Quality & Quantity

, Volume 53, Issue 3, pp 1263–1274 | Cite as

Recursive residuals for linear mixed models

  • Ahmed Bani-MustafaEmail author
  • K. M. Matawie
  • C. F. Finch
  • Amjad Al-Nasser
  • Enrico Ciavolino


This paper presents and extends the concept of recursive residuals and their estimation to an important class of statistical models, Linear Mixed Models (LMM). Recurrence formulae are developed and recursive residuals are defined. Recursive computable expressions are also developed for the model’s likelihood, together with its derivative and information matrix. The theoretical framework for developing recursive residuals and their estimation for LMM varies with the estimation method used, such as the fitting-of-constants or the Best Linear Unbiased Predictor method. These methods are illustrated through application to an LMM example drawn from a published study. Model fit is assessed through a graphical display of the developed recursive residuals and their Cumulative Sums.


BLUP Fitting-of-constant Linear mixed model Recursive estimation Recursive residuals 



This research was supported in part by an Early Career Researcher Grant (ECR); National Health and Medical Research Council (NHMRC) Principle Research Fellowship, Federation University Australia, Ballarat, Australia.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsAustralian College of KuwaitSafatKuwait
  2. 2.School of Computing, Engineering and MathematicsWestern Sydney UniversitySydneyAustralia
  3. 3.School of Medical and Health SciencesEdith Cowan UniversityJoondalupAustralia
  4. 4.Department of Statistics, Faculty of ScienceYarmouk UniversityIrbidJordan
  5. 5.University of SalentoLecceItaly

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