
A number specifying the significance level; the default is 0.05. |
CI.method |
Either "logit" or "normal", specifying the method used for calculation of the confidence intervals. |
effect |
In case of weighted, then weighted (by sample sizes) relative effects are estimated using classical ranks (mid-ranks) of the data. Otherwise, in case of effect="unweighted", unweighted relative effects are estimated with pseudo-ranks. The default option is "unweighted" resulting in pseudo-rank statistics. |
hypothesis |
The null hypothesis to be tested, either "H0F" or "H0p". The option "H0F" computes tests for testing hypotheses formulated in terms of distribution functions. Otherwise, hypotheses in relative effects are tested. The latter allows for variance heteroscedasticity even under the null hypothesis of no treatment effect and thus covers the Nonparametric Behrens-Fisher problem. |
Factor.Information |
Logical. If TRUE, descriptive statistics with point estimators, standard error as well as confidence intervals for each main and interaction effect in the model are printed. The results can furthermore be plotted with the plot function. |
contrast |
a list containing the name of the main or interaction effect (written as group1:group2), a pre-defined contrast ("Dunnett", "Tukey", "Sequen", "AVE", "Changepoint", "Williams", "Marcus", "McDermott", "UmbrellaWilliams", "GrandMean") or a user-defined contrast matrix. If the contrast coefficients do not sum up to 0, or if their sum of absolutes differs from 2, the coefficients are normalized. |
sci.method |
Either "fisher" or "multi.t" as approximation method for the multiple contrast tests and simultaneous confidence intervals. The default option is "fisher". |
info |
Logical. If TRUE, additional output information and explanation is printed to the console. |
covariance |
Logical. If TRUE, the estimated covariance matrix of the vector of relative effects is computed. |
rounds |
Number of decimals of the output values. The default option is rounds=4 (4 decimals). |
The rankFD() function calculates the Wald-type statistic (WTS), ANOVA-type
statistic (ATS) as well as multiple contrast tests for general factorial designs
for testing the null hypotheses or .
Almost every method explained in the comprehensive textbook from Brunner et al. (2019)
is implemented in rankFD. The test procedures for testing null hypotheses in distribution
functions have initially been proposed by Akritas et al. (1997), whereas methods for testing null
hypotheses formulated in relative effects have been proposed by Brunner et al. (2017).
We note that the multiple contrast test procedure using Fisher approximation computes critical and
p-values from a multivariate t-distribution with respective degrees of freedom. Simulation studies by
Konietschke et al. (2012) demonstrated an accurate control of the type-1 error rate and
the procedure is therefore recommended.
A rankFD object containing the following components:
Call |
Given response and factor names (formula) |
Descriptive |
Descriptive statistics of the data for all factor level combinations. Displayed are the number of individuals per factor level combination (size), the relative effect (Rel.Effect), Standard Error and 100*(1-alpha)% confidence intervals. |
WTS |
The value of the WTS along with degrees of freedom of the central chi-square distribution and p-value. |
ATS |
The value of the ATS, degrees of freedom of the central F distribution and the corresponding p-value. |
Kruskal-Wallis-Test |
The value of the Kruskal-Wallis test along with degrees of freedom and p-value. If effect="unweighted", the Kruskal-Wallis test using pseudo-ranks is computed. Otherwise, if effect="unweighted", the "established" Kruskal-Wallis test based on ranks is returned. |
MCTP |
Contrast matrix, local Results in terms of point estimates, standard error, value of the test statistic, (1-alpha)100 intervals as well as adjusted p-values. As a summary, the function also returns the global test decision by printing the maximum test statistic (in absolute value) as well as the (1-alpha) critical value from the multivariate T-distribution. |
Covariance.Matrix |
The estimated covariance matrix of the vector of the estimated relative effects. Note that the vector is multiplied by root N. |
Factor.Information |
Descriptive tables containing the point estimators, standard errors as well as (1-alpha)100 and interaction effects in the model. The confidence intervals are not simultaneous and for data descriptive purpose only. |
Brunner, E., Bathke, A.C., Konietschke, F. Rank and Pseudo-Rank Procedures for Independent Observations in Factorial Designs. Springer International Publishing, 2018.
Brunner, E., Konietschke, F., Pauly, M., Puri, M. L. (2017). Rank-based procedures in factorial designs: Hypotheses about non-parametric treatment effects. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(5), 1463-1485.
Akritas, M. G., Arnold, S. F., and Brunner, E. (1997). Nonparametric hypotheses and rank statistics for unbalanced factorial designs. Journal of the American Statistical Association 92, 258-265.
Brunner, E., Dette, H., and Munk, A. (1997). Box-Type Approximations in Nonparametric Factorial Designs. Journal of the American Statistical Association 92, 1494-1502.
Konietschke, F., Hothorn, L. A., Brunner, E. (2012). Rank-based multiple test procedures and simultaneous confidence intervals. Electronic Journal of Statistics, 6, 738-759.
data(Coal) model <- rankFD(Acidity ~ NaOH * Type, data = Coal, CI.method = "normal", effect = "unweighted", hypothesis = "H0F") data(Muco) model.oneway <- rankFD(HalfTime ~ Disease, data = Muco, CI.method = "logit", effect = "weighted", hypothesis = "H0p") plot(model.oneway)data(Coal) model <- rankFD(Acidity ~ NaOH * Type, data = Coal, CI.method = "normal", effect = "unweighted", hypothesis = "H0F") data(Muco) model.oneway <- rankFD(HalfTime ~ Disease, data = Muco, CI.method = "logit", effect = "weighted", hypothesis = "H0p") plot(model.oneway)
The function implements purely nonparametric Steel-type multiple contrast tests for either making many-to-one (Dunnett-type) or all pairwise (Tukey-type) comparisons. Null hypotheses are formulated in terms of the distribution functions.
steel( formula, data, control = NULL, alternative = c("two.sided", "less", "greater"), info = TRUE, correlation = TRUE )steel( formula, data, control = NULL, alternative = c("two.sided", "less", "greater"), info = TRUE, correlation = TRUE )
formula |
A model |
data |
A data.frame, list or environment containing the variables in
|
control |
Specification of the control group for making many-to-one-comparisons. If NULL, all-pairwise comparisons are performed. |
alternative |
Specification of the direction of the alternative. Default is two-sided. |
info |
Logical. If TRUE, additional output information and explanation is printed to the console. |
correlation |
Logical. If TRUE, the correlation matrix is printed. |
The steel() function calculates the Steel-type tests as explained by Munzel, U., Hothorn, L. A. (2001). A unified approach to simultaneous rank test procedures in the unbalanced one-way layout. Biometrical Journal: Journal of Mathematical Methods in Biosciences, 43(5), 553-569.
A list containing the following components:
Data.Info |
Groups and sample sizes of the data |
Analysis |
Data frame containing the test results (comparison, relative effect estimator, standard error, test statistic and p-value.) |
Correlation |
Estimated correlation matrix |
Brunner, E., Bathke, A.C., Konietschke, F. Rank and Pseudo-Rank Procedures for Independent Observations in Factorial Designs. Springer International Publishing, 2018.
Munzel, U., Hothorn, L. A. (2001). A unified approach to simultaneous rank test procedures in the unbalanced one-way layout. Biometrical Journal: Journal of Mathematical Methods in Biosciences, 43(5), 553-569.
Konietschke, F., Hothorn, L. A., Brunner, E. (2012). Rank-based multiple test procedures and simultaneous confidence intervals. Electronic Journal of Statistics, 6, 738-759.
data(Muco) model.oneway <- steel(HalfTime ~ Disease, data = Muco,info=TRUE,correlation=TRUE)data(Muco) model.oneway <- steel(HalfTime ~ Disease, data = Muco,info=TRUE,correlation=TRUE)
The function implements the sample size formula proposed by Happ et al. (see reference below). It estimates the sample size needed to detect the effect with pre-defined power at significance level alpha based on pilot data.
WMWSSP(x1, x2, alpha = 0.05, power = 0.8, t = 1/2)WMWSSP(x1, x2, alpha = 0.05, power = 0.8, t = 1/2)
x1 |
advance information for the first group |
x2 |
advance information for the second group |
alpha |
two sided type I error rate |
power |
power with the sample sizes of each group |
t |
proportion of subjects in the first group |
Returns a data frame
Brunner, E., Bathke A. C. and Konietschke, F. Rank- and Pseudo-Rank Procedures in Factorial Designs - Using R and SAS. Springer Verlag. Happ, M., Bathke, A. C., & Brunner, E. (2019). Optimal sample size planning for the Wilcoxon-Mann-Whitney test. Statistics in medicine, 38(3), 363-375.
x1 = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2) x2 = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3) WMWSSP(x1,x2,0.05,0.8,0.5)x1 = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2) x2 = c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3) WMWSSP(x1,x2,0.05,0.8,0.5)