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On composite marginal likelihoods

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Abstract

Composite marginal likelihoods are pseudolikelihoods constructed by compounding marginal densities. In several applications, they are convenient surrogates for the ordinary likelihood when it is too cumbersome or impractical to compute. This paper presents an overview of the topic with emphasis on applications.

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References

  • Aerts, M., Claeskens, G.: Bootstrapping pseudolikelihood models for clustered binary data. Ann. Inst. Stat. Math. 51, 515–530 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Allcroft, D.J., Glasbey, C.A.: A spectral estimator of ARMA parameters from thresholded data. Stat. Comput. 12, 369–376 (2002)

    Article  MathSciNet  Google Scholar 

  • Akaike, H.: Information theory and an extension of the maximum likelihood principle. In: Petrov, B.N., Caski, F. (eds.) Proc. Second International Symposium on Information Theory, pp. 267–281. Akademiai Kiado, Budapest (1973)

    Google Scholar 

  • Andersen, E.W.: Composite likelihood and two-stage estimation in family studies. Biostatistics 5, 15–30 (2004)

    Article  MATH  Google Scholar 

  • Anderson, J.A., Pemberton, J.D.: The grouped continuous model for multivariate ordered categorical variables and covariate adjustment. Biometrics 41, 875–885 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • Apanasovich, T.V., Ruppert, D., Lupton, J.R., Popovic, N., Turner, N.D., Chapkin, R.S., Carroll, R.J.: Aberrant crypt foci and semiparametric modelling of correlated binary data. Biometrics (2008, to appear)

  • Azzalini, A.: Maximum likelihood of order m for stationary stochastic processes. Biometrika 70, 381–367 (1983)

    Article  MathSciNet  Google Scholar 

  • Bellio, R., Varin, C.: A pairwise likelihood approach to generalized linear models with crossed random effects. Stat. Model. 5, 217–227 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  • Besag, J.E.: Spatial interaction and the statistical analysis of lattice systems (with discussion). J. R. Stat. Soc. B 34, 192–236 (1974)

    MathSciNet  Google Scholar 

  • Besag, J.E.: Efficiency of pseudolikelihood estimation for simple Gaussian fields. Biometrika 64, 616–618 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  • Bevilacqua, M., Gaetan, C., Mateu, J., Porcu, E.: Estimating space-time covariane functions: a composite likelihood approach (2007, submitted)

  • Breslow, N.E., Clayton, D.G.: Approximate inference in generalized linear mixed models. J. Am. Stat. Assoc. 88, 9–25 (1993)

    Article  MATH  Google Scholar 

  • Caragea, P., Smith, R.L.: Approximate likelihoods for spatial processes (2007, submitted)

  • Caragea, P., Smith, R.L.: Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models. J. Multivar. Stat. Anal. (2008, to appear)

  • Carey, V., Zeger, S.L., Diggle, P.: Modelling multivariate binary data with alternating logistic regressions. Biometrika 80, 517–526 (1993)

    Article  MATH  Google Scholar 

  • Chandler, R.E., Bate, S.: Inference for clustered data using the independence loglikelihood. Biometrika 94, 167–183 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Christensen, O., Waagepetersen, R.: Bayesian prediction of spatial count data using generalized linear mixed models. Biometrics 58, 280–286 (2002)

    Article  MathSciNet  Google Scholar 

  • Cox, D.R.: Partial likelihood. Biometrika 62, 269–276 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  • Cox, D.R., Reid, N.: A note on pseudolikelihood constructed from marginal densities. Biometrika 91, 729–737 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Cressie, N.: Statistics for Spatial Data, 2nd edn. Wiley, New York (1993)

    Google Scholar 

  • Curriero, F.C., Lele, S.: A composite likelihood approach to semivariogram estimation. J. Agric. Biol. Environ. Stat. 4, 9–28 (1999)

    Article  MathSciNet  Google Scholar 

  • Davison, A.C.: Statistical Models. Cambridge University Press, Cambridge (2003)

    MATH  Google Scholar 

  • de Leon, A.R.: Pairwise likelihood approach to grouped continuous model and its extension. Stat. Probab. Lett. 75, 49–57 (2005)

    Article  MATH  Google Scholar 

  • de Leon, A.R., Carriere, K.C.: General mixed-data model: extension of general location and grouped continuous models. Can. J. Stat. 35, 1–16 (2007)

    Google Scholar 

  • Diggle, P.J.: Binary mosaics and the spatial pattern of heather. Biometrics 37, 531–539 (1981)

    Article  Google Scholar 

  • Diggle, P.J., Tawn, J.A., Moyeed, R.A.: Model-based geostatistics (with discussion). J. R. Stat. Soc. Ser. B 47, 299–350 (1998)

    MATH  MathSciNet  Google Scholar 

  • Drum, M.L., McCullagh, P.: REML estimation with exact covariance in the logistic mixed model. Biometrics 49, 677–689 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  • Engler, D.A., Mohapatra, M., Louis, D.N., Betensky, R.A.: A pseudolikelihood approach for simultaneous analysis of array comparative genomic hybridizations. Biostatistics 7, 399–421 (2006)

    Article  MATH  Google Scholar 

  • Fearnhead, P.: Consistency of estimators of the population-scaled recombination rate. Theor. Popul. Biol. 64, 65–79 (2003)

    Article  Google Scholar 

  • Fearnhead, P., Donnely, P.: Approximate likelihood methods for estimating local recombination rates (with discussion). J. R. Stat. Soc. B 64, 657–680 (2002)

    Article  MATH  Google Scholar 

  • Fieuws, S., Verbeke, G.: Pairwise fitting of mixed models for the joint modeling of multivariate longitudinal profiles. Biometrics 62, 424–431 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  • Fieuws, S., Verbeke, G., Boen, F., Delecluse, C.: High dimensional multivariate mixed models for binary questionnaire data. Appl. Stat. 55, 449–460 (2006)

    MATH  MathSciNet  Google Scholar 

  • Fieuws, S., Verbeke, G., Molenberghs, G.: Random-effects models for multivariate repeated measures. Stat. Methods Med. Res. 16, 387–397 (2007)

    Article  MathSciNet  Google Scholar 

  • Gao, X., Song, P.X.-K., Pu, D.Q.: Composite likelihood inference for gene networks on time-course microarray data (2007, submitted)

  • Geys, H., Molenberghs, G., Ryan, L.M.: Pseudolikelihood modelling of multivariate outcomes in developmental toxicology. J. Am. Stat. Assoc. 94, 734–745 (1999)

    Article  Google Scholar 

  • Geys, H., Regan, M.M., Catalano, P.J., Molenberghs, G.: Two latent variable risk assessment approaches for mixed continuous and discrete outcomes from developmental toxicity data. J. Agric. Biol. Environ. Stat. 6, 340–355 (2001)

    Article  Google Scholar 

  • Glasbey, C.A.: Non-linear autoregressive time series with multivariate Gaussian mixtures as marginal distributions. Appl. Stat. 50, 143–154 (2001)

    MathSciNet  Google Scholar 

  • Godambe, V.P.: An optimum property of regular maximum likelihood equation. Ann. Math. Stat. 31, 1208–1211 (1960)

    Article  MathSciNet  Google Scholar 

  • Guan, Y.: A composite likelihood approach in fitting spatial point process models. J. Am. Stat. Assoc. 101, 1502–1512 (2006)

    Article  MATH  Google Scholar 

  • Guan, Y.: A composite likelihood cross-validation approach in selecting bandwidth for the estimation of the pair correlation function. Scand. J. Stat. 34, 336–346 (2007)

    Article  MATH  Google Scholar 

  • Hall, P., Jing, B.: On sample reuse methods for dependent data. J. R. Stat. Soc. Ser. B 58, 727–737 (1996)

    MATH  MathSciNet  Google Scholar 

  • Hanfelt, J.J.: Composite conditional likelihood for sparse clustered data. J. R. Stat. Soc. Ser. B 66, 259–273 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Haslett, J., Raftery, A.E.: Space-time modelling with long-memory dependence: assessing Ireland wind power resource. Appl. Stat. 38, 1–50 (1989)

    Article  Google Scholar 

  • Heagerty, P.J., Lele, S.R.: A composite likelihood approach to binary spatial data. J. Am. Stat. Assoc. 93, 1099–1111 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Heagerty, P.J., Lumley, T.: Window subsampling of estimating functions with application to regression models. J. Am. Stat. Assoc. 95, 197–211 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Heyde, C.C.: Quasi-Likelihood and its Application. Springer, New York (1997)

    MATH  Google Scholar 

  • Henderson, R., Shimakura, S.: A serially correlated gamma frailty model for longitudinal count data. Biometrika 90, 355–366 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  • Hjort, N.L., Mohn, E.: Topics in the statistical analysis of remotely sensed data (with discussion). Bull. Int. Stat. Inst. 52, 23–44 (1987) (Proceedings of the ISI Meeting, Tokyo)

    MathSciNet  Google Scholar 

  • Hjort, N.L., Omre, H.: Topics in spatial statistics (with discussion). Scand. J. Stat. 21, 289–357 (1994)

    MATH  MathSciNet  Google Scholar 

  • Hjort, N.L., Varin, C.: ML, PL and QL for Markov chain models. Scand. J. Stat. 35, 64–82 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Hougaard, P.: A class of multivariate failure time distributions. Biometrika 73, 671–678 (1986)

    MATH  MathSciNet  Google Scholar 

  • Hudson, R.R.: Two-locus sampling distributions and their application. Genetics 159, 1805–1817 (2001)

    Google Scholar 

  • Jöreskog, K.G., Moustaki, I.: Factor analysis of ordinal variables: a comparison of three approaches. Multivar. Behav. Res. 36, 347–386 (2001)

    Article  Google Scholar 

  • Kent, J.T.: Robust properties of likelihood ratio tests. Biometrika 69, 19–27 (1982)

    MATH  MathSciNet  Google Scholar 

  • Kuk, A.Y.C.: Permutation invariance of alternating logistic regression for multivariate binary data. Biometrika 91, 758–761 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Kuk, A.Y.C.: A hybrid pairwise likelihood method. Biometrika 94, 939–952 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Kuk, A.Y.C., Nott, D.J.: A pairwise likelihood approach to analyzing correlated binary data. Stat. Probab. Lett. 47, 329–335 (2000)

    Article  MATH  Google Scholar 

  • Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  • Le Cessie, S., Van Houwelingen, J.C.: Logistic regression for correlated binary data. Appl. Stat. 43, 95–108 (1994)

    Article  MATH  Google Scholar 

  • Lele, S.R.: Sampling variability and estimates of density dependence: a composite-likelihood approach. Ecology 87, 189–202 (2006)

    Article  Google Scholar 

  • Lele, S.R., Taper, M.L.: A composite likelihood approach to (co)variance components estimation. J. Stat. Plan. Inference 103, 117–135 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  • Liang, K.-Y., Qin, J.: Regression analysis under non-standard situations: a pairwise pseudolikelihood approach. J. R. Stat. Soc. B 62, 773–786 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Liang, G., Yu, B.: Maximum pseudolikelihood estimation in network tomography. IEEE Trans. Signal Process. 51, 2043–2053 (2003)

    Article  Google Scholar 

  • Liang, K.-Y., Zeger, S.L.: Longitudinal data analysis using generalized linear models. Biometrika 73, 13–22 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • Lindsay, B.G.: Composite likelihood methods. Contemp. Math. 80, 221–240 (1988)

    MathSciNet  Google Scholar 

  • Mardia, K.V., Hughes, G., Taylor, C.C.: Efficiency of the pseudolikelihood for multivariate normal and von Mises distributions (2007, submitted)

  • Mateu, J., Porcu, E., Christakos, G., Bevilacqua, M.: Fitting negative spatial covariances to geothermal field temperatures in Nea Kessani (Greece). Environmetrics 18, 759–773 (2007)

    Article  MathSciNet  Google Scholar 

  • McCullagh, P., Nelder, J.A.: Generalised Linear Models, 2nd edn. Chapman and Hall, London (1989)

    Google Scholar 

  • McVean, G., Awadalla, P., Fearnhead, P.: A coalescent-based method for detecting and estimating recombination from gene sequences. Genetics 160, 1231–1241 (2002)

    Google Scholar 

  • Molenberghs, G., Geys, H.: Multivariate clustered data analysis in developmental toxicity studies. Stat. Neerl. 55, 319–345 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Molenberghs, G., Verbeke, G.: Models for Discrete Longitudinal Data. Springer, New York (2005)

    MATH  Google Scholar 

  • Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  • Nott, D.J., Rydén, T.: Pairwise likelihood methods for inference in image models. Biometrika 86, 661–676 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  • Oakes, D., Ritz, J.: Regression in a bivariate copula model. Biometrika 87, 345–352 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Oman, S.D., Landsman, V., Carmel, Y., Kadmon, R.: Analyzing spatially distributed binary data using independent-block estimating equations. Biometrics 63, 892–900 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Parner, E.T.: A composite likelihood approach to multivariate survival data. Scand. J. Stat. 28, 295–302 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  • Parzen, M., Lipsitz, S.R., Fitzmaurice, G.M., Ibrahim, J.G., Troxel, A.: Pseudolikelihood methods for longitudinal binary data with non-ignorable missing responses and covariates. Stat. Med. 25, 2784–2796 (2006)

    Article  MathSciNet  Google Scholar 

  • Parzen, M., Lipsitz, S.R., Fitzmaurice, G.M., Ibrahim, J.G., Troxel, A., Molenberghs, G.: Pseudo-likelihood methods for the analysis of longitudinal binary data subject to nonignorable non-monotone missingness. J. Data Sci. 5, 1–21 (2007)

    Google Scholar 

  • Pinheiro, J.C., Bates, D.M.: Mixed-Effects Models in S and S-PLUS. Springer, New York (2000)

    MATH  Google Scholar 

  • Plackett, R.L.: A class of bivariate distributions. J. Am. Stat. Assoc. 60, 516–522 (1965)

    Article  MathSciNet  Google Scholar 

  • Poon, W.-Y., Lee, S.-Y.: Maximum likelihood estimation of multivariate polyserial and polychoric correlation coefficients. Psychometrika 52, 409–430 (1987) (correction in 53, p. 301)

    Article  MATH  MathSciNet  Google Scholar 

  • Porcu, E., Mateu, J., Bevilacqua, M.: Covariance functions which are stationary or nonstazionary in space and stazionary in time. Stat. Neerl. 61, 358–382 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Renard, D., Geys, H., Molenberghs, G., Burzykowski, T., Buyse, M.: Validation of surrogate endpoints in multiple randomized clinical trials with discrete outcomes. Biom. J. 44, 921–935 (2002)

    Article  MathSciNet  Google Scholar 

  • Renard, D., Molenberghs, G., Geys, H.: A pairwise likelihood approach to estimation in multilevel probit models. Comput. Stat. Data Anal. 44, 649–667 (2004)

    Article  MathSciNet  Google Scholar 

  • Rodriguez, G., Goldman, N.: An assessment of estimation procedures for multilevel models with binary responses. J. R. Stat. Soc. Ser. A 158, 79–89 (1995)

    Google Scholar 

  • Rýden, T.: Consistent and asymptotically normal parameter estimates for hidden Markov models. Ann. Stat. 22, 1841–1895 (1994)

    Article  Google Scholar 

  • Satterthwaites, F.E.: An approximate distribution of estimates of variance components. Biometrics 2, 110–114 (1946)

    Article  Google Scholar 

  • Song, P.X.-K.: Correlated Data Analysis: Modeling, Analytics and Applications. Springer, New York (2007)

    MATH  Google Scholar 

  • Stein, M.L.: Approximating likelihoods for large spatial data sets. J. R. Stat. Soc. B 66, 275–296 (2004)

    Article  MATH  Google Scholar 

  • Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and its Applications, 2nd edn. Wiley, Chichester (1995)

    MATH  Google Scholar 

  • Takeuchi, K.: Distribution of information statistics and criteria for adequacy of models. Math. Sci. 153, 12–18 (1976) (in Japanese)

    Google Scholar 

  • Tibaldi, F., Molenberghs, G., Burzykowski, T., Geys, H.: Pseudolikelihood estimation for a marginal multivariate survival model. Stat. Med. 23, 947–963 (2004a)

    Article  Google Scholar 

  • Tibaldi, F., Barbosa, F.T., Molenberghs, G.: Modelling associations between time-to-event responses in pilot cancer clinical trials using a Plackett–Dale model. Stat. Med. 23, 2173–2186 (2004b)

    Article  Google Scholar 

  • Troxel, A.B., Lipsitz, S.R., Harrington, D.P.: Marginal models for the analysis of longitudinal measurements with noningnorable non-monotone missing data. Biometrika 85, 661–672 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Vandekerkhove, P.: Consistent and asymptotically normal parameter estimates for hidden Markov mixtures of Markov models. Bernoulli 11, 103–129 (2005)

    MATH  MathSciNet  Google Scholar 

  • Vardi, Y.: Network tomography: estimating source-destination traffic intensities from link data. J. Am. Stat. Assoc. 91, 365–377 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Varin, C., Czado, C.: Modeling pain severity diaries with mixed autoregressive ordinal probit models (2008, submitted)

  • Varin, C., Vidoni, P.: A note on composite likelihood inference and model selection. Biometrika 92, 519–528 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Varin, C., Vidoni, P.: Pairwise likelihood inference for ordinal categorical time series. Comput. Stat. Data Anal. 51, 2365–2373 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Varin, C., Vidoni, P.: Pairwise likelihood inference for general state space models. Econ. Rev. (2008, to appear)

  • Varin, C., Høst, G., Skare, Ø.: Pairwise likelihood inference in spatial generalized linear mixed models. Comput. Stat. Data Anal. 49, 1173–1191 (2005)

    Article  MATH  Google Scholar 

  • Warnes, J.J., Ripley, B.D.: Problems with likelihood estimation of covariance functions of spatial Gaussian processes. Biometrika 74, 640–642 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  • Wedderburn, R.W.M.: Quasi-likelihood functions, generalized linear models and the Gauss-Newton method. Biometrika 61, 439–447 (1974)

    MATH  MathSciNet  Google Scholar 

  • West, M., Harrison, J.: Bayesian Forecasting and Dynamic Models, 2nd edn. Springer, New York (1997)

    MATH  Google Scholar 

  • Wilson, R.J., Nott, D.J.: Review of recent results on excursions set models. Image Anal. Stereol. 20, 71–78 (2001)

    Google Scholar 

  • Zhang, H.: On estimation and prediction for spatial generalized linear mixed models. Biometrics 58, 129–136 (2002)

    Article  MathSciNet  Google Scholar 

  • Zhao, L.P., Prentice, R.L.: Correlated binary regression using a quadratic exponential model. Biometrika 77, 642–648 (1990)

    Article  MathSciNet  Google Scholar 

  • Zhao, Y., Joe, H.: Composite likelihood estimation in multivariate data analysis. Can. J. Stat. 33, 335–356 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Statistics, University Ca’ Foscari, San Giobbe, Cannaregio 873, 30121, Venice, Italy

    Cristiano Varin

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Varin, C. On composite marginal likelihoods. AStA 92, 1–28 (2008). https://doi.org/10.1007/s10182-008-0060-7

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