Added-variable Plots Research Articles

Added-variable plots for panel-data estimation

In this article, I extend the theory of added-variable plots to three panel-data estimation methods: fixed effects, between effects, and random effects. An added-variable plot is an effective way to show the correlation between an independent variable and a dependent variable conditional on other independent variables. In a multivariate context, a simple scatterplot showing x versus y is not adequate to show the relationship of x with y, because it ignores the impact of the other covariates. Added-variable plots are also useful for spotting influential outliers in the data that affect the estimated regression parameters. Stata can display added-variable plots with the command avplot, but it can be used only after regress. My new command, xtavplot, is a postestimation command that creates added-variable plots after xtreg estimates. Unlike avplot, xtavplot can display a confidence interval around the fitted regression line.

Added-variable plots with confidence intervals

An added-variable plot is an effective way to show the correlation between an independent variable and a dependent variable conditional on other independent variables. For multivariate estimation, a simple scatterplot showing x versus y is not adequate to show the partial correlation of x with y, because it ignores the impact of the other covariates. Added-variable plots are especially effective for showing the correlation of a dummy x variable with y because the dummy variable conditional on other covariates becomes a continuous variable, making the relationship easier to visualize. Added-variable plots are also useful for spotting influential outliers in the data that affect the estimated regression parameters. Stata provides added-variable plots after ordinary least-squares regressions with the avplot command. I present a new command, avciplot, that adds a confidence interval and other options to the avplot command.

Application of Statistical and Machine Learning Techniques for Laboratory-Scale Pressure Filtration: Modeling and Analysis of Cake Moisture

ABSTRACTThis study deals with the modeling and analysis of the pressure filtration process using statistical and machine learning techniques. The effects of externally controllable process-influencing factors such as pressure, pH, temperature, solids concentration, filtration time, air-blow time, and cake thickness on filtration performance, measured in terms of cake moisture, were modeled. A 9-factor regression model based on an exhaustive search algorithm and a 7-6-1 artificial neural network (ANN) model based on a resilient backpropagation algorithm were developed and gave R2 values of 0.84 and 0.94, respectively. Relative importance of input variables was analyzed using novel methods such as added-variable plots based on the regression model and Olden’s method based on the ANN model. Results from both methods established a negative correlation for pressure, solids concentration, filtration time, temperature, and air-blow time and a positive correlation for cake thickness and pH. Analysis from regression and ANN models indicated pH to be the most significant process-influencing factor. Even though both models served as good interpretable models, the ANN model outperformed the regression model in terms of predictive capability, with an R2 value of 0.965 compared with the regression model’s 0.750 for the test dataset.

Toward A Diagnostic Toolkit for Linear Models with Gaussian-Process Distributed Random Effects

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).

Regressions are Commonly Misinterpreted

Much literature misinterprets results of fitting multivariable models for linear regression, logistic regression, and other generalized linear models, as well as for survival, longitudinal, and hierarchical regressions. For the leading case of multiple regression, regression coefficients can be accurately interpreted via the added-variable plot. However, a common interpretation does not reflect the way regression methods actually work. Additional support for the correct interpretation comes from examining regression coefficients in multivariate normal distributions and from the geometry of least squares. To properly implement multivariable models, one must be cautious when calculating predictions that average over other variables, as in the Stata command margins.

Using the theory of added-variable plot for linear mixed models to decompose genetic effects in family data.

Effective analytical tools are highly desirable for data analysis and for making the biological link between genotypic and phenotypic measures. In family data it is important to reconcile the methods that explain the phenotypic variability through fixed genetic effects and ones that estimate variance components using classical heritability methods. Thus, in this paper, we propose a method based on added-variable plot for polygenic linear mixed models applied to genome wide association studies in family-based designs. Our goal is to be able to discriminate genetic predictor variables in effects due to random polygenic and residual components. We also propose an index to detect influential families for each predictor variable identified with genetic effect. We assess the performance of our proposed method using our own family simulated data and the Genetic Analysis Workshop 17 family simulated data.

Added-Variable Plots and Curvature in Linear Regression

Added-variable plots are useful for a variety of data-analytic purposes but can be seriously misleading when used to investigate curvature and predictor transformations in linear regression, unless the predictors are independent.

Added-Variable Plots and Curvature in Linear Regression

Added-variable plots are useful for a variety of data-analytic purposes but can be seriously misleading when used to investigate curvature and predictor transformations in linear regression, unless the predictors are independent.

On the Interpretation of Regression Plots

Abstract 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.

On the Interpretation of Regression Plots

Abstract 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.

Regression Diagnostics With Dynamic Graphics

We develop uses for two recently proposed types of dynamic displays—rotation and animation—in regression diagnostics. Some of the general issues that we address by using these displays include checking for interactions and normality, assessing the need to transform the data, and adding predictors to a model. Animation is used in probability plotting and as an aid to understanding the effects of adding variables to a model. Rotation is used for threedimensional added-variable and residual plots, each of which may be effective for diagnosing the presence of an interaction.

Added-Variable Plots in Linear Regression

This article discusses three well-known methods for obtaining a graphical evaluation of the effect of adding an explanatory variable in linear regression. A new method is also proposed.

Residual Plots for Detecting Nonlinearity in Generalized Linear Models

The added-variable plot, partial-residual plot, and constructed-variable plot are three useful diagnostic procedures in assessing the accuracy of a normal linear model. The first two have been extended to the generalized linear model. For detecting nonlinearity of variables, I extend the last one. An example is given for illustration.

Augmented Partial Residuals

A simple way of encouraging a nonlinear effect in a regression model to show itself in a partial residual (parres) plot (component-plus-residual plot) is to include a single quadratic term in the regression. It is computationally cheap to do this for each of the independent variables. Examples show that the resulting set of augmented partial residual plots gives insights that are not available from standard parres or added-variable plots.

Diagnostic Tests for Transformations

The evidence for transformation of the response in a regression model may sometimes depend crucially on one or a few observations. Expressions are derived for estimating the effect of deletion of observations on both the score test and the estimate of the transformation parameter. The estimates are presented as index plots, which are as easy to calculate as the wellknown added-variable plots of constructed variables. One advantage of the new procedure is that the importance of individual observations is not masked by the presence of leverage. A second is that the plots refer directly to score statistics and parameter estimates, both quantities of prime inferential importance. Examples are given for the power transformation and for the shifted power transformation with unknown origin.