A Hybrid Bayesian Laplacian Approach for Generalized Linear Mixed Models

The analytical intractability of generalized linear mixed models (GLMMs) has generated a lot of research in the past two decades. Applied statisticians routinely face the frustrating prospect of widely disparate results produced by the methods that are currently implemented in commercially available software. This article is motivated by this frustration and develops guidance as well as new methods that are computationally efficient and statistically reliable. Two main classes of approximations have been developed: likelihood-based methods and Bayesian methods. Likelihood-based methods such as the penalized quasi-likelihood approach of Breslow and Clayton (1993) have been shown to produce biased estimates especially for binary clustered data with small clusters sizes. More recent methods such as the adaptive Gaussian quadrature approach perform well but can be overwhelmed by problems with large numbers of random effects, and efficient algorithms to better handle these situations have not yet been integrated in standard statistical packages. Similarly, Bayesian methods, though they have good frequentist properties when the model is correct, are known to be computationally intensive and also require specialized code, limiting their use in practice. In this article we build on our previous method (Capanu and Begg 2010) and propose a hybrid approach that provides a bridge between the likelihood-based and Bayesian approaches by employing Bayesian estimation for the variance compo- nents followed by Laplacian estimation for the regression coefficients with the goal of obtaining good statistical properties, with relatively good computing speed, and using widely available software. The hybrid approach is shown to perform well against the other competitors considered. Another impor- tant finding of this research is the surprisingly good performance of the Laplacian approximation in the difficult case of binary clustered data with small clusters sizes. We apply the methods to a real study of head and neck squamous cell carcinoma and illustrate their properties using simulations based on a widely-analyzed salamander mating dataset and on another important dataset involving the Guatemalan Child Health survey

Optimized Variable Selection Via Repeated Data Splitting

We introduce a new variable selection procedure that repeatedly splits the data into two sets, one for estimation and one for validation, to obtain an empirically optimized threshold which is then used to screen for variables to include in the final model. Simulation results show that the proposed variable selection technique enjoys superior performance compared to candidate methods, being amongst those with the lowest inclusion of noisy predictors while having the highest power to detect the correct model and being unaffected by correlations among the predictors. We illustrate the methods by applying them to a cohort of patients undergoing hepatectomy at our institution

Comparing ROC Curves Derived From Regression Models

In constructing predictive models, investigators frequently assess the incremental value of a predictive marker by comparing the ROC curve generated from the predictive model including the new marker with the ROC curve from the model excluding the new marker. Many commentators have noticed empirically that a test of the two ROC areas often produces a non-significant result when a corresponding Wald test from the underlying regression model is significant. A recent article showed using simulations that the widely-used ROC area test [1] produces exceptionally conservative test size and extremely low power [2]. In this article we show why the ROC area test is invalid in this context. We demonstrate how a valid test of the ROC areas can be constructed that has comparable statistical properties to the Wald test. We conclude that using the Wald test to assess the incremental contribution of a marker remains the best strategy. We also examine the use of derived markers from non-nested models and the use of validation samples. We show that comparing ROC areas is invalid in these contexts as well

Statistical Evaluation of Evidence for Clonal Allelic Alterations in array-CGH Experiments

In recent years numerous investigators have conducted genetic studies of pairs of tumor specimens from the same patient to determine whether the tumors share a clonal origin. These studies have the potential to be of considerable clinical significance, especially in clinical settings where the distinction of a new primary cancer and metastatic spread of a previous cancer would lead to radically different indications for treatment. Studies of clonality have typically involved comparison of the patterns of somatic mutations in the tumors at candidate genetic loci to see if the patterns are sufficiently similar to indicate a clonal origin. More recently, some investigators have explored the use of array CGH for this purpose. Standard clustering approaches have been used to analyze the data, but these existing statistical methods are not suited to this problem due to the paired nature of the data, and the fact that there exists no “gold standard” diagnosis to provide a definitive determination of which pairs are clonal and which pairs are of independent origin. In this article we propose a new statistical method that focuses on the individual allelic gains or losses that have been identified in both tumors, and a statistical test is developed that assesses the degree of matching of the locations of the markers that indicate the endpoints of the allelic change. The validity and statistical power of the test is evaluated, and it is shown to be a promising approach for establishing clonality in tumor samples

Estimating the Empirical Lorenz Curve and Gini Coefficient in the Presence of Error

The Lorenz curve is a graphical tool that is widely used to characterize the concentration of a measure in a population, such as wealth. It is frequently the case that the measure of interest used to rank experimental units when estimating the empirical Lorenz curve, and the corresponding Gini coefficient, is subject to random error. This error can result in an incorrect ranking of experimental units which inevitably leads to a curve that exaggerates the degree of concentration (variation) in the population. We explore this bias and discuss several widely available statistical methods that have the potential to reduce or remove the bias in the empirical Lorenz curve. The properties of these methods are examined and compared in a simulation study. This work is motivated by a health outcomes application which seeks to assess the concentration of black patient visits among primary care physicians. The methods are illustrated on data from this study

The first international workshop on the role and impact of mathematics in medicine: a collective account

The First International Workshop on The Role and Impact of Mathematics in Medicine (RIMM) convened in Paris in June 2010. A broad range of researchers discussed the difficulties, challenges and opportunities faced by those wishing to see mathematical methods contribute to improved medical outcomes. Finding mechanisms for inter- disciplinary meetings, developing a common language, staying focused on the medical problem at hand, deriving realistic mathematical solutions, obtainin

A Metastasis or a Second Independent Cancer? Evaluating the Clonal Origin of Tumors Using Array-CGH Data

When a cancer patient develops a new tumor it is necessary to determine if this is a recurrence (metastasis) of the original cancer, or an entirely new occurrence of the disease. This is accomplished by assessing the histo-pathology of the lesions, and it is frequently relatively straightforward. However, there are many clinical scenarios in which this pathological diagnosis is difficult. Since each tumor is characterized by a genetic fingerprint of somatic mutations, a more definitive diagnosis is possible in principle in these difficult clinical scenarios by comparing the fingerprints. In this article we develop and evaluate a statistical strategy for this comparison when the data are derived from array comparative genomic hybridization, a technique designed to identify all of the somatic allelic gains and losses across the genome. Our method involves several stages. First a segmentation algorithm is used to estimate the regions of allelic gain and loss. Then the broad correlation in these patterns between the two tumors is assessed, leading to an initial likelihood ratio for the two diagnoses. This is then further refined by comparing in detail each plausibly clonal mutation within individual chromosome arms, and the results are aggregated to determine a final likelihood ratio. The method is employed to diagnose patients from several clinical scenarios, and the results show that in many cases a strong clonal signal emerges, occasionally contradicting the clinical diagnosis. The “quality” of the arrays can be summarized by a parameter that characterizes the clarity with which allelic changes are detected. Sensitivity analyses show that most of the diagnoses are robust when the data are of high quality

Examining the common aetiology of serous ovarian cancers and basal-like breast cancers using double primaries

Background: The somatic molecular profiles of basal-like breast cancers and high-grade serous ovarian cancers share many similarities, leading to the hypothesis that they have similar aetiologies, in which case they should occur together in the same patient more often than expected. Methods: We identified 545 women with double independent primary cancers of the breast and ovary reported to the California Cancer Registry from 1999 to 2013 and examined the coincidence of subtype combinations. Results: For most subtype combinations the observed frequencies were similar to their expected frequencies, but in 103 observed cases vs 43.8 expected (O/E=2.35; 95% CI 1.90–2.81) a triple-negative breast tumour (typically basal-like) was matched with a serous ovarian tumour (typically high-grade). Conclusions: The results provide compelling evidence that basal-like breast cancer and high-grade serous ovarian cancer share a much more similar aetiology than breast and ovarian cancers more broadly. Further research is needed to clarify the influence of germ-line BRCA1 mutations and other risk factors on these results