Analyzing the dependence structure of microarray data: a copula–based approach

The main aim of this Ph.D. dissertation is the study of clustering dependent data by means of copula functions with particular emphasis on microarray data. Copula functions are a popular multivariate modeling tool in each field where the multivariate dependence is of great interest and their use in clustering has not been still investigated. The first part of this work contains the review of the literature of clustering methods, copula functions and microarray experiments. The attention focuses on the K–means (Hartigan, 1975; Hartigan and Wong, 1979), the hierarchical (Everitt, 1974) and the model–based (Fraley and Raftery, 1998, 1999, 2000, 2007) clustering techniques because their performance is compared. Then, the probabilistic interpretation of the Sklar’s theorem (Sklar’s, 1959), the estimation methods for copulas like the Inference for Margins (Joe and Xu, 1996) and the Archimedean and Elliptical copula families are presented. In the end, applications of clustering methods and copulas to the genetic and microarray experiments are highlighted. The second part contains the original contribution proposed. A simulation study is performed in order to evaluate the performance of the K–means and the hierarchical bottom–up clustering methods in identifying clusters according to the dependence structure of the data generating process. Different simulations are performed by varying different conditions (e.g., the kind of margins (distinct, overlapping and nested) and the value of the dependence parameter ) and the results are evaluated by means of different measures of performance. In light of the simulation results and of the limits of the two investigated clustering methods, a new clustering algorithm based on copula functions (‘CoClust’ in brief) is proposed. The basic idea, the iterative procedure of the CoClust and the description of the written R functions with their output are given. The CoClust algorithm is tested on simulated data (by varying the number of clusters, the copula models, the dependence parameter value and the degree of overlap of margins) and is compared with the performance of model–based clustering by using different measures of performance, like the percentage of well–identified number of clusters and the not rejection percentage of H0 on . It is shown that the CoClust algorithm allows to overcome all observed limits of the other investigated clustering techniques and is able to identify clusters according to the dependence structure of the data independently of the degree of overlap of margins and the strength of the dependence. The CoClust uses a criterion based on the maximized log–likelihood function of the copula and can virtually account for any possible dependence relationship between observations. Many peculiar characteristics are shown for the CoClust, e.g. its capability of identifying the true number of clusters and the fact that it does not require a starting classification. Finally, the CoClust algorithm is applied to the real microarray data of Hedenfalk et al. (2001) both to the gene expressions observed in three different cancer samples and to the columns (tumor samples) of the whole data matrix.

Evidence-based polytherapies and long-term mortality after acute myocardial infarction in very old subjects compared with elderly and adults: a nested case-control study

Background: Clinical trials have demonstrated that selected secondary prevention medications for patients after acute myocardial infarction (AMI) reduce mortality. Yet, these medications are generally underprescribed in daily practice, and older people are often absent from drug trials. Objectives: To examine the relationship between adherence to evidence-based (EB) drugs and post-AMI mortality, focusing on the effects of single therapy and polytherapy in very old patients (≥80 years) compared with elderly and adults (<80 years). Methods: Patients hospitalised for AMI between 01/01/2008 and 30/06/2011 and resident in the Local Health Authority of Bologna were followed up until 31/12/2011. Medication adherence was calculated as the proportion of days covered for filled prescriptions of angiotensin-converting enzyme inhibitors (ACEIs)/angiotensin receptor blockers (ARBs), β-blockers, antiplatelet drugs, and statins. We adopted a risk set sampling method, and the adjusted relationship between medication adherence (PDC≥75%) and mortality was investigated using conditional multiple logistic regression. Results: The study population comprised 4861 patients. During a median follow-up of 2.8 years, 1116 deaths (23.0%) were observed. Adherence to the 4 EB drugs was 7.1%, while nonadherence to any of the drugs was 19.7%. For both patients aged ≥80 years and those aged <80 years, rate ratios of death linearly decreased as the number of EB drugs taken increased. There was a significant inverse relationship between adherence to each of 4 medications and mortality, although its magnitude was higher for ACEIs/ARBs (adj. rate ratio=0.60, 95%CI=0.52–0.69) and statins (0.60, 0.50–0.72), and lower for β-blockers (0.75, 0.61–0.92) and antiplatelet drugs (0.73, 0.63–0.84). Conclusions: The beneficial effect of EB polytherapy on long-term mortality following AMI is evident also in nontrial older populations. Given that adherence to combination therapies is largely suboptimal, the implementation of strategies and initiatives to increase the use of post-AMI secondary preventive medications in old patients is crucial.