¹Universidade de São Paulo, Faculdade de Medicina de Ribeirão Preto, Departamento de Medicina Social, São Paulo, Brasil. Pós-graduando. Email: eacbarros@gmail.com
²Universidade de São Paulo, Faculdade de Medicina de Ribeirão Preto, Departamento de Biomecânica, Medicina e Reabilitação do Aparelho Locomotor, São Paulo, Brasil. Pós-graduando. Email: priangelottisimoes@yahoo.com.br
³Universidade de São Paulo, Faculdade de Medicina de Ribeirão Preto, Departamento de Medicina Social, São Paulo, Brasil. Professor. Email: achcar@fmrp.usp.br
⁴Universidade de São Paulo, Faculdade de Medicina de Ribeirão Preto, Departamento de Medicina Social, São Paulo, Brasil. Professor. Email: edson@fmrp.usp.br
⁵Universidade de São Paulo, Faculdade de Medicina de Ribeirão Preto, Departamento de Biomecânica, Medicina e Reabilitação do Aparelho Locomotor, São Paulo, Brasil. Professor. Email: ashimano@fmrp.usp.br

Nesse artigo, tem-se o interesse em avaliar diferentes estratégias de estimação de parâmetros para um modelo de regressão linear múltipla. Para a estimação dos parâmetros do modelo foram utilizados dados de um ensaio clínico em que o interesse foi verificar se o ensaio mecânico da propriedade de força máxima (EM-FM) está associada com a massa femoral, com o diâmetro femoral e com o grupo experimental de ratas ovariectomizadas da raça Rattus norvegicus albinus, variedade Wistar. Para a estimação dos parâmetros do modelo serão comparadas três metodologias: a metodologia clássica, baseada no método dos mínimos quadrados; a metodologia Bayesiana, baseada no teorema de Bayes; e o método Bootstrap, baseado em processos de reamostragem.

Palavras chave: regressão linear múltipla, mínimos quadrados, análise bayesiana, bootstrap.

In this paper, we show different parameters estimation forms for multiple linear regression model. We used clinical data, where the interest was to verify the relationship among the mechanical assay maximum stress with femoral mass, femoral diameter and group of ovariectomized Wistar rats. We used three inference methods: Classic inference, based on the least square method; bayesian inference, based on the Bayes theorem; and bootstrap inference, based on resampling processes.

Key words: Multiple linear regression model, Least square method, Bayesian inference, Bootstrap inference.

Texto completo disponible en PDF

Referências

1. Bain, S. D. & Rubin, C. T. (1990), `Metabolic Modulation of Disuse Osteopenia: Endocrine-Dependent Site Specificity on Bone Remodeling´, J Bone Miner Res 5, 1069-1075.

2. Bilezikian, J. P., Raisz, L. G. & Rodan, G. A. (1994), Principles of Bone Biology, Academic Press.

3. Box, G. E. P. & Tiao, G. C. (1973), Bayesian Inference in Statistical Analysis, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.. Addison-Wesley Series in Behavioral Science: Quantitative Methods.

4. Carneiro, R. A. (1996), `Osteoporose problema mundial´, ARS CVRANDII Clínica Médica 29, 5.

5. Carpenter, J. & Bithell, J. (2000), `Bootstrap Confidence Intervals: When, which, what? A Practical Guide for Medical Statistician´, Statistics in Medicine 19, 1141-1164.

6. Chen, Z. & Wang, Y. G. (2004), `Efficient Regression Analysis with Ranked-Set Sampling´, Biometrics 60(4), 997-1004.

7. Chib, S. & Greenberg, E. (1995), `Understanding the Metropolis-Hastings Algorithm´, The American Statistician 49(4), 327-335.

8. Conference, C. D. (1993), Diagnosis, Prophylaxis and Treatment of Osteoporosis, American Journal of Medicine.

9. Davison, A. C. & Hinkley, D. V. (1997), Bootstrap Methods and their Application, Vol. 1 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge. With 1 IBM-PC floppy disk (3.5 inch; HD).

10. DiCiccio, T. J. & Efron, B. (1996), `Bootstrap Confidence Intervals´, Statistical Science 11(3), 189-228.

11. Dunson, D. B. & Herring, A. H. (2005), `Bayesian Latent Variable Models For Mixed Discrete Outcomes´, Biostatistics 6(1), 11-25.

12. Eastell, R. (2003), `Management of Osteoporosis due to Ovarian Failure´, Med. Pediatric Oncol 41(3), 222-227.

13. Efron, B. (1982), The Jackknife, the Bootstrap and other Resampling Plans, Vol. 38 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, United States.

14. Efron, B. & Tibshirani, R. J. (1993), An Introduction to the Bootstrap, Vol. 57 of Monographs on Statistics and Applied Probability, Chapman and Hall, New York, United States.

15. Freedman, L. S., Fainberg, V., Kipnis, V., Midthune, D. & Carroll, R. J. (2004), `A new Method for Dealing with Measurement Error in Explanatory Variables of Regression Models´, Biometrics 60(1), 172-181.

16. Gelfand, A. E. & Smith, A. F. M. (1990), `Sampling-Based Approaches to Calculating Marginal Densities´, Journal of the American Statistical Association 85, 398-409.

17. Gelman, A. & Rubin, D. B. (1992), `Inference From Iterative Simulation Using Multiple Sequences´, Statistical Science 7, 457-511.

18. Johnell, O. (1996), `Advances in Osteoporosis: Better Identification of Risk Factures can Reduce Morbidity and Mortality´, J. Int. Med 239, 299-304.

19. Kallu, D. N. (1991), `The Ovariectomized Rat Model of Postmenopausal Bone Loss´, Endocrinology 124(1), 7-16.

20. Keller, T., Spengler, D. M. & Carter, D. R. (1986), `Geometric, Elastic, and Structural Properties of Maturing Rat Femora´, Journal of Orthopedic Research 4, 57-67.

21. Lyles, R. H. & Kupper, L. L. (1997), `A Detailed Evaluation of Adjustment Methods for Multiplicative Measurement Error in Linear Regression with Applications in Occupational Epidemiology´, Biometrics 53(3), 1008-1025.

22. Montgomery, D. C., Peck, E. A. & Vining, G. G. (2001), Introduction to Linear Regression Analysis, Wiley Series in Probability and Statistics: Texts, References, and Pocketbooks Section, Third edn, Wiley-Interscience, New York.

23. Peng, Z. Q., Vaananen, H. K. & Tuukanen, J. (1997), `Ovariectomy-Induced Bone Loss can be Affect by Different Intensities of Treadmill Running Exercise in Rats´, Calcified Tissue International 60, 441-448.

24. Pohlman, R. L., Darby, L. A. & Lechner, A. J. (1985), `Morphometry and Calcium Contents in Apendicular and Axial Bones of Exteriol Ovariectomized Rats´, American Journal Physiology 248, 12-17.

25. R Development Core Team, (2006), R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0. *http://www.R-project.org

26. Rennó, A. C. M., Driusso, P. & Ferreira, V. (2001), `Atividade física e osteoporose: uma revisão bibliográfica´, Fisioterapia em movimento 13(2).

27. Robbins, S. (1993 language es), Patologia estrutural e funcional, Guanabara.

28. Smith, E. L. & Gilligan, C. (1989), `Mechanical Forces and Bone´, J Bone Miner Res 6, 139-173.

29. Spiegelhalter, D. J., Thomas, A., Best, N. G. & Gilks, W. R. (1995), BUGS: Bayesian Inference Using Gibbs Sampling, Version 0.50., MRC Biostatistics Unit Cambridge.

30. Szejnfeld, V. L. (2000), Osteoporose: diagnóstico e tratamento, Sarvier.

31. Tibshirani, R. (1988), `Correction to Discussion of: ``Jackknife, Bootstrap and other Resampling Methods in Regression Analysis''´, Ann. Statist. 16(1), 479.

32. Turner, R. (1999), `Mechanical Signaling in the Development of Postmenopausal Osteoporosis´, Lupus 8, 388-392.

33. Tuukanen, J., Peng, Z. & Vaananen, H. K. (1994), `Effect of Running Exercise of the Bone Loss Induced by Orchidectomy in the Rats´, Calcified Tissue International 55, 33-37.

34. Van der Wiel, H. E., Lips, P., Graafmans, W. C., Danielsen, C. C., Nauta, J., Van Lingen, A. & Mosekilde, L. (1995), `Additional Weight-Bearing During Exercise is more Important than Duration of Exercise for Anabolic Stimulus of Bone: A Study of Running Exercise in Female Rats´, Bone 16(1), 73-80.

35. Wu, C. F. J. (1986), `Jackknife, Bootstrap and other Resampling Methods in Regression Analysis´, Ann. Statist. 14(4), 1261-1350. With discussion and a rejoinder by the author.

36. Yeh, J. K., Aoia, J. F., Tierney, J. M. & Sprintz, S. (1993), `Effect of Treadmill Exercise on Vertebral and Tibial Bone Mineral Content and Bone Mineral Density in the Aged Adult Rat: Determined by Dual Energy X-Ray Absorptiometry´, Calcified Tissue International 52, 234-238.

37. Zarrow, N. X., Yochim, J. M. & Mccarthy, J. L. (1964), Experimental Endocrinology: A Sourcebook of Basic Techniques, Academic Press, New York, United States.

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX: