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Alternative statistical modeling of Pharmacokinetics and Pharmacodynamics A collaboration between Aalborg University and Novo Nordisk A/S Claus Dethlefsen Center for Cardiovascular Research Participants 4 Post. Doc.’s Kim E. Andersen Claus Dethlefsen Susanne G. Bøttcher Malene Højbjerre Steering commitee Novo Nordisk A/S Judith L. Jacobsen Merete Jørgensen Aalborg University Søren Lundbye-Christensen Susanne Christensen Four different backgrounds State Space Models Inverse Problems Bayesian Networks PK/PD Graphical Models Learning Bayesian Networks Susanne Bøttcher and Claus Dethlefsen Bayesian Networks A Directed Acyclic Graph (DAG) To each node with parents there is attached a local conditional probability distribution, Lack of edges in corresponds to conditional independencies, Joint distribution Conditional Gaussian Distribution Observations of discrete variables multinomial distributed Continuous variables are Gaussian linear regressions on the continuous parents, with parameters depending on the configuration of the discrete parents. (ANCOVA) No continuous parents of discrete nodes Jointly a Conditional Gaussian (CG) distribution Advantages using Bayesian networks Qualitative representation of causal relations Compact description of the assumed independence relations among the variables Prior information is combined with data in the learning process Observations at all nodes are not needed for inference (calculation of distribution of unobserved given observed) Software Hugin: www.hugin.com Prediction in Bayesian networks R: Free software www.r-project.org Statistical software Deal: Package for R (documented) on CRAN Learning of parameters and structure. Developed by Claus Dethlefsen and Susanne Bøttcher Why Deal ? No other software learns Bayesian networks with mixed variables ! Training Data Hugin GUI Parameter priors .net Prior knowledge Parameter posteriors Network score Hugin API Posterior network Prediction of Insulin Sensitivity Index using Bayesian Networks Susanne Bøttcher and Claus Dethlefsen Insulin Sensitivity Index Insulin Sensitivity Index ( ) measures the fractional increase in glucose clearance rate during an IVGTT (Intraveneous Glucose Tolerance Test) A low is associated with risk of developing type 2 diabetes Aim Estimate insulin sensitivity index based on measurements of plasma glucose and serum insulin levels during an OGTT (Oral Glucose Tolerance Test) in individuals with normal glucose tolerance Methods 187 subjects without recognised diabetes IVGTT determines insulin sensitivity index OGTT with measurements of plasma glucose and serum insulin levels at time points 0, 30, 60, 105, 180, 240 Use 140 subjects as training data and 47 subjects as validation data Previous study Hansen et al used a multiple regression analysis Log(S.I) ~ BMI + SEX + G0 + I0 + G30 + I30 + G60 + I60 + G105 + I105 + G180 + I180 + G240 + I240 Prediction Bayesian Network Bayesian network A Bayesian Approach to the Minimal Model Kim E. Andersen and Malene Højbjerre Motivation Glucose Tolerance Test Protocols The Minimal Model of Glucose Disposal What can be done? Alternative Model Specification The Stochastic Minimal Model Results Comparison of MINMOD and Bayes References Andersen and Højbjerre. A Population-based Bayesian Approach to the Minimal Model of Glucose and Insulin Homeostasis, Statistics in Medicine, 24: 2381-2400, 2005. Andersen and Højbjerre. A Bayesian Approach to Bergman's Minimal Model, in C.M.Bishop & B.J.Frey (eds), Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, 2003. Bøttcher and Dethlefsen. deal: A package for learning Bayesian networks. Journal of Statistical Software, 8(20):1-40, 2003. Bøttcher and Dethlefsen. Prediction of the insulin sensitivity index using Bayesian networks. Technical Report R-2004-14, Aalborg University, 2004. Hansen, Drivsholm, Urhammer, Palacios, Vølund, Borch-Johnsen and Pedersen. The BIGTT test. Diabetes Care, 30:257-262, 2007.