rankFD {rankFD}

Rank-based tests for general factorial designs

Description

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.

Usage

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
)

Arguments

formula

A model formula object. The left hand side contains the response variable and the right hand side contains the factor variables of interest. An interaction term must be specified.

data

A data.frame, list or environment containing the variables in formula. The default option is NULL.

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).

Details

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 H_0^F: CF = 0 or H_0^p: Cp = 0. 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.

Value

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.

References

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.

Examples

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)