The Annals of Applied Statistics

A space–time varying coefficient model: The equity of service accessibility

Nicoleta Serban

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Research in examining the equity of service accessibility has emerged as economic and social equity advocates recognized that where people live influences their opportunities for economic development, access to quality health care and political participation. In this research paper service accessibility equity is concerned with where and when services have been and are accessed by different groups of people, identified by location or underlying socioeconomic variables. Using new statistical methods for modeling spatial-temporal data, this paper estimates demographic association patterns to financial service accessibility varying over a large geographic area (Georgia) and over a period of 13 years. The underlying model is a space–time varying coefficient model including both separable space and time varying coefficients and space–time interaction terms. The model is extended to a multilevel response where the varying coefficients account for both the within- and between-variability. We introduce an inference procedure for assessing the shape of the varying regression coefficients using confidence bands.

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Ann. Appl. Stat., Volume 5, Number 3 (2011), 2024-2051.

First available in Project Euclid: 13 October 2011

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Equity service accessibility simultaneous confidence bands spatial-temporal modeling varying coefficient model


Serban, Nicoleta. A space–time varying coefficient model: The equity of service accessibility. Ann. Appl. Stat. 5 (2011), no. 3, 2024--2051. doi:10.1214/11-AOAS473.

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  • Assuncao, R. M. (2003). Space varying coefficient models for small area data. Environmetrics 14 453–473.
  • Baladandayuthapani, V., Mallick, B. K., Hong, M. Y., Lupton, J. R., Turner, N. D. and Carroll, R. J. (2008). Bayesian hierarchical spatially correlated functional data analysis with application to colon carcinogenesis. Biometrics 64 64–73, 321–322.
  • Blackwell, A. G. and Fox, R. K. (2004). Regional equity and smart growth: Opportunities for advancing social and economic justice in America. PolicyLink and Funders’ Network for Smart Growth and Livable Communities. PolicyLink, Oakland, CA.
  • Crainiceanu, C. M., Staicu, A.-M. and Di, C.-Z. (2009). Generalized multilevel functional regression. J. Amer. Statist. Assoc. 104 1550–1561.
  • Crainiceanu, C., Ruppert, D., Claeskens, G. and Wand, M. P. (2005). Exact likelihood ratio tests for penalised splines. Biometrika 92 91–103.
  • Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York.
  • Di, C.-Z., Crainiceanu, C. M., Caffo, B. S. and Punjabi, N. M. (2009). Multilevel functional principal component analysis. Ann. Appl. Statist. 3 458–488.
  • Diggle, P. (1985). A kernel method for smoothing point process data. J. R. Stat. Soc. Ser. C Appl. Stat. 34 138–147.
  • Fan, J. and Zhang, J.-T. (2000). Two-step estimation of functional linear models with applications to longitudinal data. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 303–322.
  • Gelfand, A. E., Kim, H.-J., Sirmans, C. F. and Banerjee, S. (2003). Spatial modeling with spatially varying coefficient processes. J. Amer. Statist. Assoc. 98 387–396.
  • Graves, S. M. (2003). Landscapes of predation, landscapes of neglect: A location analysis of payday lenders and banks. The Professional Geographer 55 303–317.
  • Greven, S., Crainiceanu, C. M., Küchenhoff, H. and Peters, A. (2008). Restricted likelihood ratio testing for zero variance components in linear mixed models. J. Comput. Graph. Statist. 17 870–891.
  • Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models. J. Roy. Statist. Soc. Ser. B 55 757–796.
  • 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.
  • Huang, J. Z., Wu, C. O. and Zhou, L. (2002). Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika 89 111–128.
  • Larson, T. (2003). Why there will be no chain supermarkets in poor inner-city neighborhoods. California Politics and Policy 7 22–45.
  • Lehmann, E. L. (1997). Testing Statistical Hypotheses, 2nd ed. Springer, New York.
  • Li, Y. and Ruppert, D. (2008). On the asymptotics of penalized splines. Biometrika 95 415–436.
  • Liang, H., Wu, H. and Carroll, R. J. (2003). The relationship between virologic and immunologic responses in AIDS clinical research using mixed-effects varying-coefficient models with measurement error. Biostatistics 4 297–312.
  • Lovett, A., Haynes, R., Sunnenberg, G. and Gale, S. (2002). Car travel time and accessibility by bus to general practitioner services: A study using patient registers and GIS. Soc. Sci. Med. 55 97–111.
  • Marsh, M. T. and Schilling, D. A. (1994). Equity measurement in facility location analysis—a review and framework. European J. Oper. Res. 74 1–17.
  • Morris, J. S. and Carroll, R. J. (2006). Wavelet-based functional mixed models. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 179–199.
  • Morris, J. S., Vannucci, M., Brown, P. J. and Carroll, R. J. (2003). Wavelet-based nonparametric modeling of hierarchical functions in colon carcinogenesis. J. Amer. Statist. Assoc. 98 573–597.
  • Nychka, D. and Saltzman, N. (1998). Design of air quality monitoring networks. Lecture Notes in Statist. 132 51–76. Springer, Berlin.
  • Powell, L. M., Slater, S., Mirtcheva, D., Bao, Y. J. and Chaloupka, F. J. (2007). Food store availability and neighborhood characteristics in the United States. Preventive Medicine 44 189–195.
  • Rice, J. A. and Wu, C. O. (2001). Nonparametric mixed effects models for unequally sampled noisy curves. Biometrics 57 253–259.
  • Ruppert, D. (2002). Selecting the number of knots for penalized splines. J. Comput. Graph. Statist. 11 735–757.
  • Ruppert, D., Wand, M. P. and Carroll, R. J. (2003). Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics 12. Cambridge Univ. Press, Cambridge.
  • Serban, N. (2011). Supplement to “A space–time varying coefficient model: The equity of service accessibility.” DOI:10.1214/11-AOAS473SUPP.
  • Sim, J. and Reid, N. (1999). Statistical inference by confidence intervals: Issues of interpretation and utilization. Phys. Ther. 79 186–195.
  • Small, M. L. and McDermott, M. (2006). The presence of organizational resources in poor urban neighborhoods: An analysis of average and contextual effects. Social Forces 84 1697.
  • Staicu, A.-M., Crainiceanu, C. M. and Carroll, R. J. (2010). Fast methods for spatially correlated multilevel functional data. Biostatistics 11 177–194.
  • Talen, E. (1997). The social equity of urban service distribution: An exploration of park access in Pueblo, Co and Macon, GA. Urban Geography 18 521–541.
  • Talen, E. (2001). School, community, and spatial equity: An empirical investigation of access to elementary schools in West Virginia. Annals of the Association of American Geographers 91 465–486.
  • Talen, E. and Anselin, L. (1998). Assessing spatial equity: An evaluation of measures of accessibility to public playgrounds. Environment and Planning A 30 595–613.
  • Wahba, G. (1990). Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics 59. SIAM, Philadelphia, PA.
  • Waller, L. A., Zhu, L., Gotway, C. A., Gorman, D. M. and Gruenewald, P. J. (2007). Quantifying geographic variations in associations between alcohol distribution and violence: A comparison of geographically weighted regression and spatially varying coefficient models. Stoch. Environ. Res. Risk Assess. 21 573–588.
  • Wood, S. N. (2006). Generalized Additive Models: An Introduction with R. Chapman & Hall/CRC, Boca Raton, FL.
  • Wu, H. and Liang, H. (2004). Backfitting random varying-coefficient models with timedependent smoothing covariates. Scand. J. Statist. 31 3–20.
  • Wu, H. and Zhang, J.-T. (2002). Local polynomial mixed-effects models for longitudinal data. J. Amer. Statist. Assoc. 97 883–897.
  • Zenk, S. N., Schulz, A. J., Israel, B. A., James, S. A., Bao, S. and Wilson, M. L. (2005). Neighborhood racial composition, neighborhood poverty, and the spatial accessibility of supermarkets in metropolitan Detroit. Am. J. Public Health 95 660–667.
  • Zhang, D. (2004). Generalized linear mixed models with varying coefficients for longitudinal data. Biometrics 60 8–15.

Supplemental materials

  • Supplementary material: Supplemental Material. The supplemental materials accompanying this paper are divided into seven sections: Supplement 1. Varying-coefficient model—Decomposition of the design matrix under the tensor-product decomposition of the space–time varying coefficients. Supplement 2. Varying-coefficient model—Derivation of the confidence bands for the space and time varying coefficients. Supplement 3. Varying-coefficient model—A simulation study under multiple predictors. Supplement 4. Varying-coefficient model—Proof of Proposition 2. Supplement 5. Case study—Description of ESRI data. Supplement 6. Case study—Accessibility maps for Atlanta area. Supplement 7. Case study—Results and maps for the provider-level accessibility analysis.