Brian D. Marx

Direct generalized additive modeling with penalized likelihood

Generalized additive models (GAMs) have become an elegant and practical option in model building. Estimation of a smooth GAM component traditionally requires an algorithm that cycles through and updates each smooth, while holding other components at their current estimated fit, until specified convergence. We aim to fit all the smooth components simultaneously. This can be achieved using penalized B-spline or P-spline smoothers for every smooth component, thus transforming GAMs into the generalized linear model framework. Using a large number of equally spaced knots, P-splines purposely overfit each B-spline component. To reduce flexibility, a difference penalty on adjacent B-spline coefficients is incorporated into a penalized version of the Fisher scoring algorithm. Each component has a separate smoothing parameter, and the penalty is optimally regulated through extensions of cross validation or information criterion. An example using logistic additive models provides illustrations of the developments.

Generalized linear regression on sampled signals and curves: a P -spline approach

We consider generalized linear regression with many highly correlated regressors—for instance, digitized points of a curve on a spatial or temporal domain. We refer to this setting as signal regression, which requires severe regularization because the number of regressors is large, often exceeding the number of observations. We solve collinearity by forcing the coefficient vector to be smooth on the same domain. Dimension reduction is achieved by projecting the signal coefficient vector onto a moderate number of B splines. A difference penalty between the B-spline coefficients further increases smoothness-the P-spline framework of Eilers and Marx. The procedure is regulated by a penalty parameter chosen using information criteria or cross-validation.

Multivariate calibration with temperature interaction using two-dimensional penalized signal regression

Abstract The Penalized Signal Regression (PSR) approach to multivariate calibration (MVC) assumes a smooth vector of coefficients for weighting a spectrum to predict the unknown concentration of a chemical component. B-splines and roughness penalties, based on differences, are used to estimate the coefficients. In this paper, we extend PSR to incorporate a covariate like temperature. A smooth surface on the wavelength–temperature domain is estimated, using tensor products of B-splines and penalties along the two dimensions. A slice of this surface gives the vector of weights at an arbitrary temperature. We present the theory and apply multi-dimensional PSR to a published data set, showing good performance. We also introduce and apply a simplification based on a varying-coefficient model (VCM).

Iteratively reweighted partial least squares estimation for generalized linear regression

I extend the concept of partial least squares (PLS) into the framework of generalized linear models. A spectroscopy example in a logistic regression framework illustrates the developments. These models form a sequence of rank 1 approximations useful for predicting the response variable when the explanatory information is severely ill-conditioned. Iteratively reweighted PLS algorithms are presented with various theoretical properties. Connections to principal-component and maximum likelihood estimation are made, as well as suggestions for rules to choose the proper rank of the final model.

Multidimensional Penalized Signal Regression

We propose a general approach to regression on digitized multidimensional signals that can pose severe challenges to standard statistical methods. The main contribution of this work is to build a two-dimensional coefficient surface that allows for interaction across the indexing plane of the regressor array. We aim to use the estimated coefficient surface for reliable (scalar) prediction. We assume that the coefficients are smooth along both indices. We present a rather straightforward and rich extension of penalized signal regression using penalized B-spline tensor products, where appropriate difference penalties are placed on the rows and columns of the tensor product coefficients. Our methods are grounded in standard penalized regression, and thus cross-validation, effective dimension, and other diagnostics are accessible. Further, the model is easily transplanted into the generalized linear model framework. An illustrative example motivates our proposed methodology, and performance comparisons are made to other popular methods.

Generalized Linear Additive Smooth Structures

This article proposes a practical modeling approach that can accommodate a rich variety of predictors, united in a generalized linear model (GLM) setting. In addition to the usual ANOVA-type or covariatelinear (L) predictors, we consider modeling any combination of smooth additive (G) components, varying coefficient (V) components, and (discrete representations of) signal (S) components. We assume that G is, and the coefficients of V and S are, inherently smooth—projecting each of these onto B-spline bases using a modest number of equally spaced knots. Enough knots are used to ensure more flexibility than needed; further smoothness is achieved through a difference penalty on adjacent B-spline coefficients (P-splines). This linear re-expression allows all of the parameters associated with these components to be estimated simultaneously in one large GLM through penalized likelihood. Thus, we have the advantage of avoiding both the backfitting algorithm and complex knot selection schemes. We regulate the flexibility of each component through a separate penalty parameter that is optimally chosen based on cross-validation or an information criterion.

Seasonal and spatial water quality changes in the outflow plume of the Atchafalaya River, Louisiana, USA

The objective of this study was to examine the interaction between the Atchafalaya River and the Atchafalaya Delta estuarine complex. Measurements of suspended sediments, inorganic nutrients (NO3−, NH4+, PO43−), chlorophylla (chla), and-salinity were taken monthly from December 1996 to January 1998. These data were compiled by season, and the Atchafalaya River plume data were also analyzed using the Generalized Additive Model technique. There were significant decreases in NO3− concentrations during summer, fall, and winter as river water passed through the estuary, that were attributable to chemical and biological processes rather than dilution with ambient water. In some regions there were higher chla concentrations during summer and fall compared to winter and spring, when river discharge and the introduction of inorganic nutrients were highest, suggesting biological processes were active during this study. The presence of NH4+, as a percentage of available dissolved inorganic nitrogen, increased with distance from the Atchafalaya River, indicative of remineralization processes and NO3− reduction. Mean PO43− concentrations were often higher in the estuarine regions compared to the Atchafalaya River. During summer total suspended solid (TSS) concentrations increased with distance from the river mouth, suggesting a turbidity maximum. Highest chla concentrations were found in the bayous and shallow water bodies of the Terrebonne marshes, as were the lowest TSS concentrations. The low chla concentrations found in other areas of this study, despite high inorganic nutrient concentrations, suggest light limitation as the major control of phytoplankton growth. Salinity reached near seawater concentrations at the outer edge of the Atchafalaya River plume, but much lower salinities (<10 psu) were observed at all other regions. The Atchafalaya Delta estuarine complex buffers the impact of the Atchafalaya River on the Louisiana coastal shelf zone, with a 41% of 47% decrease in Atchafalaya River NO3− concentrations before reaching Gulf waters.

The effects of herbivory on neighbor interactions along a coastal marsh gradient.

Many current theories of community function are based on the assumption that disturbances such as herbivory act to reduce the importance of neighbor interactions among plants. In this study, we examined the effects of herbivory (primarily by nutria, Myocastor coypus) on neighbor interactions between three dominant grasses in three coastal marsh communities, fresh, oligohaline, and mesohaline. The grasses studied were Panicum virgatum, Spartina patens, and Spartina alterniflora, which are dominant species in the fresh, oligohaline, and mesohaline marshes, respectively. Additive mixtures and monocultures of transplants were used in conjunction with exclosure fences to determine the impact of herbivory on neighbor interactions in the different marsh types. Herbivory had a strong effect on all three species and was important in all three marshes. In the absence of herbivores, the impact of neighbors was significant for two of the species (Panicum virgatum and Spartina patens) and varied considerably between environments, with competition intensifying for Panicum virgatum and decreasing for Spartina patens with increasing salinity. Indications of positive neighbor effects (mutualisms) were observed for both of these species, though in contrasting habitats and to differing degrees. In the presence of herbivores, however, competitive and positive effects were eliminated. Overall, then, it was observed that in this case, intense herbivory was able to override other biotic interactions such as competition and mutualism, which were not detectable in the presence of herbivores.

Race Disparities in Psychiatric Rates in Emergency Departments

Psychiatric diagnoses based on the International Classification of Diseases--Ninth Revision were examined in the medical discharge records of 33,000 emergency department (ED) patients to determine if (a) psychiatric disorders were underdiagnosed, (b) there were race and gender disparities in psychiatric rates, and (c) psychiatric rates varied as a function of type of injury (e.g., self vs. other-inflicted injuries) and medical diagnosis. The observed psychiatric rate of 5.27% was far below the national prevalence rate of 20%-28%. Both race groups were underdiagnosed, but the underdiagnosis was larger for African Americans. Younger patients had fewer psychiatric diagnoses than older patients. Men had more psychiatric diagnoses overall, whereas women had more mood and anxiety diagnoses. Self-injury patients had much higher psychiatric rates than the other injury groups. This psychiatric underdiagnosis contributes to needless emotional suffering, especially for minorities and the poor who rely on EDs for most of their health care.

The Sage handbook of multilevel modeling

Notes on Contributors Preface Multilevel Modeling - Jeffrey S Simonoff, Marc A Scott and Brian D Marx PART ONE: MULTILEVEL MODEL SPECIFICATION AND INFERENCE The Multilevel Model Framework - Jeff Gill and Andrew Womack Multilevel Model Notation - Establishing the Commonalities - Marc A Scott, Patrick E Shrout and Sharon L Weinberg Likelihood Estimation in Multilevel Models - Harvey Goldstein Bayesian Multilevel Models - Ludwig Fahrmeir, Thomas Kneib, and Stefan Lang The Choice between Fixed and Random Effects - Zac Townsend,Jack Buckley, Masataka Harada and Marc A Scott Centering Predictors and Contextual Effects - Craig K Enders Model Selection for Multilevel Models - Russell Steele Generalized Linear Mixed Models - Overview - Geert Verbeke and Geert Molenberghs Longitudinal Data Modeling - Nan M Laird and Garrett M Fitzmaurice Complexities in Error Structures Within Individuals - Vicente Nunez-Anton and Dale L Zimmerman Design Considerations in Multilevel Studies - Gerard van Breukelen and Mirjam Moerbeek Multilevel Models and Causal Inference - Jennifer Hill PART TWO: VARIATIONS AND EXTENSIONS OF THE MULTILEVEL MODEL Multilevel Functional Data Analysis - Ciprian M Crainiceanu, Brian S Caffo and Jeffrey S Morris Nonlinear Models - Lang Wu and Wei Liu Generalized Linear Mixed Models: Estimation and Inference - Charles E McCulloch and John M Neuhaus Categorical Response Data - Jeroen Vermunt Smoothing and Semiparametric Models - Jin-Ting Zhang Penalized Splines and Multilevel Models - Goran Kauermann and Torben Kuhlenkasper Hierarchical Dynamic Models - Marina Silva Paez and Dani Gamerman Mixture and Latent Class Models in Longitudinal and Other Settings - Ryan P Browne and Paul D McNicholas Multivariate Response Data - Helena Geys and Christel Faes PART THREE: PRACTICAL CONSIDERATIONS IN MODEL FIT AND SPECIFICATION Robust Methods for Multilevel Analysis - Joop Hox and Rens van de Schoot Missing Data - Geert Molenberghs and Geert Verbeke Lack of Fit, Graphics, and Multilevel Model Diagnostics - Gerda Claeskens Multilevel Models: Is GEE a Robust Alternative in the Presence of Binary Endogenous Regressors? - Robert Crouchley Software for Fitting Multilevel Models - Andrzej T Galecki and Brady T West PART FOUR: SELECTED APPLICATIONS Meta-Analysis - Larry V Hedges and Kimberly S Maier Modeling Policy Adoption and Impact with Multilevel Methods - James E Monogan III Multilevel Models in the Social and Behavioral Sciences - David Rindskopf Survival Analysis and the Frailty Model: The effect of education on survival and disability for older men in England and Wales - Ardo van den Hout and Brian D M Tom Point-Referenced Spatial Modeling - Andrew O Finley and Sudipto Banerjee Market Research and Preference Data - Adam Sagan Multilevel Modeling for Scoial Networks and Relational Data - Marijtje A J Van Duijn Name Index Subject Index