Concordance Probability and Discriminatory Power in Proportional Hazards Regression

The concordance probability is used to evaluate the discriminatory power and the predictive accuracy of nonlinear statistical models. We derive an analytic expression for the concordance probability in the Cox proportional hazards model. The proposed estimator is a function of the regression parameters and the covariate distribution only and does not use the observed event and censoring times. For this reason it is asymptotically unbiased, unlike Harrell's c-index based on informative pairs. The asymptotic distribution of the concordance probability estimate is derived using U-statistic theory and the methodology is applied to a predictive model in lung cancer.

A Data-Analytic Strategy for Protein-Biomarker Discovery: Profiling of High-Dimensional Proteomic Data for Cancer Detection

With recent advances in mass spectrometry techniques, it is now possible to investigate proteins over a wide range of molecular weights in small biological specimens. This advance has generated data-analytic challenges in proteomics, similar to those created by microarray technologies in genetics, namely, discovery of "signature" protein profiles specific to each pathologic state (e.g., normal vs. cancer) or differential profiles between experimental conditions (e.g., treated by a drug of interest vs. untreated) from high-dimensional data. We propose a data analytic strategy for discovering protein biomarkers based on such high-dimensional mass-spectrometry data. A real biomarker-discovery project on prostate cancer is taken as a concrete example throughout the paper: the project aims to identify proteins in serum that distinguish cancer, benign hyperplasia, and normal states of prostate using the Surface Enhanced Laser Desorption/Ionization (SELDI) technology, a recently developed mass spectrometry technique. Our data analytic strategy takes properties of the SELDI mass-spectrometer into account: the SELDI output of a specimen contains about 48,000 (x, y) points where x is the protein mass divided by the number of charges introduced by ionization and y is the protein intensity of the corresponding mass per charge value, x, in that specimen. Given high coefficients of variation and other characteristics of protein intensity measures (y values), we reduce the measures of protein intensities to a set of binary variables that indicate peaks in the y-axis direction in the nearest neighborhoods of each mass per charge point in the x-axis direction. We then account for a shifting (measurement error) problem of the x-axis in SELDI output. After these pre-analysis processing of data, we combine the binary predictors to generate classification rules for cancer, benign hyperplasia, and normal states of prostate. Our approach is to apply the boosting algorithm to select binary predictors and construct a summary classifier. We empirically evaluate sensitivity and specificity of the resulting summary classifiers with a test dataset that is independent from the training dataset used to construct the summary classifiers. The proposed method performed nearly perfectly in distinguishing cancer and benign hyperplasia from normal. In the classification of cancer vs. benign hyperplasia, however, an appreciable proportion of the benign specimens were classified incorrectly as cancer. We discuss practical issues associated with our proposed approach to the analysis of SELDI output and its application in cancer biomarker discovery.

The Importance of Scale for Spatial-confounding Bias and Precision of Spatial Regression Estimators

Increasingly, regression models are used when residuals are spatially correlated. Prominent examples include studies in environmental epidemiology to understand the chronic health effects of pollutants. I consider the effects of residual spatial structure on the bias and precision of regression coefficients, developing a simple framework in which to understand the key issues and derive informative analytic results. When the spatial residual is induced by an unmeasured confounder, regression models with spatial random effects and closely-related models such as kriging and penalized splines are biased, even when the residual variance components are known. Analytic and simulation results show how the bias depends on the spatial scales of the covariate and the residual; bias is reduced only when there is variation in the covariate at a scale smaller than the scale of the unmeasured confounding. I also discuss how the scales of the residual and the covariate affect efficiency and uncertainty estimation when the residuals can be considered independent of the covariate. In an application on the association between black carbon particulate matter air pollution and birth weight, controlling for large-scale spatial variation appears to reduce bias from unmeasured confounders, while increasing uncertainty in the estimated pollution effect.

Semiparametric Theory for Causal Mediation Analysis: efficiency bounds, multiple robustness, and sensitivity analysis

Whilst estimation of the marginal (total) causal effect of a point exposure on an outcome is arguably the most common objective of experimental and observational studies in the health and social sciences, in recent years, investigators have also become increasingly interested in mediation analysis. Specifically, upon establishing a non-null total effect of the exposure, investigators routinely wish to make inferences about the direct (indirect) pathway of the effect of the exposure not through (through) a mediator variable that occurs subsequently to the exposure and prior to the outcome. Although powerful semiparametric methodologies have been developed to analyze observational studies, that produce double robust and highly efficient estimates of the marginal total causal effect, similar methods for mediation analysis are currently lacking. Thus, this paper develops a general semiparametric framework for obtaining inferences about so-called marginal natural direct and indirect causal effects, while appropriately accounting for a large number of pre-exposure confounding factors for the exposure and the mediator variables. Our analytic framework is particularly appealing, because it gives new insights on issues of efficiency and robustness in the context of mediation analysis. In particular, we propose new multiply robust locally efficient estimators of the marginal natural indirect and direct causal effects, and develop a novel double robust sensitivity analysis framework for the assumption of ignorability of the mediator variable.

On the Behaviour of Marginal and Conditional Akaike Information Criteria in Linear Mixed Models

In linear mixed models, model selection frequently includes the selection of random effects. Two versions of the Akaike information criterion (AIC) have been used, based either on the marginal or on the conditional distribution. We show that the marginal AIC is no longer an asymptotically unbiased estimator of the Akaike information, and in fact favours smaller models without random effects. For the conditional AIC, we show that ignoring estimation uncertainty in the random effects covariance matrix, as is common practice, induces a bias that leads to the selection of any random effect not predicted to be exactly zero. We derive an analytic representation of a corrected version of the conditional AIC, which avoids the high computational cost and imprecision of available numerical approximations. An implementation in an R package is provided. All theoretical results are illustrated in simulation studies, and their impact in practice is investigated in an analysis of childhood malnutrition in Zambia.