Summary
We consider estimation in a particular semiparametric regression model for the mean of a counting process under the assumption of “panel count” data. The basic model assumption is that the conditional mean function of the counting process is of the form E{ℕ(t)|itZ} = exp(θ’Z)Λ(t) where Z is a vector of covariates and Λ is the baseline mean function. The “panel count” observation scheme involves observation of the counting process ℕ for an individual at a random number K of random time points; both the number and the locations of these time points may differ across individuals.
We study maximum pseudo-likelihood and maximum likelihood estimators \( \hat \theta _n^{ps} \) and \( \mathop {\hat \theta }\nolimits_n \) of the regression parameter θ. The pseudo-likelihood estimators are fairly easy to compute, while the full maximum likelihood estimators pose more challenges from the computational perspective. We drive expressions for the asymptotic variances of both estimators under the proportional mean model. Our primary aim is to understand when the pseudo-likelihood estimators have very low efficiency relative to the full maximum likelihood estimators. The upshot is that te pseudo-likelihood estimators can have arbitrarily small efficiency relative to the full maximum likelihood estimators when the distribution of K, the number of observation time points per individual, is very heavy-tailed.
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References
Banerjee, M. and Wellner, J.A. (2001). Likelihood ratio tests for monotone functions. Ann. Statist. 29, 1699–1731.
Begun, J.M., Hall, W. J., Huang, W.M., and Wellner, J.A. (1983). Information and asymptotic efficiency in parametric-nonparametric models. Ann. Statist. 11, 432–452.
Betensky, RA., Rabinowitz, D., and Tsiatis, A.A. (2001). Computationally simple accelerated failure time regression for interval censored data. Biometrika 88, 703–711.
Bickel, P.J., Klaassen, C.A.J., Ritov, Y., and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins University Press, Baltimore.
Cook, R J., Lawless, J. F., and Nadeau, C. (1996). Robust tests for treatment comparisons based on recurrent event responses. Biometrics 52, 557–571.
Dean, C.B. and Balshaw, R. (1997). Efficiency lost by analyzing counts rather than event times in Poisson and overdispersed Poisson regression models. J. Amer. Statist. Assoc. 92, 1387–1398.
Gaver, D. P., and O’Muircheartaigh, I.G. (1987). Robust Empirical Bayes analysis of event rates, Technometrics, 29, 1–15.
Geskus, R. and Groeneboom, P. (1996). Asymptotically optimal estimation of smooth functionals for interval censoring, part 1. Statist. Neerlandica 50, 69–88.
Geskus, R. and Groeneboom, P. (1997). Asymptotically optimal estimation of smooth functionals for interval censoring, part 2. Statist. Neerlandica 51, 201–219.
Geskus, R. and Groeneboom, P. (1999). Asymptotically optimal estimation of smooth functionals for interval censoring case 2. Ann. Statist. 27, 626–674
Groeneboom, P. (1991). Nonparamet ric maximum likelihood estimators for interval censoring and deconvolution. Technical Report 378, Department of Statistics, Stanford University.
Groeneboom, P. (1996). Inverse problems in statist ics. Proceedings of the St. Flour Summer School in Probability, 1994. Lecture Notes in Math. 1648, 67–164. Springer Verlag, Berlin.
Groeneboom, P. and Wellner, J. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Basel.
Hougaard, P., Lee, M.T., and Whitmore, G. A. (1997). Analysis of overdispersed count dat a by mixtures of Poisson variables and Poisson processes. Biometrics 53, 1225–1238.
Huang, J. (1996). Efficient estimation for the Cox model with interval censoring, Annals of Statistics, 24, 540–568.
Huang, J., and Wellner, J.A. (1995). Efficient estimation for the Cox model with case 2 interval censoring, Technical Report 290, University of Washington Department of Statistics, 35 pages.
Jongbloed, G. (1998). The iterative convex minorant algorithm for nonparametric est imat ion. Journal of Computation and Graphical Statistics 7, 310–321.
Kalbfleisch, J.D. and Lawless, J.F. (1981). Statistical inference for observational plans arising in the study of life history processes. In Symposium on Statistical Inference and Applications In Honour of George Barnard’s 65th Birthday. University of Waterloo, August 5–8, 1981.
Kalbfleisch, J.D. and Lawless, J.F. (1985). The analysis of panel count data under a Markov assumption. Journal of the American Statistical Association 80, 863–871.
Lawless, J.F. (1987a). Regression methods for Poisson process data. J. Amer. Statist. Assoc. 82, 808–815.
Lawless, J.F. (1987b). Negative binomial and mixed Poisson regression. Canad. J. Statist. 15, 209–225.
Lawless, J.F. and Nadeau, C. (1995). Some simple robust methods for the analysis of recurrent events. Technometrics 37, 158–168.
Lin, D. Y., Wei, L.J., Yang, I., and Ying, Z. (2000). Semiparametric regression for the mean and rate functions of recurrent events. J. Roy. Statist. Soc. B, 711–730.
Murphy, S. and Van der Vaart, A. W. (1997). Semiparametric likelihood ratio inference. Ann. Statist. 25, 1471–1509.
Murphy, S. and Van der Vaart, A. W. (1999). Observed information in semi-parametric models. Bernoulli 5, 381–412.
Murphy, S. and Van der Vaart, A. W. (2000). On profile likelihood. J. Amer. Statist. Assoc. 95, 449–485.
Rabinowitz, D., Betensky, R. A., and Tsiatis, A. A. (2000). Using conditional logistic regression to fit proportional odds models to interval censored data. Biometrics 56, 511–518.
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
Sun, J. and Kalbfleisch, J. D. (1995). Estimation of the mean function of point processes based on panel count data. Statistica Sinice 5, 279–290.
Sun, J. and Wei, L.J. (2000). Regression analysis of panel count data with covariate-dependent observation and censoring times. J. R. Stat. Soc. Ser. B62, 293–302.
Thall, P. F., and Lachin, J. M. (1988). Analysis of Recurrent Events: Nonparametric Methods for Random-Interval Count Data. J. Amer. Statist. Assoc. 83, 339–347.
Thall, P. F. (1988). Mixed Poisson likelihood regression models for longitudinal interval count data. Biometrics 44, 197–209.
Wellner, J. A. and Zhang, Y. (2000). Two estimators of the mean of a counting process with panel count data. Ann. Statist. 28, 779–814.
Wellner, J. A., Zhang, Y., and Liu (2002). Large sample theory for two estimators in a semiparametric model for panel count data. Manuscript in progress.
Zhang, Y. (1998). Estimation for Counting Processes Based on Incomplete Data. Unpublished Ph.D. dissertation, University of Washington.
Zhang, Y. (2002). A semiparametric pseudo likelihood estimation method for panel count data. Biometrika 89, 39–48.
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Wellner, J.A., Zhang, Y., Liu, R. (2004). A Semiparametric Regression Model for Panel Count Data: When Do Pseudo-likelihood Estimators Become Badly Inefficient?. In: Lin, D.Y., Heagerty, P.J. (eds) Proceedings of the Second Seattle Symposium in Biostatistics. Lecture Notes in Statistics, vol 179. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9076-1_9
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DOI: https://doi.org/10.1007/978-1-4419-9076-1_9
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