Philip J. Brown

Measurement, regression, and calibration

3. Measurement, Regression, and Calibration. By P. J. Brown. ISBN 0 19 852245 2. Oxford University Press, Oxford, 1993. 212 pp. £27.50.

Multivariate Bayesian variable selection and prediction

The multivariate regression model is considered with p regressors. A latent vector with p binary entries serves to identify one of two types of regression coefficients: those close to 0 and those not. Specializing our general distributional setting to the linear model with Gaussian errors and using natural conjugate prior distributions, we derive the marginal posterior distribution of the binary latent vector. Fast algorithms aid its direct computation, and in high dimensions these are supplemented by a Markov chain Monte Carlo approach to sampling from the known posterior distribution. Problems with hundreds of regressor variables become quite feasible. We give a simple method of assigning the hyperparameters of the prior distribution. The posterior predictive distribution is derived and the approach illustrated on compositional analysis of data involving three sugars with 160 near infra-red absorbances as regressors.

Inference with normal-gamma prior distributions in regression problems

This paper considers the efiects of placing an absolutely continuous prior distribution on the regression coe-cients of a linear model. We show that the posterior expectation is a matrix-shrunken version of the least squares estimate where the shrinkage matrix depends on the derivatives of the prior predictive den- sity of the least squares estimate. The special case of the normal-gamma prior, which generalizes the Bayesian Lasso (Park and Casella 2008), is studied in depth. We discuss the prior interpretation and the posterior efiects of hyperparameter choice and suggest a data-dependent default prior. Simulations and a chemomet- ric example are used to compare the performance of the normal-gamma and the Bayesian Lasso in terms of out-of-sample predictive performance.

Bayesian Wavelet Regression on Curves With Application to a Spectroscopic Calibration Problem

Motivated by calibration problems in near-infrared (NIR) spectroscopy, we consider the linear regression setting in which the many predictor variables arise from sampling an essentially continuous curve at equally spaced points and there may be multiple predictands. We tackle this regression problem by calculating the wavelet transforms of the discretized curves, then applying a Bayesian variable selection method using mixture priors to the multivariate regression of predictands on wavelet coefficients. For prediction purposes, we average over a set of likely models. Applied to a particular problem in NIR spectroscopy, this approach was able to find subsets of the wavelet coefficients with overall better predictive performance than the more usual approaches. In the application, the available predictors are measurements of the NIR reflectance spectrum of biscuit dough pieces at 256 equally spaced wavelengths. The aim is to predict the composition (i.e., the fat, flour, sugar, and water content) of the dough pieces using the spectral variables. Thus we have a multivariate regression of four predictands on 256 predictors with quite high intercorrelation among the predictors. A training set of 39 samples is available to fit this regression. Applying a wavelet transform replaces the 256 measurements on each spectrum with 256 wavelet coefficients that carry the same information. The variable selection method could use subsets of these coefficients that gave good predictions for all four compositional variables on a separate test set of samples. Selecting in the wavelet domain rather than from the original spectral variables is appealing in this application, because a single wavelet coefficient can carry information from a band of wavelengths in the original spectrum. This band can be narrow or wide, depending on the scale of the wavelet selected.

Bayes model averaging with selection of regressors

When a number of distinct models contend for use in prediction, the choice of a single model can offer rather unstable predictions. In regression, stochastic search variable selection with Bayesian model averaging offers a cure for this robustness issue but at the expense of requiring very many predictors. Here we look at Bayes model averaging incorporating variable selection for prediction. This offers similar mean-square errors of prediction but with a vastly reduced predictor space. This can greatly aid the interpretation of the model. It also reduces the cost if measured variables have costs. The development here uses decision theory in the context of the multivariate general linear model. In passing, this reduced predictor space Bayes model averaging is contrasted with single-model approximations. A fast algorithm for updating regressions in the Markov chain Monte Carlo searches for posterior inference is developed, allowing many more variables than observations to be contemplated. We discuss the merits of absolute rather than proportionate shrinkage in regression, especially when there are more variables than observations. The methodology is illustrated on a set of spectroscopic data used for measuring the amounts of different sugars in an aqueous solution. Copyright 2002 Royal Statistical Society.

Wavelet-Based Nonparametric Modeling of Hierarchical Functions in Colon Carcinogenesis

In this article we develop new methods for analyzing the data from an experiment using rodent models to investigate the effect of type of dietary fat on O6-methylguanine-DNA-methyltransferase (MGMT), an important biomarker in early colon carcinogenesis. The data consist of observed profiles over a spatial variable contained within a two-stage hierarchy, a structure that we dub hierarchical functional data. We present a new method providing a unified framework for modeling these data, simultaneously yielding estimates and posterior samples for mean, individual, and subsample-level profiles, as well as covariance parameters at the various hierarchical levels. Our method is nonparametric in that it does not require the prespecification of parametric forms for the functions and involves modeling in the wavelet space, which is especially effective for spatially heterogeneous functions as encountered in the MGMT data. Our approach is Bayesian; the only informative hyperparameters in our model are effectively smoothing parameters. Analysis of this dataset yields interesting new insights into how MGMT operates in early colon carcinogenesis, and how this may depend on diet. Our method is general, so it can be applied to other settings where hierarchical functional data are encountered.

Aggregate Data, Ecological Regression, and Voting Transitions

Abstract Voting data typically comprise the marginal distributions of votes cast at two successive elections. The fact that these obtain separately for many voting precincts or areas enables one to estimate the actual transition probabilities for movements between the options available to the voter at each election. An aggregated compound multinomial model is proposed. This allows log-linear dependence on various covariates and specifies a simple and illuminating structure of random effects. The information in such aggregated data is compared with that which would obtain had all of the transitions been observed. The model is applicable to many other types of aggregated data and meets many of the difficulties inherent in “ecological regression.” It is illustrated with an analysis of the British European election of 1984.

Bayesian wavelength selection in multicomponent analysis

Multicomponent analysis attempts to simultaneously predict the ingredients of a mixture. If near‐infrared spectroscopy provides the predictor variables, then modern scanning instruments may offer absorbances at a very large number of wavelengths. Although it is perfectly possible to use whole spectrum methods (e.g. PLS, ridge and principal component regression), for a number of reasons it is often desirable to select a small number of wavelengths from which to construct the prediction equation relating absorbances to composition. This paper considers wavelength selection with a view to using the chosen wavelengths to simultaneously predict the compositional ingredients and is therefore an example of multivariate variable selection. It adopts a binary exclusion/inclusion latent variable formulation of selection and uses a Bayesian approach. Problems of search of the vast number of possible selected models are overcome by a Markov chain Monte Carlo sampling technique.

Robust, Adaptive Functional Regression in Functional Mixed Model Framework

Functional data are increasingly encountered in scientific studies, and their high dimensionality and complexity lead to many analytical challenges. Various methods for functional data analysis have been developed, including functional response regression methods that involve regression of a functional response on univariate/multivariate predictors with nonparametrically represented functional coefficients. In existing methods, however, the functional regression can be sensitive to outlying curves and outlying regions of curves, so is not robust. In this article, we introduce a new Bayesian method, robust functional mixed models (R-FMM), for performing robust functional regression within the general functional mixed model framework, which includes multiple continuous or categorical predictors and random effect functions accommodating potential between-function correlation induced by the experimental design. The underlying model involves a hierarchical scale mixture model for the fixed effects, random effect, and residual error functions. These modeling assumptions across curves result in robust nonparametric estimators of the fixed and random effect functions which down-weight outlying curves and regions of curves, and produce statistics that can be used to flag global and local outliers. These assumptions also lead to distributions across wavelet coefficients that have outstanding sparsity and adaptive shrinkage properties, with great flexibility for the data to determine the sparsity and the heaviness of the tails. Together with the down-weighting of outliers, these within-curve properties lead to fixed and random effect function estimates that appear in our simulations to be remarkably adaptive in their ability to remove spurious features yet retain true features of the functions. We have developed general code to implement this fully Bayesian method that is automatic, requiring the user to only provide the functional data and design matrices. It is efficient enough to handle large datasets, and yields posterior samples of all model parameters that can be used to perform desired Bayesian estimation and inference. Although we present details for a specific implementation of the R-FMM using specific distributional choices in the hierarchical model, 1D functions, and wavelet transforms, the method can be applied more generally using other heavy-tailed distributions, higher dimensional functions (e.g., images), and using other invertible transformations as alternatives to wavelets. Supplementary materials for this article are available online.

The complete family of epidermal growth factor receptors and their ligands are co-ordinately expressed in breast cancer

The levels of expression of the four receptors and eleven ligands composing the epidermal growth factor family were measured using immunohistochemical staining in one hundred cases of breast cancer. All of the family were expressed to some degree in some cases; however, individual cases showed a very wide range of expression of the family from essentially none to all the factors at high levels. The highest aggregate level of expression of a receptor was HER2 followed by HER1, then HER3, then HER4. The ligands (including two splice variants of the NRG1 and NRG2 genes) broadly fell into three groups, those with the highest aggregate expression were Epigen, Epiregulin, Neuregulin 1α, Neuregulin 2α, Neuregulin 2β, Neuregulin 4 and TGFα, moderate expression was seen with EGF, Neuregulin 1β and Neuregulin 3, and relatively low levels of expression were seen of HB-EGF, Betacellulin and Amphiregulin. Statistical analysis using Spearman’s Rank Correlation showed a positive correlation of expression between each of the factors. Analysing the data using the Cox Proportional Hazards model showed that, in this dataset, the most powerful predictors of relapse free interval and overall survival were the combined measurement of only Epigen and Neuregulin 4.