Homogeneity analysis combines maximizing the correlations between variables of a multivariate data set with that of optimal scaling. In this article we present methodological and practical issues of the R package homals which performs homogeneity analysis and various extensions. By setting rank constraints nonlinear principal component analysis can be performed. The variables can be partitioned into sets such that homogeneity analysis is extended to nonlinear canonical correlation analysis or to predictive models which emulate discriminant analysis and regression models. For each model the scale level of the variables can be taken into account by setting level constraints. All algorithms allow for missing values.

In a series of papers de Leeuw developed a general framework for multivariate analysis with optimal scaling. The basic idea of optimal scaling is to transform the observed variables (categories) in terms of quanti�cations. In the approach presented here the multivariate data are collected into a multivariable. An aspect of a multivariable is a function that is used to measure how well the multivariable satis�es some criterion. Basically we can think of two di�erent families of aspects which unify many well-known multivariate methods: Correlational aspects based on sums of correlations, eigenvalues and determinants which unify multiple regression, path analysis, correspondence analysis, nonlinear PCA, etc. Non-correlational aspects which linearize bivariate regressions and can be used for SEM preprocessing with categorical data. Additionally, other aspects can be established that do not correspond to classical techniques at all. By means of the R package aspect we provide a uni�ed majorization-based implementation of this methodology. Using various data examples we will show the flexibility of this approach and how the optimally scaled results can be represented using graphical tools provided by the package.

This paper presents the R package anacor for the computation of simple and canonical correspondence analysis with missing values. The canonical correspondence analysis is specified in a rather general way by imposing covariates on the rows and/or the columns of the two-dimensional frequency table. The package allows for scaling methods such as standard, Benz'ecri, centroid, and Goodman scaling. In addition, along with well-known two- and three-dimensional joint plots including confidence ellipsoids, it offers alternative plotting possibilities in terms of transformation plots, Benz'ecri plots, and regression plots.

In this paper we present the methodology of multidimensional scaling problems (MDS) solved by means of the majorization algorithm. The objective function to be minimized is known as stress and functions which majorize stress are elaborated. This strategy to solve MDS problems is called SMACOF and it is implemented in an R package of the same name which is presented in this article. We extend the basic SMACOF theory in terms of configuration constraints, three-way data, unfolding models, and projection of the resulting configurations onto spheres and other quadratic surfaces. Various examples are presented to show the possibilities of the SMACOF approach offered by the corresponding package.

The model family proposed by Gifi (1990) is a flexible framework for the analysis of multivariate data. The common properties shared by all Gifi-models are the specification of a loss function solved by alternating least squares and transformations of the variables which lead to quantifications of the categories. The latter issue implies the concept of "optimal scaling" and allows to account for the scaling level of the variables. Starting from the basic model of homogeneity analysis, we present various extensions in terms of rank restrictions (nonlinear principal component analysis) and restrictions on sets of variables (nonlinear canonical correlation analysis). We focus on recent methodological developments and on the R package "homals" that allows for the computation of these models and provides new visualization techniques of the results.