RW {RobustANOVA}

Robust Welch Test based on MML Estimators

Description

Computes the observed value of robust Welch (RW) test, degrees of freedoms (numerator and denominator) and the corresponding p-value for the equality of means of several long-tailed symmetric (LTS) distributions when the variances are unknown and arbitrary.

Usage

RW(formula, data, alpha=0.05, verbose = TRUE, p_shape)

Arguments

formula

a formula of the form left-hand-side(lhs) ~ right-hand-side(rhs). lhs shows the observed values and rhs shows the group corresponding to the observed values.

data

data frame containing the variables in the formula.

alpha

the level of significance. Default is set to alpha = 0.05.

verbose

a logical for printing output to R console.

p_shape

shape parameter of the LTS distribution

Details

RW test based on modifed maximum likelihood (MML) estimators is proposed as a robust alternative to Welch's F test (Welch, 1951). The test statistic is formulated as follows

RW= \frac{T(\hat{\mu}_1, \dots, \hat{\mu}_a;\hat{\sigma}_1^{2},\dots,\hat{\sigma}_a^{2})/(a-1)}{1+(2(a-2)/(3\nu_1))}

where

T(\hat{\mu}_1,\dots,\hat{\mu}_a; \hat{\sigma}_1^{2},\dots,\hat{\sigma}_a^{2})=\sum\limits_{i=1}^a \frac{M_i}{\hat{\sigma}_i^{2}} \hat{\mu}_i^{2}- \frac{(\sum\limits_{i=1}^a M_i\hat{\mu}_i/\hat{\sigma}_i^{2})^2}{\sum\limits_{i=1}^a M_i/\hat{\sigma}_i^{2}},

\nu_1= [\frac{3}{a^2-1} \sum\limits_{i = 1}^a \frac{1}{n_i-1}(1-( M_i/\hat{\sigma}_i^2)/( \sum\limits_{j= 1}^a M_j/\hat{\sigma}_j^2))^{2}]^{-1},

\hat{\mu}_{i} and \hat{\sigma}_{i} (i=1,2,...,a) are the MML estimators of the location and scale parameters, respectively, see Tiku (1967, 1968) for the details of MML estimators.

The null hypothesis is rejected if the computed RW statistic is higher than the (1-\alpha)th quantile of the F distribution with a-1 and \nu_{1} degrees of freedom.

For further details, see Guven et al. (2022).

Value

A list with class "htest" containing the following components:

statistic

the observed value of the RW test statistic.

dfs

the numerator and the denominator degrees of freedom of the approximate F distribution.

p.value

the p-value for the RW test.

alpha

the level of significance.

method

a character string "Robust Welch Test based on MML Estimators" indicating which test is used.

data

a data frame containing the variables.

formula

a formula of the form left-hand-side(lhs) ~ right-hand-side(rhs). lhs shows the observed values and rhs shows the group corresponding to the observed values.

Author(s)

Gamze Guven <[e-mail]>

References

G. Guven, S. Acitas, and B. Senoglu, B. RobustANOVA: An R Package for one-way ANOVA under heteroscedasticity and nonnormality. Under review, 2022.

M. L. Tiku. Estimating the mean and standard deviation from a censored normal sample. Biometrika, 54:155-165, 1967.

M. L. Tiku. Estimating the parameters of log-normal distribution from censored samples. Journal of the American Statistical Association, 63(321): 134-140, 1968.

B. L. Welch. On the comparison of several mean values: an alternative approach. Biometrika, 38(3): 330-336, 1951.

Examples

library(RobustANOVA)

RW(obs ~ methods, data = peak_discharge, alpha = 0.05, verbose = TRUE, p_shape=2.3)