Many data sources, including tracking social behavior to election polling to testing studies for understanding disease spread, are subject to sampling bias whose implications are not fully yet understood. In this paper we study estimation of a given feature (such as disease, or behavior at social media platforms) from biased samples, treating non-respondent individuals as missing data. Prevalence of the feature among sampled individuals has an upward bias under the assumption of individuals’ willingness to be sampled. This can be viewed as a regression model with symptoms as covariates and the feature as outcome. It is assumed that the outcome is unknown at the time of sampling, and therefore the missingness mechanism only depends on the covariates. We show that data, in spite of this, is missing at random only when the sizes of symptom classes in the population are known; otherwise data is missing not at random. With an information theoretic viewpoint, we show that sampling bias corresponds to external information due to individuals in the population knowing their covariates, and we quantify this external information by active information. The reduction in prevalence, when sampling bias is adjusted for, similarly translates into active information due to bias correction, with opposite sign to active information due to testing bias. We develop unified results that show that prevalence and active information estimates are asymptotically normal under all missing data mechanisms, when testing errors are absent and present respectively. The asymptotic behavior of the estimators is illustrated through simulations.

Principal components analysis has been used to reduce the dimensionality of datasets for a long time. In this paper, we will demonstrate that in mode detection the components of smallest variance, the pettiest components, are more important. We prove that for a multivariate normal or Laplace distribution, we obtain boxes of optimal volume by implementing "pettiest component analysis," in the sense that their volume is minimal over all possible boxes with the same number of dimensions and fixed probability. This reduction in volume produces an information gain that is measured using active information. We illustrate our results with a simulation and a search for modal patterns of digitized images of hand-written numbers using the famous MNIST database; in both cases pettiest components work better than their competitors. In fact, we show that modes obtained with pettiest components generate better written digits for MNIST than principal components.

BACKGROUND: DNA microarrays open up a new horizon for studying the genetic determinants of disease. The high throughput nature of these arrays creates an enormous wealth of information, but also poses a challenge to data analysis. Inferential problems become even more pronounced as experimental designs used to collect data become more complex. An important example is multigroup data collected over different experimental groups, such as data collected from distinct stages of a disease process. We have developed a method specifically addressing these issues termed Bayesian ANOVA for microarrays (BAM). The BAM approach uses a special inferential regularization known as spike-and-slab shrinkage that provides an optimal balance between total false detections and total false non-detections. This translates into more reproducible differential calls. Spike and slab shrinkage is a form of regularization achieved by using information across all genes and groups simultaneously. RESULTS: BAMarray is a graphically oriented Java-based software package that implements the BAM method for detecting differentially expressing genes in multigroup microarray experiments (up to 256 experimental groups can be analyzed). Drop-down menus allow the user to easily select between different models and to choose various run options. BAMarraycan also be operated in a fully automated mode with preselected run options. Tuning parameters have been preset at theoretically optimal values freeing the user from such specifications. BAMarray provides estimates for gene differential effects and automatically estimates data adaptive, optimal cutoff values for classifying genes into biological patterns of differential activity across experimental groups. A graphical suite is a core feature of the product and includes diagnostic plots for assessing model assumptions and interactive plots that enable tracking of prespecified gene lists to study such things as biological pathway perturbations. The user can zoom in and lasso genes of interest that can then be saved for downstream analyses. CONCLUSION: BAMarray is user friendly platform independent software that effectively and efficiently implements the BAM methodology. Classifying patterns of differential activity is greatly facilitated by a data adaptive cutoff rule and a graphical suite. BAMarray is licensed software freely available to academic institutions. More information can be found at http://www.bamarray.com.