A new R package is presented for dealing with non-normality and variance heterogeneity of sample data when conducting hypothesis tests of main effects and interactions in mixed models. The proposal departs from an existing SAS program which implements Johansen's general formulation of Welch-James's statistic with approximate degrees of freedom, which makes it suitable for testing any linear hypothesis concerning cell means in univariate and multivariate mixed model designs when the data pose non-normality and non-homogeneous variance. Improved type I error rate control is obtained using bootstrapping for calculating an empirical critical value, whereas robustness against non-normality is achieved through trimmed means and Winsorized variances. A wrapper function eases the application of the test in common situations, such as performing omnibus tests on all effects and interactions, pairwise contrasts, and tetrad contrasts of two-way interactions. The package is demonstrated in several problems including unbalanced univariate and multivariate designs.

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In this contribution we deal with the problem of doing computations with a Markov chain when the information about transition probabilities is expressed linguistically. This could be the case, for instance, if the process we are modeling is described by a human expert, for whom the use of linguistic labels is easier than being forced to give inexact numerical probabilities which, in turn, may yield an unstable chain. We address the uncertainty of linguistic judgments by introducing fuzzy probabilities, and carry on the calculation of the linguistic stationary distribution of the chain by resorting to an existing fuzzy approach with restricted matrix multiplication. Preliminary results are very promising and deserve further research.

Markov chains are well-established probabilistic models of a wide variety of real systems that evolve along time. Countless examples of applications of Markov chains that successfully capture the probabilistic nature of real problems include areas as diverse as biology, medicine, social science, and engineering. One interesting feature which characterizes certain kinds of Markov chains is their stationary distribution, which stands for the global fraction of time the system spends in each state. The computation of the stationary distribution requires precise knowledge of the transition probabilities. When the only information available is a sequence of observations drawn from the system, such probabilities have to be estimated. Here we review an existing method to estimate fuzzy transition probabilities from observations and, with them, obtain the fuzzy stationary distribution of the resulting fuzzy Markov chain. The method also works when the user directly provides fuzzy transition probabilities. We provide an implementation in the R environment that is the first available to the community and serves as a proof of concept. We demonstrate the usefulness of our proposal with computational experiments on a toy problem, namely a time-homogeneous Markov chain that guides the randomized movement of an autonomous robot that patrols a small area.

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