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Inferences for Fixed Effects Based Regression Parameters in a Finite Population Setup Using Two-stage Cluster Sample

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  • Brajendra C. Sutradhar

    (Memorial University)

Abstract

In a clusters based infinite/super population (SP) setup, the inference complexity for regression parameters depend on the cluster correlations structure oriented marginal fixed or mixed effects based mean models, where as opposed to the fixed effects based models the mixed models exhibit both regression and cluster variance/correlation parameters in the mean structure especially for discrete such as clustered counts and binary data. On the other hand, as recently discussed by Sutradhar (2022, Sankhya B, 84, 259-302), in the fixed models setup the modelling of cluster correlations can be complex. In the finite population (FP) setup where FP data follow a SP model but the regression effects estimation is done using a two-stage cluster sample, the inferences become much more complex. As opposed to the recent mixed models based FP inferences (Sutradhar 2023b, Annals of the Institute of Statistical Mathematics, 75, 425-462), there does not appear any studies for fixed models based analysis for exponential family data where true correlation structures play important role in defining FP regression parameters. To resolve this inference issue, in this paper we make following specific contributions. First, the cluster correlations for linear, counts and binary data are developed in a SP setup such that the marginal mean models depend only on the fixed regression effects. Second the FP data which are hypothetical or unobserved until a sample is taken to observe a part, are utilized to develop hypothetical estimating equations for the SP regression parameters, which subsequently define the FP regression and correlation parameters. Third, the design weighted unbiased estimators are obtained for the FP regression and correlation parameters, where the specific formulas for correlation estimators depend on the nature of the response data whether linear, counts or binary. Also, as an additional application of the regression effects estimation, the FP total prediction is discussed. Next, the design consistency of the regression estimators is developed in details.

Suggested Citation

  • Brajendra C. Sutradhar, 2024. "Inferences for Fixed Effects Based Regression Parameters in a Finite Population Setup Using Two-stage Cluster Sample," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 86(2), pages 951-991, August.
    • Handle: RePEc:spr:sankha:v:86:y:2024:i:2:d:10.1007_s13171-024-00362-w
    DOI: 10.1007/s13171-024-00362-w
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    References listed on IDEAS

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    1. Zengri Wang, 2003. "Matching conditional and marginal shapes in binary random intercept models using a bridge distribution function," Biometrika, Biometrika Trust, vol. 90(4), pages 765-775, December.
    2. Sutradhar, Brajendra C. & Rao, R. Prabhakar, 2001. "On Marginal Quasi-Likelihood Inference in Generalized Linear Mixed Models," Journal of Multivariate Analysis, Elsevier, vol. 76(1), pages 1-34, January.
    3. Sutradhar, Brajendra C. & Jowaheer, Vandna, 2003. "On familial longitudinal Poisson mixed models with gamma random effects," Journal of Multivariate Analysis, Elsevier, vol. 87(2), pages 398-412, November.
    4. Brajendra C. Sutradhar, 2023. "Regression analysis for exponential family data in a finite population setup using two-stage cluster sample," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(3), pages 425-462, June.
    5. Brajendra C. Sutradhar, 2023. "Cluster Correlations and Complexity in Binary Regression Analysis Using Two-stage Cluster Samples," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 829-884, February.
    6. Brajendra C. Sutradhar, 2022. "Fixed versus Mixed Effects Based Marginal Models for Clustered Correlated Binary Data: an Overview on Advances and Challenges," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 259-302, May.
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