Using cross-classified multivariate mixed response models with application to life history traits in great tits (Parus major)

Abstract

Longitudinal observations on known individuals are an important source of data with which to test evolutionary theory within natural populations, in particular, the evolution and maintenance of life-history traits. In this paper, we concentrate on the reproductive behaviour and survival of a small passerine bird, the great tit (Parus major). The dataset we consider is taken from the long-term study of great tits in Wytham Woods in Oxfordshire. The models we consider are designed to relate variation in several phenotypic response variables that are linked to evolutionary fitness, alongside the correlations between them, to both general environmental and individual-specific factors. We fit multivariate cross-classified random effects models using a Markov chain Monte Carlo (MCMC) estimation algorithm described in the paper. Our results show for which traits variability is influenced by environmental factors and for which traits individual bird factors are more important. The partitioning of correlations is particularly illuminating, producing some pairs of ‘antagonistic’ correlations which are biologically meaningful.

Keywords

complex data structures cross-classified models Markov chain Monte Carlo (MCMC)mixed responses multilevel modelling

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References

Albert J and Chib S (1993) Bayesian analysis of binary and polychotomous response data . Journal of the American Statistical Association, 88, 669–79 .

Barry SC , Brooks SP , Catchpole EA and Morgan BJT (2003) The analysis of ring-recovery data using random effects . Biometrics, 59, 54–65 .

Besbeas P , Lebreton JD and Morgan BJT (2003) The efficient integration of abundance and demographic data . Journal of the Royal Statistical Society, Series C, 52, 95–102 .

Browne WJ (2004) An illustration of the use of reparameterisation methods for improving MCMC efficiency in crossed random effect models . Multilevel Modelling Newsletter, 16 (1), 13–25 .

Browne WJ (2006) MCMC Algorithms for constrained variance matrices . Computational Statistics and Data Analysis, 50, 1655–77 .

Browne WJ and Draper D (2000) Implementation and performance issues in the Bayesian and likelihood fitting of multilevel models . Computational Statistics, 15, 391–420 .

Browne WJ and Draper D (2006) A comparison of Bayesian and likelihood methods for fitting multilevel models (with discussion) . Bayesian Analysis, 1, 473–550 .

Browne WJ , Goldstein H and Rasbash J (2001) Multiple membership multiple classification (MMMC) models . Statistical Modelling, 1, 103–24 .

Catalona PJ and Ryan LM (1992) Bivariate latent variable models for clustered discrete and continuous outcomes . Journal of the American Statistical Association, 87, 651–58 .

10.

Catchpole EA , Freeman SN , Morgan BJT and Harris MP (1998) Integrated recovery/recapture data analysis . Biometrics, 54, 33–46 .

11.

Chib S and Greenburg E (1998) Analysis of multivariate probit models . Biometrika, 85, 347–61 .

12.

Chib S and Hamilton BH (2000) Bayesian analysis of cross-sectional and clustered data selection models . Journal of Econometrics, 51, 79–99 .

13.

Dunson DB (2000) Bayesian latent variable models for clustered mixed outcomes . Journal of the Royal Statistical Society, Series B, 62, 355–66 .

14.

Gelman A and Rubin D (1992) Inference from iterative simulation using multiple sequences . Statistical Science, 7, 457–511 .

15.

Gilmour AR , Thompson R and Cullis B (1995) AIREML, an efficient algorithm for variance parameter estimation in linear mixed models . Biometrics, 51, 1440–50 .

16.

Goldstein H , Browne WJ and Rasbash J (2002) Partitioning variation in multilevel models . Understanding Statistics, 1, 223–32 .

17.

Gueorguieva RV and Agresti A (2001) A Correlated probit model for joint modeling of clustered binary and continuous responses . Journal of the American Statistical Association, 96, 1102–12 .

18.

Henderson CR (1953) Estimation of variance and covariance components . Biometrics, 9, 226–52 .

19.

King R and Brooks SP (2002) Model selection for integrated recovery/recapture data . Biometrics, 58, 841–51 .

20.

Lynch M and Walsh B (1998 Genetics and analysis of quantitative traits. Sunderland, MA: Sinauer Associates .

21.

Natarajan R and McCulloch CE (1999) Modelling heterogeneity in nest survival data . Biometrics, 55, 553–59 .

22.

Patterson HD and Thompson R (1971) Recovery of inter-block information when block sizes are unequal . Biometrika, 58, 545–54 .

23.

Perrins CM (1965) Population fluctuations and clutch-size in great tit (Parus major) . Journal of Animal Ecology, 34, 601–47 .

24.

Perrins CM (1979 British Tits. London: Collins .

25.

Pettifor RA , Perrins CM and McCleery RH (1988) Variation in clutch-size in great tits: evidence for the individual optimisation hypothesis . Nature, 336, 160–62 .

26.

Pettifor RA , Perrins CM and McCleery RH (2001) The individual optimisation of fitness: variation in reproductive output, including clutch-size, mean nestling mass and offspring recruitment, in manipulated broods of great tits, Parus major . Journal of Animal Ecology, 70, 62–79 .

27.

Rasbash J , Browne WJ , Goldstein H , Yang M et al. (2000 A User's guide to MLwiN (version 2.1). London: Institute of Education, University of London .

28.

Spiegelhalter DJ , Thomas A , Best NG and Lunn D (2002a WinBUGS version 1.4: user manual. Cambridge: Medical Research Council Biostatistics Unit .

29.

Spiegelhalter DJ , Best NG , Carlin BP and van der Linde A (2002b) Bayesian measures of model complexity and fit (with discussion) . Journal of the Royal Statistical Society, Series B, 64, 583–640 .