The above discussion and examples have emphasised parametric smoothing methods based on exchangeable sample members. The fully Bayes approach to combining infor-mation over exchangeable units using exponential family densities is exemplified by George et al. (1994), and stresses the benefits (e.g. in fully expressing uncertainty) as compared to parametric empirical Bayes smoothing (see, for example, Morris 1983).
The fully Bayes method implemented through repeated sampling allows the derivation of complex inferences concerning the relationships among the units, such as the density of the maximum or the density of the rank attached to each sample unit (Marshall and Spiegelhalter, 1998, p. 237).
If the simplest model based on exchangeable units assumes smoothing to a common global mean and a unique variance readily made modifications may be more realistic, for example allowing asymmetric skewed densities (Branco and Dey, 2001) or allowing the data to be distributed as a mixture of two or three distributions with different means and/or variances. Adopting discrete mixtures of parametric densities leads into semi and
non-parametric Bayesian methods. Further flexibility is provided by the range of approaches based on the Dirichlet process priors (illustrated in Examples 2.7 and 2.8), and discussed by authors such as Deyet al. (1999) and Walkeret al. (1999). These are more natural approaches if clustering of sub-groups within the sample is expected, or as providing a sensitivity analysis against baseline unimodal smoothing model. Sometimes the latter may suffice: for example, Marshall and Spiegelhalters analysis of 33 transplant centres failed to confirm a two cluster division of the centres. More specialised depart-ures from exchangeability occur in the analysis of spatially correlated data where the clustering is based on spatial contiguity (Chapter 7).
Whether a unimodal symmetric density is appropriate or not as a basis for combining information, a further major element to the process of joint inferences about sample units is the presence of further relevant information, possibly over different levels of data hierarchies (pupils, schools, etc.). Hence inferences about means or ranks for sample units may need to take account of covariates: for instance, severity or casemix indices may be relevant to rankings of medical institutions (see Example 4.6). The next two chapters accordingly consider the modelling of covariate effects in single and multi-level data.
REFERENCES
Akaike, H. (1973) Information theory and an extension of the maximum likelihood principle.2nd Internat. Sympos. Inform. Theory, Tsahkadsor 1971, 267±281.
Aitchison, J. and Shen, S. M. (1980) Logistic-normal distributions: Some properties and uses.
Biometrika67, 261±272.
Aitkin, M. (1991) Posterior Bayes factors.J. Roy. Stat. Soc., Ser. B53(1), 111±114.
Berger, J. and Pericchi, L. (1996) The intrinsic Bayes factor for model selection and prediction.J.
Am. Stat. Assoc.91(433), 109±122.
Berkhof, J., van Mechelen, I. and Hoijtink, H. (2000) Posterior predictive checks: principles and discussion,Computational Stat.,15(3), 337±354.
Besag, J. (1989) A candidates formula: a curious result in Bayesian prediction.Biometrika76, 183.
Bohning, D. (2000)Computer Assisted Analysis of Mixtures and Applications. Monographs on Statistics and Applied Probability, 81. London: Chapman & Hall.
Box, G. and Tiao, G. (1973)Bayesian Inference in Statistical Analysis. Addison-Wesley.
Bozdogan, H. (2000) Akaike's Information Criterion and recent developments in information complexity.J. Math. Psychol.,44(1), 62±91.
Branco, M. and Dey, D. (2001) A general class of multivariate skew elliptical distributions.J.
Multivariate Analysis79, 99±113.
Brooks, S. and Gelman, A. (1998) General methods for monitoring convergence of iterative simulations.J. Comp. Graph. Statist.7, 434±455.
Carlin, B. and Chib, S. (1995) Bayesian model choice via Markov chain Monte Carlo methods.J.
Roy. Stat. Soc., Ser. B57(3), 473±484.
Carlin, B. and Louis, T. (2000)Bayes and Empirical Bayes Methods for Data Analysis, 2nd edn.
Texts in Statistical Sciences. Boca Raton: Chapman and Hall/ RCR.
Chen, M., Shao, Q. and Ibrahim, J. (2000) Monte Carlo Methods in Bayesian Computation.
Springer Series in Statistics. New York, NY: Springer.
Chen, H., Chen, J. and Kalbfleisch, J. (2001) A modified likelihood ratio test for homogeneity in finite mixture models.J. Roy. Stat. Soc,Ser B,63(1), 19±30.
Chib, S. (1995) Marginal likelihood from the Gibbs output.J. Am. Stat. Assoc.90, 1313±1321.
Clayton, D. and Kaldor, J. (1987) Empirical Bayes estimates of age-standardised relative risks for use in disease mapping.Biometrics43, 671±681.
Davison, A. C. and Hinkley, D. (1997)Bootstrap Methods and their Application. Cambridge Series on Statistical and Probabilistic Mathematics. Cambridge: Cambridge University Press.
REFERENCES 75
De Finetti, B. (1961) The Bayesian approach to the rejection of outliers.Proc. 4th Berkeley Symp.
Math. Stat. Probab.1, 199±210.
DeSantis, F. and Spezzaferri, F. (1997) Alternative Bayes factors for model selection.Can. J. Stat.
25, 503±515.
Dey, D., Muller, P. and Sinha, D. (1999)Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statistics 133. New York, NY: Springer.
DiCiccio, T., Kass, R., Raftery, A. and Wasserman, L. (1997) Computing Bayes factors by combining simulation and asymptotic approximations. J. Am. Stat. Assoc. 92(439), 903±915.
DuMouchel, W. (1990) Bayesian meta-analysis. In: Berry, D. (ed.),Statistical Methodology in the Pharmaceutical Sciences. Marcel Dekker.
Fernandez, C. and Steel, M. (1998) On Bayesian modelling of fat tails and skewness.J. Am. Stat.
Assoc,93, 359±367.
Fruhwirth-Schattner, S. (2001) Markov Chain Monte Carlo estimation of classical and dynamic switching and mixture models.J. Am. Stat. Assoc.96, 194±209.
Geisser, S. and Eddy, W. (1979) A predictive approach to model selection.J. Am. Stat. Assoc.74, 153±160.
Gelfand, A. (1996) Model determination using sampling based methods. In: W. Gilks, S. Richard-son and D. Spieglehalter (eds.), Markov Chain Monte Carlo in Practice, Boca Raton:
Chapman & Hall/CRC.
Gelfand, A. and Dey, D. (1994) Bayesian model choice: asymptotics and exact calculations.
J. Roy. Stat. Soc., Ser. B56(3), 501±514.
Gelman, A., Carlin, J., Stern, H. and Rubin, D. (1995)Bayesian Data Analysis. CRC Press.
George E., Makov, U. and Smith, A. (1994) Fully Bayesian hierarchical analysis for exponential families via Monte Carlo Computation. In: Freeman, P. and Smith, A. (eds.),Aspects of Uncertainty ± A Tribute to D. V. Lindley. New York: Wiley.
Geweke, J. (1993) Bayesian treatment of the independent Student-t linear model. J. Appl.
Econometrics8, S19±S40.
Green, P. (1995) Reversible jump Markov Chain Monte Carlo computation and Bayesian model determination.Biometrika82(4), 711±732.
Harrison, G. and Millard, P. (1991) Balancing acute and long-term care: the mathematics of throughput in Departments of Geriatric Medicine.Meth. Infor. Medicine30, 221±228.
Hsiao, C., Tzeng, J. and Wang, C. (2000) Comparing the performance of two indices for spatial model selection: application to two mortality data sets.Stat. in Medicine19, 1915±1930.
Ishwaran, H. and James, L. (2001) Gibbs sampling methods for stick-breaking priors.J. Am. Stat.
Assoc.96, 161±173.
Kashiwagi, N. and Yanagimoto, T. (1992) Smoothing serial count data through a state-space model.Biometrics48, 1187±1194.
Kass, R. and Raftery, A. (1995) Bayes factors.J. Am. Stat. Assoc.90, 773±795.
Kass, R. and Steffey, D. (1989) Approximate Bayesian inference in conditionally independent hierarchical models.J. Am. Stat. Assoc.84, 717±726.
Kennett, S. (1983) Migration within and between labour markets. In: Goddard, J. and Champion, A. (eds.),The Urban and Regional Transformation of Britain. London: Methuen.
Kim, S. and Ibrahim, J. (2000) Default Bayes factors for generalized linear models. J. Stat. Plan.
Inference87(2), 301±315.
Kitagawa, G. and Gersch, W. (1996)Smoothness Priors Analysis of Time Series. Lecture Notes in Statistics 116. New York, NY: Springer.
Laird, N. (1982) Empirical Bayes estimates using the nonparametric maximum likelihood esti-mate for the prior.J. Stat. Comput. Simulation15, 211±220.
Laud, P. and Ibrahim, J. (1995) Predictive model selection.J. Roy. Stat. Soc. Ser. B57(1), 247±
Lee, P. (1997)262. Bayesian Statistics: An Introduction.2nd ed. London: Arnold.
Leonard, T. (1980) The roles of inductive modelling and coherence in Bayesian statistics. In:
Bernardo, J., DeGroot, M., Lindley, D. and Smith, A. (eds.),Bayesian Statistics I. Valencia:
University Press, pp. 537±555.
Lenk, P. and Desarbo, W. (2000) Bayesian inference for finite mixtures of generalized linear models with random effects.Psychometrika65, 93±119.
Leonard, T. (1973) A Bayesian method for histograms.Biometrika60, 297±308.
Leonard, T. and Hsu, J. (1999)Bayesian Methods: An Analysis for Statisticians and Interdisciplin-ary Researchers. Cambridge: Cambridge University Press.
Lindley, D. (1957) A statistical paradox.Biometrika44, 187±192.
McClean, S. and Millard, P. (1993) Patterns of length of stay after admission in geriatric-medicine
± an event history approach.The Statistician42(3), 263±274.
McCullagh, P. and Nelder, J. (1989)Generalized Linear Models. Boca Raton: Chapman & Hall/
Marshall, E. and Spiegelhalter, D. (1998) Comparing institutional performance using MarkovCRC.
chain Monte Carlo methods. In: Everitt, B. and Dunn, G. (eds.),Recent Advances in the Statistical Analysis of Medical Data. London: Arnold, pp. 229±250.
Morgan, B. (2000)Applied Stochastic Modelling. London: Arnold.
Morris, C. A. (1983) Parametric empirical Boyesian inference: theory and applications.J. Am.
Stat. Assoc.78, 47±65.
Newbold, E. (1926) A contribution to the study of the human factor in the causation of accidents.
Industrial Health Research Board, Report 34, London.
Raftery, A. (1995) Bayesian model selection in social research.Sociological Methodology25, 111±
Rao, C. (1975) Simultaneous estimation of parameters in different linear models and applications163.
to biometric problems.Biometrics31, 545±554.
Richardson, S. and Green, P. (1997) On Bayesian analysis of mixtures with an unknown number of components.J. Roy. Stat. Soc. Ser. B59(4), 731±758.
Robert, C. (1996) Mixtures of distributions: inferences and estimation. In:Markov Chain Monte Carlo in Practice, Gilks, W., Richardson, S. and Spieglehalter, D. (eds.), Boca Raton:
Chapman & Hall/CRC.
Sethuraman, J. (1994) A constructive definition of Dirichlet priors.Stat. Sin.4(2), 639±650.
Schwarz, G. (1978) Estimating the dimension of a model.Ann. Stat,6, 461±464.
Snedecor, G. and Cochran, W. (1989)Statistical Methods, 8th ed. Ames, IA: Iowa State Univer-sity Press.
Spiegelhalter, D. (1999) An initial synthesis of statistical sources concerning the nature and outcomes of paediatric cardiac surgical services at Bristol relative to other specialist centres from 1984 to 1995. Bristol Royal Infirmary Inquiry (http://www.bristol-inquiry.org.uk/
brisdsanalysisfinal.htm# Background Papers).
Spiegelhalter, D., Best, N., Carlin, B., and van der Linde, A. (2002) Bayesian measures of model complexity and fit,J. Royal Statistical Society,64B, 1±34.
Stephens, M. (2000) Bayesian analysis of mixture models with an unknown number of compon-ents ± An alternative to reversible jump methods.Ann. Stat.28(1), 40±74.
Symons, M., Grimson, R. C. and Yuan, Y. (1983) Clustering of rare events.Biometrics39, 193±205.
Tsutakawa, R. (1985) Estimation of cancer mortality rates: A Bayesian analysis of small frequen-cies.Biometrics41, 69±79.
Upton, G. (1991) The exploratory analysis of survey data using log-linear models.The Statistician 40, 169±182.
Walker, S., Damine, P., Laud, P. and Smith, A. (1999) Bayesian nonparametric inference for random distributions and related functions (with discussion).J. Roy. Stat. Soc., Ser. B,61, 485±527.
Wasserman, L. (2000)Asymptotic inference for mixture models using data-dependent priors.J.
Roy. Stat. Soc., Ser. B, Stat. Methodol.62(1), 159±180.
Weinberg, C. and Gladen, B. (1986) The Beta-geometric distribution applied to comparative fecundability studies.Biometrics42(3), 547±560.
Welham, J., McLachlan, G. and Davies, G. (2000) Heterogeneity in schizophrenia; mixture modelling of age-at-first-admission, gender and diagnosis. Acta Psychiat. Scand.101 (4), 312±317.
West, M. (1984) Outlier models and prior distributions in Bayesian linear regression.J. Roy. Stat.
Soc., Ser. B46, 431±439.
West, M. (1992) Modelling with mixtures. pp 503±524 In: Bernardo, J., Berger, J., David, A. and Smith, A. (eds.),Bayesian Statistics 4. New York, NY: OUP.
Williams, D. (1982) Extra-binomial variation in logistic linear models.J. Roy. Stat. Soc., Ser. C 31, 144±148.
Woodward, M. (1999)Epidemiology. London: Chapman & Hall.
REFERENCES 77
EXERCISES
1. Using the data in Example 2.1, consider comparing model fit (e.g. via the DIC approach) between the fixed effects and gamma-Poisson mixture models. The fixed effects model allows the underlying relative risks li to be different, but does not relate them to an overall hyperdensity. Identify the largest rate under each approach and the probability that it exceeds the average rate (by using the sample to assess this probability).
2. Also in Example 2.1, again try to analyse via random effects, but using Normal and Student t mixtures applied in the log scale forli. How far does the robust Studentt alternative (with degrees of freedom an unknown parameter) make a difference to the smoothed relative risks?
3. In Example 2.2, apply the DIC procedure to discriminate between binomial and beta-binomial models.
4. Repeat the beta-geometric mixture analysis of Example 2.3 for couples with non-smoking female partners. Calculate the chi square statistic for comparing actual and predicted cycles to conception counts (as in Table 2.4). Also, consider how to use this statistic in a predictive check fashion (see Equation (2.15)).
5. In Example 2.8, try the DPP analysis with a Dirichlet precision parametera of 5.
How does this compare with the results when taking a1, and what are the implications for the number of sub-groups apparent in the data. Also, try to identify a four group mixture by `conventional' discrete mixture methods and consider how identifiability is compromised.
6. In Example 2.11, program the sampling of replicate frequenciesZi for ages 15±84, and so compare the predictive criterion G2 in Equation (2.14) between the two models.