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Cholesky Residuals for Assessing Normal Errors in a Linear Model with Correlated Outcomes: Technical Report

Author

Listed:
  • E. Andres Houseman

    (Harvard School of Public Health)

  • Louise Ryan

    (Harvard School of Public Health and Dana-Farber Cancer Institute)

  • Brent Coull

    (Harvard School of Public Health)

Abstract

Despite the widespread popularity of linear models for correlated outcomes (e.g. linear mixed models and time series models), distribution diagnostic methodology remains relatively underdeveloped in this context. In this paper we present an easy-to-implement approach that lends itself to graphical displays of model fit. Our approach involves multiplying the estimated margional residual vector by the Cholesky decomposition of the inverse of the estimated margional variance matrix. The resulting "rotated" residuals are used to construct an empirical cumulative distribution function and pointwise standard errors. The theoretical framework, including conditions and asymptotic properties, involves technical details that are motivated by Lange and Ryan (1989), Pierce (1982), and Randles (1982). Our method appears to work well in a variety of circumstances, including models having independent units of sampling (clustered data) and models for which all observations are correlated (e.g., a single time series). Our methods can produce satisfactory results even for models that do not satisfy all of the technical conditions stated in our theory.

Suggested Citation

  • E. Andres Houseman & Louise Ryan & Brent Coull, 2004. "Cholesky Residuals for Assessing Normal Errors in a Linear Model with Correlated Outcomes: Technical Report," Harvard University Biostatistics Working Paper Series 1019, Berkeley Electronic Press.
  • Handle: RePEc:bep:hvdbio:1019
    Note: oai:bepress.com:harvardbiostat-1019
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    References listed on IDEAS

    as
    1. Nicholas T. Longford, 2001. "Simulation‐based diagnostics in random‐coefficient models," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 164(2), pages 259-273.
    2. Wand M. P., 2002. "Vector Differential Calculus in Statistics," The American Statistician, American Statistical Association, vol. 56, pages 55-62, February.
    3. Coull B.A. & Hobert J.P. & Ryan L.M. & Holmes L.B., 2001. "Crossed Random Effect Models for Multiple Outcomes in a Study of Teratogenesis," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1194-1204, December.
    4. Fraccaro, R. & Hyndman, R. & Veevers, A., 1998. "Residual Diagnostic Plots for Checking for model Mis-Specification in Time Series Regression," Monash Econometrics and Business Statistics Working Papers 12/98, Monash University, Department of Econometrics and Business Statistics.
    5. J. S. Hodges, 1998. "Some algebra and geometry for hierarchical models, applied to diagnostics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(3), pages 497-536.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. E. Andres Houseman, 2004. "A Robust Regression Model for a First-Order Autoregressive Time Series with Unequal Spacing: Technical Report," Harvard University Biostatistics Working Paper Series 1016, Berkeley Electronic Press.
    2. Agresti, Alan & Caffo, Brian & Ohman-Strickland, Pamela, 2004. "Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies," Computational Statistics & Data Analysis, Elsevier, vol. 47(3), pages 639-653, October.
    3. Ahmed Bani-Mustafa & K. M. Matawie & C. F. Finch & Amjad Al-Nasser & Enrico Ciavolino, 2019. "Recursive residuals for linear mixed models," Quality & Quantity: International Journal of Methodology, Springer, vol. 53(3), pages 1263-1274, May.
    4. Mark Reiser & Silvia Cagnone & Junfei Zhu, 2023. "An Extended GFfit Statistic Defined on Orthogonal Components of Pearson’s Chi-Square," Psychometrika, Springer;The Psychometric Society, vol. 88(1), pages 208-240, March.
    5. Hejazi, Taha-Hossein & Badri, Hossein & Yang, Kai, 2019. "A Reliability-based Approach for Performance Optimization of Service Industries: An Application to Healthcare Systems," European Journal of Operational Research, Elsevier, vol. 273(3), pages 1016-1025.
    6. B. N. Sánchez & E. A. Houseman & L. M. Ryan, 2009. "Residual-Based Diagnostics for Structural Equation Models," Biometrics, The International Biometric Society, vol. 65(1), pages 104-115, March.

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    Keywords

    Cumulative distribution function; Goodness-of-fit; Linear mixed model; Random effects; Residual diagnostics;
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