Extending Mosaic Displays: Marginal,Conditional, and Partial Views of Categorical Data

Abstract

This article first illustrates the use of mosaic displays for the analysis of multiway contingency tables. We then introduce several extensions of mosaic displays designed to integrate graphical methods for categorical data with those used for quantitative data. The scatterplot matrix shows all pairwise (bivariate marginal) views of a set of variables in a coherent display. One analog for categorical data is a matrix of mosaic displays showing some aspect of the bivariate relation between all pairs of variables. The simplest case shows the bivariate marginal relation for each pair of variables. Another case shows the conditional relation between each pair, with all other variables partialled out. For quantitative data this represents (a) a visualization of the conditional independence relations studied by graphical models, and (b) a generalization of partial residual plots. The conditioning plot, or coplot shows a collection of partial views of several quantitative variables, conditioned by the values of one or more other variables. A direct analog of the coplot for categorical data is an array of mosaic plots of the dependence among two or more variables, stratified by the values of one or more given variables. Each such panel then shows the partial associations among the foreground variables; the collection of such plots shows how these associations change as the given variables vary.     

Clustering Visualizations of Multidimensional Data

Many graphical methods for displaying multivariate data consist of arrangements of multiple displays of one or two variables; scatterplot matrices and parallel coordinates plots are two such methods. In principle these methods generalize to arbitrary numbers of variables but become difficult to interpret for even moderate numbers of variables. This article demonstrates that the impact of high dimensions is much less severe when the component displays are clustered together according to some index of merit. Effectively, this clustering reduces the dimensionality and makes interpretation easier. For scatterplot matrices and parallel coordinates plots clustering of component displays is achieved by finding suitable permutations of the variables. I discuss algorithms based on cluster analysis for finding permutations, and present examples using various indices of merit.     

The heteroscedastic method: Multivariate implementation

Summary Over the past decade, procedures have been developed which allow one (in the univariate case) to make inferences about means even in the presence of unknown and unequal variances. A general method (called The Heteroscedastic Method) allowing this in all statistical problems simultaneously was formulated in 1979 and allowed specifically for the multivariate case (e.g., MANOVA and other multivariate inferences). While in the univariate case The Heteroscedastic Method is readily implemented, in the multivariate case practical implementation was not heretofore possible since a certain problem in construction of matrices required by the method had not been solved. In this paper we solve that problem and give a computer algorithm allowing for use of the solution in The Heteroscedastic Method.     

The Generalized Ridge Trace Plot: Visualizing Bias and Precision

In ridge regression and related shrinkage methods, the ridge trace plot, a plot of estimated coefficients against a shrinkage parameter, is a common graphical adjunct to help determine a favorable trade-off of bias against precision (inverse variance) of the estimates. However, standard unidimensional versions of this plot are ill-suited for this purpose because they show only bias directly and ignore the multidimensional nature of the problem.

A generalized version of the ridge trace plot is introduced, showing covariance ellipsoids in parameter space, whose centers show bias and whose size and shape show variance and covariance, respectively, in relation to the criteria for which these methods were developed. These provide a direct visualization of both bias and precision. Even two-dimensional bivariate versions of this plot show interesting features not revealed in the standard univariate version. Low-rank versions of this plot, based on an orthogonal transformation of predictor space extend these ideas to larger numbers of predictor variables, by focusing on the dimensions in the space of predictors that are likely to be most informative about the nature of bias and precision. Two well-known datasets are used to illustrate these graphical methods. The genridge package for R implements computation and display.     

A nonparametric lack-of-fit test for heteroscedastic regression models

A simple test is proposed for examining the correctness of a given completely specified response function against unspecified general alternatives in the context of univariate regression. The usual diagnostic tools based on residual plots are useful but heuristic. We introduce a formal statistical test supplementing the graphical analysis. Technically, the test statistic is the maximum length of the sequences of ordered (with respect to the covariate) observations that are consecutively overestimated or underestimated by the candidate regression function. Note that the testing procedure can cope with heteroscedastic errors and no replicates. Recursive formulae allowing one to calculate the exact distribution of the test statistic under the null hypothesis and under a class of alternative hypotheses are given.     

A modified two-factor multivariate analysis of variance: asymptotics and small sample approximations

In this paper, we present results for testing main, simple and interaction effects in heteroscedastic two factor MANOVA models. In particular, we suggest modifications to the MANOVA sum of squares and cross product matrices to account for heteroscedasticity. Based on these modified matrices, we define some multivariate test statistics and derive their asymptotic distributions under non-normality for the null as well as non-null cases. Derivation of these results relies on the perturbation method and limit theorems for independently distributed random matrices. Based on the asymptotic distributions, we devise small sample approximations for the quantiles of the null distributions. The numerical accuracy of the large sample as well as small sample approximations are favorable. A real data set from a Smoking Cessation Trial is analyzed to illustrate the application of the methods.     

On Affine Equivariant Multivariate Quantiles

An extension of univariate quantiles in the multivariate set-up has been proposed and studied. The proposed approach is affine equivariant, and it is based on an adaptive transformation retransformation procedure. Behadur type linear representations of the proposed quantiles are established and consequently asymptotic distributions are also derived. As applications of these multivariate quantiles, we develop some affine equivariant quantile contour plots which can be used to study the geometry of the data cloud as well as the underlying probability distribution and to detect outliers. These quantiles can also be used to construct affine invariant versions of multivariate Q-Q plots which are useful in checking how well a given multivariate probability distribution fits the data and for comparing the distributions of two data sets. We illustrate these applications with some simulated and real data sets. We also indicate a way of extending the notion of univariate L-estimates and trimmed means in the multivariate set-up using these affine equivariant quantiles.     

Normal Linear Regression Models With Recursive Graphical Markov Structure

A multivariate normal statistical model defined by the Markov properties determined by an acyclic digraph admits a recursive factorization of its likelihood function (LF) into the product of conditional LFs, each factor having the form of a classical multivariate linear regression model (≡WMANOVA model). Here these models are extended in a natural way to normal linear regression models whose LFs continue to admit such recursive factorizations, from which maximum likelihood estimators and likelihood ratio (LR) test statistics can be derived by classical linear methods. The central distribution of the LR test statistic for testing one such multivariate normal linear regression model against another is derived, and the relation of these regression models to block-recursive normal linear systems is established. It is shown how a collection of nonnested dependent normal linear regression models (≡Wseemingly unrelated regressions) can be combined into a single multivariate normal linear regression model by imposing a parsimonious set of graphical Markov (≡Wconditional independence) restrictions.     

几种新的回归诊断图示模型及其医学应用

本文在变量选择问题的基础上，提出了一种新的图示模型──减变残差图。并给出它的两种推广形式：均值平移异常值检验图和部分影响诊断图。通过它们不但可以容易地考察一个变量在模型中的作用和检验异常值，而且可以诊断样本点对模型和变量的影响大小。     

寿命分布的PP图

本文介绍了三种寿命分布的PP图,并比较了PP图和QQ图,得出PP图是一种更加直观的、更具解释性的拟合优度检验图;     

多个相异性矩阵的数据分析

本文给出了分析多个相异性矩阵的三种方法.首先找到了一种图表示,使我们对所有相异性矩阵有一个总体的了解;其次定义了一个新的相异性矩阵,它可以看作是对所有原始相异性矩阵的一个折衷处理;最后提出了一种MIMU方法.在文中我们还对由上述方法得到的坐标图进行了比较.     

Analysis of means approach in advanced designs

Analysis of means (ANOM), similar to Shewhart control chart that exhibits individual mean effects on a graphical display, is an attractive alternative mean testing procedure for the analysis of variance (ANOVA). The procedure is primarily used to analyze experimental data from designs with only fixed effects. Recently introduced, the ANOM procedure based on the q‐distribution (ANOMQ procedure) generalizes the ANOM approach to random effects models. This article reveals that the application of ANOM and ANOMQ procedures in advanced designs such as hierarchically nested and split‐plot designs with fixed, random, and mixed effects enhances the data visualization aspect in graphical testing. Data from two real‐world experiments are used to illustrate the proposed procedure; furthermore, these experiments exhibit the ANOM procedures' visualization ability compared with ANOVA from the point of view of the practitioner.     

Visualising concordance

Concordance describes the agreement between m rankings of k objects. Despite the long history of measures of concordance and the recently revived interest in comparison of concordance (c.f. Legendre in J Agric Biol Environ Stat 10(2):226–245, 2005), the task of visualising concordance remained virtually unaddressed. We first show how to depict concordance by simply plotting raw data in parallel coordinates. Then we review further possibilities for depicting concordance using the recently developed plots of inter-rater variability in ordinal ratings (Nelson and Pepe in Stat Methods Med Res 9:475–496, 2000) and plots of correlation matrices (Trosset in J Comput Graph Stat 14(1):1–19, 2005). Next, we propose two novel concordance plots. The concordance bubble-plot is based on raw rank data, while the pin-cushion plot depicts rank differences in polar coordinates. We present visualisations of artificial and real-life datasets with different degree of concordance and identify strong and weak points of the proposed plots. In conclusion, we review some other work related to visualisation of concordance and discuss some other options for constructing novel concordance plots.     

Residual Diagnostics for Covariate Effects in Spatial Point Process Models

For a spatial point process model in which the intensity depends on spatial covariates, we develop graphical diagnostics for validating the covariate effect term in the model, and for assessing whether another covariate should be added to the model. The diagnostics are point-process counterparts of the well-known partial residual plots (component-plus-residual plots) and added variable plots for generalized linear models. The new diagnostics can be derived as limits of these classical techniques under increasingly fine discretization, which leads to efficient numerical approximations. The diagnostics can also be recognized as integrals of the point process residuals, enabling us to prove asymptotic results. The diagnostics perform correctly in a simulation experiment. We demonstrate their utility in an application to geological exploration, in which a point pattern of gold deposits is modeled as a point process with intensity depending on the distance to the nearest geological fault. Online supplementary materials include technical proofs, computer code, and results of a simulation study.     

A Bayesian Sampling Approach to Regression Model Checking

A necessary step in any regression analysis is checking the fit of the model to the data. Graphical methods are often employed to allow visualization of features that the data should exhibit if the model holds. Judging whether such features are present or absent in any particular diagnostic plot can be problematic. In this article I take a Bayesian approach to aid in this task. The “unusualness” of some data with respect to a model can be assessed using the predictive distribution of the data under the model; an alternative is to use the posterior predictive distribution. Both approaches can be given a sampling interpretation that can then be used to enhance regression diagnostic plots such as marginal model plots.     

Multivariate Polynomials,Duality, and Structured Matrices

We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlation to multivariate polynomials. We describe the correlation in terms of the associated operators of multiplication in the polynomial ring and its dual space, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues, and relations between them are studied. Finally, we show some applications of this study to rootfinding problems for a system of multivariate polynomial equations, where the dual space, algebraic residues, Bezoutians, and other structured matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial computations and to improve substantially the known techniques of the study of these problems. In particular, we simplify and/or generalize the known reduction of the multivariate polynomial systems to the matrix eigenproblem, the derivation of the Bézout and Bernshtein bounds on the number of the roots, and the construction of multiplication tables. From the algorithmic and computational complexity point, we yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving multivariate polynomial systems of equations.     

CARTscans: A Tool for Visualizing Complex Models

We present CARTscans, a graphical tool that displays predicted values across a fourdimensional subspace. We show how these plots are useful for understanding the structure and relationships between variables in a wide variety of models, including (but not limited to) regression trees, ensembles of trees, and linear regressions with varying degrees of interactions. In addition, the common visualization framework allows diverse complex models to be visually compared in a way that illuminates the similarities and differences in the underlying methods, facilitates the choice of a particular model structure, and provides a useful check for implausible predictions of future observations in regions with little or no data.     

Asymptotic expansions for the distributions of some functions of the latent roots of matrices in three situations

In this paper we derive asymptotic expansions for the distributions of some functions of the latent roots of the matrices in three situations in multivariate normal theory, i.e., (i) principal component analysis, (ii) MANOVA model and (iii) canonical correlation analysis. These expansions are obtained by using a perturbation method. Confidence intervals for the functions of the corresponding population roots are also obtained.     

Visualization of Multivariate Density Estimates With Shape Trees

This article introduces graphical tools for visualizing multivariate functions, specializing to the case of visualizing multivariate density estimates. We visualize a density estimate by visualizing a series of its level sets. From each connected part of a level set a shape tree is formed. A shape tree is a tree whose nodes are associated with regions of the level set. With the help of a shape tree we define a transformation of a multivariate set to a univariate function. The shape trees are visualized with the shape plots and the location plot. By studying these plots one may identify the regions of the Euclidean space where the probability mass is concentrated. An application of shape trees to visualize the distribution of stock index returns is presented.