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Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the virtues of the ellipse and her higher-dimensional cousins for both these purposes in a variety of contexts, including linear models, multivariate linear models and mixed-effect models. We emphasize the strong relationships among statistical methods, matrix-algebraic solutions and geometry that can often be easily understood in terms of ellipses.

Gaussian processes (GPs) are widely used as distributions of random effects in linear mixed models, which are fit using the restricted likelihood or the closely related Bayesian analysis. This article addresses two problems. First, we propose tools for understanding how data determine estimates in these models, using a spectral basis approximation to the GP under which the restricted likelihood is formally identical to the likelihood for a gamma-errors GLM with identity link. Second, to examine the data's support for a covariate and to understand how adding that covariate moves variation in the outcome y out of the GP and error parts of the fit, we apply a linear-model diagnostic, the added variable plot (AVP), both to the original observations and to projections of the data onto the spectral basis functions. The spectral- and observation-domain AVPs estimate the same coefficient for a covariate but emphasize low- and high-frequency data features respectively and thus highlight the covariate's effect on the GP and error parts of the fit, respectively. The spectral approximation applies to data observed on a regular grid; for data observed at irregular locations, we propose smoothing the data to a grid before applying our methods. The methods are illustrated using the forest-biomass data of Finley et al. (2008).

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.

We show how to construct multiple linear regression datasets with the property that the plot of residuals versus predicted values from the least squares fit of the correct model reveals a hidden image or message. In the full PDF version of this article, the abstract itself is one such plot.

A framework is developed for the interpretation of regression plots, including plots of the response against selected covariates, residual plots, added-variable plots, and detrended added-variable plots. It is shown that many of the common interpretations associated with these plots can be misleading. The framework also allows for the generalization of standard plots and the development of new plotting paradigms. A paradigm for graphical exploration of regression problems is sketched.

Local influence diagnostics can be used to assess the influence of predictor values in multiple linear regression. For n observations and k regressors, an eigenanalysis of an nk ×nk matrix is required to assess the influence on the estimated coefficients. We provide the analytic expressions for the eigenvectors and show that they are easily computed, describe influence on the parameter estimates of a principal components regression, and are related to leverage, outliers, and added-variables plots. The results indicate that multicollinearity and overfitting contribute to a fitted model's sensitivity, leading to strategies for model assessment and selection.

The marginal dependence structure of a dataset y is displayed by a scatterplot matrix, a matrix whose elements are scatterplots of each pair of variables. A natural way to view the conditional dependence structure of y is through a partial scatterplot matrix, which contains scatterplots of partial residuals. We discuss its construction and give two examples of its use.

Start with a multiple regression in which each predictor enters linearly. How can we tell if there is a curve so that the model is not valid? Possibly for one of the predictors an additional square or square-root term is needed. We focus on the case in which an additional term is needed rather than the monotonic case in which a power transformation or logarithm might be sufficient. Among the plots that have been used for diagnostic purposes, nine methods are applied here. All nine methods work fine when the predictors are not related to each other, but two of them are designed to work even when the predictors are arbitrary noisy functions of each other. These two are recent methods, Cook's CERES plot and the plot for an additive model with nonparametric smoothing applied to one predictor. Even these plots, however, can miss a curve in some cases and show a false curve in others. To give a measure of curve detection, the curve can be fitted nonparametrically, and this fit can be used in place of the predictor in the multiple regression. When a curve is detected, it can be approximated with a parametric curve such as a polynomial in an arbitrary power.

We suggest various modifications to make scatterplots more informative when used with data obtained from a sample survey. Aspects of survey data leading to the plot modifications include the sample weights associated with the observations, imputed data for item nonresponse, and large sample sizes. Examples are given using data from the 1988 National Maternal Infant and Health Survey, the second National Health and Nutrition Examination Survey, and the epidemiologic follow-up of the first National Health and Nutrition Examination Survey.

This article concerns three-dimensional (3-D) added variable and partial residual plots. We show that the two-dimensional (2-D) added variable and partial residual plots are included as views in the 3-D plots, and these views are along (parallel to) the regression plane. That is, the 2-D plots are limited views, restricted to a sort of flatland, so they have a limited ability to show what is going on. In the case of just two predictors, recent improvements in the 2-D partial residual plot, such as the Mallows augmented plot, are also included as views in the 3-D plot, but these views are not along the regression plane.