Generalized Beta Regression to Elicit Conditional Distributions of Medical Variables

Abstract

Univariate conditional models are of core importance in supporting medical reasoning, as they allow to decompose a joint probability distribution using the chain rule. Although several methods are available for the elicitation of the joint prior distribution of parameters when the response is a medical categorical variable, the case of a medical continuous response is typically difficult to address, because its sample space is often bounded to an interval and its relationship with explanatory variables may be not linear. In these situations, the elicitation of an informative prior distribution on parameters of a univariate conditional model is challenging, because some level of statistical training is required to a medical expert for interpreting parameters and for retrieving appropriate quantitative information about them. The task can be eased and made efficient by recognizing that physicians typically distinguish among values involving medically normal and pathological patient conditions on the grounds of their personal clinical experience. In this paper, we propose a Generalized Beta regression where parameter elicitation is performed by establishing a correspondence among measured values expressed as relative positions within intervals with a clinical interpretation, regardless the original scales of variables. Software implementing the elicitation procedure is freely available.

References

Bagley SC, White H, Golom BA (2001). “Logistic Regression in the Medical Literature: Standards for Use and Reporting, with Particular Attention to One Medical Domain.” Journal of Clinical

Epidemiology, 54(10), 979–985.

Banner MJ, Kirby RR, Kirton OC, DeHaven CB, Blanch PB (1995). “Breathing Frequency and Pattern are Poor Predictors of Work of Breathing in Patients Receiving Pressure Support Ventilation.” Chest, 108(5), 1338–1344.

Bedrick EJ, Christensen R, Johnson W (1996). “A New Perspective on Priors for Generalized Linear Models.” Journal of the American Statistical Association, 91, 1450–1460.

Bové DS, Held L (2011). “Hyper g-Priors for Generalized Linear Models.” Bayesian Analysis, 6(3), 387–410.

Braun SR (1990). “Respiratory Rate and Pattern.” In HK Walker, WD Hall, JW Hurst (eds.), Clinical Methods: The History, Physical, and Laboratory Examinations, 3rd edition. Chapter 43, Butterworths, Boston, US-MA.

Chen MH, Ibrahim JG, Yiannoutsos C (1999). “Prior Elicitation, Variable Selection and Bayesian Computation for Logistic Regression Models.” Journal of the Royal Statistical Society. Series B (Methodological), 61, 223–242.

Cretikos MA, Bellomo R, Hillman K, Chen J, Finfer S, Flabouris A (2008). “Respiratory Rate: the Neglected Vital Sign.” Medical Journal of Australia, 188(11), 657–659.

DeMaris A (2004). Regression with Social Data: Modeling Continuous and Limited Response Variables. John Wiley & Sons, Hoboken, US-NJ.

Díez FJ, Druzdzel MJ (2006). “Canonical Probabilistic Models for Knowledge Engineering.” Technical Report CISIAD-06-01, UNED, Madrid, E.

Díez FJ, Mira J, Iturralde E, Zubillaga S (1997). “DIAVAL, a Bayesian Expert System for Echocardiography.” Artificial Intelligence in Medicine, 10, 59–73.

Druzdzel MJ, van der Gaag LC (2000). “Building Probabilistic Networks: Where Do the Numbers Come From?” IEEE Transactions on Knowledge and Data Engineering, 12(4), 481–486.

Ferrari SLP, Cribari-Neto F (2004). “Beta Regression for Modelling Rates and Proportions.” Journal of Applied Statistics, 31(7), 799–815.

Garthwaite PH, Al-Awadhib SA, Fadlalla GE, Jenkinsonc DJ (2013). “Prior Distribution Elicitation for Generalized Linear and Piecewise-Linear Models.” Journal of Applied Statistics, 40(1), 59–75.

Geweke J (1986). “On Assessing Prior Distributions and Bayesian Regression Analysis with g-Prior Distributions.” In DA Berry, KM Chaloner, JK Geweke (eds.), Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti, pp. 233–243. Elsevier, Amsterdam, NL.

Gigerenzer G (2002). Calculated Risks: How to Know When Numbers Deceive You. Simon & Schuster, New York, NY.

Guyatt G, Cairns J, Churchill D (1992). “Evidence-Based Medicine. A New Approach to Teaching the Practice of Medicine.” Journal of the American Medical Association, 268(17), 2420–2425.

Hamid Q, Shannon J, Martin J (2005). Physiological Basis of Respiratory Disease. BC Decker Inc., Hamilton, CN-ON.

Hastie T, Tibshirani R, Friedman J (2009). The Elements of Statistical Learning - Data Mining, Inference, and Prediction. 2nd edition. Springer, Stanford, US-CA.

Irwin RS, Rippe JM (2011). Irwin and Rippe’s Intensive Care Medicine. 7nd edition. Lippincott Williams & Wilkins, Philadelphia, US-PA.

Jacobs DS, Oxley DK, DeMott WR (2001). Jacobs & DeMott Laboratory Test Handbook. 5nd edition. Lexi-Comp, Cleveland, US-OH.

Johnson NL, Kotz S, Balakrishnan N (1995). Continuous Univariate Distributions. 2nd edition. Wiley, Hoboken, US-NJ.

Kadane JB, Dickey JM, Winkler RL, Smith WS, Peters SC (1980). “Interactive Elicitation of Opinion for a Normal Linear Model.” Journal of the American Statistical Association, 75, 845–854.

Kadane JB, Wolfson LJ (1998). “Experiences in Elicitation.” The Statistician, 47, 3–19.

Kahneman D, Slovic P, Tversky A (1982). Judgement under Uncertainty: Heuristics and Biases. Cambridge Univeristy Press, Cambridge, UK.

Koller D, Friedman N (2009). Probabilistic Graphical Models. Principles and Techniques. The MIT Press, Cambridge, US-MA.

Kynn M (2008). “The ‘Heuristics and Biases’ Bias in Expert Elicitation.” Journal of the Royal Statistical Society ser. A, 171, 239–264.

Luciani D, Cavuto S, Antiga L, Miniati M, Monti S, Pistolesi M, Bertolini G (2007). “Bayes Pulmonary Embolism Assisted Diagnosis: A New Expert System for Clinical Use.” Journal of Emergency Medicine, 24(3), 157–164.

Luciani D, Marchesi M, Bertolini G (2003). “The Role of Bayesian Networks in the Diagnosis of Pulmonary Embolism.” Journal of Thrombosis and Haemostasis, 1(4), 698–707.

Luciani D, Stefanini FM (2012). “Automated Interviews on Clinical Case Reports to Elicit Directed

Acyclic Graphs.” Artificial Intelligence in Medicine, 55(1), 1–11.

O’Hagan A, Buck CE, Daneshkhah A, Eiser JR, Garthwaite PH, Jenkinson DJ, Oakley JF, Rakow T (2006). Uncertain Judgements: Eliciting Experts’ Probabilities. John Wiley & Sons, Chichester, UK.

Onisko A, Druzdzel MJ, Wasyluk H (2001). “Learning Bayesian Networks Parameters from Small Data Sets: Applications of Noisy-OR Gates.” International Journal of Approximate Reasoning, 27, 165–182.

Papiris S, Kotanidou A, Malagari K, Roussos C (2002). “Clinical review: Severe Asthma.” Critical Care, 6, 30–44.

Papp LA, Martinez JM, Klein DF, Coplan JD, Norman RG, Cole R, de Jesus MJ, Ross D, Goetz R, Gorman JM (1997). “Respiratory Psychophysiology of Panic Disorder: Three Respiratory Challenges in 98 Subjects.” American Journal of Psychiatry, 154, 1557–1565.

Pearl J (2010). “An Introduction to Causal Inference.” Journal of Machine Learning Research, 11, 1643–1662.

Rhoades R, Bell DR (2012). Medical Physiology: Principles for Clinical Medicine. 3rd edition. Lippincott Williams & Wilkins, Philadelphia, US-PA.

Schneider A, Hommel G, Blettner M (2010). “Linear Regression Analysis: Part 14 of a Series on Evaluation of Scientific Publications.” Deutsches Arzteblatt International, 107(44), 776–782.

Published
2018-05-27
How to Cite
Magrini, A., Luciani, D., & Stefanini, F. M. (2018). Generalized Beta Regression to Elicit Conditional Distributions of Medical Variables. Austrian Journal of Statistics, 47(3), 20-38. https://doi.org/10.17713/ajs.v47i3.629