Bayesian inference for partially identified models: Exploring the limits of limited data

Bayesian Inference for Partially Identified Models: Exploring the Limits of Limited Data

Volumn , Issue , 2015, Pages 1-173

  Gustafson, Paul ^a  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84991399540     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1201/b18308     Document Type: Book

References (120)

1. Aldrich, J. (2002), ‘How likelihood and identification went Bayesian’, International Statistical Review 70, 79–98.
2. Barankin, E. W. (1960), ‘Sufficient parameters: Solution of the minimal dimensionality problem’, Annals of the Institute of Mathematical Statistics 12, 91–118.
3. Berger, J. O. (1985), Statistical Decision Theory and Bayesian Analysis, Springer.
4. Bernardo, J. M. and Smith, A. F. M. (1994), Bayesian Theory, Wiley.
5. Bolstad, W. M. (2007), Introduction to Bayesian Statistics, 2nd edn, Wiley.
6. Burstyn, I., Kim, H.-M., Yasui, Y. and Cherry, N. M. (2009), ‘The virtues of a deliberately misspecified disease model in demonstrating a gene-environment interaction’, Occupational and Environmental Medicine 66, 374–380.
7. Carlin, B. P. and Louis, T. A. (2011), Bayesian Methods for Data Analysis, 3rd edn, Chapman & Hall, CRC Press.
8. Carroll, R. J., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. M. (2006), Measurement Error in Nonlinear Models: A Modern Perspective, 2nd edn, Chapman & Hall, CRC Press.
9. Chatterjee, N. and Carroll, R. (2005), ‘Semiparametric maximum likelihood estimation exploiting gene-environment independence in case-control studies’, Biometrika 92, 399–418.
10. Chen, H. and Chen, J. (2011), ‘On information coded in gene-environment independence in case-control studies’, American Journal of Epidemiology 174, 736–743.
11. Chickering, D. and Pearl, J. (1996), A clinician’s tool for analyzing noncompliance, in ‘Proceedings of the Thirteenth National Conference on Artificial Intelligence (AAAI-96), Portland, OR’, Vol. 2, pp. 1269–1276.
12. Christensen, R., Johnson, W. O., Branscum, A. J. and Hanson, T. E. (2011), Bayesian Ideas and Data Analysis: An Introduction for Scientists and Statisticians, Chapman & Hall, CRC Press.
13. Cook, S., Gelman, A. and Rubin, D. B. (2006), ‘Validation of software for Bayesian models using posterior quantiles’, Journal of Computational and Graphical Statistics 15, 675–692.
14. Dawid, A. (1979), ‘Conditional independence in statistical theory’, Journal of the Royal Statistical Society, Series B 41, 1–31.
15. Dendukuri, N., Bélisle, P. and Joseph, L. (2010), ‘Bayesian sample size for diagnostic test studies in the absence of a gold standard: Comparing identifiable with non-identifiable models’, Statistics in Medicine 29, 2688–2697.
16. Dendukuri, N., Rahme, E., Bélisle, P. and Joseph, L. (2004), ‘Bayesian sample size determination for prevalence and diagnostic test studies in the absence of a gold standard test’, Biometrics 60, 388–397.
17. Doucet, A., De Freitas, N., Gordon, N. et al. (2001), Sequential Monte Carlo Methods in Practice, Vol. 1, Springer.
18. Frangakis, C. E. and Rubin, D. B. (2002), ‘Principal stratification in causal inference’, Biometrics 58, 21–29.
19. Fréchet, M. (1951), ‘Sur les tableaux de corrélation dont les marges sont données’, Annales de l’Université de Lyon, Section A 9, 53–77.
20. Gelfand, A. and Smith, A. (1990), ‘Sampling-based approaches to calculating marginal densities’, Journal of the American Statistical Association 85, 398–409.
21. Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. and Rubin, D. B. (2013), Bayesian Data Analysis, Chapman & Hall, CRC Press.
22. Ghosal, S. (1996), A review of consistency and convergence rates of posterior distributions, in ‘Proceedings of Varanashi Symposium in Bayesian Infer-ence’, Banaras Hindu University.
23. Gile, K. J. and Handcock, M. S. (2010), ‘Respondent-driven sampling: An assessment of current methodology’, Sociological Methodology 40, 285– 327.
24. Goetghebeur, E., Liinev, J., Boelaert, M. and Van der Stuyft, P. (2000), ‘Di-agnostic test analyses in search of their gold standard: Latent class analyses with random effects’, Statistical Methods in Medical Research 9, 231–248.
25. Goodman, L. A. (1974), ‘Exploratory latent structure analysis using both identifiable and unidentifiable models’, Biometrika 61, 215–231.
26. Greenland, S. (2000), ‘An introduction to instrumental variables for epidemi-ologists’, International Journal of Epidemiology 29, 722–729.
27. Greenland, S. (2003), ‘The impact of prior distributions for uncontrolled confounding and response bias: A case study of the relation of wire codes and magnetic fields to childhood leukemia’, Journal of the American Statistical Association 98, 47–55.
28. Greenland, S. (2005), ‘Multiple-bias modelling for analysis of observational data’, Journal of the Royal Statistical Society, Series A 168, 267–306.
29. Greenland, S. (2009a), ‘Bayesian perspectives for epidemiological research: III. Bias analysis via missing data methods’, International Journal of Epidemiology 38, 1662–1673.
30. Greenland, S. (2009b), ‘Relaxation penalties and priors for plausible modeling of nonidentified bias sources’, Statistical Science 24, 195–210.
31. Gustafson, P. (2005a), ‘On model expansion, model contraction, identifiability, and prior information: two illustrative scenarios involving mismeasured variables (with discussion)’, Statistical Science 20, 111–140.
32. Gustafson, P. (2005b), ‘The utility of prior information and stratification for parameter estimation with two screening tests but no gold standard’, Statistics in Medicine 24, 1203–1217.
33. Gustafson, P. (2006), ‘Sample size implications when biases are modelled rather than ignored’, Journal of the Royal Statistical Society, Series A 169, 883–902.
34. Gustafson, P. (2007), ‘Measurement error modeling with an approximate instrumental variable’, Journal of the Royal Statistical Society, Series B 69, 797–815.
35. Gustafson, P. (2009), ‘What are the limits of posterior distributions arising from nonidentified models, and why should we care?’, Journal of the American Statistical Association 104, 1682–1695.
36. Gustafson, P. (2010), ‘Bayesian inference for partially identified models’, International Journal of Biostatistics 6, issue 2 article 17.
37. Gustafson, P. (2011), Comment on ‘Transparent parameterizations of models for potential outcomes,’ by Richardson, Evans, and Robins, in J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West, eds, ‘Bayesian Statistics 9: Proceedings of the Ninth Valencia International Meeting’, Oxford University Press.
38. Gustafson, P. (2012), ‘On the behaviour of bayesian credible intervals in partially identified models’, Electronic Journal of Statistics 6, 2107–2124.
39. Gustafson, P. (2014), ‘Bayesian inference in partially identified models: Is the shape of the posterior distribution useful?’, Electronic Journal of Statistics 8, 476–496.
40. Gustafson, P. and Burstyn, I. (2011), ‘”Bayesian inference of gene-environment interaction from incomplete data: What happens when information on environment is disjoint from data on gene and disease?’, Statistics in Medicine 30, 877–889.
41. Gustafson, P., Gilbert, M., Xia, M., Michelow, W., Robert, W., Trussler, T., McGuire, M., Paquette, D., Moore, D. M. and Gustafson, R. (2013), ‘Impact of statistical adjustment for frequency of venue attendance in a venue-based survey of men who have sex with men’, American Journal of Epidemiology 177, 1157–1164.
42. Gustafson, P. and Greenland, S. (2006), ‘The performance of random coefficient regression in accounting for residual confounding’, Biometrics 62, 760–768.
43. Gustafson, P. and Greenland, S. (2009), ‘Interval estimation for messy observational data’, Statistical Science 24, 328–342.
44. Gustafson, P., Le, N. D. and Saskin, R. (2001), ‘Case-control analysis with partial knowledge of exposure misclassification probabilities’, Biometrics 57, 598–609.
45. Gustafson, P., McCandless, L. C., Levy, A. R. and Richardson, S. (2010), ‘Simplified Bayesian sensitivity analysis for mismeasured and unobserved confounders’, Biometrics 66, 1129–1137.
46. Hanson, T. and Johnson, W. (2005), ‘Comment on ‘On model expansion, model contraction, identifiability and prior information: Two illustrative scenarios involving mismeasured variables,’ by Gustafson’, Statistical Science 20, 131–134.
47. Hastings, W. K. (1970), ‘Monte Carlo sampling methods using Markov chains and their applications’, Biometrika 57, 97–109.
48. Hernán, M. A. and Robins, J. M. (2006), ‘Instruments for causal inference: an epidemiologist’s dream?’, Epidemiology 17, 360–372.
49. Hoff, P. D. (2009), A First Course in Bayesian Statistical Methods, Springer.
50. Hopper, J. L. and Seeman, E. (1994), ‘The bone density of female twins discordant for tobacco use’, New England Journal of Medicine 330, 387–392.
51. Huber, P. (1967), The behavior of maximum likelihood estimates under nonstandard conditions, in ‘Proceedings of the fifth Berkeley symposium on mathematical statistics and probability’, Vol. 1, pp. 221–233.
52. Hui, S. L. and Walter, S. D. (1980), ‘Estimating the error rates of diagnostic tests’, Biometrics 36, 167–171.
53. Hui, S. L. and Zhou, X. H. (1998), ‘Evaluation of diagnostic tests without gold standards’, Statistical Methods in Medical Research 7, 354–370.
54. Imbens, G. W. and Manski, C. F. (2004), ‘Confidence intervals for partially identified parameters’, Econometrica 72, 1845–1857.
55. Imbens, G. W. and Rubin, D. B. (1997), ‘Bayesian inference for causal effects in randomized experiments with noncompliance’, Annals of Statistics 25, 305–327.
56. Johnston, L. G. and Sabin, K. (2010), ‘Sampling hard-to-reach populations with respondent driven sampling’, Methodological Innovations Online 5, 38–48.
57. Jones, G., Johnson, W. O., Hanson, T. E. and Christensen, R. (2010), ‘Identifiability of models for multiple diagnostic testing in the absence of a gold standard’, Biometrics 66, 855–863.
58. Joseph, L. and Bélisle, P. (2013), ‘Bayesian sample size determination for case-control studies when exposure may be misclassified’, American Journal of Epidemiology 178, 1673–1679.
59. Joseph, L., Gyorkos, T. and Coupal, L. (1995), ‘Bayesian estimation of disease prevalence and the parameters of diagnostic tests in the absence of a gold standard’, American Journal of Epidemiology 141, 263–272.
60. Kadane, J. B. (1974), The role of identification in Bayesian theory, in S. E. Fienberg and A. Zellner, eds, ‘Studies in Bayesian Econometrics and Statistics, In Honor of Leonard J. Savage’, p. 175.
61. Karon, J. M. and Wejnert, C. (2012), ‘Statistical methods for the analysis of time–location sampling data’, Journal of Urban Health 89, 565–586.
62. King, G. (2013), A solution to the ecological inference problem: Reconstructing individual behavior from aggregate data, Princeton University Press.
63. Kosinski, A. S. and Flanders, W. D. (1999), ‘Evaluating the exposure and disease relationship with adjustment for different types of exposure misclassification: a regression approach’, Statistics in Medicine 18, 2795–2808.
64. Kraft, P., Yen, Y.-C., Stram, D. O., Morrison, J. and Gauderman, W. J. (2007), ‘Exploiting gene-environment interaction to detect genetic associations’, Human Heredity 63, 111–119.
65. Lash, T. L., Fox, M. P. and Fink, A. K. (2009), Applying Quantitative Bias Analysis to Epidemiologic Data, Springer.
66. Lawlor, D. A., Harbord, R. M., Sterne, J. A., Timpson, N. and Davey Smith, G. (2008), ‘Mendelian randomization: using genes as instruments for making causal inferences in epidemiology’, Statistics in Medicine 27, 1133–1163.
67. Lee, P. M. (2012), Bayesian Statistics: An Introduction, 4th edn, Wiley.
68. Liao, Y. and Jiang, W. (2010), ‘Bayesian analysis in moment inequality models’, Annals of Statistics 38, 275–316.
69. Lindley, D. V. (1972), Bayesian Statistics: A Review, Society for Industrial and Applied Mathematics.
70. Little, R. J. A. and Rubin, D. B. (2002), Statistical Analysis with Missing Data, 2nd edn, Wiley.
71. Lumley, T. (2004), ‘Analysis of complex survey samples’, Journal of Statistical Software 9, 1–19.
72. Lunn, D., Spiegelhalter, D., Thomas, A. and Best, N. (2009), ‘The BUGS project: Evolution, critique and future directions’, Statistics in Medicine 28, 3049–3067.
73. Lunn, D., Thomas, A., Best, N. and Spiegelhalter, D. (2000), ‘WinBUGS-a Bayesian modelling framework: Concepts, structure, and extensibility’, Statistics and Computing 10, 325–337.
74. Luo, H., Burstyn, I. and Gustafson, P. (2013), ‘Investigations of gene-disease associations: Costs and benefits of environmental data’, Epidemiology 24, 562–568.
75. Manski, C. F. (2003), Partial Identification of Probability Distributions, Springer.
76. Manski, C. F. (2010), ‘Unlearning and discovery’, American Economist 55, 9– 18.
77. McCandless, L. C., Gustafson, P. and Levy, A. R. (2007), ‘Bayesian sensitivity analysis for unmeasured confounding in observational studies’, Statistics in Medicine 26, 2331–2347.
78. McCandless, L. C., Gustafson, P. and Levy, A. R. (2008), ‘A sensitivity analysis using information about measured confounders yielded improved assessments of uncertainty from unmeasured confounding’, Journal of Clinical Epidemiology 61, 247–255.
79. McCandless, L. C., Gustafson, P., Levy, A. R. and Richardson, S. (2012), ‘Hi-erarchical priors for bias parameters in Bayesian sensitivity analysis for unmeasured confounding’, Statistics in Medicine 31, 383–396.
80. M’Lan, C. E., Joseph, L. and Wolfson, D. B. (2006), ‘Bayesian sample size determination for case-control studies’, Journal of the American Statistical Association 101, 760–772.
81. Moon, H. R. and Schorfheide, F. (2012), ‘Bayesian and frequentist inference in partially identified models’, Econometrica 80, 755–782.
82. Morgenstern, H. (1995), ‘Ecologic studies in epidemiology: Concepts, principles, and methods’, Annual Review of Public Health 16, 61–81.
83. Neath, A. and Samaniego, F. (1997), ‘On the efficacy of Bayesian inference for nonidentifiable models’, American Statistician 51, 225–232.
84. Newhouse, J. P. and McClellan, M. (1998), ‘Econometrics in outcomes research: the use of instrumental variables’, Annual Review of Public Health 19, 17–34.
85. Palatini, P., Pessina, A. and Dal Palù, C. (1993), ‘The hypertension and ambulatory recording venetia study (harvest): a trial on the predictive value of ambulatory blood pressure monitoring for the development of fixed hypertension in patients with borderline hypertension’, High Blood Pressure 2, 11–18.
86. Pasek, J., with some assistance from Alex Tahk and some code modified from R-core (2012), weights: Weighting and Weighted Statistics. R package version 0.75. URL: http://CRAN.R-project.org/package=weights
87. Pearl, J. (2000), Causality: Models, Reasoning, and Inference, Cambridge University Press.
88. Pepe, M. S. and Janes, H. (2007), ‘Insights into latent class analysis of diag nostic test performance’, Biostatistics 8, 474–484.
89. Poirier, D. (1998), ‘Revising beliefs in nonidentified models’, Econometric Theory 14, 483–509.
90. Qu, Y., Tan, M. and Kutner, M. H. (1996), ‘Random effects models in latent class analysis for evaluating accuracy of diagnostic tests’, Biometrics 52, 797–810.
91. Rahme, E., Joseph, L. and Gyorkos, T. W. (2000), ‘Bayesian sample size determination for estimating binomial parameters from data subject to misclassi-fication’, Journal of the Royal Statistical Society: Series C 49, 119–128.
92. Rassen, J. A., Brookhart, M. A., Glynn, R. J., Mittleman, M. A. and Schneeweiss, S. (2009), ‘Instrumental variables I: Instrumental variables exploit natural variation in nonexperimental data to estimate causal relation-ships’, Journal of Clinical Epidemiology 62, 1226–1232.
93. Richardson, T. S., Evans, R. J. and Robins, J. M. (2011), Transparent parameterizations of models for potential outcomes, in J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West, eds, ‘Bayesian Statistics 9: Proceedings of the Ninth Valencia International Meeting’, Oxford University Press, pp. 569–610.
94. Romano, J. P. and Shaikh, A. M. (2008), ‘Inference for identifiable parameters in partially identified econometric models’, Journal of Statistical Planning and Inference 138, 2786–2807.
95. Rosin, M. P., Cheng, X., Poh, C., Lam, W. L., Huang, Y., Lovas, J., Berean, K., Epstein, J. B., Priddy, R., Le, N. D. et al. (2000), ‘Use of allelic loss to predict malignant risk for low-grade oral epithelial dysplasia’, Clinical Cancer Research 6, 357–362.
96. Rosner, B. (2000), Fundamentals of Biostatistics, 5th edn, Duxbury Press.
97. Rothenberg, T. J. (1971), ‘Identification in parametric models’, Econometrica 39, 577–591.
98. Sacerdote, C., Guarrera, S., Smith, G. D., Grioni, S., Krogh, V., Masala, G., Mattiello, A., Palli, D., Panico, S., Tumino, R. et al. (2007), ‘Lactase persistence and bitter taste response: instrumental variables and Mendelian randomization in epidemiologic studies of dietary factors and cancer risk’, American Journal of Epidemiology 166, 576–581.
99. San Martın, E. and González, J. (2010), ‘Bayesian identifiability: Contributions to an inconclusive debate’, Chilean Journal of Statistics 1, 69–91.
100. Schork, M. A. and Remington, R. D. (2000), Statistics with Applications to the Biological and Health Sciences, Prentice Hall.
101. Semaan, S. (2010), ‘Time-space sampling and respondent-driven sampling with hard-to-reach populations’, Methodological Innovations Online 5, 60– 75.
102. Shapiro, A. (1986), ‘Asymptotic theory of overparameterized structural models’, Journal of the American Statistical Association 81, 142–149.
103. Smith, G. D. and Ebrahim, S. (2003), ‘Mendelian randomization: Can genetic epidemiology contribute to understanding environmental determinants of disease?’, International Journal of Epidemiology 32, 1–22.
104. Smith, G. D. and Ebrahim, S. (2004), ‘Mendelian randomization: prospects, potentials, and limitations’, International Journal of Epidemiology 33, 30– 42.
105. Stamey, J. D., Seaman, J. W. and Young, D. M. (2005), ‘Bayesian sample-size determination for inference on two binomial populations with no gold standard classifier’, Statistics in Medicine 24, 2963–2976.
106. Stan Development Team (2013), ‘Stan: A c++ library for probability and sampling, version 1.3’. URL http://mc-stan.org/
107. Steenland, K. and Greenland, S. (2004), ‘Monte-Carlo sensitivity analysis and Bayesian analysis of smoking as an unmeasured confounder in a study of silica and lung cancer’, American Journal of Epidemiology 160, 384–392.
108. Stoye, J. (2009), ‘More on confidence intervals for partially identified parameters’, Econometrica 77, 1299–1315.
109. Sturges, H. A. (1926), ‘The choice of a class interval’, Journal of the American Statistical Association 21, 65–66.
110. Szpiro, A. A., Rice, K. M. and Lumley, T. (2010), ‘Model-robust regression and a Bayesian sandwich estimator’, Annals of Applied Statistics 4, 2099– 2113.
111. Tierney, L. (1994), ‘Markov chains for exploring posterior distributions’, Annals of Statistics 22, 1701–1728.
112. Umbach, D. and Weinberg, C. (1998), ‘Designing and analysing case-control studies to exploit independence of genotype and exposure’, Statistics in Medicine 16, 1731–1743.
113. Vansteelandt, S., Goetghebeur, E., Kenward, M. G. and Molenberghs, G. (2006), ‘Ignorance and uncertainty regions as inferential tools in a sensitivity analysis’, Statistica Sinica 16, 953–979.
114. Wakefield, J. (2004), ‘Ecological inference for 2× 2 tables (with discussion)’, Journal of the Royal Statistical Society, Series A 167, 385–445.
115. Wakefield, J. (2008), ‘Ecologic studies revisited’, Annual Review of Public Health 29, 75–90.
116. Wang, D., Shen, T. and Gustafson, P. (2012), ‘Partial identification arising from nondifferential exposure misclassification: How informative are data on the unlikely, maybe, and likely exposed?’, International Journal of Biostatistics 8, issue 1, article 31.
117. Wang, F. and Gelfand, A. E. (2002), ‘A simulation-based approach to Bayesian sample size determination for performance under a given model and for separating models’, Statistical Science 17, 193–208.
118. White, H. (1982), ‘Maximum likelihood estimation of misspecified models’, Econometrica 50, 1–25.
119. Xia, M. and Gustafson, P. (2012), ‘A Bayesian method for estimating prevalence in the presence of a hidden sub-population’, Statistics in Medicine 31, 2386–2398.
120. Xia, M. and Gustafson, P. (2014), ‘Bayesian sensitivity analyses for hidden sub-populations in weighted sampling’, Canadian Journal of Statistics 42, 436–450.