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Bayesian Nonparametric Biostatistics

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Nonparametric Bayesian Inference in Biostatistics

Part of the book series: Frontiers in Probability and the Statistical Sciences ((FROPROSTAS))

Abstract

We discuss some typical applications of Bayesian nonparametrics in biostatistics. The chosen applications highlight how Bayesian nonparametrics can contribute to addressing some fundamental questions that arise in biomedical research. In particular, we review some modern Bayesian semi- and nonparametric approaches for modeling longitudinal, survival, and medical diagnostic outcome data. Our discussion includes methods for longitudinal data analysis, non-proportional hazards survival analysis, joint modeling of longitudinal and survival data, longitudinal diagnostic test outcome data, and receiver operating characteristic curves. Throughout, we make comparisons among competing BNP models for the various data types considered.

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Notes

  1. 1.

    See also Müller and Mitra (2013) for a recent survey.

  2. 2.

    Zeger and Diggle (1994) used \(\rho (s) =\alpha +(1-\alpha )\rho ^{s}\). There are additional choices, including the possibility that σ w 2 could depend on t, resulting in a nonhomogeneous Ornstein–Uhlenbeck process (Zhang et al. 1998). Taylor et al. (1994) used an integrated Ornstein–Uhlenbeck process (integrating over an Ornstein–Uhlenbeck with exponential covariance function) that results in a covariance function that depends on both t and s. With structured covariance functions, the marginal covariance matrix for Y i is Cov\((Y _{i}) =\varSigma _{i}(\xi,\phi,\sigma ^{2}) = Z_{i}D(\xi )Z_{i}^{\mathrm{T}} + H_{i}(\phi ) +\sigma ^{2}I_{n_{ i}}\).

  3. 3.

    According to Ryan and Woodall (2005); Cox (1972) and Kaplan and Meier (1958) are the two most-cited statistical papers.

  4. 4.

    For ease of notation, we often write x β to denote of \(x^{\boldsymbol{T}}\beta\).

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Acknowledgements

We thank the Editors for the invitation, and we are indebted to our ‘partners in crime,’ including Adam Branscum, Ron Christensen, Ian Gardner, Maria De Iorio, Alejandro Jara, Prakash Laud, Michelle Norris, Fernando Quintana, Gary Rosner, and Mark Thurmond. Special thanks go to Vanda Inácio de Carvalho, Tim Hanson, and Peter Müller, who made substantive contributions to the penultimate draft of this paper, in addition to their contributions to the work presented. M. de. C was supported by Fondecyt grant 11121186.

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Authors and Affiliations

  1. University of California, Irvine, CA, USA

    Wesley O. Johnson

  2. Departamento de Estadística, Pontifica Universidad Católica de Chile, Santiago, Chile

    Miguel de Carvalho

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Editors and Affiliations

  1. Department of Bioinformatics and Biostatistics, University of Louisville, Louisville, Kentucky, USA

    Riten Mitra

  2. Department of Mathematics, University of Texas, Austin, Texas, USA

    Peter Müller

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Johnson, W.O., de Carvalho, M. (2015). Bayesian Nonparametric Biostatistics. In: Mitra, R., Müller, P. (eds) Nonparametric Bayesian Inference in Biostatistics. Frontiers in Probability and the Statistical Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-19518-6_2

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