Influence analysis of generalized least squares estimators

Victor De Gruttola, James H. Ware, Thomas A. Louis

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


Influence analysis and leverage analysis are important and well-established adjuncts to ordinary least squares (OLS) regression, but analogous regression diagnostics are not generally available for multivariate regression problems. This article describes measures of influence and leverage for a generalized least squares (GLS) estimator of the regression coefficients in a class of multivariate linear models for repeated measurements. When the covariance matrix of the observation vectors is known, the GLS estimator can be expressed as an OLS estimator calculated from transformed variables. In this situation, the multivariate regression diagnostics are directly related to the univariate measures. When the covariance matrix is unknown, however, multistage estimation procedures are required. We describe measures of influence and leverage for a three-step estimator that uses the residuals from an initial OLS regression to estimate the covariance matrix of the response vectors. These diagnostics account for the influence of individual data values on the estimated covariance matrix. The article reports on two different measures of influence and the relation between them: (a) derivative influence, the differential change in an estimated parameter or predicted value resulting from a slight perturbation in the weight assigned to a given observation or vector of observations, and (b) deletion influence, the change in a parameter estimate resulting from the deletion of the vector of observations. Both of these measures can be represented as the sum of components that correspond to the stages of the GLS estimation. Examination of the relative asymptotic sizes of these components and of the differences between derivative and deletion influence leads to modified versions of these diagnostics that require less computation than deletion influence but are more appropriate than derivative influence for small samples. This approach yields computationally feasible diagnostics for the three-step estimator under study and for a larger class of non-iterative GLS estimators. An example illustrates their usefulness.

Original languageEnglish (US)
Pages (from-to)911-917
Number of pages7
JournalJournal of the American Statistical Association
Issue number399
StatePublished - Sep 1987
Externally publishedYes


  • GLS estimation
  • Leverage
  • Outliers
  • Score function

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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