Local Empirical Likelihood Diagnosis of Varying Coefficient Density-Ratio Models Based on Case-Control Data


In this paper, a varying-coefficient density-ratio model for case-control studies is developed. We investigate the local empirical likelihood diagnosis of varying coefficient density-ratio model for case-control data. The local empirical log-likelihood ratios for the nonparametric coefficient functions are introduced. First, the estimation equations based on empirical likelihood method are established. Then, a few of diagnostic statistics are proposed. At last, we also examine the performance of proposed method for finite sample sizes through simulation studies.

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Wang, S. , Zheng, L. and Dai, J. (2014) Local Empirical Likelihood Diagnosis of Varying Coefficient Density-Ratio Models Based on Case-Control Data. Open Journal of Statistics, 4, 751-756. doi: 10.4236/ojs.2014.49070.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Shumway, R.H. (1988) Applied Statistical Time Series Analysis. Prentice-Hall, Englewood Cliffs.
[2] Hastie, T.J. and Tibshirani, T. (1993) Varying-Coefficient Models. Journal of the Royal Statistical Society, 55, 757-796.
[3] Hoover, D.R., Rice, J.A., Wu, C.O. and Yang, L.P. (1998) Nonparametric Smoothing Estimates of Time-Varying Coefficient Models with Longitudinal Data. Biometrika, 85, 809-822.
[4] Fan, J.Q. and Zhang, W.Y. (1999) Statistical Estimation in Varying-Coefficient Models. Annals of Statistics, 27, 1491-1518.
[5] Huang, J.Z., Wu, C.O. and Zhou, L. (2004) Polynomial Spline Estimation and Inference for Varying Coefficient Models with Longitudinal Data. Statistica Sinica, 14, 763-788.
[6] Thomas, D.C. (1981) General Relative-Risk Models for Survival Time and Matched Case-Control Analysis. Biometrics, 37, 673-686.
[7] Lustbader, E.D., Moolgavkar, S.H. and Venzon, D.J. (1984) Tests of the Null Hypothesis in Case-Control Studies. Biometrics, 40, 1017-1024.
[8] Kay, R. and Little, S. (1987) Transformations of the Explanatory Variables in the Logistic Regression Model for Binary Data. Biometrika, 74, 495-501.
[9] Qin, J. and Zhang, B. (1997) A Goodness-of-It Test for Logistic Regression Models Based on Case-Control Data. Biometrika, 84, 609-618.
[10] Qin, J., Berwick, M., Ashbolt, R., et al. (2002) Quantifying the Change of Melanoma Incidence by Breslow Thickness. Biometrics, 58, 665-670.
[11] Zhang, B. (2001) A Information Matrix Test for Logistic Regression Models Based on Case-Control Data. Biometrika, 88, 921-932.
[12] Zou, F., Fine, J.P. and Yandell, B.S. (2002) On Empirical Likelihood for a Semiparametric Mixture Model. Biometrika, 89, 61-75.
[13] White, I.R. and Thompson, S.G. (2003) Choice of Test for Comparing Two Groups, with Particular Application to Skewed Outcomes. Statistics in Medicine, 22, 1205-1215.
[14] Liu, X., Jiang, H. and Zhou, Y. (2013) Local Empirical Likelihood Inference for Varying-Coefficient Density-Ratio Models Based on Case-Control Data. Journal of the American Statistical Association, 109, 635-646.
[15] Thomas, D.R. and Grunkemeier, G.L. (1975) Confidence Interval Estimation of Survival Interval Estimation of Survival Probabilities for Censored Data. Journal of the American Statistical Association, 70, 865-871.
[16] Owen, A. (2001) Empirical Likelihood. Chapman and Hall, New York.
[17] Zhu, H.T., Ibrahim, J.G., Tang, N.S and Zhang, H. (2008) Diagnostic Measures for Empirical Likelihood of Generalized Estimating Equations. Biometrika, 95, 489-507.
[18] Xue, L. and Zhu, L. (2010) Empirical Likelihood in Nonparametric and Semiparametric Models. Science Press, Beijing.
[19] Cook, R.D. and Weisberg, S. (1982) Residuals and Influence in Regression. Chapman and Hall, New York.
[20] Wei, B., Lu, G. and Shi, J. (1990) Statistical Diagnostics. Publishing House of Southeast University, Nanjing.
[21] Cai, Z., Fan, J. and Li, R. (2000) Efficient Estimation and Inferences for Varying-Coefficient Models. Journal of American Statistical Association, 95, 888-902.
[22] Fan, J. and Zhang, W. (2008) Statistical Methods with Varying Coefficient Models. Statistics and Its Interface, 1, 179-195.
[23] Prentice, R.L. and Pyke, R. (1979) Logistic Disease Incidence Models and Case-Control Studies. Biometrika, 66, 403-411.

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